Let  $a,b,c,d$  be distinct digits such that the product of the  $2$ -digit numbers  $\overline{ab}$  and  $\overline{cb}$  is of the form  $\overline{ddd}$ . Find all possible values of  $a+b+c+d$ .


A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these  $4$  numbers?


Suppose that  $a, b, c$ , and  $d$  are real numbers simultaneously satisfying $a + b - c - d = 3$  $ab - 3bc + cd - 3da = 4$  $3ab - bc + 3cd - da = 5$ Find  $11(a - c)^2 + 17(b -d)^2$ .


 $ABCDEF$  is a hexagon inscribed in a circle such that the measure of  $\angle{ACE}$  is  $90^{\circ}$ . What is the average of the measures, in degrees, of  $\angle{ABC}$  and  $\angle{CDE}$ ?

*2018 CCA Math Bonanza Lightning Round #1.3*


Let  $M$  be a  set with  $n$  elements. How many pairs  $(A, B)$  of subsets of  $M$  are there such that  $A$  is a subset of  $B?$


Find the least three digit number that is equal to the sum of its digits plus twice the product of its digits.


$p, q, r$  are distinct prime numbers which satisfy  $$ 2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A $$  for natural number  $A$ . Find all values of  $A$ .


Solution: \boxed{ 2^{n-1} }


Let  $x, y$  be positive integers such that  $$ x^4=(x-1)\left(y^3-23\right)-1 . $$  Find the maximum possible value of  $x+y$ .


Regular tetrahedron  $ABCD$  has center  $O$  and side length  $1$ . Points  $A'$ ,  $B'$ ,  $C'$ , and  $D'$  are defined by reflecting  $A$ ,  $B$ ,  $C$ , and  $D$  about  $O$ . Compute the volume of the polyhedron with vertices  $ABCDA'B'C'D'$ .


Let  $S$  be a finite set of points in the plane, such that for each  $2$  points  $A$  and  $B$  in  $S$ , the segment  $AB$  is a side of a regular polygon all of whose vertices are contained in  $S$ . Find all possible values for the number of elements of  $S$ .

Proposed by *Viktor Simjanoski, Macedonia*

Solution: \boxed{ 3 }


Which point on the circle  $ (x \minus{} 11)^2 \plus{} (y \minus{} 13)^2 \equal{} 116$  is farthest from the point  $ (41, 25)$ ?

Solution: \boxed{(1, 9)}


Suppose an integer  $x$ , a natural number  $n$  and a prime number  $p$  satisfy the equation  $7x^2-44x+12=p^n$ . Find the largest value of  $p$ .


Compute the least possible value of  $ABCD - AB \times CD$ , where  $ABCD$  is a 4-digit positive integer, and  $AB$  and  $CD$  are 2-digit positive integers.  (Here  $A$ ,  $B$ ,  $C$ , and  $D$  are digits, possibly equal.  Neither  $A$  nor  $C$  can be zero.)


Consider a polynomial  $P(x) \in \mathbb{R}[x]$ , with degree  $2023$ , such that  $P(\sin^2(x))+P(\cos^2(x)) =1$  for all  $x \in \mathbb{R}$ . If the sum of all roots of  $P$  is equal to  $\dfrac{p}{q}$  with  $p, q$  coprime, then what is the product  $pq$ ?


Find the biggest natural number  $m$  that has the following property: among any five 500-element subsets of  $\{ 1,2,\dots, 1000\}$  there exist two sets, whose intersection contains at least  $m$  numbers.


In a tournament each player played exactly one game against each of the other players.  In each game the winner was awarded 1 point, the loser got 0 points, and each of the two players earned 1/2 point if the game was a tie.  After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points.  (In particular, each of the ten lowest scoring players earned half of her/his points against the other nine of the ten).  What was the total number of players in the tournament?


A right triangle  $ABC$  is inscribed in the circular base of a cone. If two of the side lengths of  $ABC$  are  $3$  and  $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is  $3$ , then the minimum possible volume of the cone can be written as  $\frac{m\pi\sqrt{n}}{p}$ , where  $m$ ,  $n$ , and  $p$  are positive integers,  $m$  and  $p$  are relatively prime, and  $n$  is squarefree. Find  $m + n + p$ .


For any positive integer  $n$ , let  $a_n$  denote the closest integer to  $\sqrt{n}$ , and let  $b_n=n+a_n$ . Determine the increasing sequence  $(c_n)$  of positive integers which do not occur in the sequence  $(b_n)$ .

Solution: \boxed{c_n = n^2}


A circle is inscribed in a unit square, and a diagonal of the square is drawn. Find the total length of the segments of the diagonal not contained within the circle.

Solution: \boxed{ \sqrt{2} - 1 }


Consider a regular polygon with  $2^n$  sides, for  $n \ge 2$ , inscribed in a circle of radius  $1$ . Denote the area of this polygon by  $A_n$ . Compute  $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$


Yukihira is counting the minimum number of lines  $m$ , that can be drawn on the plane so that they intersect in exactly  $200$  distinct points.What is  $m$ ?


There are  $n$  sheep and a wolf in sheep's clothing . Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. First, the wolf chooses some sheep to make friend's with. In each of the following days, the wolf eats one of its friends. Whenever the wolf eats a sheep  $A$ :
(a) If a friend of  $A$  is originally a friend of the wolf, it un-friends the wolf.
(b) If a friend of  $A$  is originally not a friend of the wolf, it becomes a friend of the wolf.
Repeat the procedure until the wolf has no friend left.
Find the largest integer  $m$  in terms of  $n$  satisfying the following: There exists an initial friendsheep structure such that the wolf has  $m$  different ways of choosing initial sheep to become friends, so that the wolf has a way to eat all of the sheep.

Solution: \boxed{ 2^{n-1} }


Let  $f : \mathbb{N} \to \mathbb{N}$  be a function such that the following conditions hold: $\qquad\ (1) \; f(1) = 1.$  $\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$  $\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$  $\qquad\ (4) \; f(4n + 3) \le 2f(2n + 1) \; \forall \; n \ge 0.$ Find the sum of all possible values of  $f(2023)$ .


Jack is jumping on the number line. He first jumps one unit and every jump after that he jumps one unit more than the previous jump. What is the least amount of jumps it takes to reach exactly  $19999$  from his starting place?

*2017 CCA Math Bonanza Lightning Round #2.3*


Consider the acute angle  $ABC$ . On the half-line  $BC$  we consider the distinct points  $P$  and  $Q$  whose projections onto the line  $AB$  are the points  $M$  and  $N$ . Knowing that  $AP=AQ$  and  $AM^2-AN^2=BN^2-BM^2$ , find the angle  $ABC$ .


Each pair of vertices of a regular  $67$ -gon is joined by a line segment. Suppose  $n$  of these segments are selected, and each of them is painted one of ten available colors. Find the minimum possible value of  $n$  for which, regardless of which  $n$  segments were selected and how they were painted, there will always be a vertex of the polygon that belongs to seven segments of the same color.

Solution: \boxed{ n = 2011 }


Let  $ABC$  be an acute-angled triangle with altitude  $AT = h$ . The line passing through its circumcenter  $O$  and incenter  $I$  meets the sides  $AB$  and  $AC$  at points  $F$  and  $N$ , respectively. It is known that  $BFNC$  is a cyclic quadrilateral. Find the sum of the distances from the orthocenter of  $ABC$  to its vertices.


Let  $ABCD$  be a unit square. For any interior points  $M,N$  such that the line  $MN$  does not contain a vertex of the square, we denote by  $s(M,N)$  the least area of the triangles having their vertices in the set of points  $\{ A,B,C,D,M,N\}$ . Find the least number  $k$  such that  $s(M,N)\le k$ , for all points  $M,N$ .

*Dinu Șerbănescu*


Find the number of ordered triplets  $(a,b,c)$  of positive integers such that  $abc=2008$  (the product of  $a$ ,  $b$ , and  $c$  is  $2008$ ).


Given two points  $P,\ Q$  on the parabola  $C: y=x^2-x-2$  in the  $xy$  plane.
Note that the  $x$  coodinate of  $P$  is less than that of  $Q$ .

(a) If the origin  $O$  is the midpoint of the lines　egment  $PQ$ , then find the equation of the line  $PQ$ .

(b) If the origin  $O$  divides internally the line segment  $PQ$  by 2:1, then find the equation of  $PQ$ .

(c) If the origin  $O$  divides internally the line segment  $PQ$  by 2:1, find the area of the figure bounded by the parabola  $C$  and the line  $PQ$ .


Find all pairs of positive integers  $(m, n)$  such that  $m^2-mn+n^2+1$  divides both numbers  $3^{m+n}+(m+n)!$  and  $3^{m^3+n^3}+m+n$ .

*Proposed by Dorlir Ahmeti*

Solution: \boxed{ (2, 2) }


Find all prime numbers of the form  $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$ , where  $n$  is a natural number.


Find the smallest prime which is not the difference (in some order) of a power of  $2$  and a power of  $3$ .


The positive integer  $m$  is a multiple of  $101$ , and the positive integer  $n$  is a multiple of  $63$ . Their sum is  $2018$ . Find  $m - n$ .


Let  $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$ .

(1) Show that  $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$ .

(2) Find the value of  $ nx_nx_{n \minus{} 1}$ .

(3) Show that a sequence  $ \{x_n\}$  is monotone decreasing.

(4) Find  $ \lim_{n\to\infty} nx_n^2$ .


Consider all sums that add up to  $2015$ . In each sum, the addends are consecutive positive integers, and all sums have less than  $10$  addends. How many such sums are there?


If  $ x\neq1$ ,  $ y\neq1$ ,  $ x\neq y$  and
\[ \frac{yz\minus{}x^{2}}{1\minus{}x}\equal{}\frac{xz\minus{}y^{2}}{1\minus{}y}\]
show that both fractions are equal to  $ x\plus{}y\plus{}z$ .

Solution: \boxed{ x + y + z }


To each positive integer  $ n$  it is assigned a non-negative integer  $f(n)$  such that the following conditions are satisfied:

(1)  $ f(rs) \equal{} f(r)\plus{}f(s)$ (2)  $ f(n) \equal{} 0$ , if the first digit (from right to left) of  $ n$  is 3.
(3)  $ f(10) \equal{} 0$ .

Find  $f(1985)$ . Justify your answer.


Justine has two fair dice, one with sides labeled  $1,2,\ldots, m$  and one with sides labeled  $1,2,\ldots, n.$  She rolls both dice once. If  $\tfrac{3}{20}$  is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of  $m+n$ .

*Proposed by Justin Stevens*


Parallelogram  $ABCD$  has  $AB=CD=6$  and  $BC=AD=10$ , where  $\angle ABC$  is obtuse. The circumcircle of  $\triangle ABD$  intersects  $BC$  at  $E$  such that  $CE=4$ . Compute  $BD$ .


Let  $B(n)$  be the number of ones in the base two expression for the positive integer  $n.$  Determine whether  $$ \exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right) $$  is a rational number.

Solution: \boxed{ 4 }


Compute the square of the distance between the incenter (center of the inscribed circle) and circumcenter (center of the circumscribed circle) of a 30-60-90 right triangle with hypotenuse of length 2.

Solution: \boxed{2 - \sqrt{3}}


The integer  $n$ , between 10000 and 99999, is  $abcde$  when written in decimal notation.  The digit  $a$  is the remainder when  $n$  is divided by 2, the digit  $b$  is the remainder when  $n$  is divided by 3, the digit  $c$  is the remainder when  $n$  is divided by 4, the digit  $d$  is the remainder when  $n$  is divied by 5, and the digit  $e$  is the reminader when  $n$  is divided by 6.  Find  $n$ .


A function  $f: \N\rightarrow\N$  is circular if for every  $p\in\N$  there exists  $n\in\N,\ n\leq{p}$  such that  $f^n(p)=p$  ( $f$  composed with itself  $n$  times) The function  $f$  has repulsion degree  $k>0$  if for every  $p\in\N$   $f^i(p)\neq{p}$  for every  $i=1,2,\dots,\lfloor{kp}\rfloor$ . Determine the maximum repulsion degree can have a circular function.**Note:** Here  $\lfloor{x}\rfloor$  is the integer part of  $x$ .

Solution: \boxed{ \frac{1}{2} }


Nine positive integers  $a_1,a_2,...,a_9$  have their last  $2$ -digit part equal to  $11,12,13,14,15,16,17,18$  and  $19$  respectively. Find the last  $2$ -digit part of the sum of their squares.


Laurie loves multiplying numbers in her head. One day she decides to multiply two  $2$ -digit numbers  $x$  and  $y$  such that  $x\leq y$  and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering  $51$  she remembers  $15$ ). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?


Suppose that  $P(x, y, z)$  is a homogeneous degree 4 polynomial in three variables such that  $P(a, b, c) = P(b, c, a)$  and  $P(a, a, b) = 0$  for all real  $a$ ,  $b$ , and  $c$ . If  $P(1, 2, 3) = 1$ , compute  $P(2, 4, 8)$ .

Note:  $P(x, y, z)$  is a homogeneous degree  $4$  polynomial if it satisfies  $P(ka, kb, kc) = k^4P(a, b, c)$  for all real  $k, a, b, c$ .


For how many integer values of  $m$ ,
(i)  $1\le m \le 5000$ (ii)  $[\sqrt{m}] =[\sqrt{m+125}]$ Note:  $[x]$  is the greatest integer function


Three real numbers  $x$ ,  $y$ , and  $z$  are such that  $(x+4)/2=(y+9)/(z-3)=(x+5)/(z-5)$ .  Determine the value of  $x/y$ .


For any positive integer  $a$ , let  $\tau(a)$  be the number of positive divisors of  $a$ . Find, with proof, the largest possible value of  $4\tau(n)-n$  over all positive integers  $n$ .


A positive integer  $n\geq 4$  is called *interesting* if there exists a complex number  $z$  such that  $|z|=1$  and \[1+z+z^2+z^{n-1}+z^n=0.\] Find how many interesting numbers are smaller than  $2022.$


An equilateral triangle  $ABC$  shares a side with a square  $BCDE$ . If the resulting pentagon has a perimeter of  $20$ , what is the area of the pentagon? (The triangle and square do not overlap).

Solution: \boxed{16 + 4\sqrt{3}}


Let  $f:\mathbb{N}\mapsto\mathbb{R}$  be the function \[f(n)=\sum_{k=1}^\infty\dfrac{1}{\operatorname{lcm}(k,n)^2}.\] It is well-known that  $f(1)=\tfrac{\pi^2}6$ .  What is the smallest positive integer  $m$  such that  $m\cdot f(10)$  is the square of a rational multiple of  $\pi$ ?


Let  $\triangle A_1B_1C_1$  be an equilateral triangle of area  $60$ .  Chloe constructs a new triangle  $\triangle A_2B_2C_2$  as follows.  First, she flips a coin.  If it comes up heads, she constructs point  $A_2$  such that  $B_1$  is the midpoint of  $\overline{A_2C_1}$ .  If it comes up tails, she instead constructs  $A_2$  such that  $C_1$  is the midpoint of  $\overline{A_2B_1}$ .  She performs analogous operations on  $B_2$  and  $C_2$ .  What is the expected value of the area of  $\triangle A_2B_2C_2$ ?


Let  $ABCD$  be a trapezoid with  $AB\parallel DC$ . Let  $M$  be the midpoint of  $CD$ . If  $AD\perp CD, AC\perp BM,$  and  $BC\perp BD$ , find  $\frac{AB}{CD}$ .

[i]Proposed by Nathan Ramesh


How many numbers  $\overline{abcd}$  with different digits satisfy the following property:
if we replace the largest digit with the digit  $1$  results in a multiple of  $30$ ?


$p(m)$  is the number of distinct prime divisors of a positive integer  $m>1$  and  $f(m)$  is the  $\bigg \lfloor \frac{p(m)+1}{2}\bigg \rfloor$  th smallest prime divisor of  $m$ . Find all positive integers  $n$  satisfying the equation:   $$ f(n^2+2) + f(n^2+5) = 2n-4 $$

Solution: \boxed{n = 5}


3 points  $ O(0,\ 0),\ P(a,\ a^2), Q( \minus{} b,\ b^2)\ (a > 0,\ b > 0)$  are on the parabpla  $ y \equal{} x^2$ .
Let  $ S_1$  be the area bounded by the line  $ PQ$  and the parabola and let  $ S_2$  be the area of the triangle  $ OPQ$ .

Find the minimum value of  $ \frac {S_1}{S_2}$ .

Solution: \boxed{ \frac{4}{3} }


In some primary school there were  $94$  students in  $7$ th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies  $40$  students outside school program, german  $27$  students and  $60$  students do sports. Out of the students doing sports,  $24$  of them also goes to spanish language.  $10$  students who study spanish also study german.  $12$  students who study german also do sports. Only  $4$  students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?


Given triangle  $ABC$  and a point  $P$  inside it,  $\angle BAP=18^\circ$ ,  $\angle CAP=30^\circ$ ,  $\angle ACP=48^\circ$ , and  $AP=BC$ .  If  $\angle BCP=x^\circ$ , find  $x$ .


Let  $ABC$  be a triangle with circumcenter  $O$  such that  $AC = 7$ . Suppose that the circumcircle of  $AOC$  is tangent to  $BC$  at  $C$  and intersects the line  $AB$  at  $A$  and  $F$ . Let  $FO$  intersect  $BC$  at  $E$ . Compute  $BE$ .


Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?


Prime  $p>2$  and a polynomial  $Q$  with integer coefficients are such that there are no integers  $1 \le i < j \le p-1$  for which  $(Q(j)-Q(i))(jQ(j)-iQ(i))$  is divisible by  $p$ . What is the smallest possible degree of  $Q$ ?

*(Proposed by Anton Trygub)*

Solution: \boxed{ p-2 }


 $99$  identical balls lie on a table.  $50$  balls are made of copper, and  $49$  balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to  $2$  balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?

*Proposed by N. Vlasova, S. Berlov*


Given  $w$  and  $z$  are complex numbers such that  $|w+z|=1$  and  $|w^2+z^2|=14$ , find the smallest possible value of  $|w^3+z^3|$ . Here  $| \cdot |$  denotes the absolute value of a complex number, given by  $|a+bi|=\sqrt{a^2+b^2}$  whenever  $a$  and  $b$  are real numbers.


Evaluate   $ \int_0^1 (1 \plus{} x \plus{} x^2 \plus{} \cdots \plus{} x^{n \minus{} 1})\{1 \plus{} 3x \plus{} 5x^2 \plus{} \cdots \plus{} (2n \minus{} 3)x^{n \minus{} 2} \plus{} (2n \minus{} 1)x^{n \minus{} 1}\}\ dx.$


Let  $\triangle ABC$  be a right-angled triangle and  $BC > AC$ .  $M$  is a point on  $BC$  such that  $BM = AC$  and  $N$  is a point on  $AC$  such that  $AN = CM$ . Find the angle between  $BN$  and  $AM$ .


Suppose we have  $10$  balls and  $10$  colors. For each ball, we (independently) color it one of the  $10$  colors, then group the balls together by color at the end. If  $S$  is the expected value of the square of the number of distinct colors used on the balls, find the sum of the digits of  $S$  written as a decimal.

*Proposed by Michael Kural*


We call a pair  $(a,b)$  of positive integers,  $a<391$ , *pupusa* if   $$ \textup{lcm}(a,b)>\textup{lcm}(a,391) $$  Find the minimum value of  $b$  across all *pupusa* pairs.

Fun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortilla filled with cheese, meat, etc.


Evaluate  $ \int _ { 0 } ^ { a } d x / ( 1 + \cos x + \sin x ) $  for  $ - \pi / 2 < a < \pi $ . Use your answer to show that  $ \int _ { 0 } ^ { \pi / 2 } d x / ( 1 + \cos x + \sin x ) = \ln 2 $ .

Solution: \boxed{ \ln 2 }


Let  $A = {1, 2, 3, 4, 5}$ . Find the number of functions  $f$  from the nonempty subsets of  $A$  to  $A$ , such that  $f(B) \in B$  for any  $B \subset A$ , and  $f(B \cup C)$  is either  $f(B)$  or  $f(C)$  for any  $B$ ,  $C \subset A$


Let  $a$  and  $b$  be perfect squares whose product exceeds their sum by  $4844$ .  Compute the value of \[\left(\sqrt a + 1\right)\left(\sqrt b + 1\right)\left(\sqrt a - 1\right)\left(\sqrt b - 1\right) - \left(\sqrt{68} + 1\right)\left(\sqrt{63} + 1\right)\left(\sqrt{68} - 1\right)\left(\sqrt{63} - 1\right).\]


Determine all real numbers  $a$  such that \[4\lfloor an\rfloor =n+\lfloor a\lfloor an\rfloor \rfloor \; \text{for all}\; n \in \mathbb{N}.\]

Solution: \boxed{ a = 2 + \sqrt{3} }


What is the sum of the squares of the roots of the equation  $x^2 -7 \lfloor x\rfloor +5=0$  ?


Determine the largest natural number  $m$  such that for each non negative real numbers  $a_1 \ge a_2 \ge ... \ge a_{2014} \ge 0$  , it is true that   $$ \frac{a_1+a_2+...+a_m}{m}\ge \sqrt{\frac{a_1^2+a_2^2+...+a_{2014}^2}{2014}} $$

Solution: \boxed{m = 44}


With about six hours left on the van ride home from vacation, Wendy looks for something to do.  She starts working on a project for the math team.

There are sixteen students, including Wendy, who are about to be sophomores on the math team.  Elected as a math team officer, one of Wendy's jobs is to schedule groups of the sophomores to tutor geometry students after school on Tuesdays.  The way things have been done in the past, the same number of sophomores tutor every week, but the same group of students never works together.  Wendy notices that there are even numbers of groups she could select whether she chooses  $4$  or  $5$  students at a time to tutor geometry each week:
\begin{align*}\dbinom{16}4&=1820,\dbinom{16}5&=4368.\end{align*}
Playing around a bit more, Wendy realizes that unless she chooses all or none of the students on the math team to tutor each week that the number of possible combinations of the sophomore math teamers is always even.  This gives her an idea for a problem for the  $2008$  Jupiter Falls High School Math Meet team test:

\[\text{How many of the 2009 numbers on Row 2008 of Pascal's Triangle are even?}\]

Wendy works the solution out correctly.  What is her answer?


Find the least positive integer of the form a b a a b a, where a and b are distinct digits, such that the integer can be written as a product of six distinct primes


Sierpinski's triangle is formed by taking a triangle, and drawing an upside down triangle inside each upright triangle that appears. A snake sees the fractal, but decides that the triangles need circles inside them. Therefore, she draws a circle inscribed in every upside down triangle she sees (assume that the snake can do an infinite amount of work). If the original triangle had side length  $1$ , what is the total area of all the individual circles?

*2015 CCA Math Bonanza Lightning Round #4.4*


Find the least positive integer  $N$  which is both a multiple of 19 and whose digits add to 23.


Let  $A$  and  $B$  be digits between  $0$  and  $9$ , and suppose that the product of the two-digit numbers  $\overline{AB}$  and  $\overline{BA}$  is equal to  $k$ . Given that  $k+1$  is a multiple of  $101$ , find  $k$ .

*Proposed by Andrew Wu*


Suppose  $f$  and  $g$  are differentiable functions such that \[xg(f(x))f^\prime(g(x))g^\prime(x)=f(g(x))g^\prime(f(x))f^\prime(x)\] for all real  $x$ .  Moreover,  $f$  is nonnegative and  $g$  is positive.  Furthermore, \[\int_0^a f(g(x))dx=1-\dfrac{e^{-2a}}{2}\] for all reals  $a$ .  Given that  $g(f(0))=1$ , compute the value of  $g(f(4))$ .


Let  $\{a, b, c, d, e, f, g, h\}$  be a permutation of  $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . What is the probability that  $\overline{abc} +\overline{def}$  is even?


Let  $n>1$  be an integer and  $S_n$  the set of all permutations  $\pi : \{1,2,\cdots,n \} \to \{1,2,\cdots,n \}$  where  $\pi$  is bijective function. For every permutation  $\pi \in S_n$  we define:

\[ F(\pi)= \sum_{k=1}^n  |k-\pi(k)|  \ \ \text{and} \ \  M_{n}=\frac{1}{n!}\sum_{\pi \in S_n} F(\pi) \]
where  $M_n$  is taken with all permutations  $\pi \in S_n$ . Calculate the sum  $M_n$ .

Solution: \boxed{\frac{n^2 - 1}{3}}


Alexis imagines a  $2008\times 2008$  grid of integers arranged sequentially in the following way:

\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-2008+3,&\ldots,&2008^2\end{array}\]
She picks one number from each row so that no two numbers she picks are in the same column.  She them proceeds to add them together and finds that  $S$  is the sum.  Next, she picks  $2008$  of the numbers that are distinct from the  $2008$  she picked the first time.  Again she picks exactly one number from each row and column, and again the sum of all  $2008$  numbers is  $S$ .  Find the remainder when  $S$  is divided by  $2008$ .


Given is a triangle  $ABC$  with the property that  $|AB| + |AC| = 3|BC|$ . Let  $T$  be the point on segment  $AC$  such that  $|AC| = 4|AT|$ . Let  $K$  and  $L$  be points on the interior of line segments  $AB$  and  $AC$  respectively such that  $KL \parallel BC$  and  $KL$  is tangent to the inscribed circle of  $\vartriangle ABC$ . Let  $S$  be the intersection of  $BT$  and  $KL$ . Determine the ratio  $\frac{|SL|}{|KL|}$


Let  $ABC$  be an equilateral triangle with side length  $1$ . Points  $A_1$  and  $A_2$  are chosen on side  $BC$ , points  $B_1$  and  $B_2$  are chosen on side  $CA$ , and points  $C_1$  and  $C_2$  are chosen on side  $AB$  such that  $BA_1<BA_2$ ,  $CB_1<CB_2$ , and  $AC_1<AC_2$ .
Suppose that the three line segments  $B_1C_2$ ,  $C_1A_2$ ,  $A_1B_2$  are concurrent, and the perimeters of triangles  $AB_2C_1$ ,  $BC_2A_1$ , and  $CA_2B_1$  are all equal. Find all possible values of this common perimeter.

*Ankan Bhattacharya*


Determine the least real number  $k$  such that the inequality  $$ \left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right) $$  holds for all real numbers  $a,b,c$ .

*Proposed by Mohammad Jafari*

Solution: \boxed{k = 8}


In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long.  What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's?


Find all increasing sequences  $a_1,a_2,a_3,...$  of natural numbers such that for each  $i,j\in \mathbb N$ , number of the divisors of  $i+j$  and  $a_i+a_j$  is equal. (an increasing sequence is a sequence that if  $i\le j$ , then  $a_i\le a_j$ .)

Solution: \boxed{ a_n = n }


A triangle  $ABC$  with  $AC=20$  is inscribed in a circle  $\omega$ . A tangent  $t$  to  $\omega$  is drawn through  $B$ . The distance  $t$  from  $A$  is  $25$  and that from  $C$  is  $16$ .If  $S$  denotes the area of the triangle  $ABC$ , find the largest integer not exceeding  $\frac{S}{20}$


What is the greatest positive integer  $m$  such that  $ n^2(1+n^2-n^4)\equiv 1\pmod{2^m} $  for all odd integers  $n$ ?

Solution: \boxed{ m = 7 }


João calculated the product of the non zero digits of each integer from  $1$  to  $10^{2009}$  and then he summed these  $10^{2009}$  products. Which number did he obtain?


All the squares of a board of  $(n+1)\times(n-1)$  squares
are painted with **three  colors** such that, for any two different
columns and any two different rows, the 4 squares in their
intersections they don't have all the same color. Find the
greatest possible value of  $n$ .


Find all pairs of real numbers  $a, b$  for which the equation in the domain of the real numbers
\[\frac{ax^2-24x+b}{x^2-1}=x\]
has two solutions and the sum of them equals  $12$ .

Solution: \boxed{ (35, -5819) }


If  $x$  is a real number and  $k$  is a nonnegative integer, recall that the binomial coefficient  $\binom{x}{k}$  is defined by the formula
\[
  \binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, .
\]
Compute the value of
\[
  \frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, .
\]

Solution: \boxed{ -\frac{1}{4027} }


Say that a 4-digit positive integer is *mixed* if it has 4 distinct digits, its leftmost digit is neither the biggest nor the smallest of the 4 digits, and its rightmost digit is not the smallest of the 4 digits.  For example, 2013 is mixed.  How many 4-digit positive integers are mixed?


Given are the lines  $l_1,l_2,\ldots ,l_k$  in the plane, no two of which are parallel and no three of which are concurrent. For which  $k$  can one label the intersection points of these lines by  $1, 2,\ldots , k-1$  so that in each of the given lines all the labels appear exactly once?

Solution: \boxed{ k }


For a sequence  $(a_{n})_{n\geq 1}$  of real numbers it is known that  $a_{n}=a_{n-1}+a_{n+2}$  for  $n\geq 2$ .
What is the largest number of its consecutive elements that can all be positive?


Consider a circle centered at  $O$ . Parallel chords  $AB$  of length  $8$  and  $CD$  of length  $10$  are of distance  $2$  apart such that  $AC < AD$ . We can write  $\tan \angle BOD =\frac{a}{b}$  , where  $a, b$  are positive integers such that gcd  $(a, b) = 1$ . Compute  $a + b$ .


Let  $ABCDEF$  be a regular hexagon. Points  $P$  and  $Q$  on tangents to its circumcircle at  $A$  and  $D$  respectively are such that  $PQ$  touches the minor arc  $EF$  of this circle. Find the angle between  $PB$  and  $QC$ .


Find the number of maps  $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$  such that  $f(i) \le f(j)$  whenever  $i < j$ .


The work team was working at a rate fast enough to process  $1250$  items in ten hours. But after working for six hours, the team was given an additional  $165$  items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?


For a positive integer  $ n$ , let  $ \theta(n)$  denote the number of integers  $ 0 \leq x < 2010$  such that  $ x^2 \minus{} n$  is divisible by  $ 2010$ . Determine the remainder when  $ \displaystyle \sum_{n \equal{} 0}^{2009} n \cdot \theta(n)$  is divided by  $ 2010$ .


One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is 1000 km. Bob similarly totaled his distances to everyone else. What is the largest total that Bob could have obtained? (Consider the cities as points on the plane; if two people live in the same city, the distance between their cities is considered zero).


On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let  $ M$  be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when  $ M$  is divided by 10.


For some fixed positive integer  $n>2$ , suppose  $x_1$ ,  $x_2$ ,  $x_3$ ,  $\ldots$  is a nonconstant sequence of real numbers such that  $x_i=x_j$  if  $i \equiv j \pmod{n}$ . Let  $f(i)=x_i + x_i x_{i+1} + \dots + x_i x_{i+1} \dots x_{i+n-1}$ . Given that    $$ f(1)=f(2)=f(3)=\cdots $$   find all possible values of the product  $x_1 x_2 \ldots x_n$ .


$N $  different numbers are written on blackboard and one of these numbers is equal to  $0$ .One may take any polynomial such that each of its coefficients is equal to one of written numbers ( there may be some equal coefficients ) and write all its roots on blackboard.After some of these operations all integers between  $-2016$  and  $2016$  were written on blackboard(and some other numbers maybe). Find  the smallest possible value of  $N $ .

Solution: \boxed{ N = 2 }


Cyclic pentagon  $ ABCDE$  has side lengths  $ AB\equal{}BC\equal{}5$ ,  $ CD\equal{}DE\equal{}12$ , and  $ AE \equal{} 14$ . Determine the radius of its circumcircle.


Find all positive integers  $n$  satisfying the following conditions simultaneously:

(a) the number of positive divisors of  $n$  is not a multiple of  $8$ ;
(b) for all integers  $x$ , we have
\[x^n \equiv x \mod n.\]
*Proposed by usjl*

Solution: \boxed{ n = 1 }


In how many ways can one choose distinct numbers a and b from {1, 2, 3, ..., 2005} such that a + b is a multiple of 5?


For a positive integer  $n$ , let  $\langle n \rangle$  denote the perfect square integer closest to  $n$ . For example,  $\langle 74 \rangle = 81$ ,  $\langle 18 \rangle = 16$ . If  $N$  is the smallest positive integer such that  $$  \langle 91 \rangle \cdot \langle 120 \rangle \cdot \langle 143 \rangle \cdot \langle 180 \rangle \cdot \langle N \rangle = 91 \cdot 120 \cdot 143 \cdot 180 \cdot N  $$  find the sum of the squares of the digits of  $N$ .


Consider a  $(2m-1)\times(2n-1)$  rectangular region, where  $m$  and  $n$  are integers such that  $m,n\ge 4.$  The region is to be tiled using tiles of the two types shown:
\[
\begin{picture}(140,40)

\put(0,0){\line(0,1){40}}
\put(0,0){\line(1,0){20}}
\put(0,40){\line(1,0){40}}
\put(20,0){\line(0,1){20}}
\put(20,20){\line(1,0){20}}
\put(40,20){\line(0,1){20}}
\multiput(0,20)(5,0){4}{\line(1,0){3}}
\multiput(20,20)(0,5){4}{\line(0,1){3}}

\put(80,0){\line(1,0){40}}
\put(120,0){\line(0,1){20}}
\put(120,20){\line(1,0){20}}
\put(140,20){\line(0,1){20}}
\put(80,0){\line(0,1){20}}
\put(80,20){\line(1,0){20}}
\put(100,20){\line(0,1){20}}
\put(100,40){\line(1,0){40}}
\multiput(100,0)(0,5){4}{\line(0,1){3}}
\multiput(100,20)(5,0){4}{\line(1,0){3}}
\multiput(120,20)(0,5){4}{\line(0,1){3}}

\end{picture}
\]
(The dotted lines divide the tiles into  $1\times 1$  squares.) The tiles may be rotated and reflected, as long as their sides are parallel to the sides of the rectangular region. They must all fit within the region, and they must cover it completely without overlapping.

What is the minimum number of tiles required to tile the region?

Solution: \boxed{mn}


The product  $20! \cdot 21! \cdot 22! \cdot \cdot \cdot 28!$  can be expressed in the form  $m$   $\cdot$   $n^3$ , where m and n are positive integers, and m is not divisible by the cube of any prime. Find m.


$2$  darts are thrown randomly at a circular board with center  $O$ , such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked  $A$  and  $B$ . What is the probability that  $\angle AOB$  is acute?


For real numbers  $a, b$ , and  $c$  the polynomial  $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$  has  $7$  real roots whose sum is  $97$ . Find the sum of the reciprocals of those  $7$  roots.


Determine the smallest  $n$  such that  $n \equiv (a - 1)$  mod  $a$  for all  $a \in \{2,3,..., 10\}$ .


In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from   $1$  to  $n$  and let  $a_i$  be the number of friends of the  $i^{\text{th}}$  resident. Suppose that
\[ \sum_{i=1}^{n}a_i^2=n^2-n \]
Let  $k$  be the smallest number of residents (at least three) who can be seated at a round table in such a way that any two neighbors are friends. Determine all possible values of  $k.$

Solution: \boxed{k = 5}


In the universe of Pi Zone, points are labeled with  $2 \times 2$  arrays of positive reals. One can teleport from point  $M$  to point  $M'$  if  $M$  can be obtained from  $M'$  by multiplying either a row or column by some positive real. For example, one can teleport from  $\left( \begin{array}{cc} 1 & 2  3 & 4 \end{array} \right)$  to  $\left( \begin{array}{cc} 1 & 20  3 & 40 \end{array} \right)$  and then to  $\left( \begin{array}{cc} 1 & 20  6 & 80 \end{array} \right)$ .

A *tourist attraction* is a point where each of the entries of the associated array is either  $1$ ,  $2$ ,  $4$ ,  $8$  or  $16$ .  A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?

*Proposed by Michael Kural*


Let  $A$  be a positive integer which is a multiple of 3, but isn't a multiple of 9. If adding the product of each digit of  $A$  to  $A$  gives a multiple of 9, then find the possible minimum value of  $A$ .


The numbers  $1, 2\ldots11$  are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.


Let  $n$  be an integer greater than  $1$ . For a positive integer  $m$ , let  $S_{m}= \{ 1,2,\ldots, mn\}$ . Suppose that there exists a  $2n$ -element set  $T$  such that
(a) each element of  $T$  is an  $m$ -element subset of  $S_{m}$ ;
(b) each pair of elements of  $T$  shares at most one common element;
and
(c) each element of  $S_{m}$  is contained in exactly two elements of  $T$ .

Determine the maximum possible value of  $m$  in terms of  $n$ .

Solution: \boxed{ m = 2n - 1 }


Three couples sit for a photograph in  $2$  rows of three people each such that no couple is sitting in the same row next to each other or in the same column one behind the other. How many such arrangements are possible?


Two sides of a regular polygon of  $n$  sides when extended meet at  $28$  degrees. What is smallest possible value of  $n$


Find the least positive integer, which may not be represented as  ${2^a-2^b\over 2^c-2^d}$ , where  $a,\,b,\,c,\,d$  are positive integers.


Find the smallest positive integer  $k$  which is representable in the form  $k=19^n-5^m$  for some positive integers  $m$  and  $n$ .


Triangle  $GRT$  has  $GR=5,$   $RT=12,$  and  $GT=13.$  The perpendicular bisector of  $GT$  intersects the extension of  $GR$  at  $O.$  Find  $TO.$


For what digit  $A$  is the numeral  $1AA$  a perfect square in base- $5$  and a perfect cube in base- $6$ ? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }3\qquad\text{(E) }4$


Let  $a$  be positive numbers.For  $|x|\leq a,$  find the maximum and minimum value of  $\int_{x-a}^{x+a}\sqrt{4a^{2}-t^{2}}\ dt.$

Solution: \boxed{\pi a^2}


Find the smallest positive integer  $n$  for which the polynomial \[x^n-x^{n-1}-x^{n-2}-\cdots -x-1\] has a real root greater than  $1.999$ .

[i]Proposed by James Lin


For each positive integer  $n$ , let  $f(n)$  be the sum of the digits in the base-four representation of  $n$  and let  $g(n)$  be the sum of the digits in the base-eight representation of  $f(n)$ . For example,  $f(2020) = f(133210_\text{four}) = 10 = 12_\text{eight}$ , and  $g(2020) = \text{the digit sum of } 12_\text{eight} = 3$ . Let  $N$  be the least value of  $n$  such that the base-sixteen representation of  $g(n)$  cannot be expressed using only the digits  $0$  through  $9.$  Find the remainder when  $N$  is divided by  $1000.$


Once in a restaurant ***Dr. Strange*** found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he didn’t tried on the 1st day. Similarly, on the 3rd day, he eats either item 4 or the item he didn’t tried on the 2nd day. If someday he's not able to choose items that way, he eats the item that remained uneaten from the menu. In how many ways can he eat the items for 12 days?


Find all triples  $(a,b,c)$  of positive integers such that the product of any two of them when divided by the third leaves the remainder  $1$ .

Solution: \boxed{ (2, 3, 5) }


Let  $A=(a_{ij})$  be the  $n\times n$  matrix, where  $a_{ij}$  is the remainder of the division of  $i^j+j^i$  by  $3$  for  $i,j=1,2,\ldots,n$ . Find the greatest  $n$  for which  $\det A\ne0$ .


Across the face of a rectangular post-it note, you idly draw lines that are parallel to its edges.  Each time you draw a line, there is a  $50\%$  chance it'll be in each direction and you never draw over an existing line or the edge of the post-it note.  After a few minutes, you notice that you've drawn 20 lines.  What is the expected number of rectangles that the post-it note will be partitioned into?


Find the period of the repetend of the fraction  $\frac{39}{1428}$  by using *binary* numbers, i.e. its binary decimal representation.

(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form  $0.b_1b_2\cdots b_sa_1a_2\cdots a_ka_1a_2\cdots a_ka_1a_2 \cdots a_k \cdots$ . Here the repeated sequence  $a_1a_2\cdots a_k$  is called the *repetend* of the fraction, and the smallest length of the repetend,  $k$ , is called the *period* of the decimal number.)


The function  $f: \mathbb{R}\rightarrow \mathbb{R}$  is such that  $f(x+1)=2f(x)$  for  $\forall$   $x\in \mathbb{R}$  and  $f(x)=x(x-1)$  for  $\forall$   $x\in (0,1]$ . Find the greatest real number  $m$ , for which the inequality  $f(x)\geq -\frac{8}{9}$  is true for  $\forall$   $x\in (-\infty , m]$ .


Let  $S$  be the set of all natural numbers with the property: the sum of the biggest three divisors of number  $n$ , different from  $n$ , is bigger than  $n$ . Determine the largest natural number  $k$ , which divides any number from  $S$ .
(A natural number is a positive integer)


Yang has a standard  $6$ -sided die, a standard  $8$ -sided die, and a standard  $10$ -sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as  $\frac{m}{n}$ , for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by Yannick Yao*


Let  $L_1$  and  $L_2$  be perpendicular lines, and let  $F$  be a point at a distance  $18$  from line  $L_1$  and a distance  $25$  from line  $L_2$ . There are two distinct points,  $P$  and  $Q$ , that are each equidistant from  $F$ , from line  $L_1$ , and from line  $L_2$ . Find the area of  $\triangle{FPQ}$ .


Circle  $\omega_1$  with radius 3 is inscribed in a strip  $S$  having border lines  $a$  and  $b$ .  Circle  $\omega_2$  within  $S$  with radius 2 is tangent externally to circle  $\omega_1$  and is also tangent to line  $a$ .  Circle  $\omega_3$  within  $S$  is tangent externally to both circles  $\omega_1$  and  $\omega_2$ , and is also tangent to line  $b$ .  Compute the radius of circle  $\omega_3$ .


Call a positive integer  $N$  a  $\textit{7-10 double}$  if the digits of the base-7 representation of  $N$  form a base-10 number that is twice  $N.$   For example, 51 is a 7-10 double because its base-7 representation is 102.  What is the largest 7-10 double?


Let  $n$  be a positive integer.
Consider sequences  $a_0, a_1, ..., a_k$  and  $b_0, b_1,,..,b_k$  such that  $a_0 = b_0 = 1$  and  $a_k = b_k = n$  and such that for all  $i$  such that  $1 \le  i \le k $ , we have that  $(a_i, b_i)$  is either equal to  $(1 + a_{i-1}, b_{i-1})$  or  $(a_{i-1}; 1 + b_{i-1})$ .
Consider for  $1 \le  i \le k$  the number  $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1}
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$ Show that  $c_1 + c_2 + ... + c_k = n^2 - 1$ .

Solution: \boxed{n^2 - 1}


Determine the sum of all possible positive integers  $n, $  the product of whose digits equals  $n^2 -15n -27$ .


Find all prime numbers  $p$  such that there exists a unique  $a \in \mathbb{Z}_p$  for which  $a^3 - 3a + 1 = 0.$

Solution: \boxed{p = 3}


Find the minimum value of  $m$  such that any  $m$ -element subset of the set of integers  $\{1,2,...,2016\}$  contains at least two distinct numbers  $a$  and  $b$  which satisfy  $|a - b|\le  3$ .


A square is inscribed in a circle of radius  $6$ . A quarter circle is inscribed in the square, as shown in the diagram below. Given the area of the region inside the circle but outside the quarter circle is  $n\pi$  for some positive integer  $n$ , what is  $n$ ?
[asy]
        size(5 cm);
        draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
        draw(circle((1,1),1.41));
        draw(arc((0,0),2,0,90));[/asy]

*2021 CCA Math Bonanza Lightning Round #1.2*


For any natural number  $n$ , expressed in base  $10$ , let  $S(n)$  denote the sum of all digits of  $n$ . Find all positive integers  $n$  such that  $n^3 = 8S(n)^3+6S(n)n+1$ .


A plane cuts a sphere of radius  $ 1$  into two pieces, one of which has three times the surface area of the other. What is the area of the disk that the sphere cuts out of the plane?


Determine all prime numbers  $p$  such that  $p^2 - 6$  and  $p^2 + 6$  are both prime numbers.

Solution: \boxed{ p = 5 }


Let  $a,b$  and  $c$  be real numbers such that  $| (a-b) (b-c) (c-a) | = 1$ . Find the smallest value of the expression  $| a | + | b | + | c |$ . (K.Satylhanov )


Find the smallest integer  $k\ge3$  with the property that it is possible to choose two of the number  $1,2,...,k$  in such a way that their product is equal to the sum of the remaining  $k-2$  numbers.


The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is  $x$  (there are no ties).  The expected value for the total number of games played can be written as  $f(x)$ , with  $f$  a polynomial.  Find  $f(-1)$ .


In the coordinate plane, let  $A = (-8, 0)$ ,  $B = (8, 0)$ , and  $C = (t, 6)$ .  What is the maximum value of  $\sin m\angle CAB \cdot \sin m\angle CBA$ , over all real numbers  $t$ ?


a) Consider\[f\text{:}\left[ \text{0,}\infty  \right)\to \left[ \text{0,}\infty  \right)\] a differentiable and convex function .Show that  $f\left( x \right)\le x$ ,  for every  $x\ge 0$ , than  ${f}'\left( x \right)\le 1$  ,for every  $x\ge 0$ b) Determine \[f\text{:}\left[ \text{0,}\infty  \right)\to \left[ \text{0,}\infty  \right)\] differentiable and convex functions which have the property that  $f\left( 0 \right)=0\,$ , and  ${f}'\left( x \right)f\left( f\left( x \right) \right)=x$ , for every  $x\ge 0$

Solution: \boxed{ f(x) = x }


On rectangular coordinates, point  $A = (1,2)$ ,  $B = (3,4)$ .  $P = (a, 0)$  is on  $x$ -axis. Given that  $P$  is chosen such that  $AP + PB$  is minimized, compute  $60a$ .


Jerry purchased some stock for  $ \$ 14,400 $ at the same time that Susan purchased a bond for $  \ $6,250$ . Jerry’s investment went up  $20$  percent the first year, fell  $10$  percent the second year, and rose another  $20$  percent the third year. Susan’s investment grew at a constant rate of compound interest for three years. If both investments are worth the same after three years, what was the annual percentage increase of Susan’s investment?


Determine all triples  $(p, q, r)$  of positive integers, where  $p, q$  are also primes, such that  $\frac{r^2-5q^2}{p^2-1}=2$ .

Solution: \boxed{ (3, 2, 6) }


A subset  $S$  of the set  $M=\{1,2,.....,p-1\}$ ,where  $p$  is a prime number of the kind
 $12n+11$ ,is *essential*,if the product  ${\Pi}_s$  of all elements of the subset
is not less than the product  $\bar{{\Pi}_s}$  of all other elements of the set.The
**difference**  $\bigtriangleup_s=\Pi_s-\bar{{\Pi}_s}$  is called *the deviation*
of the subset  $S$ .Define the least possible remainder of division by  $p$  of the deviation of an essential subset,containing  $\frac{p-1}{2}$  elements.


Let  $AB$  be the diameter of a circle  $k$  and let  $E$  be a point in the interior of  $k$ . The line  $AE$  intersects  $k$  a second time in  $C \ne A$  and the line  $BE$  intersects  $k$  a second time in  $D \ne B$ .

Show that the value of  $AC \cdot AE+BD\cdot BE$  is independent of the choice of  $E$ .


For each lattice point on the Cartesian Coordinate Plane, one puts a positive integer at the lattice such that **for any given rectangle with sides parallel to coordinate axes, the sum of the number inside the rectangle is not a prime.** Find the minimal possible value of the maximum of all numbers.


Let  $p$  be an odd prime number, and let  $\mathbb{F}_p$  denote the field of integers modulo  $p$ . Let  $\mathbb{F}_p[x]$  be the ring of polynomials over  $\mathbb{F}_p$ , and let  $q(x) \in \mathbb{F}_p[x]$  be given by  $q(x) = \sum_{k=1}^{p-1} a_k x^k$  where  $a_k = k^{(p-1)/2}$  mod  $p$ . Find the greatest nonnegative integer  $n$  such that  $(x-1)^n$  divides  $q(x)$  in  $\mathbb{F}_p[x]$ .


Find all triples of positive integers  $(a,b,c)$  such that the following equations are both true:
I-   $a^2+b^2=c^2$ II-  $a^3+b^3+1=(c-1)^3$

Solution: \boxed{ (6, 8, 10) }


Determine the greatest real number  $ C $ , such that for every positive integer  $ n\ge 2 $ , there exists  $ x_1, x_2,..., x_n  \in [-1,1]$ , so that  $$ \prod_{1\le i<j\le n}(x_i-x_j) \ge C^{\frac{n(n-1)}{2}} $$  .


For each integer  $n\geq0$ , let  $S(n)=n-m^2$ , where  $m$  is the greatest integer with  $m^2\leq n$ . Define a sequence by  $a_0=A$  and  $a_{k+1}=a_k+S(a_k)$  for  $k\geq0$ . For what positive integers  $A$  is this sequence eventually constant?

Solution: \boxed{ A }


Solution: \boxed{ 0 }


A triangle  $\vartriangle A_0A_1A_2$  in the plane has sidelengths  $A_0A_1 = 7$ , $A_1A_2 = 8$ , $A_2A_0 = 9$ . For  $i \ge 0$ , given  $\vartriangle A_iA_{i+1}A_{i+2}$ , let  $A_{i+3}$  be the midpoint of  $A_iA_{i+1}$  and let Gi be the centroid of  $\vartriangle A_iA_{i+1}A_{i+2}$ . Let point  $G$  be the limit of the sequence of points  $\{G_i\}^{\infty}_{i=0}$ . If the distance between  $G$  and  $G_0$  can be written as  $\frac{a\sqrt{b}}{c}$  , where  $a, b, c$  are positive integers such that  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime, find  $a^2 + b^2 + c^2$ .


In a sequence of natural numbers  $ a_1 $ ,  $ a_2 $ ,  $ \dots $ ,  $ a_ {1999} $ ,  $ a_n-a_ {n-1} -a_ {n-2} $  is divisible by  $ 100 (3 \leq n \leq 1999) $ . It is known that  $ a_1 =  19$  and  $ a_2 =  99$ . Find the remainder of  $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $  by  $8$ .


Each of  $n$  boys and  $n$  girls chooses a random number from the set  $\{ 1, 2, 3, 4, 5 \}$ , uniformly and independently.  Let  $p_n$  be the probability that every boy chooses a different number than every girl.  As  $n$  approaches infinity, what value does  $\sqrt[n]{p_n}$  approach?


Bob has  $30$  identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid?

*Proposed by Nathan Xiong*


Given a triangle  $ABC$  with angles  $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let  $H$  be its orthocentre, and  $O$  be its circumcenter. Let  $F$  be the midpoint of side  $AB$ , and  $Q$  be the foot of the perpendicular from  $B$  onto  $AC$ . Denote by  $X$  the intersection point of the lines  $FH$  and  $QO$ . Suppose the ratio of the length of  $FX$  and the circumradius of the triangle is given by  $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of  $1000a + 100b + 10c + d$ .


Find the number of positive integers  $x$  such that \[ \left[ \frac{x}{99} \right] = \left[ \frac{x}{101} \right] .  \]


In the convex quadrilateral  $ABCD$ , point  $X$  is selected on side  $AD$ , and the diagonals intersect at point  $E$ . It is known that  $AC = BD$ ,  $\angle ABX = \angle AX B = 50^o$ ,  $\angle CAD = 51^o$ ,  $\angle AED = 80^o$ . Find the value of angle  $\angle AXC$ .


Let  $M$  be the intersection of two diagonal,  $AC$  and  $BD$ , of a rhombus  $ABCD$ , where angle  $A<90^\circ$ . Construct  $O$  on segment  $MC$  so that  $OB<OC$  and let  $t=\frac{MA}{MO}$ , provided that  $O \neq M$ . Construct a circle that has  $O$  as centre and goes through  $B$  and  $D$ . Let the intersections between the circle and  $AB$  be  $B$  and  $X$ . Let the intersections between the circle and  $BC$  be  $B$  and  $Y$ . Let the intersections of  $AC$  with  $DX$  and  $DY$  be  $P$  and  $Q$ , respectively. Express  $\frac{OQ}{OP}$  in terms of  $t$ .

Solution: \boxed{\frac{t + 1}{t - 1}}


A *site* is any point  $(x, y)$  in the plane such that  $x$  and  $y$  are both positive integers less than or equal to 20.

Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to  $\sqrt{5}$ . On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone.

Find the greatest  $K$  such that Amy can ensure that she places at least  $K$  red stones, no matter how Ben places his blue stones.

*Proposed by Gurgen Asatryan, Armenia*

Solution: \boxed{ 100 }


Find the smallest integer  $n$  satisfying the following condition: regardless of how one colour the vertices of a regular  $n$ -gon with either red, yellow or blue, one can always find an isosceles trapezoid whose vertices are of the same colour.

Solution: \boxed{ n = 17 }


In an exam every question is solved by exactly four students, every pair of questions is solved by exactly one student, and none of the students solved all of the questions. Find the maximum possible number of questions in this exam.


Given is a prime number  $p$  and natural  $n$  such that  $p \geq n \geq 3$ . Set  $A$  is made of sequences of lenght  $n$  with elements from the set  $\{0,1,2,...,p-1\}$  and have the following property:
  For arbitrary two sequence  $(x_1,...,x_n)$  and  $(y_1,...,y_n)$  from the set  $A$  there exist three different numbers  $k,l,m$  such that:

 $x_k \not = y_k$ ,  $x_l \not = y_l$ ,  $x_m \not = y_m$ .
Find the largest possible cardinality of  $A$ .


Let  $N$  be the number of ordered pairs of integers  $(x, y)$  such that
\[
  4x^2 + 9y^2 \le 1000000000.
\]
Let  $a$  be the first digit of  $N$  (from the left) and let  $b$  be the second digit of  $N$ .  What is the value of  $10a + b$  ?


Consider real numbers  $A$ ,  $B$ , \dots,  $Z$  such that \[
  EVIL = \frac{5}{31}, \;
  LOVE = \frac{6}{29}, \text{ and }
  IMO  = \frac{7}{3}.
\] If  $OMO = \tfrac mn$  for relatively prime positive integers  $m$  and  $n$ , find the value of  $m+n$ .

*Proposed by Evan Chen*


Show that the expression  $(a + b + 1) (a + b - 1) (a - b + 1) (- a + b + 1)$ , where  $a =\sqrt{1 + x^2}$ ,  $b =\sqrt{1 + y^2}$  and  $x + y = 1$  is constant ¸and be calculated that constant value.


For how many natural numbers  $n$  between  $1$  and  $2014$  (both inclusive) is  $\frac{8n}{9999-n}$  an integer?


Find the largest real number  $a$  such that \[\left\{ \begin{array}{l}
x - 4y = 1
ax + 3y = 1
 \end{array} \right.
\] has an integer solution.


Find the maximum number of different integers that can be selected from the set  $ \{1,2,...,2013\}$  so that no two exist that their difference equals to  $17$ .


The radius  $r$  of a circle with center at the origin is an odd integer.
There is a point ( $p^m, q^n$ ) on the circle, with  $p,q$  prime numbers and  $m,n$  positive integers.
Determine  $r$ .


Let  $ABC$  be an isosceles right triangle with  $\angle A=90^o$ . Point  $D$  is the midpoint of the side  $[AC]$ , and point  $E \in [AC]$  is so that  $EC = 2AE$ . Calculate  $\angle AEB + \angle ADB$  .


There is a  $2n\times 2n$  rectangular grid and a chair in each cell of the grid. Now, there are  $2n^2$  pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at  $(3,3)$  and  $(2,5)$  respectively, then the distance between them is  $|3-2|+|3-5|=3$ . Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.


The area of the region in the  $xy$ -plane satisfying the inequality \[\min_{1 \le n \le 10} \max\left(\frac{x^2+y^2}{4n^2}, \, 2 - \frac{x^2+y^2}{4n^2-4n+1}\right) \le 1\] is  $k\pi$ , for some integer  $k$ . Find  $k$ .

*Proposed by Michael Tang*


Let  $f(x) = x-\tfrac1{x}$ , and define  $f^1(x) = f(x)$  and  $f^n(x) = f(f^{n-1}(x))$  for  $n\ge2$ . For each  $n$ , there is a minimal degree  $d_n$  such that there exist polynomials  $p$  and  $q$  with  $f^n(x) = \tfrac{p(x)}{q(x)}$  and the degree of  $q$  is equal to  $d_n$ . Find  $d_n$ .

Solution: \boxed{ d_n = 2^n - 1 }


A store offers packages of  $12$  pens for  $\$ 10 $ and packages of $ 20 $ pens for $ \ $15$ . Using only these two types of packages of pens, find the greatest number of pens  $\$ 173$ can buy at this store.

*Proposed by James Lin*


Find all positive integers  $n$  such that the inequality   $$ \left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i $$   holds for any  $n$  positive numbers  $a_1, \dots, a_n$ .

Solution: \boxed{ n = 3 }


Let  $b$  be a real number randomly sepected from the interval  $[-17,17]$ .  Then,  $m$  and  $n$  are two relatively prime positive integers such that  $m/n$  is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has  $\textit{at least}$  two distinct real solutions.  Find the value of  $m+n$ .


Points  $E$  and  $F$  are chosen on sides  $BC$  and  $CD$  respectively of rhombus  $ABCD$  such that  $AB=AE=AF=EF$ , and  $FC,DF,BE,EC>0$ . Compute the measure of  $\angle ABC$ .


Let  $\alpha\geq 1$  be a real number. Define the set  $$ A(\alpha)=\{\lfloor \alpha\rfloor,\lfloor 2\alpha\rfloor, \lfloor 3\alpha\rfloor,\dots\} $$  Suppose that all the positive integers that **does not belong** to the  $A(\alpha)$  are exactly the positive integers that have the same remainder  $r$  in the division by  $2021$  with  $0\leq r<2021$ . Determine all the possible values of  $\alpha$ .


Let  $\mathcal{F}$  be the set of continuous functions  $f: \mathbb{R} \to \mathbb{R}$  such that   $$ e^{f(x)}+f(x) \geq x+1, \: \forall x \in \mathbb{R} $$  For  $f \in \mathcal{F},$  let   $$ I(f)=\int_0^ef(x) dx $$  Determine  $\min_{f \in \mathcal{F}}I(f).$ *Liviu Vlaicu*


What is the smallest number that can be written as a sum of  $2$  squares in  $3$  ways?


Let ABC be a right triangle with  $\angle B = 90^{\circ}$ .Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.


$56$  lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly  $594$  points, what is the maximum number of them that could have the same slope?


A polyhedron has faces that all either triangles or squares. No two square faces share an edge, and no two triangular faces share an edge. What is the ratio of the number of triangular faces to the number of square faces?


Let  $S$  be the set of all integers  $k$ ,  $1\leq k\leq n$ , such that  $\gcd(k,n)=1$ . What is the arithmetic mean of the integers in  $S$ ?

Solution: \boxed{ \frac{n}{2} }


Let  $H$  be a rectangle with angle between two diagonal  $\leq 45^{0}$ . Rotation  $H$  around the its center with angle  $0^{0}\leq x\leq 360^{0}$  we have rectangle  $H_{x}$ . Find  $x$  such that  $[H\cap H_{x}]$  minimum, where  $[S]$  is area of  $S$ .

Solution: \boxed{ x = 45^\circ }


Let  $a$ ,  $b$ ,  $c$  be positive integers such that  $abc + bc + c = 2014$ . Find the minimum possible value of  $a + b + c$ .


There are  $n$  rooms in a sauna, each has unlimited capacity. No room may be attended by a female and a male simultaneously. Moreover, males want to share a room only with males that they don't know and females want to share a room only with females that they know. Find the biggest number  $k$  such that any  $k$  couples can visit the sauna at the same time, given that two males know each other if and only if their wives know each other.


Find the integer  $n$  such that
\[n + \left\lfloor\sqrt{n}\right\rfloor + \left\lfloor\sqrt{\sqrt{n}}\right\rfloor = 2017.\] Here, as usual,  $\lfloor\cdot\rfloor$  denotes the floor function.


In a trapezoid  $ABCD$ , the internal bisector of angle  $A$  intersects the base  $BC$ (or its extension) at the point  $E$ . Inscribed in the triangle  $ABE$  is a circle touching the side  $AB$  at  $M$  and side  $BE$  at the point  $P$ . Find the angle  $DAE$  in degrees, if  $AB:MP=2$ .


For all pairs  $(m, n)$  of positive integers that have the same number  $k$  of divisors we define the operation  $\circ$ . Write all their divisors in an ascending order:  $1=m_1<\ldots<m_k=m$ ,  $1=n_1<\ldots<n_k=n$  and set  $$ m\circ n= m_1\cdot n_1+\ldots+m_k\cdot n_k.  $$  Find all pairs of numbers  $(m, n)$ ,  $m\geqslant n$ , such that  $m\circ n=497$ .

Solution: \boxed{ (18, 20) }


Juca has decided to call all positive integers with 8 digits as  $sextalternados$  if it is a multiple of 30 and its consecutive digits have different parity. At the same time, Carlos decided to classify all  $sextalternados$  that are multiples of 12 as  $super sextalternados$ .

a) Show that  $super sextalternados$  numbers don't exist.

b) Find the smallest  $sextalternado$  number.


Find the largest positive integer  $n$  such that  $n$  is divisible by all the positive integers less than  $\sqrt[3]{n}$ .


Find all  $3$ -digit numbers  $\overline{abc}$  ( $a,b \ne 0$ ) such that  $\overline{bcd} \times  a = \overline{1a4d}$  for some integer  $d$  from  $1$  to  $9$


A square grid  $100 \times 100$  is tiled in two ways - only with dominoes and only with squares  $2 \times 2$ . What is the least number of dominoes that are entirely inside some square  $2 \times 2$ ?


The side lengths  $a,b,c$  of a triangle  $ABC$  are positive integers. Let:

\[T_{n}=(a+b+c)^{2n}-(a-b+c)^{2n}-(a+b-c)^{2n}+(a-b-c)^{2n}\]

for any positive integer  $n$ . If  $\frac{T_{2}}{2T_{1}}=2023$  and  $a>b>c$  , determine all possible perimeters of the triangle  $ABC$ .


Find the integer  $n \ge 48$  for which the number of trailing zeros in the decimal representation of  $n!$  is exactly  $n-48$ .

*Proposed by Kevin Sun*


The product of  $10$  integers is  $1024$ . What is the greatest possible sum of these  $10$  integers?


Let  $n$  be a nonnegative integer less than  $2023$  such that  $2n^2 + 3n$  is a perfect square. What is the sum of all possible  $n$ ?

*Proposed by Giacomo Rizzo*


Shenelle has some square tiles. Some of the tiles have side length  $5\text{ cm}$  while the others have side length  $3\text{ cm}$ . The total area that can be covered by the tiles is exactly  $2014\text{ cm}^2$ . Find the least number of tiles that Shenelle can have.


Find the mathematical expectation of the area of the projection of a cube with edge of length  $1$  onto a plane with an isotropically distributed random direction of projection.


Find  $k$  where  $2^k$  is the largest power of  $2$  that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]


For a positive integer  $n$ , let  $\omega(n)$  denote the number of positive prime divisors of  $n$ . Find the smallest positive tinteger  $k$  such that  $2^{\omega(n)}\leq k\sqrt[4]{n}\forall n\in\mathbb{N}$ .


Determine the smallest prime  $p$  such that  $2018!$  is divisible by  $p^{3}$  , but not divisible by  $p^{4}$ .


Find all monic polynomials  $f$  with integer coefficients satisfying the following condition: there exists a positive integer  $N$  such that  $p$  divides  $2(f(p)!)+1$  for every prime  $p>N$  for which  $f(p)$  is a positive integer.

*Note: A monic polynomial has a leading coefficient equal to 1.*

*(Greece - Panagiotis Lolas and Silouanos Brazitikos)*

Solution: \boxed{f(x) = x - 3}


Let  $ABC$  be a triangle with sides  $51, 52, 53$ . Let  $\Omega$  denote the incircle of  $\bigtriangleup ABC$ . Draw tangents to  $\Omega$  which are parallel to the sides of  $ABC$ . Let  $r_1, r_2, r_3$  be the inradii of the three corener triangles so formed, Find the largest integer that does not exceed  $r_1 + r_2 + r_3$ .


Find the derivative of the solution of the equation  $\ddot{x} = \dot{x}^2 + x^3$  with initial condition  $x(0) = 0$ ,  $\dot{x}(0) = A$  with respect to  $A$  for  $A = 0$ .

Solution: \boxed{t}


In a certain tournament bracket, a player must be defeated three times to be eliminated. If 512 contestants enter the tournament, what is the greatest number of games that could be played?


On  $x-y$  plane,  let  $C: y=2006x^{3}-12070102x^{2}+\cdots.$
Find the area of the region surrounded by the tangent line of  $C$  at  $x=2006$  and the curve  $C.$


Circles  $\mathcal{P}$  and  $\mathcal{Q}$  have radii  $1$  and  $4$ , respectively, and are externally tangent at point  $A$ . Point  $B$  is on  $\mathcal{P}$  and point  $C$  is on  $\mathcal{Q}$  so that line  $BC$  is a common external tangent of the two circles. A line  $\ell$  through  $A$  intersects  $\mathcal{P}$  again at  $D$  and intersects  $\mathcal{Q}$  again at  $E$ . Points  $B$  and  $C$  lie on the same side of  $\ell$ , and the areas of  $\triangle DBA$  and  $\triangle ACE$  are equal. This common area is  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

[asy]
import cse5;
pathpen=black; pointpen=black;
size(6cm);

pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689);

filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7));
filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7));
D(CR((0,1),1)); D(CR((4,4),4,150,390));
D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5));
D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0));
D(MP("E",E,N));
[/asy]


Let  $f : [0, 1] \to \mathbb R$  be continuous and satisfy:
\[ \begin{cases}bf(2x) = f(x), &\mbox{ if } 0 \leq x \leq 1/2, f(x) = b + (1 - b)f(2x - 1), &\mbox{ if } 1/2 \leq x \leq 1,\end{cases}\]
where  $b = \frac{1+c}{2+c}$ ,  $c > 0$ . Show that  $0 < f(x)-x < c$  for every  $x, 0 < x < 1.$

Solution: \boxed{ 0 < f(x) - x < c }


Bianca has a rectangle whose length and width are distinct primes less than 100.  Let  $P$  be the perimeter of her rectangle, and let  $A$  be the area of her rectangle.  What is the least possible value of  $\frac{P^2}{A}$ ?


For positive integers  $n,$  let  $s(n)$  be the sum of the digits of  $n.$  Over all four-digit positive integers  $n,$  which value of  $n$  maximizes the ratio  $\frac{s(n)}{n}$ ?

*Proposed by Michael Tang*

Solution: \boxed{ 1099 }


The positive numbers  $a, b, c,d,e$  are such that the following identity hold for all real number  $x$ : $(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$ . Find the smallest value of  $d$ .


Let  $n$  be a positive integer. A sequence of  $n$  positive integers (not necessarily distinct) is called **full** if it satisfies the following condition: for each positive integer  $k\geq2$ , if the number  $k$  appears in the sequence then so does the number  $k-1$ , and moreover the first occurrence of  $k-1$  comes before the last occurrence of  $k$ . For each  $n$ , how many full sequences are there ?

Solution: \boxed{n!}


For any real number  $t$ , let  $\lfloor t \rfloor$  denote the largest integer  $\le t$ . Suppose that  $N$  is the greatest integer such that   $$ \left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4 $$  Find the sum of digits of  $N$ .


At a meeting of  $ 12k$  people, each person exchanges greetings with exactly  $ 3k\plus{}6$  others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?


Find the number of trailing zeros at the end of the base- $10$  representation of the integer  $525^{25^2}
\cdot 252^{52^5}$  .


In  $ISI$  club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does  $ISI$  club have????


Integers  $1, 2, 3, ... ,n$ , where  $n > 2$ , are written on a board. Two numbers  $m, k$  such that  $1 < m < n, 1 < k < n$  are removed and the average of the remaining numbers is found to be  $17$ . What is the maximum sum of the two removed numbers?


Find the least possible cardinality of a set  $A$  of natural numbers, the smallest and greatest of which are  $1$  and  $100$ , and having the property that every element of  $A$  except for  $1$  equals the sum of two elements of  $A$ .

Solution: \boxed{ 9 }


Let  $a$  be a complex number, and set  $\alpha$ ,  $\beta$ , and  $\gamma$  to be the roots of the polynomial  $x^3 - x^2 + ax - 1$ .  Suppose \[(\alpha^3+1)(\beta^3+1)(\gamma^3+1) = 2018.\] Compute the product of all possible values of  $a$ .


Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.

Solution: \boxed{ 2 \cdot 3^{662} }


Find largest possible constant  $M$  such that, for any sequence  $a_n$ ,  $n=0,1,2,...$  of real numbers, that satisfies the conditions :
i)  $a_0=1$ ,  $a_1=3$ ii)  $a_0+a_1+...+a_{n-1} \ge 3 a_n - a_{n+1}$  for any integer  $n\ge 1$ to be true that  $$ \frac{a_{n+1}}{a_n} >M $$   for any integer  $n\ge 0$ .

Solution: \boxed{ M = 2 }


Let  $P(x) = x^2 + ax + b$  be a quadratic polynomial. For how many pairs  $(a, b)$  of positive integers where  $a, b < 1000$  do the quadratics  $P(x+1)$  and  $P(x) + 1$  have at least one root in common?


 $n \ge 4$  real numbers are arranged in a circle. It turned out that for any four consecutive numbers  $a, b, c, d$ , that lie on the circle in this order, holds  $a+d = b+c$ . For which  $n$  does it follow that all numbers on the circle are equal?

*Proposed by Oleksiy Masalitin*

Solution: \boxed{ n }


Let  $a_1,a_2,...,a_9$  be nine real numbers, not necessarily distinct, with average  $m$ . Let  $A$  denote the number of triples  $1 \le i < j < k \le 9$  for which  $a_i + a_j + a_k \ge 3m$ .  What is the minimum possible value of  $A$ ?

*Proposed by Ray Li*


I have  $8$  unit cubes of different colors, which I want to glue together into a  $2\times 2\times 2$  cube.  How many distinct  $2\times 2\times 2$  cubes can I make?  Rotations of the same cube are not considered distinct, but reflections are.


Determine the maximal value of  $k$  such that the inequality  $$ \left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left(  \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right) $$  holds for all positive reals  $a, b, c$ .

Solution: \boxed{ \sqrt[3]{9} - 1 }


Let  $a$  and  $b$  be the two possible values of  $\tan\theta$  given that \[\sin\theta + \cos\theta = \dfrac{193}{137}.\] If  $a+b=m/n$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $m+n$ .


Given  $a,x\in\mathbb{R}$  and  $x\geq 0$ , $a\geq 0$  . Also  $\sin(\sqrt{x+a})=\sin(\sqrt{x})$  . What can you say about  $a$ ??? Justify your answer.

Solution: \boxed{ a = 0 }


Show that there exists an infinite arithmetic progression of natural numbers such that the first term is  $16$  and the number of positive divisors of each term is divisible by  $5$ . Of all such sequences, find the one with the smallest possible common difference.


Points  $E$  and  $F$  lie inside rectangle  $ABCD$  with  $AE=DE=BF=CF=EF$ . If  $AB=11$  and  $BC=8$ , find the area of the quadrilateral  $AEFB$ .


In how many ways can six marbles be placed in the squares of a  $6$ -by- $6$  grid such that no two marbles lie in the same row or column?


Let  $n$  be an odd integer greater than  $11$ ;  $k\in \mathbb{N}$ ,  $k \geq 6$ ,  $n=2k-1$ .
We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for  $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$  and  $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$ .
Show that  $n=23$  if  $T$  has a subset  $S$  satisfying
[list=i]
[*] $|S|=2^k$ [*]For each  $x \in T$ , there exists exacly one  $y\in S$  such that  $d(x,y)\leq 3$ [/list]

Solution: \boxed{ n = 23 }


Determine the least integer  $n$  such that for any set of  $n$  lines in the 2D plane, there exists either a subset of  $1001$  lines that are all parallel, or a subset of  $1001$  lines that are pairwise nonparallel.

*Proposed by Samuel Wang*

Solution*Solution.*  $\boxed{1000001}$ Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most  $1001$  groups of lines, all pairwise intersecting, with each group containing at most  $1001$  lines. Thus,  $n = 1000^2 + 1 =
\boxed{1000001}$ .


An equilateral triangle of side  $n$  is divided into equilateral triangles of side  $1$ .  Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.

Solution: \boxed{ n(n+1) }


The numbers  $\frac{1}{1}, \frac{1}{2}, \cdots , \frac{1}{2012}$  are written on the blackboard. Aïcha chooses any two numbers from the blackboard, say  $x$  and  $y$ , erases them and she writes instead the number  $x + y + xy$ . She continues to do this until only one number is left on the board. What are the possible values of the final number?


 $p$  is a prime number that is greater than  $2$ . Let  $\{ a_{n}\}$  be a sequence such that  $ na_{n+1}= (n+1) a_{n}-\left( \frac{p}{2}\right)^{4}$ .

Show that if  $a_{1}=5$ , the  $16 \mid a_{81}$ .

Solution: \boxed{ 16 \mid a_{81} }


A sequence  $a_1, a_2, a_3, \ldots$  of positive integers satisfies  $a_1 > 5$  and  $a_{n+1} = 5 + 6 + \cdots + a_n$  for all positive integers  $n$ . Determine all prime numbers  $p$  such that, regardless of the value of  $a_1$ , this sequence must contain a multiple of  $p$ .

Solution: \boxed{ p = 2 }


The student population at one high school consists of freshmen, sophomores, juniors, and seniors.  There are 25 percent more freshmen than juniors, 10 percent fewer sophomores than freshmen, and 20 percent of the students are seniors. If there are 144 sophomores, how many students attend the school?


Let  $BE$  and  $CF$  be altitudes in triangle  $ABC$ . If  $AE = 24$ ,  $EC = 60$ , and  $BF = 31$ , determine  $AF$ .


Suppose the side lengths of triangle  $ABC$  are the roots of polynomial  $x^3 - 27x^2 + 222x - 540$ . What is the product of its inradius and circumradius?


Let  $S$  be the locus of all points  $(x,y)$  in the first quadrant such that  $\dfrac{x}{t}+\dfrac{y}{1-t}=1$  for some  $t$  with  $0<t<1$ .  Find the area of  $S$ .


$22$  football players took part in the football training. They were divided into teams of equal size for each game ( $11:11$ ). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.


Given the sequence  $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$  find  $n$  such that the sum of the first  $n$  terms is  $2008$  or  $2009$ .


Two tangents  $AT$  and  $BT$  touch a circle at  $A$  and  $B$ , respectively, and meet perpendicularly at  $T$ .  $Q$  is on  $AT$ ,  $S$  is on  $BT$ , and  $R$  is on the circle, so that  $QRST$  is a rectangle with  $QT = 8$  and  $ST = 9$ . Determine the radius of the circle.


Let  $n$  be a positive integer and let  $d_{1},d_{2},,\ldots ,d_{k}$  be its divisors, such that  $1=d_{1}<d_{2}<\ldots <d_{k}=n$ . Find all values of  $n$  for which  $k\geq 4$  and  $n=d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}$ .


A triangle  $\triangle ABC$  satisfies  $AB = 13$ ,  $BC = 14$ , and  $AC = 15$ . Inside  $\triangle ABC$  are three points  $X$ ,  $Y$ , and  $Z$  such that:

- $Y$  is the centroid of  $\triangle ABX$
- $Z$  is the centroid of  $\triangle BCY$
- $X$  is the centroid of  $\triangle CAZ$

    What is the area of  $\triangle XYZ$ ?

*Proposed by Adam Bertelli*


Consider an  $8\times 8$  grid of squares. A rook is placed in the lower left corner, and every minute it moves to a square in the same row or column with equal probability (the rook must move; i.e. it cannot stay in the same square). What is the expected number of minutes until the rook reaches the upper right corner?


The positive integers  $ a$  and  $ b$  are such that the numbers   $ 15a \plus{} 16b$  and  $ 16a \minus{} 15b$  are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?


Let  $X$  be a set with  $n\ge 2$  elements. Define  $\mathcal{P}(X)$  to be the set of all subsets of  $X$ . Find the number of functions  $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$  such that  $$ |f(A)\cap f(B)|=|A\cap B| $$  whenever  $A$  and  $B$  are two distinct subsets of  $X$ .

*(Sergiu Novac)*

Solution: \boxed{n!}


A positive integer is *bold* iff it has  $8$  positive divisors that sum up to  $3240$ . For example,  $2006$  is bold because its  $8$  positive divisors,  $1$ ,  $2$ ,  $17$ ,  $34$ ,  $59$ ,  $118$ ,  $1003$  and  $2006$ , sum up to  $3240$ . Find the smallest positive bold number.


There are  $9$  cards with the numbers  $1, 2, 3, 4, 5, 6, 7, 8$  and  $9$ . What is the largest number of these cards can be decomposed in a certain order in a row, so that in any two adjacent cards, one of the numbers is divided by the other?


Let the complex number  $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$  Find the smallest positive integer  $n$  so that  $z^n$  has an imaginary part which exceeds  $\tfrac{1}{2}.$


Determine all integers  $a$  for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions  $x, \;y$ .

Solution: \boxed{ a = 2 }


If  $x > 10$ , what is the greatest possible value of the expression
\[
  {( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10.


Deﬁne the determinant  $D_1$  =  $|1|$ , the determinant  $D_2$  =      $|1 1|$   $|1 3|$
, and the determinant  $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant  $D_n$  have 1s in every position of its ﬁrst row and ﬁrst
column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that  $D_n$   $\geq$  2015.


Let  $C_1$  be a circle with centre  $O$ , and let  $AB$  be a chord of the circle that is not a diameter.  $M$  is the midpoint of  $AB$ . Consider a point  $T$  on the circle  $C_2$  with diameter  $OM$ . The tangent to  $C_2$  at the point  $T$  intersects  $C_1$  at two points. Let  $P$  be one of these points. Show that  $PA^2+PB^2=4PT^2$ .

Solution: \boxed{PA^2 + PB^2 = 4PT^2}


We define the polynomial   $$ P (x) = 2014x^{2013} + 2013x^{2012} +... + 4x^3 + 3x^2 + 2x. $$   Find the largest prime divisor of  $P (2)$ .


Let  $A,B,C$  be nodes of the lattice  $Z\times Z$  such that inside the triangle  $ABC$  lies a unique node  $P$  of the lattice. Denote  $E = AP \cap BC$ . Determine max  $\frac{AP}{PE}$  , over all such configurations.


For positive integers  $n$ , let  $S_n$  be the set of integers  $x$  such that  $n$  distinct lines, no three concurrent, can divide a plane into  $x$  regions (for example,  $S_2=\{3,4\}$ , because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise). What is the minimum  $i$  such that  $S_i$  contains at least 4 elements?


Consider the sum   $$ S =\sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|. $$  The value of  $S$  can be written as  $\tan \left( \frac{c\pi}{d} \right)$  for some relatively prime positive integers  $c, d$ , satisfying  $2c < d$ . Find the value of  $c + d$ .


Let  $P(X,Y)=X^2+2aXY+Y^2$  be a real polynomial where  $|a|\geq 1$ . For a given positive integer  $n$ ,  $n\geq 2$  consider the system of equations:  \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions  $(x_1,x_2,\ldots,x_n)$  and  $(y_1,y_2,\ldots,y_n)$  of the system to be equivalent if there exists a real number  $\lambda \neq 0$ ,  $x_1=\lambda y_1$ ,  $\ldots$ ,  $x_n= \lambda y_n$ . How many nonequivalent solutions does the system have?

*Mircea Becheanu*


Numbers  $x,y,z$  are positive integers and satisfy the equation  $x+y+z=2013$ . (E)
a) Find the number of the triplets  $(x,y,z)$  that are solutions of the equation (E).
b) Find the number of the solutions of the equation (E) for which  $x=y$ .
c) Find the solution  $(x,y,z)$   of the equation (E) for which the product  $xyz$  becomes maximum.

Solution: \boxed{(671, 671, 671)}


On  $5\times 5$  squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile.


There are  $n\leq 99$  people around a circular table. At every moment everyone can either be truthful (always says the truth) or a liar (always lies). Initially some of people (possibly none) are truthful and the rest are liars. At every minute everyone answers at the same time the question "Is your left neighbour truthful or a liar?" and then becomes the same type of person as his answer. Determine the largest  $n$  for which, no matter who are the truthful people in the beginning, at some point everyone will become truthful forever.


Point  $P$  and equilateral triangle  $ABC$  satisfy  $|AP|=2$ ,  $|BP|=3$ . Maximize  $|CP|$ .


We wish to write  $n$  distinct real numbers  $(n\geq3)$  on the circumference of a circle in such a way that each number is equal to the product of its immediate neighbors to the left and right. Determine all of the values of  $n$  such that this is possible.

Solution: \boxed{ n = 6 }


A jar contains  $2$  yellow candies,  $4$  red candies, and  $6$  blue candies. Candies are randomly drawn out of the jar one-by-one and eaten. The probability that the  $2$  yellow candies will be eaten before any of the red candies are eaten is given by the fraction  $\frac{m}{n}$  where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


Find all  $\{a_n\}_{n\ge 0}$  that satisfies the following conditions.

(1)  $a_n\in \mathbb{Z}$ (2)  $a_0=0, a_1=1$ (3) For infinitly many  $m$ ,  $a_m=m$ (4) For every  $n\ge2$ ,  $\{2a_i-a_{i-1} | i=1, 2, 3, \cdots , n\}\equiv \{0, 1, 2, \cdots , n-1\}$   $\mod n$

Solution: \boxed{a_n = n}


Given the polynomial   $$ P(x) = 1+3x + 5x^2 + 7x^3 + ...+ 1001x^{500}. $$  Express the numerical value of its derivative of order  $325$  for  $x = 0$ .

Solution: \boxed{651 \cdot 325!}


Each rational number is painted either white or red. Call such a coloring of the rationals *sanferminera* if for any distinct rationals numbers  $x$  and  $y$  satisfying one of the following three conditions: [list=1][*] $xy=1$ ,
[*] $x+y=0$ ,
[*] $x+y=1$ ,[/list]we have  $x$  and  $y$  painted different colors. How many sanferminera colorings are there?


Let  $\alpha\in\mathbb R\backslash \{ 0 \}$  and suppose that  $F$  and  $G$  are linear maps (operators) from  $\mathbb R^n$  into  $\mathbb R^n$  satisfying  $F\circ G - G\circ F=\alpha F$ .

a) Show that for all  $k\in\mathbb N$  one has  $F^k\circ G-G\circ F^k=\alpha kF^k$ .

b) Show that there exists  $k\geq 1$  such that  $F^k=0$ .

Solution: \boxed{ F^k = 0 }


From the positive integers,  $m,m+1,\dots,m+n$ , only the sum of digits of  $m$  and the sum of digits of  $m+n$  are divisible by  $8$ . Find the maximum value of  $n$ .


There are  $100$  white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.


Crisp All, a basketball player, is *dropping dimes* and nickels on a number line. Crisp drops a dime on every positive multiple of  $10$ , and a nickel on every multiple of  $5$  that is not a multiple of  $10$ . Crisp then starts at  $0$ . Every second, he has a  $\frac{2}{3}$  chance of jumping from his current location  $x$  to  $x+3$ , and a  $\frac{1}{3}$  chance of jumping from his current location  $x$  to  $x+7$ . When Crisp jumps on either a dime or a nickel, he stops jumping. What is the probability that Crisp *stops on a dime*?


How many distinct lines pass through the point  $(0, 2016)$  and intersect the parabola  $y = x^2$  at two lattice points?  (A lattice point is a point whose coordinates are integers.)


A certain function  $f$  has the properties that  $f(3x)=3f(x)$  for all positive real values of  $x$ , and that  $f(x)=1-\mid x-2 \mid$  for  $1\leq x \leq 3$ . Find the smallest  $x$  for which  $f(x)=f(2001)$ .


Consider the polynomials
\[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\]
\[g(p) = p^3 + 2p + m.\]
Find all integral values of  $m$  for which  $f$  is divisible by  $g$ .

Solution: \boxed{ m = 3 }


Define  $f\left(n\right)=\textrm{LCM}\left(1,2,\ldots,n\right)$ . Determine the smallest positive integer  $a$  such that  $f\left(a\right)=f\left(a+2\right)$ .

*2017 CCA Math Bonanza Lightning Round #2.4*


Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of  $2$ .

*Proposed by Harun Khan*


A positive integer  $n$  has the property that there are three positive integers  $x, y, z$  such that  $\text{lcm}(x, y) = 180$ ,  $\text{lcm}(x, z) = 900$ , and  $\text{lcm}(y, z) = n$ , where  $\text{lcm}$  denotes the lowest common multiple. Determine the number of positive integers  $n$  with this property.


Tony plays a game in which he takes  $40$  nickels out of a roll and tosses them one at a time toward his desk where his change jar sits.  He awards himself  $5$  points for each nickel that lands in the jar, and takes away  $2$  points from his score for each nickel that hits the ground.  After Tony is done tossing all  $40$  nickels, he computes  $88$  as his score.  Find the greatest number of nickels he could have successfully tossed into the jar.


Call the Graph the set which composed of several vertices  $P_1,\ \cdots P_2$  and several edges  $($ segments $)$  connecting two points among these vertices. Now let  $G$  be a graph with 9 vertices and satisfies the following condition.

Condition: Even if we select any five points from the vertices in  $G,$  there exist at least two edges whose endpoints are included in the set of 5 points.

What is the minimum possible numbers of edges satisfying the condition?


In an international meeting of  $n \geq 3$  participants, 14 languages are spoken. We know that:

- Any 3 participants speak a common language.

- No language is spoken more that by the half of the participants.

What is the least value of  $n$ ?


Let  $ f(n)$  be the number of times you have to hit the  $ \sqrt {\ }$  key on a calculator to get a number less than  $ 2$  starting from  $ n$ . For instance,  $ f(2) \equal{} 1$ ,  $ f(5) \equal{} 2$ . For how many  $ 1 < m < 2008$  is  $ f(m)$  odd?


Let  $a \star b = ab + a + b$  for all integers  $a$  and  $b$ .  Evaluate  $1 \star ( 2 \star ( 3 \star (4 \star \ldots ( 99 \star 100 ) \ldots )))$ .

Solution: \boxed{101! - 1}


The numbers  $2^{1989}$  and  $5^{1989}$  are written out one after the other (in decimal notation). How many digits are written altogether?

(G. Galperin)


Two (not necessarily different) numbers are chosen independently and at random from  $\{1, 2, 3, \dots, 10\}$ . On average, what is the product of the two integers? (Compute the expected product. That is, if you do this over and over again, what will the product of the integers be on average?)


The points  $A, B$  and  $C$  lie on the surface of a sphere with center  $O$  and radius 20.  It is given that  $AB=13, BC=14, CA=15,$  and that the distance from  $O$  to triangle  $ABC$  is  $\frac{m\sqrt{n}}k,$  where  $m, n,$  and  $k$  are positive integers,  $m$  and  $k$  are relatively prime, and  $n$  is not divisible by the square of any prime.  Find  $m+n+k.$


Point  $P$  is  $\sqrt3$  units away from plane  $A$ . Let  $Q$  be a region of  $A$  such that every line through  $P$  that intersects  $A$  in  $Q$  intersects  $A$  at an angle between  $30^o$  and  $60^o$  . What is the largest possible area of  $Q$ ?


Find all positive integers  $n$  such that the number  $n^5+79$  has all the same digits when it is written in decimal represantation.

Solution: \boxed{ n = 2 }


A computer network is formed by connecting  $2004$  computers by cables. A set  $S$  of these computers is said to be independent if no pair of computers of  $S$  is connected by a cable. Suppose that the number of cables used is the minimum number possible such that the size of any independent set is at most  $50$ . Let  $c(L)$  be the number of cables connected to computer  $L$ . Show that for any distinct computers  $A$  and  $B$ ,  $c(A)=c(B)$  if they are connected by a cable and  $|c(A)-c(B)| \le 1$  otherwise. Also, find the number of cables used in the network.


We glue together  $990$  one inch cubes into a  $9$  by  $10$  by  $11$  inch rectangular solid. Then we paint the outside of the solid. How many of the original  $990$  cubes have just one of their sides painted?


$12$  knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.  $5$  knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?


For a natural number  $n \ge 3$ , we draw  $n - 3$  internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the  $n$ -gon into  $n - 2$  triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural number and no two of these angle values are equal. What is the largest possible value of  $n$ ?


Let  $S = \{1, 2,..., 8\}$ . How many ways are there to select two disjoint subsets of  $S$ ?


Determine all real values of  $x$  for which
\[\frac{1}{\sqrt{x} + \sqrt{x - 2}} + \frac{1}{\sqrt{x} + \sqrt{x + 2}} = \frac{1}{4}.\]


There are  $2^{2n+1}$  towns with  $2n+1$  companies and each two towns are connected with airlines from one of the companies. What’s the greatest number  $k$  with the following property:
We can close  $k$  of the companies and their airlines in such way that we can still reach each town from any other (connected graph).

Solution: \boxed{ k = n }


For any positive integer  $n$ , let  $a_n=\sum_{k=1}^{\infty}[\frac{n+2^{k-1}}{2^k}]$ , where  $[x]$  is the largest integer that is equal or less than  $x$ . Determine the value of  $a_{2015}$ .


Let  $S$  be a finite set of real numbers such that given any three distinct elements  $x,y,z\in\mathbb{S}$ , at least one of  $x+y$ ,  $x+z$ , or  $y+z$  is also contained in  $S$ . Find the largest possible number of elements that  $S$  could have.


The natural number  $n>1$  is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?


Find the maximum number of elements which can be chosen from the set  $ \{1,2,3,\ldots,2003\}$  such that the sum of any two chosen elements is not divisible by 3.


Let  $ P$  be a convex  $ n$  polygon each of which sides and diagnoals is colored with one of  $ n$  distinct colors. For which  $ n$  does: there exists a coloring method such that for any three of  $ n$  colors, we can always find one triangle whose vertices is of  $ P$ ' and whose sides is colored by the three colors respectively.

Solution: \boxed{ n }


Call an ordered triple  $(a, b, c)$  of integers feral if  $b -a, c - a$  and  $c - b$  are all prime.
Find the number of feral triples where  $1  \le a < b < c \le 20$ .


Find the number of unordered pairs  $\{ A,B \}$  of subsets of an n-element set  $X$  that satisfies the following:
(a)  $A \not= B$
(b)  $A \cup B = X$

Solution: \boxed{\frac{3^n - 1}{2}}


Let  $F:(1,\infty) \rightarrow \mathbb{R}$  be the function defined by  $$ F(x)=\int_{x}^{x^{2}} \frac{dt}{\ln(t)}. $$  Show that  $F$  is injective and find the set of values of  $F$ .

Solution: \boxed{ (\ln(2), \infty) }


The vertices of a regular  $2012$ -gon are labeled  $A_1,A_2,\ldots, A_{2012}$  in some order. It is known that if  $k+\ell$  and  $m+n$  leave the same remainder when divided by  $2012$ , then the chords  $A_kA_{\ell}$  and  $A_mA_n$  have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled  $A_1$  and  $A_4$ . How is the tenth vertex labeled?

*Proposed by A. Golovanov*


Given a (fixed) positive integer  $N$ , solve the functional equation
\[f \colon \mathbb{Z} \to \mathbb{R}, \ f(2k) = 2f(k) \textrm{ and } f(N-k) =  f(k), \ \textrm{for all } k \in \mathbb{Z}.\]

*(Dan Schwarz)*

Solution: \boxed{ f(a) = 0 }


There are  $n\geq 3$  cities in a country and between any two cities  $A$  and  $B$ , there is either a one way road from  $A$  to  $B$ , or a one way road from  $B$  to  $A$  (but never both). Assume the roads are built such that it is possible to get from any city to any other city through these roads, and define  $d(A,B)$  to be the minimum number of roads you must go through to go from city  $A$  to  $B$ . Consider all possible ways to build the roads. Find the minimum possible average value of  $d(A,B)$  over all possible ordered pairs of distinct cities in the country.

Solution: \boxed{ \frac{3}{2} }


Determine the largest odd positive integer  $n$  such that every odd integer  $k$  with  $1<k<n$  and  $\gcd(k, n)=1$  is a prime.


Find all pair of integers  $(m,n)$  and  $m \ge  n$  such that there exist a positive integer  $s$  and
a) Product of all divisor of  $sm, sn$  are equal.
b) Number of divisors of  $sm,sn$  are equal.

Solution: \boxed{ (m, n) }


Malmer Pebane's apartment uses a six-digit access code, with leading zeros allowed. He noticed that his fingers leave  that reveal which digits were pressed. He decided to change his access code to provide the largest number of possible combinations for a burglar to try when the digits are known. For each number of distinct digits that could be used in the access code, calculate the number of possible combinations when the digits are known but their order and frequency are not known. For example, if there are smudges on  $3$  and  $9,$  two possible codes are  $393939$  and  $993999.$  Which number of distinct digits in the access code offers the most combinations?


How many positive integers  $k$  are there such that \[\dfrac k{2013}(a+b)=lcm(a,b)\] has a solution in positive integers  $(a,b)$ ?


In the following diagram (not to scale),  $A$ ,  $B$ ,  $C$ ,  $D$  are four consecutive vertices of an 18-sided regular polygon with center  $O$ . Let  $P$  be the midpoint of  $AC$  and  $Q$  be the midpoint of  $DO$ . Find  $\angle OPQ$  in degrees.
[asy]
pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15;
pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd);
[/asy]


Let  $x \neq y$  be positive reals satisfying  $x^3+2013y=y^3+2013x$ , and let  $M = \left( \sqrt{3}+1 \right)x + 2y$ .  Determine the maximum possible value of  $M^2$ .

*Proposed by Varun Mohan*


Let  $a$  and  $b$  be nonzero real numbers such that  $\tfrac{1}{3a}+\tfrac{1}{b}=2011$  and  $\tfrac{1}{a}+\tfrac{1}{3b}=1$ .  What is the quotient when  $a+b$  is divided by  $ab$ ?


The function  $g$ , with domain and real numbers, fulfills the following: $\bullet$   $g (x) \le x$ , for all real  $x$  $\bullet$   $g (x + y)  \le g (x) + g (y)$  for all  real  $x,y$ Find  $g (1990)$ .


Let  $p$  be an odd prime of the form  $p=4n+1$ . [list=a][*] Show that  $n$  is a quadratic residue  $\pmod{p}$ . [*] Calculate the value  $n^{n}$    $\pmod{p}$ . [/list]

Solution: \boxed{ 1 }


Determine the number of integers  $2 \le n \le 2016$  such that  $n^n-1$  is divisible by  $2$ ,  $3$ ,  $5$ ,  $7$ .


Find the sum of all positive integers  $n$  where the mean and median of  $\{20, 42, 69, n\}$  are both integers.

*Proposed by bissue*


Let  $\{x_1, x_2, x_3, ..., x_n\}$  be a set of  $n$  distinct positive integers, such that the sum of any  $3$  of them is a prime number. What is the maximum value of  $n$ ?


Let  $G$  be the centroid of triangle  $ABC$  with  $AB=13,BC=14,CA=15$ . Calculate the sum of the distances from  $G$  to the three sides of the triangle.

Note: The *centroid* of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.

*2019 CCA Math Bonanza Individual Round #11*


A mouse has a wheel of cheese which is cut into  $2018$  slices. The mouse also has a  $2019$ -sided die, with faces labeled  $0,1,2,\ldots, 2018$ , and with each face equally likely to come up. Every second, the mouse rolls the dice. If the dice lands on  $k$ , and the mouse has at least  $k$  slices of cheese remaining, then the mouse eats  $k$  slices of cheese; otherwise, the mouse does nothing. What is the expected number of seconds until all the cheese is gone?

*Proposed by Brandon Wang*


For how many ordered pairs of positive integers  $(x, y)$  is the least common multiple of  $x$  and  $y$  equal to  $1{,}003{,}003{,}001$ ?


Find all triples  $ (x,y,z)$  of real numbers that satisfy the system of equations
\[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50,  y^3 \equal{} 12y\plus{}3z\minus{}2,  z^3 \equal{} 27z \plus{} 27x. \end{cases}\]

*Razvan Gelca.*

Solution: \boxed{ (2, 4, 6) }


 $m$  and  $n$  are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:

Starting from one of the hexagons, the Donkey moves  $m$  cells in one of the  $6$  directions, then it turns  $60$  degrees clockwise and after that moves  $n$  cells in this new direction until it reaches it's final cell.

At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?

*Proposed by Shayan Dashmiz*

Solution: \boxed{ m^2 + mn + n^2 }


Let  $ABCDE$  be a convex pentagon such that: $\angle ABC=90,\angle BCD=135,\angle DEA=60$  and  $AB=BC=CD=DE$ . Find angle  $\angle DAE$ .


Find all triples  $(p,q,n)$  that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where  $p,q$  are odd primes and  $n$  is an positive integer.

Solution: \boxed{ (3, 3, n) }


Determine the length of  $BC$  in an acute triangle  $ABC$  with  $\angle ABC = 45^{\circ}$ ,  $OG = 1$  and  $OG \parallel BC$ . (As usual  $O$  is the circumcenter and  $G$  is the centroid.)