Four mathletes and two coaches sit at a circular table. How many distinct arrangements are there of these six people if the two coaches sit opposite each other?


Find all positive integers  $x$  such that  $2x+1$  is a perfect square but none of the integers  $2x+2, 2x+3, \ldots, 3x+2$  are perfect squares.


Given a  $15\times 15$  chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is  $\leq 200$ .


We define a sequence  $a_n$  so that  $a_0=1$  and
\[a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2,  a_n + d & \textrm{ otherwise. } \end{cases}  \]
for all postive integers  $n$ .
Find all positive integers  $d$  such that there is some positive integer  $i$  for which  $a_i=1$ .

Solution: \boxed{ d }


Find the least value of  $k$  such that for all  $a,b,c,d \in \mathbb{R}$  the inequality
\[ \begin{array}  c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)}   \  \ge 2( ab+bc+cd+da+ac+bd)-k  \end{array}\]
holds.


The temperatures  $ f^\circ \text{F}$  and  $ c^\circ \text{C}$  are equal when  $ f \equal{} \frac {9}{5}c \plus{} 32$ . What temperature is the same in both  $ ^\circ \text{F}$  and  $ ^\circ \text{C}$ ?


Find the smallest positive value of  $36^k - 5^m$ , where  $k$  and  $m$  are positive integers.


A positive integer  $n$  between  $1$  and  $N=2007^{2007}$  inclusive is selected at random.  If  $a$  and  $b$  are natural numbers such that  $a/b$  is the probability that  $N$  and  $n^3-36n$  are relatively prime, find the value of  $a+b$ .


A tetrahedron is a polyhedron composed of four triangular faces. Faces  $ABC$  and  $BCD$  of a tetrahedron  $ABCD$  meet at an angle of  $\pi/6$ . The area of triangle  $\triangle ABC$  is  $120$ . The area of triangle  $\triangle BCD$  is  $80$ , and  $BC = 10$ . What is the volume of the tetrahedron? We call the volume of a tetrahedron as one-third the area of it's base times it's height.


For each positive natural number  $n$  let  $d (n)$  be the number of its divisors including  $1$  and  $n$ .
For which positive natural numbers  $n$ , for every divisor  $t$  of  $n$ , that  $d (t)$  is a divisor of  $d (n)$ ?

Solution: \boxed{ n }


In triangle  $ABC$  with  $AB=8$  and  $AC=10$ , the incenter  $I$  is reflected across side  $AB$  to point  $X$  and across side  $AC$  to point  $Y$ . Given that segment  $XY$  bisects  $AI$ , compute  $BC^2$ . (The incenter  is the center of the inscribed circle of triangle .)


On Monday a school library was attended by 5 students, on Tuesday, by 6, on Wednesday, by 4, on Thursday, by 8, and on Friday, by 7. None of them have attended the library two days running. What is the least possible number of students who visited the library during a week?


Determine all three-digit numbers  $N$  having the property that  $N$  is divisible by 11, and  $\dfrac{N}{11}$  is equal to the sum of the squares of the digits of  $N$ .


Michael, David, Evan, Isabella, and Justin compete in the NIMO Super Bowl, a round-robin cereal-eating tournament. Each pair of competitors plays exactly one game, in which each competitor has an equal chance of winning (and there are no ties). The probability that none of the five players wins all of his/her games is  $\tfrac{m}{n}$  for relatively prime positive integers  $m$ ,  $n$ . Compute  $100m + n$ .

*Proposed by Evan Chen*


Let  $a, b, c$ , and  $d$  be real numbers such that  $a^2 + b^2 + c^2 + d^2 = 3a + 8b + 24c + 37d = 2018$ . Evaluate  $3b + 8c + 24d + 37a$ .


Find all primes that can be written both as a sum of two primes and as a difference of two primes.


Given triangle  $ ABC$  of area 1. Let  $ BM$  be the perpendicular from  $ B$  to the bisector of angle  $ C$ . Determine the area of triangle  $ AMC$ .


Points  $P_1, P_2, P_3,$  and  $P_4$  are  $(0,0), (10, 20), (5, 15),$  and  $(12, -6)$ , respectively. For what point  $P \in \mathbb{R}^2$  is the sum of the distances from  $P$  to the other  $4$  points minimal?

Solution: \boxed{ (6, 12) }


Let  $S$  be the set of all lattice points  $(x, y)$  in the plane satisfying  $|x|+|y|\le 10$ . Let  $P_1,P_2,\ldots,P_{2013}$  be a sequence of 2013 (not necessarily distinct) points such that for every point  $Q$  in  $S$ , there exists at least one index  $i$  such that  $1\le i\le 2013$  and  $P_i = Q$ . Suppose that the minimum possible value of  $|P_1P_2|+|P_2P_3|+\cdots+|P_{2012}P_{2013}|$  can be expressed in the form  $a+b\sqrt{c}$ , where  $a,b,c$  are positive integers and  $c$  is not divisible by the square of any prime. Find  $a+b+c$ . (A *lattice point* is a point with all integer coordinates.)
<details><summary>Clarifications</summary>

- $k = 2013$ , i.e. the problem should read, ``... there exists at least one index  $i$  such that  $1\le i\le 2013$  ...''.  An earlier version of the test read  $1 \le i \le k$ .
</details>

*Anderson Wang*


Consider all the positive integers  $N$  with the property that all of the divisors of  $N$  can be written as  $p-2$  for some prime number  $p$ . Then, there exists an integer  $m$  such that  $m$  is the maximum possible number of divisors of all
numbers  $N$  with such property. Find the sum of all possible values of  $N$  such that  $N$  has  $m$  divisors.

*Proposed by **FedeX333X***


In the diagram below, three squares are inscribed in right triangles. Their areas are  $A$ ,  $M$ , and  $N$ , as indicated in the diagram. If  $M = 5$  and  $N = 12$ , then  $A$  can be expressed as  $a + b\sqrt{c}$ , where  $a$ ,  $b$ , and  $c$  are positive integers and  $c$  is not divisible by the square of any prime. Compute  $a + b + c$ .

[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
		 pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3);
		 draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0));
		 label(" $A$ ", (.5,.5));
		 label(" $M$ ", (7/6, 1/6));
		 label(" $N$ ", (1/3, 4/3));[/asy]

*Proposed by Aaron Lin*


Let  $\triangle ABC$  have  $M$  as the midpoint of  $BC$  and let  $P$  and  $Q$  be the feet of the altitudes from  $M$  to  $AB$  and  $AC$  respectively. Find  $\angle BAC$  if  $[MPQ]=\frac{1}{4}[ABC]$  and  $P$  and  $Q$  lie on the segments  $AB$  and  $AC$ .


Find the smallest positive integer  $n$  that is divisible by  $100$  and has exactly  $100$  divisors.


Say that an integer  $n \ge 2$  is *delicious* if there exist  $n$  positive integers adding up to 2014 that have distinct remainders when divided by  $n$ .  What is the smallest delicious integer?


The tangent lines to the circumcircle of triangle ABC passing through vertices  $B$  and  $C$  intersect at point  $F$ . Points  $M$ ,  $L$  and  $N$  are the feet of the perpendiculars from vertex  $A$  to the lines  $FB$ ,  $FC$  and  $BC$  respectively. Show that  $AM+AL \geq 2AN$

Solution: \boxed{ AM + AL \geq 2AN }


Euler's formula states that for a convex polyhedron with  $V$  vertices,  $E$  edges, and  $F$  faces,  $V - E + F = 2$ .  A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon.  At each of its  $V$  vertices,  $T$  triangular faces and  $P$  pentagonal faces meet.  What is the value of  $100P + 10T + V$ ?


Equilateral triangle  $ABC$  has point  $P$  inside and point  $Q$  on side  $BC$ , satisfying  $PB = PQ = QC$  and  $\angle PBC = 20^\circ$ .
Find the measure of  $\angle AQP$ .


Compute \[\left\lfloor \dfrac{2007!+2004!}{2006!+2005!}\right\rfloor.\] (Note that  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .)


The real numbers  $x$ ,  $y$ ,  $z$ , and  $t$  satisfy the following equation:
\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for  $t$ .


Pratyya and Payel have a number each,  $n$  and  $m$  respectively, where  $n>m.$  Everyday, Pratyya multiplies his number by  $2$  and then subtracts  $2$  from it, and Payel multiplies his number by  $2$  and then add  $2$  to it. In other words, on the first day their numbers will be  $(2n-2)$  and  $(2m+2)$  respectively. Find minimum integer  $x$  with proof such that if  $n-m\geq x,$  then Pratyya's number will be larger than Payel's number everyday.


Consider the addition problem:
\begin{tabular}{ccccc}
&C&A&S&H
+&&&M&E
\hline
O&S&I&D&E
\end{tabular}
where each letter represents a base-ten digit, and  $C,M,O \ne 0.$  (Distinct letters are allowed to represent
the same digit.) How many ways are there to assign values to the letters so that the addition problem
is true?


Find the sum of all the integers   $N  >  1$   with the properties that the each prime factor of   $N $  is either   $2, 3,$  or  $5,$  and   $N$   is not divisible by any perfect cube greater than  $1.$


Given a point  $p$  and a line segment  $l$ , let  $d(p,l)$  be the distance between them.  Let  $A$ ,  $B$ , and  $C$  be points in the plane such that  $AB=6$ ,  $BC=8$ ,  $AC=10$ .  What is the area of the region in the  $(x,y)$ -plane formed by the ordered pairs  $(x,y)$  such that there exists a point  $P$  inside triangle  $ABC$  with  $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$


Let  $ABC$  be a triangle with  $\angle A = 60^o$ . Line  $\ell$  intersects segments  $AB$  and  $AC$  and splits triangle  $ABC$  into an equilateral triangle and a quadrilateral. Let  $X$  and  $Y$  be on  $\ell$  such that lines  $BX$  and  $CY$  are perpendicular to ℓ. Given that  $AB = 20$  and  $AC = 22$ , compute  $XY$  .


The  $\textit{arithmetic derivative}$   $D(n)$  of a positive integer  $n$  is defined via the following rules:

- $D(1) = 0$ ;
- $D(p)=1$  for all primes  $p$ ;
- $D(ab)=D(a)b+aD(b)$  for all positive integers  $a$  and  $b$ .

Find the sum of all positive integers  $n$  below  $1000$  satisfying  $D(n)=n$ .


How many ordered triples  $(a,b,c)$  of integers with  $1\le a\le b\le c\le 60$  satisfy  $a\cdot b=c$ ?


Let  $\triangle ABC$  with  $AB=AC$  and  $BC=14$  be inscribed in a circle  $\omega$ . Let  $D$  be the point on ray  $BC$  such that  $CD=6$ . Let the intersection of  $AD$  and  $\omega$  be  $E$ . Given that  $AE=7$ , find  $AC^2$ .

*Proposed by Ephram Chun and Euhan Kim*


Find all pairs of real number  $x$  and  $y$  which simultaneously satisfy the following 2 relations: $x+y+4=\frac{12x+11y}{x^2+y^2}$  $y-x+3=\frac{11x-12y}{x^2+y^2}$

Solution: \boxed{ (2, 1) }


Based on a city's rules, the buildings of a street may not have more than  $9$  stories. Moreover, if the number of stories of two buildings is the same, no matter how far they are from each other, there must be a building with a higher number of stories between them. What is the maximum number of buildings that can be built on one side of a street in this city?


A solid rectangular block is formed by gluing together  $N$  congruent 1-cm cubes face to face. When the block is viewed so that three of its faces are visible, exactly 231 of the 1-cm cubes cannot be seen. Find the smallest possible value of  $N$ .


Let  $ ABC$  be a triangle with circumradius  $ R$ , perimeter  $ P$  and area  $ K$ . Determine the maximum value of:  $ \frac{KP}{R^3}$ .


Find all real numbers  $x, y, z$  that satisfy the following system  $$ \sqrt{x^3 - y} = z - 1 $$    $$ \sqrt{y^3 - z} = x - 1 $$    $$ \sqrt{z^3 - x} = y - 1 $$

Solution: \boxed{ (1, 1, 1) }


I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.


Define  $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$ . Let the sum of all  $H_n$  that are terminating in base 10 be  $S$ . If  $S = m/n$  where m and n are relatively prime positive integers, find  $100m+n$ .

*Proposed by Lewis Chen*


Compute the sum of all positive integers  $a\leq 26$  for which there exist integers  $b$  and  $c$  such that  $a+23b+15c-2$  and  $2a+5b+14c-8$  are both multiples of  $26$ .


Let  $k$  be the product of every third positive integer from  $2$  to  $2006$ , that is  $k = 2\cdot 5\cdot 8\cdot 11 \cdots 2006$ . Find the number of zeros there are at the right end of the decimal representation for  $k$ .


The five-digit number  $12110$  is divisible by the sum of its digits  $1 + 2 + 1 + 1 + 0 = 5.$  Find the greatest five-digit number which is divisible by the sum of its digits


We delete the four corners of a  $8 \times 8$  chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares?

*Proposed by Evan Chen*


If  $a$  and  $b$  are the roots of  $x^2 - 2x + 5$ , what is  $|a^8 + b^8|$ ?


Let  $ABCDEF$  be a regular hexagon with side length 10 inscribed in a circle  $\omega$ .  $X$ ,  $Y$ , and  $Z$  are points on  $\omega$  such that  $X$  is on minor arc  $AB$ ,  $Y$  is on minor arc  $CD$ , and  $Z$  is on minor arc  $EF$ , where  $X$  may coincide with  $A$  or  $B$  (and similarly for  $Y$  and  $Z$ ). Compute the square of the smallest possible area of  $XYZ$ .

*Proposed by Michael Ren*


Find all positive integers  $n$  for which both numbers \[1\;\;\!\!\!\!\underbrace{77\ldots 7}_{\text{ $n$  sevens}}\!\!\!\!\quad\text{and}\quad 3\;\; \!\!\!\!\underbrace{77\ldots 7}_{\text{ $n$  sevens}}\] are prime.

Solution: \boxed{ n = 1 }


$12$  students need to form five study groups. They will form three study groups with  $2$  students each and two study groups with  $3$  students each. In how many ways can these groups be formed?


An increasing arithmetic sequence with infinitely many terms is determined as follows.  A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms.  Determine with proof how many of the 36 possible sequences formed in this way contain at least one perfect square.


(F.Nilov) Given right triangle  $ ABC$  with hypothenuse  $ AC$  and  $ \angle A \equal{} 50^{\circ}$ . Points  $ K$  and  $ L$  on the cathetus  $ BC$  are such that  $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$ . Determine the ratio  $ CK/LB$ .


A set  $M$  consists of  $n$  elements. Find the greatest  $k$  for which there is a collection of  $k$  subsets of  $M$  such that for any subsets  $A_{1},...,A_{j}$  from the collection, there is an element belonging to an odd number of them

Solution: \boxed{k = n}


Let  $ n$  be a positive integer and  $ [ \ n ] = a.$  Find the largest integer  $ n$  such that the following two conditions are satisfied:

 $ (1)$   $ n$  is not a perfect square;

 $ (2)$   $ a^{3}$  divides  $ n^{2}$ .


Let  $k$  be a positive integer with the following property: For every subset  $A$  of  $\{1,2,\ldots, 25\}$  with  $|A|=k$ , we can find distinct elements  $x$  and  $y$  of  $A$  such that  $\tfrac23\leq\tfrac xy\leq\tfrac 32$ .  Find the smallest possible value of  $k$ .


Two circles with radius  $2$  and radius  $4$  have a common center at P.  Points  $A, B,$  and  $C$  on the larger circle are the vertices of an equilateral triangle.  Point  $D$  is the intersection of the smaller circle and the line segment  $PB$ . Find the square of the area of triangle  $ADC$ .


A set of  $8$  problems was prepared for an examination. Each student was given  $3$  of them. No two students received more than one common problem. What is the largest possible number of students?

Solution: \boxed{ 8 }


The Binomial Expansion is valid for exponents that are not integers.  That is, for all real numbers  $ x, y,$  and  $ r$  with  $ |x| > |y|,$
\[ (x \plus{} y)^r \equal{} x^r \plus{} rx^{r \minus{} 1}y \plus{} \frac {r(r \minus{} 1)}2x^{r \minus{} 2}y^2 \plus{} \frac {r(r \minus{} 1)(r \minus{} 2)}{3!}x^{r \minus{} 3}y^3 \plus{} \cdots
\]
What are the first three digits to the right of the decimal point in the decimal representation of  $ \left(10^{2002} \plus{} 1\right)^{10/7}?$


Let  $V$  be a rectangular prism with integer side lengths. The largest face has area  $240$  and the smallest face has area  $48$ . A third face has area  $x$ , where  $x$  is not equal to  $48$  or  $240$ . What is the sum of all possible values of  $x$ ?


Let  $f$  be a real-valued function defined on the positive integers satisfying the following condition: For all  $n>1$  there exists a prime divisor  $p$  of  $n$  such that  $f(n)=f\left(\frac{n}{p}\right)-f(p)$ . Given that  $f(2001)=1$ , what is the value of  $f(2002)$ ?


A semicircle with diameter  $d$  is contained in a square whose sides have length  $8$ . Given the maximum value of  $d$  is  $m- \sqrt{n}$ , find  $m+n$ .


Suppose  $\overline{a_1a_2...a_{2009}}$  is a  $2009$ -digit integer such that for each  $i = 1,2,...,2007$ , the  $2$ -digit integer  $\overline{a_ia_{i+1}}$  contains  $3$  distinct prime factors. Find  $a_{2008}$
(Note:  $\overline{xyz...}$  denotes an integer whose digits are  $x, y,z,...$ .)


Consider a random permutation of the set  $\{1, 2, . . . , 2015\}$ . In other words, for each  $1 \le i \le 2015$ ,  $i$  is sent to the element  $a_i$  where  $a_i \in \{1, 2, . . . , 2015\}$  and if  $i \neq j$ , then  $a_i \neq a_j$ . What is the expected number of ordered pairs  $(a_i, a_j )$  with  $i - j > 155$  and  $a_i - a_j > 266$ ?


In a school there are 1200 students. Each student must join exactly  $k$  clubs. Given that there is a common club joined by every 23 students, but there is no common club joined by all 1200 students, find the smallest possible value of  $k$ .

Solution: \boxed{ k = 23 }


Find the triplets of primes  $(a,\ b,\ c)$  such that  $a-b-8$  and  $b-c-8$  are primes.

Solution: \boxed{(23, 13, 3)}


How many integers  $x$ , from  $10$  to  $99$  inclusive, have the property that the remainder of  $x^2$  divided by  $100$  is equal to the square of the units digit of  $x$ ?


Let  $ u_1$ ,  $ u_2$ ,  $ \ldots$ ,  $ u_{1987}$  be an arithmetic progression with  $ u_1 \equal{} \frac {\pi}{1987}$  and the common difference  $ \frac {\pi}{3974}$ . Evaluate
\[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u_{1987}\right)
\]


A rectangular pool table has a hole at each of three of its corners. The lengths of sides of the table are the real numbers  $a$  and  $b$ . A billiard ball is shot from the fourth corner along its angle bisector. The ball falls in one of the holes. What should the relation between  $a$  and  $b$  be for this to happen?

Solution: \boxed{ \frac{a}{b} }


Determine for what  $n\ge 3$  integer numbers,  it is possible to find positive integer numbers  $a_1 < a_2 < ...< a_n$  such  $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1$  and  $a_1 a_2\cdot\cdot\cdot a_n$  is a perfect square.

Solution: \boxed{ n \geq 3 }


In  $\Delta ABC$ ,  $AC=kAB$ , with  $k>1$ . The internal angle bisector of  $\angle BAC$  meets  $BC$  at  $D$ . The circle with  $AC$  as diameter cuts the extension of  $AD$  at  $E$ . Express  $\dfrac{AD}{AE}$  in terms of  $k$ .

Solution: \boxed{\dfrac{2}{k + 1}}


If  $a$  and  $b$  are nonzero real numbers such that  $\left| a \right| \ne \left| b \right|$ , compute the value of the expression
\[
  \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
  \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
  \left(
    \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
      - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
  \right).
\]


For integers  $a, b$ , call the lattice point with coordinates  $(a,b)$  **basic** if  $gcd(a,b)=1$ . A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between  $(a_1,b_1)$  and  $(a_2,b_2)$  if and only if  $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$  or  $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$ . Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?


Find all positive integer solutions  $(x,y,z,n)$  of equation  $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$ , where  $n\ge 2$  and  $z \le 5\times 2^{2n}$ .

Solution: \boxed{ (3, 1, 70, 2) }


The Stromquist Comet is visible every 61 years. If the comet is visible in 2017, what is the next leap year when the comet will be visible?


Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the *square* of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is  $\frac{1734}{274}$ , then you would submit 1734274).

Solution: \boxed{ 52972 }


Suppose  $ABCD$  is a trapezoid with  $AB\parallel CD$  and  $AB\perp BC$ . Let  $X$  be a point on segment  $\overline{AD}$  such that  $AD$  bisects  $\angle BXC$  externally, and denote  $Y$  as the intersection of  $AC$  and  $BD$ . If  $AB=10$  and  $CD=15$ , compute the maximum possible value of  $XY$ .


A circle has radius  $52$  and center  $O$ .  Points  $A$  is on the circle, and point  $P$  on  $\overline{OA}$  satisfies  $OP = 28$ .  Point  $Q$  is constructed such that  $QA = QP = 15$ , and point  $B$  is constructed on the circle so that  $Q$  is on  $\overline{OB}$ . Find  $QB$ .

*Proposed by Justin Hsieh*


$(BEL 1)$  A parabola  $P_1$  with equation  $x^2 - 2py = 0$  and parabola  $P_2$  with equation  $x^2 + 2py = 0, p > 0$ , are given. A line  $t$  is tangent to  $P_2.$  Find the locus of pole  $M$  of the line  $t$  with respect to  $P_1.$

Solution: \boxed{ x^2 + 2py = 0 }


Find the area bounded by the graph of  $ y\equal{}\sqrt[3]{x\plus{}\sqrt{x^2\plus{}1}}\plus{}\sqrt[3]{x\minus{}\sqrt{x^2\plus{}1}}$ , the line  $ x\minus{}y\minus{}1\equal{}0$  and the  $ x$  axis.


What is the number of ways in which one can color the squares of a  $4\times 4$  chessboard with colors red and blue such that each row as well as each column has exactly two red squares and two blue squares?


All positive divisors of a positive integer  $N$  are written on a blackboard. Two players  $A$  and  $B$  play the following game taking alternate moves. In the firt move, the player  $A$  erases  $N$ . If the last erased number is  $d$ , then the next player erases either a divisor of  $d$  or a multiple of  $d$ . The player who cannot make a move loses. Determine all numbers  $N$  for which  $A$  can win independently of the moves of  $B$ .

*(4th Middle European Mathematical Olympiad, Individual Competition, Problem 2)*

Solution: \boxed{ N }


There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?

[i]Proposed by Yiming Zheng


Find all odd positive integers  $n>1$  such that there is a permutation  $a_1, a_2, a_3, \ldots, a_n$  of the numbers  $1, 2,3, \ldots, n$  where  $n$  divides one of the numbers  $a_k^2 - a_{k+1} - 1$  and  $a_k^2 - a_{k+1} + 1$  for each  $k$ ,  $1 \leq k \leq n$  (we assume  $a_{n+1}=a_1$ ).

Solution: \boxed{ n = 3 }


For how many values of  $k$  is  $12^{12}$  the least common multiple of the positive integers  $6^6, 8^8,$  and  $k$ ?


The set  $X$  of  $N$  four-digit numbers formed from the digits  $1,2,3,4,5,6,7,8$  satisfies the following condition:

*for any two different digits from  $1,2,3,4,,6,7,8$  there exists a number in  $X$  which contains both of them.*

Determine the smallest possible value of  $N$ .


Let the triangle  $ABC$  have area  $1$ . The interior bisectors of the angles  $\angle BAC,\angle ABC, \angle BCA$  intersect the sides  $(BC), (AC), (AB) $  and the circumscribed circle of the respective triangle  $ABC$  at the points  $L$  and  $G, N$  and  $F, Q$  and  $E$ . The lines  $EF, FG,GE$  intersect the bisectors  $(AL), (CQ) ,(BN)$  respectively at points  $P, M, R$ . Determine the area of the hexagon  $LMNPR$ .


For a positive integer  $n$ , let

\[S_n=\int_0^1 \frac{1-(-x)^n}{1+x}dx,\ \ T_n=\sum_{k=1}^n \frac{(-1)^{k-1}}{k(k+1)}\]

Answer the following questions:

(1) Show the following inequality.

\[\left|S_n-\int_0^1 \frac{1}{1+x}dx\right|\leq \frac{1}{n+1}\]

(2) Express  $T_n-2S_n$  in terms of  $n$ .

(3) Find the limit  $\lim_{n\to\infty} T_n.$

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


Let  $ABC$  be a right-angled triangle with  $\angle ABC=90^\circ$ , and let  $D$  be on  $AB$  such that  $AD=2DB$ . What is the maximum possible value of  $\angle ACD$ ?


What is the maximum number of subsets of  $S = {1, 2, . . . , 2n}$  such that no one is contained in another and no two cover whole  $S$ ?

*Proposed by Fedor Petrov*

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


Describe how to place the vertices of a triangle in the faces of a cube in such a way that the shortest side of the triangle is the biggest possible.

Solution: \boxed{ \sqrt{2} }


Find all positive integers  $n$  for which all positive divisors of  $n$  can be put into the cells of a rectangular table under the following constraints:

- each cell contains a distinct divisor;
- the sums of all rows are equal; and
- the sums of all columns are equal.

Solution: \boxed{n = 1}


Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?


It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$  has exactly one solution. Find  $a$ .


Let  $m$  and  $n$  be integers such that  $m + n$  and  $m - n$  are prime numbers less than  $100$ . Find the maximal possible value of  $mn$ .


The new PUMaC tournament hosts  $2020$  students, numbered by the following set of labels  $1, 2, . . . , 2020$ . The students are initially divided up into  $20$  groups of  $101$ , with each division into groups equally likely. In each of the groups, the contestant with the lowest label wins, and the winners advance to the second round. Out of these  $20$  students, we chose the champion uniformly at random. If the expected value of champion’s number can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime integers, determine  $a + b$ .


A tetrahedron  $ABCD$  satisfies the following conditions: the edges  $AB,AC$  and  $AD$  are pairwise orthogonal,  $AB=3$  and  $CD=\sqrt2$ . Find the minimum possible value of  $$ BC^6+BD^6-AC^6-AD^6. $$


On a table there are  $100$  red and  $k$  white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least once during the moves, the amount of water in these buckets is the same. Find all the possible values of  $k$ .


Estimate the range of the submissions for this problem. Your answer must be between  $[0, 1000]$ . An estimate  $E$  earns  $\frac{2}{1+0.05|A-E|}$  points, where  $A$  is the actual answer.

*2022 CCA Math Bonanza Lightning Round 5.2*


Let  $M$  be a finite subset of the plane such that for any two different points  $A,B\in M$  there is a point  $C\in M$  such that  $ABC$  is equilateral. What is the maximal number of points in  $M?$

Solution: \boxed{ 3 }


Points  $A$ ,  $B$ ,  $C$ , and  $D$  lie in the plane with  $AB=AD=7$ ,  $CB=CD=4$ , and  $BD=6$ . Compute the sum of all possible values of  $AC$ .

Solution: \boxed{ 4\sqrt{10} }


Alexandra draws a letter A which stands on the  $x$ -axis.
[list=a][*]The left side of the letter A lies along the line with equation  $y=3x+6$ . What is the  $x$ -intercept of the line with equation  $y=3x+6$ ?
[*]The right side of the letter A lies along the line  $L_2$  and the leter is symmetric about the  $y$ -axis. What is the equation of line  $L_2$ ?
[*]Determine the are of the triangle formed by the  $x$  axis and the left and right sides of the letter A.
[*]Alexandra completes the letter A by adding to Figure 1. She draws the horizontal part of the letter A along the line  $y=c$ , as in Figure 2. The area of the shaded region inside the letter A and above the line with equation  $y=c$  is  $\frac49$  of the total area of the region above the  $x$  axis and between the left and right sides. Determine the value of  $c$ .[/list]**Figure 1**
[asy]
 /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.8408113739622465, xmax = 5.491811096383217, ymin = -3.0244242161812847, ymax = 8.241467380517944;  /* image dimensions */
pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);

Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, EndArrow(6), above = true);
yaxis(ymin, ymax, EndArrow(8.25),above = true); /* draws axes; NoZero hides '0' label */
 /* draw figures */
draw((0,6)--(-2,0), linewidth(2));
draw((-2,0)--(2,0), linewidth(2));
draw((2,0)--(0,6), linewidth(2));
label(" $y=3x+6$ ",(-2.874280000573916,3.508459668295191),SE*labelscalefactor);
label(" $L_2$ ",(1.3754276283584919,3.5917872688624928),SE*labelscalefactor);
label(" $O$ ",(0,0),SW*labelscalefactor);
 /* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 /* end of picture */
[/asy]**Figure 2**
[asy]
 /* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(10cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -4.707487213054563, xmax = 5.6251352572909, ymin = -3.4577277391312538, ymax = 7.808163857567977;  /* image dimensions */
pen cqcqcq = rgb(0.7529411764705882,0.7529411764705882,0.7529411764705882);

draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--(0,6)--cycle, linewidth(2));

Label laxis; laxis.p = fontsize(10);
xaxis(xmin, xmax, EndArrow(6), above = true);
yaxis(ymin, ymax, EndArrow(6), above = true); /* draws axes; NoZero hides '0' label */
 /* draw figures */
draw((0,6)--(-2,0), linewidth(2));
draw((-2,0)--(2,0), linewidth(2));
draw((2,0)--(0,6), linewidth(2));
label(" $O$ ",(0,0),SW*labelscalefactor);
draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2));
draw((-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678), linewidth(2));
draw((1.1148845961134441,2.6553462116596678)--(0,6), linewidth(2));
draw((0,6)--(-1.114884596113444,2.6553462116596678), linewidth(2));
fill((0,6)--(-1.114884596113444,2.6553462116596678)--(1.1148845961134441,2.6553462116596678)--cycle,black);
label(" $y=c$ ",(1.4920862691527148,3.1251527056856054),SE*labelscalefactor);
 /* dots and labels */
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 /* yes i used geogebra fight me*/
[/asy]


The difference of fractions  $\frac{2024}{2023} - \frac{2023}{2024}$  was represented as an irreducible fraction  $\frac{p}{q}$ . Find the value of  $p$ .


There are three bags. One bag contains three green candies and one red candy. One bag contains two green candies and two red candies. One bag contains one green candy and three red candies. A child randomly selects one of the bags, randomly chooses a first candy from that bag, and eats the candy. If the first candy had been green, the child randomly chooses one of the other two bags and randomly selects a second candy from that bag. If the first candy had been red, the child randomly selects a second candy from the same bag as the first candy. If the probability that the second candy is green is given by the fraction  $m/n$  in lowest terms, find  $m + n$ .


While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments  $AB = 10$  and  $CD = 7$  on a plane. Across all possible configurations of  those two segments, determine the maximum possible area of quadrilateral  $ACBD$ .


A class of  $10$  students took a math test. Each problem was solved by exactly  $7$  of the students. If the first nine students each solved  $4$  problems, how many problems did the tenth student solve?


Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says  $0$ . In each turn except the first the possible moves are determined from the previous number  $n$  in the following way: write
\[n =\sum_{m\in O_n}2^m;\]
the valid numbers are the elements  $m$  of  $O_n$ . That way, for example, after Arnaldo says  $42= 2^5 + 2^3 + 2^1$ , Bernaldo must respond with  $5$ ,  $3$  or  $1$ .

We define the sets  $A,B\subset \mathbb{N}$  in the following way. We have  $n\in A$  iff Arnaldo, saying  $n$  in his first turn, has a winning strategy; analogously, we have  $n\in B$  iff Bernaldo has a winning strategy if Arnaldo says  $n$  during his first turn. This way,
\[A =\{0, 2, 8, 10,\cdots\}, B = \{1, 3, 4, 5, 6, 7, 9,\cdots\}\]
Define  $f:\mathbb{N}\to \mathbb{N}$  by  $f(n)=|A\cap \{0,1,\cdots,n-1\}|$ . For example,  $f(8) = 2$  and  $f(11)=4$ .
Find
\[\lim_{n\to\infty}\frac{f(n)\log(n)^{2005}}{n}\]

Solution: \boxed{ 0 }


Determine the number of real roots of the equation ${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$

Solution: \boxed{ 0 }


The diagram below shows an isosceles triangle with base  $21$  and height  $28$ .  Inscribed in the triangle is a square.  Find the area of the shaded region inside the triangle and outside of the square.

[asy]
size(170);
defaultpen(linewidth(0.8));
draw((0,0)--(1,1));
pair A=(5,0),B=(-5,0),C=(0,14), invis[]={(1,2),(-1,2)};
pair intsquare[]={extension(origin,invis[0],A,C),extension(origin,invis[1],B,C)};
path triangle=A--B--C--cycle,square=(intsquare[0]--intsquare[1]--(intsquare[1].x,0)--(intsquare[0].x,0)--cycle);
fill(triangle,gray);
unfill(square);
draw(triangle^^square);
[/asy]


Each cell of a  $29 \times 29$  table contains one of the integers  $1, 2, 3, \ldots , 29$ , and each of these integers appears  $29$  times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.


A lucky number is a number whose digits are only  $4$  or  $7.$  What is the  $17$ th smallest lucky number?

*Author: Ray Li*

Clarifications[list=1][*]Lucky numbers are positive.
[*]"only 4 or 7" includes combinations of 4 and 7, as well as only 4 and only 7. That is, 4 and 47 are both lucky numbers.[/list]


Dave arrives at an airport which has twelve gates arranged in a straight line with exactly  $ 100$  feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks  $ 400$  feet or less to the new gate be a fraction  $ \frac{m}{n}$ , where  $ m$  and  $ n$  are relatively prime positive integers. Find  $ m\plus{}n$ .


The function  $f_n (x)\ (n=1,2,\cdots)$  is defined  as follows.

\[f_1 (x)=x,\ f_{n+1}(x)=2x^{n+1}-x^n+\frac{1}{2}\int_0^1 f_n(t)\ dt\ \ (n=1,2,\cdots)\]

Evaluate

\[\lim_{n\to\infty} f_n \left(1+\frac{1}{2n}\right)\]


Fuzzy draws a segment of positive length in a plane. How many locations can Fuzzy place another point in the same plane to form a non-degenerate isosceles right triangle with vertices consisting of his new point and the endpoints of the segment?

*Proposed by Timothy Qian*


At the Intergalactic Math Olympiad held in the year 9001, there are 6 problems, and on each problem you can earn an integer score from 0 to 7. The contestant's score is the *product* of the scores on the 6 problems, and ties are broken by the sum of the 6 problems. If 2 contestants are still tied after this, their ranks are equal. In this olympiad, there are  $8^6=262144$  participants, and no two get the same score on every problem. Find the score of the participant whose rank was  $7^6 = 117649$ .

*Proposed by Yang Liu*


Let  $1 = x_{1} < x_{2} < \dots < x_{k} = n$  denote the sequence of all divisors  $x_{1}, x_{2} \dots x_{k}$  of  $n$  in increasing order. Find the smallest possible value of  $n$  such that  $$ n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}. $$  *Proposed by Justin Lee*


Except for the first two terms, each term of the sequence  $1000, x, 1000-x,\ldots$  is obtained by subtracting the preceding term from the one before that.  The last term of the sequence is the first negative term encounted.  What positive integer  $x$  produces a sequence of maximum length?


Find the largest interval  $ M \subseteq \mathbb{R^ \plus{} }$ , such that for all  $ a$ ,  $ b$ ,  $ c$ ,  $ d \in M$  the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}\]
holds. Does the inequality
\[ \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}\]
hold too for all  $ a$ ,  $ b$ ,  $ c$ ,  $ d \in M$ ?
( $ \mathbb{R^ \plus{} }$  denotes the set of positive reals.)

Solution: \boxed{ M = [2, \infty) }


Find a positive integer  $n$  with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of  $n$ .


Among  $100$  points in the plane, no three collinear, exactly  $4026$  pairs are connected by line segments. Each point is then randomly assigned an integer from  $1$  to  $100$  inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points whose labels differ by at least  $50$ .

*Proposed by Evan Chen*


Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck,     the cards that were in the deck of size \(n\)     appear in order, as do the cards that were in the deck of size \(52-n\).)

Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee?

*Proposed by Espen Slettnes*


Circles  $\mathcal{C}_1$ ,  $\mathcal{C}_2$ , and  $\mathcal{C}_3$  have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line  $t_1$  is a common internal tangent to  $\mathcal{C}_1$  and  $\mathcal{C}_2$  and has a positive slope, and line  $t_2$  is a common internal tangent to  $\mathcal{C}_2$  and  $\mathcal{C}_3$  and has a negative slope. Given that lines  $t_1$  and  $t_2$  intersect at  $(x,y)$ , and that  $x=p-q\sqrt{r}$ , where  $p$ ,  $q$ , and  $r$  are positive integers and  $r$  is not divisible by the square of any prime, find  $p+q+r$ .


A number is called *capicua* if when it is written in decimal notation, it can be read equal from left to right as from right to left; for example:  $8, 23432, 6446$ . Let  $x_1<x_2<\cdots<x_i<x_{i+1},\cdots$  be the sequence of all capicua numbers. For each  $i$  define  $y_i=x_{i+1}-x_i$ . How many distinct primes contains the set  $\{y_1,y_2, \ldots\}$ ?

Solution: \boxed{ 2 }


Find the number of partitions of the set  $\{1,2,3,\cdots ,11,12\}$  into three nonempty subsets such that no subset has two elements which differ by  $1$ .

[i]Proposed by Nathan Ramesh


On Day  $1$ , Alice starts with the number  $a_1=5$ . For all positive integers  $n>1$ , on Day  $n$ , Alice randomly selects a positive integer  $a_n$  between  $a_{n-1}$  and  $2a_{n-1}$ , inclusive. Given that the probability that all of  $a_2,a_3,\ldots,a_7$  are odd can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $m+n$ .

*2021 CCA Math Bonanza Lightning Round #1.4*


Given a positive integer  $x>1$  with  $n$  divisors, define  $f(x)$  to be the product of the smallest  $\lceil\tfrac{n}{2}\rceil$  divisors of  $x$ . Let  $a$  be the least value of  $x$  such that  $f(x)$  is a multiple of  $X$ , and  $b$  be the least value of  $n$  such that  $f(y)$  is a multiple of  $y$  for some  $y$  that has exactly  $n$  factors. Compute  $a+b$ .


Compute the determinant of the  $n\times n$  matrix  $A=(a_{ij})_{ij}$ ,  $$ a_{ij}=\begin{cases}
(-1)^{|i-j|} & \text{if}\, i\ne j,
2 & \text{if}\, i= j.
\end{cases} $$

Solution: \boxed{n + 1}


Find the least positive integer  $n$ , such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
      - For  $i=0,1,\dots,2n$  it is  $2014 \leq a_i \leq 2015$ .
      - There is a real number  $\xi$  with  $P(\xi)=0$ .


Find the greatest common divisor of all numbers of the form  $(2^{a^2}\cdot 19^{b^2} \cdot 53^{c^2} + 8)^{16} - 1$  where  $a,b,c$  are integers.


Let  $p_{1}$ ,  $p_{2}$ , ...,  $p_{k}$  be different prime numbers. Determine the number of positive integers of the form  $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}...p_{k}^{\alpha_{k}}$ ,  $\alpha_{i}$   $\in$   $\mathbb{N}$  for which  $\alpha_{1} \alpha_{2}...\alpha_{k}=p_{1}p_{2}...p_{k}$ .

Solution: \boxed{k^k}


Find the total length of the set of real numbers satisfying \[\frac{x^2 - 80x + 1500}{x^2 - 55x + 700} < 0.\]


The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to  $1, 1$  and  $2$ , respectively. Determine the length of this hexagon's side.

(Mikhail Evdokimov)


Determine all positive integers  $n$  satisfying the following condition: for every monic polynomial  $P$  of degree at most  $n$  with integer coefficients, there exists a positive integer  $k\le n$  and  $k+1$  distinct integers  $x_1,x_2,\cdots ,x_{k+1}$  such that \[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\].

*Note.* A polynomial is *monic* if the coefficient of the highest power is one.

Solution: \boxed{ n = 2 }


Albrecht writes numbers on the points of the first quadrant with integer coordinates in the following way: If at least one of the coordinates of a point is 0, he writes 0; in all other cases the number written on point  $(a, b)$  is one greater than the average of the numbers written on points  $ (a+1 , b-1) $  and  $ (a-1,b+1)$  . Which numbers could he write on point  $(121, 212)$ ?
Note: The elements of the first quadrant are points where both of the coordinates are non- negative.

Solution: \boxed{121 \times 212}


Each number from the set  $\{1, 2, 3, 4, 5, 6, 7\}$  must be written in each circle of the diagram, so that the sum of any three *aligned* numbers is the same (e.g.,  $A+D+E = D+C+B$ ). What number cannot be placed on the circle  $E$ ?


What is the smallest positive integer  $t$  such that there exist integers  $x_{1},x_{2}, \cdots, x_{t}$  with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]


Find all real numbers  $k$  for which the inequality  $(1+t)^k (1-t)^{1-k} \leq 1$

is true for every real number  $t \in (-1, 1)$ .

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


The expression   $$ \dfrac{(1+2+\cdots + 10)(1^3+2^3+\cdots + 10^3)}{(1^2+2^2+\cdots + 10^2)^2} $$   reduces to  $\tfrac mn$ , where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .

Solution: \boxed{ 104 }


For each  $k$ ,  $\mathcal{C}_k$  is biased so that, when tossed, it has probability  $\tfrac{1}{(2k+1)}$  of falling heads. If the  $n$  coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function  $n$ .


Find all triples of three-digit positive integers  $x < y < z$  with  $x,y,z$  in arithmetic progression and  $x, y, z + 1000$  in geometric progression.

*For this problem, you may use calculators or computers to gain an intuition about how to solve the problem. However, your final submission should include mathematical derivations or proofs and should not be a solution by exhaustive search.*

Solution: \boxed{(160, 560, 960)}


Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is  $2005$ . What is the number that goes into the leftmost box?

[asy]
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));

draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
[/asy]


Let  $n$  be a positive integer. Determine the smallest positive integer  $k$  with the following property: it is possible to mark  $k$  cells on a  $2n \times 2n$  board so that there exists a unique partition of the board into  $1 \times 2$  and  $2 \times 1$  dominoes, none of which contain two marked cells.

Solution: \boxed{ k = 2n }


Let  $N$  be the number of functions  $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$  that have the property that for  $1\leq x\leq 5$  it is true that  $f(f(x))=x$ . Given that  $N$  can be written in the form  $5^a\cdot b$  for positive integers  $a$  and  $b$  with  $b$  not divisible by  $5$ , find  $a+b$ .

[i]Proposed by Nathan Ramesh


We define the weight  $W$  of a positive integer as follows:  $W(1) = 0$ ,  $W(2) = 1$ ,  $W(p) = 1 + W(p + 1)$  for every odd prime  $p$ ,  $W(c) = 1 + W(d)$  for every composite  $c$ , where  $d$  is the greatest proper factor of  $c$ .  Compute the greatest possible weight of a positive integer less than 100.


Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.


Let  $ABC$  be a triangle where $\angle$ **B=55** and  $\angle$  **C = 65**. **D** is the mid-point of **BC**. Circumcircle of **ACD** and**ABD** cuts **AB** and**AC** at point **F** and **E** respectively. Center of circumcircle of **AEF** is**O**.  $\angle$ **FDO** = ?


Let  $S(M)$  denote the sum of the digits of a positive integer  $M$  written in base  $10$ . Let  $N$  be the smallest positive integer such that  $S(N) = 2013$ . What is the value of  $S(5N + 2013)$ ?


Zadam Heng bets Saniel Dun that he can win in a free throw contest. Zadam shoots until he has made  $5$  shots. He wins if this takes him  $9$  or fewer attempts. The probability that Zadam makes any given attempt is  $\frac{1}{2}$ . What is the probability that Zadam Heng wins the bet?

*2018 CCA Math Bonanza Individual Round #4*


Determine all real numbers  $q$  for which the equation  $x^4 -40x^2 +q = 0$  has four real solutions which form an arithmetic progression


A  $1 \times n$  rectangle ( $n \geq 1 $ ) is divided into  $n$  unit ( $1 \times 1$ ) squares. Each square of this rectangle is colored red, blue or green. Let  $f(n)$  be the number of colourings of the rectangle in which there are an even number of red squares. What is the largest prime factor of  $f(9)/f(3)$ ? (The number of red squares can be zero.)


A town-planner has built an isolated city whose road network consists of  $2N$  roundabouts, each connecting exactly three roads. A series of tunnels and bridges ensure that all roads in the town meet only at roundabouts. All roads are two-way, and each roundabout is oriented clockwise.
Vlad has recently passed his driving test, and is nervous about roundabouts. He starts driving from his house, and always takes the first edit at each roundabout he encounters. It turns out his journey incluldes every road in the town in both directions before he arrives back at the starting point in the starting direction. For what values of  $N$  is this possible?

Solution: \boxed{ N }


Let  $A$ ,  $B$ ,  $C$ ,  $D$ ,  $E$ ,  $F$  be  $6$  points on a circle in that order. Let  $X$  be the intersection of  $AD$  and  $BE$ ,  $Y$  is the intersection of  $AD$  and  $CF$ , and  $Z$  is the intersection of  $CF$  and  $BE$ .  $X$  lies on segments  $BZ$  and  $AY$  and  $Y$  lies on segment  $CZ$ . Given that  $AX = 3$ ,  $BX = 2$ ,  $CY = 4$ ,  $DY = 10$ ,  $EZ = 16$ , and  $FZ = 12$ , find the perimeter of triangle  $XYZ$ .


Farmer John has  $ 5$  cows,  $ 4$  pigs, and  $ 7$  horses. How many ways can he pair up the animals so that every pair consists of animals of different species? (Assume that all animals are distinguishable from each other.)


In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a *box*. Two boxes *intersect* if they have a common point in their interior or on their boundary. Find the largest  $ n$  for which there exist  $ n$  boxes  $ B_1$ ,  $ \ldots$ ,  $ B_n$  such that  $ B_i$  and  $ B_j$  intersect if and only if  $ i\not\equiv j\pm 1\pmod n$ .

*Proposed by Gerhard Woeginger, Netherlands*


Given a positive integer  $c$ , we construct a sequence of fractions  $a_1, a_2, a_3,...$  as follows: $\bullet$   $a_1 =\frac{c}{c+1} $  $\bullet$  to get  $a_n$ , we take  $a_{n-1}$  (in its most simplified form, with both the numerator and denominator chosen to be positive) and we add  $2$  to the numerator and  $3$  to the denominator. Then we simplify the result again as much as possible, with positive numerator and denominator.
For example, if we take  $c = 20$ , then  $a_1 =\frac{20}{21}$  and  $a_2 =\frac{22}{24} = \frac{11}{12}$  . Then we find that  $a_3 =\frac{13}{15}$  (which is already simplified) and  $a_4 =\frac{15}{18} =\frac{5}{6}$ .
(a) Let  $c = 10$ , hence  $a_1 =\frac{10}{11}$  . Determine the largest  $n$  for which a simplification is needed in the construction of  $a_n$ .
(b) Let  $c = 99$ , hence  $a_1 =\frac{99}{100}$  . Determine whether a simplification is needed somewhere in the sequence.
(c) Find two values of  $c$  for which in the first step of the construction of  $a_5$  (before simplification) the numerator and denominator are divisible by  $5$ .

Solution: \boxed{ n = 7 }


The USAMO is a  $6$  question test. For each question, you submit a positive integer number  $p$  of pages on which your solution is written. On the  $i$ th page of this question, you write the fraction  $i/p$  to denote that this is the  $i$ th page out of  $p$  for this question. When you turned in your submissions for the  $2017$  USAMO, the bored proctor computed the sum of the fractions for all of the pages which you turned in. Surprisingly, this number turned out to be  $2017$ . How many pages did you turn in?

*Proposed by Tristan Shin*


Let  $P$  be a regular  $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of  $P$  into two parts, each composed of an odd number of sides of  $P$ . The sides of  $P$  are also called *good*.
Suppose  $P$  has been dissected into triangles by  $2003$  diagonals, no two of which have a common point in the interior of  $P$ . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.


The sum  $$ \frac{1^2-2}{1!} + \frac{2^2-2}{2!} + \frac{3^2-2}{3!} + \cdots + \frac{2021^2 - 2}{2021!} $$    $ $
can be expressed as a rational number  $N$ . Find the last 3 digits of  $2021! \cdot N$ .


Suppose  $x$  and  $y$  are real numbers which satisfy the system of equations \[x^2-3y^2=\frac{17}x\qquad\text{and}\qquad 3x^2-y^2=\frac{23}y.\] Then  $x^2+y^2$  can be written in the form  $\sqrt[m]{n}$ , where  $m$  and  $n$  are positive integers and  $m$  is as small as possible.  Find  $m+n$ .


Find all ordered pairs of positive integers  $(m,n)$  for which there exists a set  $C=\{c_1,\ldots,c_k\}$  ( $k\ge1$ ) of colors and an assignment of colors to each of the  $mn$  unit squares of a  $m\times n$  grid such that for every color  $c_i\in C$  and unit square  $S$  of color  $c_i$ , exactly two direct (non-diagonal) neighbors of  $S$  have color  $c_i$ .

*David Yang.*

Solution: \boxed{ (m, n) }


The triangle  $ABC$  is right in  $A$  and  $R$  is the midpoint of the hypotenuse  $BC$  . On the major leg  $AB$  the point  $P$  is marked such that  $CP = BP$  and on the segment  $BP$  the point  $Q$  is marked such that the triangle  $PQR$  is equilateral.  If the area of triangle  $ABC$  is  $27$ , calculate the area of triangle  $PQR$  .


For what integers  $ n\ge 3$  is it possible to accommodate, in some order, the numbers  $ 1,2,\cdots, n$  in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?


Show that among the vertices of any area  $1$  convex polygon with  $n > 3$  sides there exist four such that the quadrilateral formed by these four has area at least  $1/2$ .


Suppose that for some positive integer  $n$ , the first two digits of  $5^n$  and  $2^n$  are identical. Suppose the first two digits are  $a$  and  $b$  in this order. Find the two-digit number  $\overline{ab}$ .


Suppose that  $a_1, a_2, a_3, \ldots$  is an infinite geometric sequence such that for all  $i \ge 1$ ,  $a_i$  is a positive integer. Suppose furthermore that  $a_{20} + a_{21} = 20^{21}$ . If the minimum possible value of  $a_1$  can be expressed as  $2^a 5^b$  for positive integers  $a$  and  $b$ , find  $a + b$ .

*Proposed by Andrew Wu*


Determine the number of permutations  $a_1, a_2, \dots, a_n$  of  $1, 2, \dots, n$  such that for every positive integer  $k$  with  $1 \le k \le n$ , there exists an integer  $r$  with  $0 \le r \le n - k$  which satisfies
\[ 1 + 2 + \dots + k = a_{r+1} + a_{r+2} + \dots + a_{r+k}. \]


Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is  $social$  if it knows at least  $20$  other persons and at least  $2$  of those  $20$  know each other. For a person, we say that it is  $shy$  if it doesn't know at least  $20$  other persons and at least  $2$  of those  $20$  don't know each other. Find the maximal number of people in that group, if we know that group doesn't have any  $social$  nor  $shy$  persons.


An *up-right path* from  $(a, b) \in \mathbb{R}^2$  to  $(c, d) \in \mathbb{R}^2$  is a finite sequence  $(x_1, y_z), \dots, (x_k, y_k)$  of points in  $ \mathbb{R}^2 $  such that  $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$ , and for each  $1 \le i < k$  we have that either  $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$  or  $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$ . Two up-right paths are said to intersect if they share any point.

Find the number of pairs  $(A, B)$  where  $A$  is an up-right path from  $(0, 0)$  to  $(4, 4)$ ,  $B$  is an up-right path from  $(2, 0)$  to  $(6, 4)$ , and  $A$  and  $B$  do not intersect.


Find the smallest possible  $\alpha\in \mathbb{R}$  such that if  $P(x)=ax^2+bx+c$  satisfies  $|P(x)|\leq1 $  for  $x\in [0,1]$  , then we also have  $|P'(0)|\leq \alpha$ .

Solution: \boxed{ \alpha = 8 }


Given that  $\tbinom{n}{k}=\tfrac{n!}{k!(n-k)!}$ , the value of   $$ \sum_{n=3}^{10}\frac{\binom{n}{2}}{\binom{n}{3}\binom{n+1}{3}} $$   can be written in the form  $\tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Compute  $m+n$ .


Find all pairs  $(a,b)$  of positive integers, such that for **every**  $n$  positive integer, the equality  $a^n+b^n=c_n^{n+1}$  is true, for some  $c_n$  positive integer.

Solution: \boxed{ (2, 2) }


Find the natural number  $A$  such that there are  $A$  integer solutions to  $x+y\geq A$  where  $0\leq x \leq 6$  and  $0\leq y \leq 7$ .

*Proposed by David Tang*


 $p(x)$  is a polynomial of degree  $n$  with leading coefficient  $c$ , and  $q(x)$  is a polynomial of degree  $m$  with leading coefficient  $c$ , such that
\[
p(x)^2 = \left(x^2 - 1\right)q(x)^2 + 1
\]
Show that  $p'(x) = nq(x)$ .

Solution: \boxed{ p'(x) = nq(x) }


Let  $S$  be the set of integers between  $1$  and  $2^{40}$  whose binary expansions have exactly two  $1$ 's. If a number is chosen at random from  $S$ , the probability that it is divisible by  $9$  is  $p/q$ , where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .


Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $CA=15$ . The incircle of  $ABC$  meets  $BC$  at  $D$ . Line  $AD$  meets the circle through  $B$ ,  $D$ , and the reflection of  $C$  over  $AD$  at a point  $P\neq D$ . Compute  $AP$ .

*2020 CCA Math Bonanza Tiebreaker Round #4*


Define a sequence  $(a_n)$  recursively by  $a_1=0, a_2=2, a_3=3$  and  $a_n=\max_{0<d<n} a_d \cdot a_{n-d}$  for  $n \ge 4$ . Determine the prime factorization of  $a_{19702020}$ .


Sequences  $a_n$  and  $b_n$  are defined for all positive integers  $n$  such that  $a_1 = 5,$   $b_1 = 7,$    $$ a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2}, $$  and   $$ b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}. $$   $ $
How many integers  $n$  from 1 to 1000 satisfy the property that  $a_n, b_n$  form the legs of a right triangle with a hypotenuse that has integer length?


The *rank* of a rational number  $q$  is the unique  $k$  for which  $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$ , where each  $a_i$  is the smallest positive integer  $q$  such that  $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$ .  Let  $q$  be the largest rational number less than  $\frac{1}{4}$  with rank  $3$ , and suppose the expression for  $q$  is  $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$ .  Find the ordered triple  $(a_1,a_2,a_3)$ .

Solution: \boxed{(5, 21, 421)}


Point  $B$  is on  $\overline{AC}$  with  $AB = 9$  and  $BC = 21$ .  Point  $D$  is not on  $\overline{AC}$  so that  $AD = CD$ , and  $AD$  and  $BD$  are integers.  Let  $s$  be the sum of all possible perimeters of  $\triangle ACD$ .  Find  $s$ .


Find the greatest value of the expression \[ \frac{1}{x^2-4x+9}+\frac{1}{y^2-4y+9}+\frac{1}{z^2-4z+9} \] where  $x$ ,  $y$ ,  $z$  are nonnegative real numbers such that  $x+y+z=1$ .


For  $n\geq 2$  , an equilateral triangle is divided into  $n^2$  congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the  $\frac{(n+1)(n+2)}{2}$  vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the big triangle.


Bobbo starts swimming at  $2$  feet/s across a  $100$  foot wide river with a current of  $5$  feet/s.  Bobbo doesn’t know that there is a waterfall  $175$  feet from where he entered the river. He realizes his predicament midway across the river. What is the minimum speed that Bobbo must increase to make it to the other side of the river safely?


Find  all functions   $f:Q\rightarrow Q$  such that  \[ f(x+y)+f(y+z)+f(z+t)+f(t+x)+f(x+z)+f(y+t)\ge 6f(x-3y+5z+7t) \]  for all   $x,y,z,t\in Q.$

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


Using only once each of the digits  $1, 2, 3, 4, 5, 6, 7$  and  $ 8$ , write the square and the cube of a positive integer. Determine what that number can be.


In the interior of an equilateral triangle  $ ABC $  a point  $ P $  is chosen such that  $ PA ^2 = PB ^2 + PC ^2 $ . Find the measure of  $ \angle BPC $ .


Let  $f(x)=\sum_{i=1}^{2014}|x-i|$ . Compute the length of the longest interval  $[a,b]$  such that  $f(x)$  is constant on that interval.


Find maximum of the expression  $(a -b^2)(b - a^2)$ , where  $0 \le  a,b  \le 1$ .


Let  $a$  and  $b$  be real numbers such that  $ \frac {ab}{a^2 + b^2} = \frac {1}{4} $ . Find all possible values of  $ \frac {|a^2-b^2|}{a^2+b^2} $ .


Determine the minimum number of lines that can be drawn on the plane so that they intersect in exactly  $200$  distinct points.

(Note that for  $3$  distinct points, the minimum number of lines is  $3$  and for  $4$  distinct points, the minimum is  $4$ )

Solution: \boxed{ 21 }


Determine all polynomial  $P(x)\in \mathbb{R}[x]$  satisfying the following two conditions:
(a)  $P(2017)=2016$  and
(b)  $(P(x)+1)^2=P(x^2+1)$  for all real number  $x$ .

Solution: \boxed{P(x) = x - 1}


Find the least real number  $k$  with the following property: if the real numbers  $x$ ,  $y$ , and  $z$  are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]

Solution: \boxed{ \frac{16}{9} }


On an infinite chessboard, a solitaire game is played as follows: at the start, we have  $n^2$  pieces occupying a square of side  $n.$  The only allowed move is to jump over an occupied square to an unoccupied one, and the piece which has been jumped over is removed. For which  $n$  can the game end with only one piece remaining on the board?

Solution: \boxed{ n }


Find all positive integers  $n$  such that  $n^4 - n^3 + 3n^2 + 5$  is a perfect square.

Solution: \boxed{ n = 2 }


Find the smallest positive integer  $k$  for which there exists a colouring of the positive integers  $\mathbb{Z}_{>0}$  with  $k$  colours and a function  $f:\mathbb{Z}_{>0}\to \mathbb{Z}_{>0}$  with the following two properties:

  $(i)$  For all positive integers  $m,n$  of the same colour,  $f(m+n)=f(m)+f(n).$   $(ii)$  There are positive integers  $m,n$  such that  $f(m+n)\ne f(m)+f(n).$ *In a colouring of  $\mathbb{Z}_{>0}$  with  $k$  colours, every integer is coloured in exactly one of the  $k$  colours. In both  $(i)$  and  $(ii)$  the positive integers  $m,n$  are not necessarily distinct.*

Solution: \boxed{ k = 3 }


A subset of  $X$  of  $\{1,2,3, \ldots 10000 \}$  has the following property: If  $a,b$  are distinct elements of  $X$ , then  $ab\not\in X$ . What is the maximal number of elements in  $X$ ?


Find all value of  $ n$  for which there are nonzero real numbers  $ a, b, c, d$  such that after expanding and collecting similar terms, the polynomial  $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$  has exactly  $ n$  nonzero coefficients.

Solution: \boxed{0, 50, 100, 101}


Let  $ AB$  be the diameter of a circle with a center  $ O$  and radius  $ 1$ . Let  $ C$  and  $ D$  be two points on the circle such that  $ AC$  and  $ BD$  intersect at a point  $ Q$  situated inside of the circle, and  $ \angle AQB\equal{} 2 \angle COD$ . Let  $ P$  be a point that intersects the tangents to the circle that pass through the points  $ C$  and  $ D$ .

Determine the length of segment  $ OP$ .


Find the number of subsets  $ S$  of  $ \{1,2, \dots 63\}$  the sum of whose elements is  $ 2008$ .


We call a number greater than  $25$ , *semi-prime* if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be *semi-prime*?


The triangle  $ABC$  is  $| BC | = a$  and  $| AC | = b$ . On the ray starting from vertex  $C$  and passing the midpoint of side  $AB$  , choose any point  $D$  other than vertex  $C$ . Let  $K$  and  $L$  be the projections of  $D$  on the lines  $AC$  and  $BC$ , respectively,  $K$  and  $L$ . Find the ratio  $| DK | : | DL |$ .

Solution: \boxed{\frac{a}{b}}


A kite is inscribed in a circle with center  $O$  and radius  $60$ . The diagonals of the kite meet at a point  $P$ , and  $OP$  is an integer. The minimum possible area of the kite can be expressed in the form  $a\sqrt{b}$ , where  $a$  and  $b$  are positive integers and  $b$  is squarefree. Find  $a+b$ .

Solution: \boxed{ 239 }


Let  $a_0$ ,  $a_1$ ,  $a_2$ ,  $\dots$  be an infinite sequence of real numbers such that  $a_0 = \frac{4}{5}$  and
\[
  a_{n} = 2 a_{n-1}^2 - 1
\]
for every positive integer  $n$ .  Let  $c$  be the smallest number such that for every positive integer  $n$ , the product of the first  $n$  terms satisfies the inequality
\[
  a_0 a_1 \dots a_{n - 1} \le \frac{c}{2^n}.
\]
What is the value of  $100c$ , rounded to the nearest integer?


Three lines are drawn parallel to each of the three sides of  $\triangle ABC$  so that the three lines intersect in the interior of  $ABC$ . The resulting three smaller triangles have areas  $1$ ,  $4$ , and  $9$ . Find the area of  $\triangle ABC$ .

[asy]
defaultpen(linewidth(0.7)); size(120);
pair relpt(pair P, pair Q, real a, real b) { return (a*Q+b*P)/(a+b); }
pair B = (0,0), C = (1,0), A = (0.3, 0.8), D = relpt(relpt(A,B,3,3),relpt(A,C,3,3),1,2);
draw(A--B--C--cycle);
label(" $A$ ",A,N); label(" $B$ ",B,S); label(" $C$ ",C,S);
filldraw(relpt(A,B,2,4)--relpt(A,B,3,3)--D--cycle, gray(0.7));
filldraw(relpt(A,C,1,5)--relpt(A,C,3,3)--D--cycle, gray(0.7));
filldraw(relpt(C,B,2,4)--relpt(B,C,1,5)--D--cycle, gray(0.7));[/asy]


A *T-tetromino* is formed by adjoining three unit squares to form a  $1 \times 3$  rectangle, and adjoining on top of the middle square a fourth unit square.
Determine the least number of unit squares that must be removed from a  $202 \times 202$  grid so that it can be tiled using T-tetrominoes.


The initial number of inhabitants of a city of more than  $150$  inhabitants is a perfect square. With an increase of  $1000$  inhabitants it becomes a perfect square plus a unit. After from another increase of  $1000$  inhabitants it is again a perfect square. Determine the quantity of inhabitants that are initially in the city.


In a tetrahedral  $ABCD$ , given that  $\angle ADB=\angle BDC =\angle CDA=\frac{\pi}{3}$ ,  $AD=BD=3$ , and  $CD=2$ . Find the radius of the circumsphere of  $ABCD$ .


Find all  $10$ -digit whole numbers  $N$ , such that first  $10$  digits of  $N^2$  coincide with the digits of  $N$  (in the same order).


Define an operation  $\Diamond$  as  $ a \Diamond b = 12a - 10b.$  Compute the value of  $((((20 	\Diamond 22) 	\Diamond 22) 	\Diamond 22) 	\Diamond22).$


Let  $P$  denote the set of all ordered pairs  $ \left(p,q\right)$  of nonnegative integers. Find all functions  $f: P \rightarrow \mathbb{R}$  satisfying
\[ f(p,q) \equal{} \begin{cases} 0 & \text{if} \; pq \equal{} 0,
1 \plus{} \frac{1}{2} f(p+1,q-1) \plus{}  \frac{1}{2} f(p-1,q+1) & \text{otherwise} \end{cases}
\]

Compare IMO shortlist problem 2001, algebra A1 for the three-variable case.

Solution: \boxed{ f(p, q) = pq }


There are  $2016$  costumers who entered a shop on a particular day. Every customer entered the shop exactly once. (i.e. the customer entered the shop, stayed there for some time and then left the shop without returning back.)
Find the maximal  $k$  such that the following holds:
There are  $k$  customers such that either all of them were in the shop at a specic time instance or no two of them were both in the shop at any time instance.


Let  $T$  be the answer to question  $18$ . Rectangle  $ZOMR$  has  $ZO = 2T$  and  $ZR = T$ . Point  $B$  lies on segment  $ZO$ ,  $O'$  lies on segment  $OM$ , and  $E$  lies on segment  $RM$  such that  $BR = BE = EO'$ , and  $\angle BEO' = 90^o$ . Compute  $2(ZO + O'M + ER)$ .

PS. You had better calculate it in terms of  $T$ .


Let  $p(x)$  and  $q(x)$  be two cubic polynomials such that  $p(0)=-24$ ,  $q(0)=30$ , and \[p(q(x))=q(p(x))\] for all real numbers  $x$ .  Find the ordered pair  $(p(3),q(6))$ .

Solution: \boxed{ (3, -24) }


One of the receipts for a math tournament showed that  $72$  identical trophies were purchased for  $\$  $-$ 99.9$-, where the first and last digits were illegible.  How much did each trophy cost?


A wall made of mirrors has the shape of  $\triangle ABC$ , where  $AB = 13$ ,  $BC = 16$ , and  $CA = 9$ . A laser positioned at point  $A$  is fired at the midpoint  $M$  of  $BC$ . The shot reflects about  $BC$  and then strikes point  $P$  on  $AB$ . If  $\tfrac{AM}{MP} = \tfrac{m}{n}$  for relatively prime positive integers  $m, n$ , compute  $100m+n$ .

*Proposed by Michael Tang*


Consider all ordered pairs  $(m, n)$  of positive integers satisfying  $59 m - 68 n = mn$ . Find the sum of all the possible values of  $n$  in these ordered pairs.


Three parallel lines  $L_1, L_2, L_2$  are drawn in the plane such that the perpendicular distance between  $L_1$  and  $L_2$  is  $3$  and the perpendicular distance between lines  $L_2$  and  $L_3$  is also  $3$ . A square  $ABCD$  is constructed such that  $A$  lies on  $L_1$ ,  $B$  lies on  $L_3$  and  $C$  lies on  $L_2$ . Find the area of the square.


Suppose that the roots of the quadratic  $x^2 + ax + b$  are  $\alpha$  and  $\beta$ . Then  $\alpha^3$  and   $\beta^3$  are the roots of some quadratic  $x^2 + cx + d$ . Find  $c$  in terms of  $a$  and  $b$ .

Solution: \boxed{a^3 - 3ab}


The Fibonacci numbers are defined by  $F_1=F_2=1$  and  $F_n=F_{n-1}+F_{n-2}$  for  $n>2$ .  It is well-known that the sum of any  $10$  consecutive Fibonacci numbers is divisible by  $11$ .  Determine the smallest integer  $N$  so that the sum of any  $N$  consecutive Fibonacci numbers is divisible by  $12$ .


A wheel is rolled without slipping through  $15$  laps on a circular race course with radius  $7$ . The wheel is perfectly circular and has radius  $5$ . After the three laps, how many revolutions around its axis has the wheel been turned through?


Two 10-digit integers are called neighbours if they diﬀer in exactly one digit (for example, integers  $1234567890$  and  $1234507890$  are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.

Solution: \boxed{9 \cdot 10^8}


A rectangle  $ABEF$  is drawn on the leg  $AB$  of a right triangle  $ABC$ , whose apex  $F$  is on the leg  $AC$ . Let  $X$  be the intersection of the diagonal of the rectangle  $AE$  and the hypotenuse  $BC$  of the triangle. In what ratio does point  $X$  divide the hypotenuse  $BC$  if it is known that  $| AC | = 3 | AB |$  and  $| AF | = 2 | AB |$ ?


Each person in Cambridge drinks a (possibly different)  $12$  ounce mixture of water and apple juice,
where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks  $\frac{1}{6}$  of the total amount of water drunk and  $\frac{1}{8}$  of the total amount of apple juice drunk. How many people are in Cambridge?


A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.

How many three-element subsets of the set of integers  $\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}$  are arithmetic and harmonic?

(Remark: The arithmetic mean  $A(a,b)$  and the harmonic mean  $H(a,b)$  are defined as
\[A(a,b)=\frac{a+b}{2}\quad\mbox{and}\quad H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,}\]
respectively, where  $H(a,b)$  is not defined for some  $a$ ,  $b$ .)


In  $\triangle ABC$ ,  $AB= 425$ ,  $BC=450$ , and  $AC=510$ . An interior point  $P$  is then drawn, and segments are drawn through  $P$  parallel to the sides of the triangle. If these three segments are of an equal length  $d$ , find  $d$ .


Find all pairs of positive integers  $(a,b)$  such that  $a^2 + b^2$  divides both  $a^3 + 1$  and  $b^3 + 1$ .

Solution: \boxed{(1, 1)}


Let  $p$  be a prime number of the form  $4k+1$ . Show that \[\sum^{p-1}_{i=1}\left( \left \lfloor \frac{2i^{2}}{p}\right \rfloor-2\left \lfloor \frac{i^{2}}{p}\right \rfloor \right) = \frac{p-1}{2}.\]

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


Positive real numbers  $a$ ,  $b$ ,  $c$  satisfy  $a+b+c=1$ . Find the smallest possible value of   $$ E(a,b,c)=\frac{a^3}{1-a^2}+\frac{b^3}{1-b^2}+\frac{c^3}{1-c^2}. $$


Let  $ \left( a_n \right)_{n\ge 1} $  be a sequence of real numbers such that  $ a_1>2 $  and  $ a_{n+1} =a_1+\frac{2}{a_n} , $  for all natural numbers  $ n. $ **a)** Show that  $ a_{2n-1} +a_{2n} >4 , $  for all natural numbers  $ n, $  and  $ \lim_{n\to\infty} a_n =2. $ **b)** Find the biggest real number  $ a $  for which the following inequality is true:  $$  \sqrt{x^2+a_1^2} +\sqrt{x^2+a_2^2} +\sqrt{x^2+a_3^2} +\cdots +\sqrt{x^2+a_n^2} > n\sqrt{x^2+a^2}, \quad\forall x\in\mathbb{R} ,\quad\forall n\in\mathbb{N} .  $$

Solution: \boxed{ 2 }


Let  $n$  be a positive integer. A sequence  $(a, b, c)$  of  $a, b, c \in \{1, 2, . . . , 2n\}$  is called *joke* if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences  $(4, 5, 3)$  and  $(3, 8, 3)$  are jokes, but  $(3, 2, 7)$  and  $(3, 8, 8)$  are not. Determine the number of joke sequences in terms of  $n$ .


On the coordinate plane is given the square with vertices  $T_1(1,0),T_2(0,1),T_3(-1,0),T_4(0,-1)$ . For every  $n\in\mathbb N$ , point  $T_{n+4}$  is defined as the midpoint of the segment  $T_nT_{n+1}$ . Determine the coordinates of the limit point of  $T_n$  as  $n\to\infty$ , if it exists.

Solution: \boxed{ (0, 0) }


The world-renowned Marxist theorist *Joric* is obsessed with both mathematics and social egalitarianism. Therefore, for any decimal representation of a positive integer  $n$ , he tries to partition its digits into two groups, such that the difference between the sums of the digits in each group be as small as possible. Joric calls this difference the *defect* of the number  $n$ . Determine the average value of the defect (over all positive integers), that is, if we denote by  $\delta(n)$  the defect of  $n$ , compute \[\lim_{n \rightarrow \infty}\frac{\sum_{k = 1}^{n}\delta(k)}{n}.\]
*Iurie Boreico*


For every positive integeer  $n>1$ , let  $k(n)$  the largest positive integer  $k$  such that there exists a positive integer  $m$  such that  $n = m^k$ .

Find   $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$

Solution: \boxed{ 1 }


Determine the least positive integer  $n{}$  for which the following statement is true: the product of any  $n{}$  odd consecutive positive integers is divisible by  $45$ .


The number of positive integer pairs  $(a,b)$  that have  $a$  dividing  $b$  and  $b$  dividing  $2013^{2014}$  can be written as  $2013n+k$ , where  $n$  and  $k$  are integers and  $0\leq k<2013$ .  What is  $k$ ?  Recall  $2013=3\cdot 11\cdot 61$ .


The sides and vertices of a pentagon are labelled with the numbers  $1$  through  $10$  so that the sum of the numbers on every side is the same. What is the smallest possible value of this sum?


Let  $n$  be a positive integer. Each of the numbers  $1,2,3,\ldots,100$  is painted with one of  $n$  colors in such a way that two distinct numbers with a sum divisible by  $4$  are painted with different colors. Determine the smallest value of  $n$  for which such a situation is possible.


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


Let  $A$  be a subset of  $\{1,2,\ldots,2020\}$  such that the difference of any two distinct elements in  $A$  is not prime. Determine the maximum number of elements in set  $A$ .


**Problem 2**
Determine all pairs  $(n, m)$  of positive integers satisfying the equation  $$ 5^n = 6m^2 + 1\ .  $$

Solution: \boxed{ (2, 2) }


For any positive integer  $a$ , define  $M(a)$  to be the number of positive integers  $b$  for which  $a+b$  divides  $ab$ . Find all integer(s)  $a$  with  $1\le a\le 2013$  such that  $M(a)$  attains the largest possible value in the range of  $a$ .


Triangle  $ABC$  has a right angle at  $C$ , and  $D$  is the foot of the altitude from  $C$  to  $AB$ . Points  $L, M,$  and  $N$  are the midpoints of segments  $AD, DC,$  and  $CA,$  respectively. If  $CL = 7$  and  $BM = 12,$  compute  $BN^2$ .


For every n = 2; 3; : : : , we put  $$ A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right)  $$  Determine all positive integer  $ n (n \geq 2)$  such that  $\frac{1}{A_n}$  is an integer.


How many subsets of  $\{1, 2, ... , 2n\}$  do not contain two numbers with sum  $2n+1$ ?


A week ago, Sandy’s seasonal Little League batting average was  $360$ . After five more at bats this week, Sandy’s batting average is up to  $400$ . What is the smallest number of hits that Sandy could have had this season?


Let  $\mathcal F = \{ f: [0,1] \to [0,\infty) \mid f$  continuous  $\}$  and  $n$  an integer,  $n\geq 2$ . Find the smallest real constant  $c$  such that for any  $f\in \mathcal F$  the following inequality takes place \[ \int^1_0 f \left( \sqrt [n] x \right) dx \leq c \int^1_0 f(x) dx.  \]

Solution: \boxed{ c = n }


At the round table,  $10$  people are sitting, some of them are knights, and the rest are liars (knights always say pride, and liars always lie) .  It is clear thath I have at least one knight and at least one liar.
What is the largest number of those sitting at the table can say: ''Both of my neighbors are knights '' ?
(A statement that is at least partially false is considered false.)


Integer  $n>2$  is given. Find the biggest integer  $d$ , for which holds, that from any set  $S$  consisting of  $n$  integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by  $d$ .

Solution: \boxed{ n-1 }


Given a line segment  $PQ$  with endpoints on the parabola  $y=x^2$  such that the area bounded by  $PQ$  and the parabola always equal to  $\frac 43.$  Find the equation of the locus of the midpoint  $M$ .

Solution: \boxed{y = x^2 + 1}


Let  $ p \geq 3$  be a prime, and let  $ p$  points  $ A_{0}, \ldots, A_{p-1}$  lie on a circle in that order. Above the point  $ A_{1+\cdots+k-1}$  we write the number  $ k$  for  $ k=1, \ldots, p$  (so  $ 1$  is written above  $ A_{0}$ ). How many points have at least one number written above them?


An object moves in two dimensions according to
\[\vec{r}(t) = (4.0t^2 - 9.0)\vec{i} + (2.0t - 5.0)\vec{j}\]
where  $r$  is in meters and  $t$  in seconds. When does the object cross the  $x$ -axis? $ \textbf{(A)}\ 0.0 \text{ s}\qquad\textbf{(B)}\ 0.4 \text{ s}\qquad\textbf{(C)}\ 0.6 \text{ s}\qquad\textbf{(D)}\ 1.5 \text{ s}\qquad\textbf{(E)}\ 2.5 \text{ s}$

Solution: \boxed{2.5 \text{ s}}


Given positive integer  $n$ , find the biggest real number  $C$  which satisfy the condition that if the sum of the reciprocals of a set of integers (They can be the same.) that are greater than  $1$  is less than  $C$ , then we can divide the set of numbers into no more than  $n$  groups so that the sum of reciprocals of every group is less than  $1$ .


Let  $n$  be a positive integer, set  $S_n = \{ (a_1,a_2,\cdots,a_{2^n}) \mid a_i=0 \ \text{or} \ 1, 1 \leq i \leq 2^n\}$ . For any two elements  $a=(a_1,a_2,\cdots,a_{2^n})$  and  $b=(b_1,b_2,\cdots,b_{2^n})$  of  $S_n$ , define
\[ d(a,b)= \sum_{i=1}^{2^n} |a_i - b_i| \]
We call  $A \subseteq S_n$  a  $\textsl{Good Subset}$  if  $d(a,b) \geq 2^{n-1}$  holds for any two distinct elements  $a$  and  $b$  of  $A$ . How many elements can the  $\textsl{Good Subset}$  of  $S_n$  at most have?

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


Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola  $ xy\equal{}1$  and both branches of the hyperbola  $ xy\equal{}\minus{}1.$  (A set  $ S$  in the plane is called *convex* if for any two points in  $ S$  the line segment connecting them is contained in  $ S.$ )


Find the number of positive integers  $k \le 2018$  for which there exist integers  $m$  and  $n$  so that  $k = 2^m + 2^n$ .
For example,  $64 = 2^5 + 2^5$ ,  $65 = 2^0 + 2^6$ , and  $66 = 2^1 + 2^6$ .


  Let  $F(0)=0$ ,  $F(1)=\frac32$ , and  $F(n)=\frac{5}{2}F(n-1)-F(n-2)$   for  $n\ge2$ .

Determine whether or not  $\displaystyle{\sum_{n=0}^{\infty}\,
    \frac{1}{F(2^n)}}$  is a rational number.

(Proposed by Gerhard Woeginger, Eindhoven University of Technology)


Find the largest value of n, such that there exists a polygon with n sides, 2 adjacent sides of length 1, and all his diagonals have an integer length.

Solution: \boxed{ n = 4 }


Suppose  $100$  points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?


Let  $n$  be a positive integer,  $n \ge 2$ , and put  $\theta=\frac{2\pi}{n}$ . Define points  $P_k=(k,0)$  in the *xy*-plane, for  $k=1,2,\dots,n$ . Let  $R_k$  be the map that rotates the plane counterclockwise by the angle  $\theta$  about the point  $P_k$ . Let  $R$  denote the map obtained by applying in order,  $R_1$ , then  $R_2$ , ..., then  $R_n$ . For an arbitrary point  $(x,y)$ , find and simplify the coordinates of  $R(x,y)$ .

Solution: \boxed{(x+n, y)}


Hamilton Avenue has eight houses. On one side of the street are the houses
numbered 1,3,5,7 and directly opposite are houses 2,4,6,8 respectively. An
eccentric postman starts deliveries at house 1 and delivers letters to each of
the houses, finally returning to house 1 for a cup of tea. Throughout the
entire journey he must observe the following rules. The numbers of the houses
delivered to must follow an odd-even-odd-even pattern throughout, each house
except house 1 is visited exactly once (house 1 is visited twice) and the postman
at no time is allowed to cross the road to the house directly opposite. How
many different delivery sequences are possible?


Let  $a$  be real numbers.Find the following limit value.

\[ \lim_{T\rightarrow \infty} \frac{1}{T} \int_0^T (\sin x+\sin ax)^2 dx.  \]

Solution: \boxed{ 1 }


For all positive integers  $m,n$  define  $f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3$ . Find all numbers  $n$  with the property that  $f(m, n)$  is divisible by 1992 for every  $m$ .

*Bulgaria*

Solution: \boxed{ n }


Find the magnitude of the product of all complex numbers  $c$  such that the recurrence defined by  $x_1 = 1$ ,  $x_2 = c^2 - 4c + 7$ , and  $x_{n+1} = (c^2 - 2c)^2 x_n x_{n-1} + 2x_n - x_{n-1}$  also satisfies  $x_{1006} = 2011$ .

*Author: Alex Zhu*


How many positive integers  $n\leq100$  satisfy  $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$ ? Here  $\left\lfloor x\right\rfloor$  is the greatest integer less than or equal to  $x$ ; for example,  $\left\lfloor\pi\right\rfloor=3$ .

*2018 CCA Math Bonanza Lightning Round #3.2*


Let  $x,y,z$  is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}
{y\left( {6 - z} \right) = 9}
{z\left( {6 - x} \right) = 9}\end{cases}$

Solution: \boxed{ x = 3 }


Consider a  $2 \times n$  grid where each cell is either black or white, which we attempt to tile with  $2 \times 1$  black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer  $N$  where  $N$  takes the value  $n$  with probability  $\frac{1}{2^n}$ . We then take a  $2 \times N$  grid and randomly color each cell black or white independently with equal probability. Compute the probability the resulting grid has a valid tiling.


If  $a,b,c,d$  are Distinct Real no. such that  $a = \sqrt{4+\sqrt{5+a}}$  $b = \sqrt{4-\sqrt{5+b}}$  $c = \sqrt{4+\sqrt{5-c}}$  $d = \sqrt{4-\sqrt{5-d}}$ Then  $abcd = $


Let triangle  $ABC$  with incenter  $I$  satisfy  $AB = 10$ ,   $BC = 21$ , and   $CA = 17$ . Points  $D$  and E lie on side  $BC$  such that  $BD = 4$ ,  $DE = 6$ , and  $EC = 11$ . The circumcircles of triangles  $BIE$  and  $CID$  meet again at point  $P$ , and line  $IP$  meets the altitude from  $A$  to  $BC$  at  $X$ . Find  $(DX \cdot EX)^2$ .


The equation is given $x^2-(m+3)x+m+2=0$ .
If  $x_1$  and  $x_2$  are its solutions find all  $m$  such that  $\frac{x_1}{x_1+1}+\frac{x_2}{x_2+1}=\frac{13}{10}$ .

Solution: \boxed{ m = 2 }


Determine all positive integers  $n$  such that it is possible to fill the  $n \times n$  table with numbers  $1, 2$  and  $-3$  so that the sum of the numbers in each row and each column is  $0$ .

Solution: \boxed{ n \geq 3 }


Let  $A(x)=\lfloor\frac{x^2-20x+16}{4}\rfloor$ ,  $B(x)=\sin\left(e^{\cos\sqrt{x^2+2x+2}}\right)$ ,  $C(x)=x^3-6x^2+5x+15$ ,  $H(x)=x^4+2x^3+3x^2+4x+5$ ,  $M(x)=\frac{x}{2}-2\lfloor\frac{x}{2}\rfloor+\frac{x}{2^2}+\frac{x}{2^3}+\frac{x}{2^4}+\ldots$ ,  $N(x)=\textrm{the number of integers that divide }\left\lfloor x\right\rfloor$ ,  $O(x)=|x|\log |x|\log\log |x|$ ,  $T(x)=\sum_{n=1}^{\infty}\frac{n^x}{\left(n!\right)^3}$ , and  $Z(x)=\frac{x^{21}}{2016+20x^{16}+16x^{20}}$  for any real number  $x$  such that the functions are defined. Determine   $$ C(C(A(M(A(T(H(B(O(N(A(N(Z(A(2016)))))))))))))). $$  *2016 CCA Math Bonanza Lightning #5.3*


While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the “blue screen of death” (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk. He had computed three positive integers  $a$ ,  $b$ , and  $c$ , and recalled that their product is  $24$ , but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than  $25$  with fewer than  $6$  divisors. Help James by computing  $a+b+c$ .


Let  $ABC$  be a triangle and let  $P$  be a point in its interior. Suppose  $ \angle B A P = 10 ^ { \circ } , \angle A B P = 20 ^ { \circ } , \angle P C A = 30 ^ { \circ } $  and  $ \angle P A C = 40 ^ { \circ } $ . Find  $ \angle P B C $ .


A random walk is a process in which something moves from point to point, and where the direction of movement at each step is randomly chosen. Suppose that a person conducts a random walk on a line: he starts at  $0$  and each minute randomly moves either  $1$  unit in the positive direction or  $1$  unit in the negative direction. What is his expected distance from the origin after  $6$  moves?

*2017 CCA Math Bonanza Lightning Round #3.4*


Let  $\overline{abc}$  be a three digit number with nonzero digits such that  $a^2 + b^2 = c^2$ . What is the largest possible prime factor of  $\overline{abc}$


A sequence of seven digits is randomly chosen in a weekly lottery. Every digit can be any of the digits  $0, 1, 2, 3, 4, 5, 6, 7, 8, 9.$
What is the probability of having at most five different digits in the sequence?


A sequence of real numbers  $\{a_n\}_{n = 1}^\infty (n=1,2,...)$  has the following property:
\begin{align*}
6a_n+5a_{n-2}=20+11a_{n-1}\  (\text{for }n\geq3).
\end{align*}
The first two elements are  $a_1=0, a_2=1$ . Find the integer closest to  $a_{2011}$ .


Throw 6 dice at a time, find the probability, in the lowest form, such that there will be exactly four kinds of the outcome.

Solution: \boxed{ \frac{325}{648} }


A sequence  $(a_n)$  of positive integers is defined by  $a_0=m$  and  $a_{n+1}= a_n^5 +487$  for all  $n\ge 0$ .
Find all positive integers  $m$  such that the sequence contains the maximum possible number of perfect squares.


The sequence of natural numbers  $1, 5, 6, 25, 26, 30, 31,...$  is made up of powers of  $5$  with natural exponents or sums of powers of  $5$  with different natural exponents, written in ascending order. Determine the term of the string written in position  $167$ .


Find the best approximation of  $\sqrt{3}$  by a rational number with denominator less than or equal to  $15$


Camila creates a pattern to write the following numbers: $2, 4$  $5, 7, 9, 11$  $12, 14, 16, 18, 20, 22$  $23, 25, 27, 29, 31, 33, 35, 37$  $…$ Following the same pattern, what is the sum of the numbers in the tenth row?


Find the number of ordered triples of sets  $(T_1, T_2, T_3)$  such that
1. each of  $T_1, T_2$ , and  $T_3$  is a subset of  $\{1, 2, 3, 4\}$ ,
2.  $T_1 \subseteq T_2 \cup T_3$ ,
3.  $T_2 \subseteq T_1 \cup T_3$ , and
4.  $T_3\subseteq T_1 \cup T_2$ .


Let  $ABC$  be a triangle with  $AB = 5$ ,  $AC = 8$ , and  $BC = 7$ . Let  $D$  be on side  $AC$  such that  $AD = 5$  and  $CD = 3$ . Let  $I$  be the incenter of triangle  $ABC$  and  $E$  be the intersection of the perpendicular bisectors of  $\overline{ID}$  and  $\overline{BC}$ .  Suppose  $DE = \frac{a\sqrt{b}}{c}$  where  $a$  and  $c$  are relatively prime positive integers, and  $b$  is a positive integer not divisible by the square of any prime. Find  $a+b+c$ .

*Proposed by Ray Li*


The sequence  ${a_0, a_1, a_2, ...}$  of real numbers satisfies the recursive relation   $$ n(n+1)a_{n+1}+(n-2)a_{n-1} = n(n-1)a_n $$   for every positive integer  $n$ , where  $a_0 = a_1 = 1$ . Calculate the sum   $$ \frac{a_0}{a_1} + \frac{a_1}{a_2} + ... + \frac{a_{2008}}{a_{2009}} $$  .

Solution: \boxed{2009 \times 1005}


An alphabet consists of  $n$  letters. What is the maximal length of a word if we know that any two consecutive letters  $a,b$  of the word are different and that the word cannot be reduced to a word of the kind  $abab$  with  $a\neq b$  by removing letters.

Solution: \boxed{ 2n - 1 }


In a convex pentagon  $ABCDE$  the sides have lengths  $1,2,3,4,$  and  $5$ , though not necessarily in that order.  Let  $F,G,H,$  and  $I$  be the midpoints of the sides  $AB$ ,  $BC$ ,  $CD$ , and  $DE$ , respectively.  Let  $X$  be the midpoint of segment  $FH$ , and  $Y$  be the midpoint of segment  $GI$ .  The length of segment  $XY$  is an integer.  Find all possible values for the length of side  $AE$ .


There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later).
After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?
*I. Izmest’ev*


How many nonzero coefficients can a polynomial  $ P(x)$  have if its coefficients are integers and  $ |P(z)| \le 2$  for any complex number  $ z$  of unit length?


Let  $S={1,2,\ldots,100}$ . Consider a partition of  $S$  into  $S_1,S_2,\ldots,S_n$  for some  $n$ , i.e.  $S_i$  are nonempty, pairwise disjoint and  $\displaystyle S=\bigcup_{i=1}^n S_i$ . Let  $a_i$  be the average of elements of the set  $S_i$ . Define the score of this partition by
\[\dfrac{a_1+a_2+\ldots+a_n}{n}.\]

Among all  $n$  and partitions of  $S$ , determine the minimum possible score.


The positive integers  $N$  and  $N^2$  both end in the same sequence of four digits  $abcd$  when written in base 10, where digit  $a$  is not zero. Find the three-digit number  $abc$ .


Let  $f(x) = x^3 + 3x - 1$  have roots  $ a, b, c$ . Given that  $\frac{1}{a^3 + b^3}+\frac{1}{b^3 + c^3}+\frac{1}{c^3 + a^3}$  can be written as  $\frac{m}{n}$ , where  $m, n$  are positive integers and  $gcd(m, n) = 1$ , find  $100m + n$ .

Solution: \boxed{ 3989 }


In convex quadrilateral  $ABCD$ ,  $\angle BAD = \angle BCD = 90^o$ , and  $BC = CD$ . Let  $E$  be the intersection of diagonals  $\overline{AC}$  and  $\overline{BD}$ . Given that  $\angle AED = 123^o$ , find the degree measure of  $\angle ABD$ .


In an  $n$ -by- $m$  grid,  $1$  row and  $1$  column are colored blue, the rest of the cells are white. If precisely  $\frac{1}{2010}$  of the cells in the grid are blue, how many values are possible for the ordered pair  $(n,m)$


Find the number of pairs  $(m,n)$  of integers with  $-2014\le m,n\le 2014$  such that  $x^3+y^3 = m + 3nxy$  has infinitely many integer solutions  $(x,y)$ .

*Proposed by Victor Wang*


Consider a fair coin and a fair 6-sided die.  The die begins with the number 1 face up.  A *step* starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step.  After 5 such steps, what is the probability that the number 1 is face up on the die?


While there do not exist pairwise distinct real numbers  $a,b,c$  satisfying  $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let  $a,b,c$  be complex numbers such that  $a^2+b^2+c^2 = ab+bc+ca$  and  $|a+b+c| = 21$ . Given that  $|a-b| = 2\sqrt{3}$ ,  $|a| = 3\sqrt{3}$ , compute  $|b|^2+|c|^2$ .
<details><summary>Clarifications</summary>

- The problem should read  $|a+b+c| = 21$ .  An earlier version of the test read  $|a+b+c| = 7$ ; that value is incorrect.
- $|b|^2+|c|^2$  should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers  $m$  and  $n$ .  Find  $m+n$ .''
</details>

*Ray Li*


A wizard thinks of a number from  $1$  to  $n$ . You can ask the wizard any number of yes/no questions about the number. The wizard must answer all those questions, but not necessarily in the respective order. What is the least number of questions that must be asked in order to know what the number is for sure. (In terms of  $n$ .)

Fresh translation.

Solution: \boxed{ n-1 }


If  $m=2+2\sqrt{44n^2+1}$  is an integer then show that it is also a perfect square. Here  $n$  is a natural number.

Solution: \boxed{ m }


How many different solutions does the congruence  $x^3+3x^2+x+3 \equiv 0 \pmod{25}$  have? $
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 2
\qquad\textbf{(C)}\ 4
\qquad\textbf{(D)}\ 5
\qquad\textbf{(E)}\ 6
$


Let the sequence  $\{a_i\}^\infty_{i=0}$  be defined by  $a_0 =\frac12$  and  $a_n = 1 + (a_{n-1} - 1)^2$ . Find the product   $$ \prod_{i=0}^\infty a_i=a_0a_1a_2\ldots $$


For a positive integer  $n$ ,  $\sigma(n)$  denotes the sum of the positive divisors of  $n$ . Determine   $$ \limsup\limits_{n\rightarrow \infty} \frac{\sigma(n^{2023})}{(\sigma(n))^{2023}} $$  **Note:** Given a sequence ( $a_n$ ) of real numbers, we say that  $\limsup\limits_{n\rightarrow \infty} a_n = +\infty$  if ( $a_n$ ) is not upper bounded, and, otherwise,  $\limsup\limits_{n\rightarrow \infty} a_n$  is the smallest constant  $C$  such that, for every real  $K > C$ , there is a positive integer  $N$  with  $a_n < K$  for every  $n > N$ .


For a positive integer  $a$ , define a sequence of integers  $x_1,x_2,\ldots$  by letting  $x_1=a$  and  $x_{n+1}=2x_n+1$  for  $n\geq 1$ . Let  $y_n=2^{x_n}-1$ . Determine the largest possible  $k$  such that, for some positive integer  $a$ , the numbers  $y_1,\ldots,y_k$  are all prime.

Solution: \boxed{ k = 2 }


The incircle of  $ABC$  touches the sides  $BC,CA,AB$  at  $A' ,B' ,C'$  respectively. The line  $A' C'$   meets the angle bisector of  $\angle A$  at  $D$ . Find  $\angle ADC$ .


Let  $ABCD$  be a trapezium in which  $AB //CD$  and  $AD \perp AB$ . Suppose  $ABCD$  has an incircle which touches  $AB$  at  $Q$  and  $CD$  at  $P$ . Given that  $PC = 36$  and  $QB = 49$ , find  $PQ$ .


Consider the set  $ S_n$  of all the  $ 2^n$  numbers of the type  $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$  where number  $ 2$  appears  $ n\plus{}1$  times.
 $ (a)$  Show that all members of  $ S_n$  are real.
 $ (b)$  Find the product  $ P_n$  of the elements of  $ S_n$ .


You know that the binary function  $\diamond$  takes in two non-negative integers and has the following properties:

\begin{align*}0\diamond a&=1 a\diamond a&=0\end{align*}  $\text{If } a<b, \text{ then } a\diamond b\&=(b-a)[(a-1)\diamond (b-1)].$ Find a general formula for  $x\diamond y$ , assuming that  $y\gex>0$ .

Solution: \boxed{ (y - x)^x }


Suppose that  $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$  for positive integers  $a,b,c,d,e,$  and  $f$ . Find  $\frac{100bdf}{ace}$ .

*Proposed by Andrew Wu*


In the ﬁgure on the right,  $O$  is the center of the circle,  $OK$  and  $OA$  are perpendicular to one another,  $M$  is the midpoint of  $OK$ ,  $BN$   is parallel to  $OK$ , and  $\angle AMN=\angle NMO$ . Determine the measure of  $\angle A B N$  in degrees.

[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K=dir(180), M=0.5*dir(180), N=2/5*dir(90), B=dir(degrees((2/5, sqrt(21/25)))+90);
draw(K--O--A--M--N--B--A^^Circle(origin,1));
label(" $A$ ", A, dir(O--A));
label(" $K$ ", K, dir(O--K));
label(" $B$ ", B, dir(O--B));
label(" $N$ ", N, E);
label(" $M$ ", M, S);
label(" $O$ ", O, SE);[/asy]


Determine the positive integers expressible in the form  $\frac{x^2+y}{xy+1}$ , for at least  $2$  pairs  $(x,y)$  of positive integers


Let  $ABCD$  be a square of side  $a$ . On side  $AD$  consider points  $E$  and  $Z$  such that  $DE=a/3$  and  $AZ=a/4$ . If the lines  $BZ$  and  $CE$  intersect at point  $H$ , calculate the area of the triangle  $BCH$  in terms of  $a$ .


Let  $R$  be the rectangle in the Cartesian plane with vertices at  $(0,0)$ ,  $(2,0)$ ,  $(2,1)$ , and  $(0,1)$ .  $R$  can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7));
draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point  $P$  at random in the interior of  $R$ . Find the probability that the line through  $P$  with slope  $\frac{1}{2}$  will pass through both unit squares.


In the convex pentagon  $ABCDE$ :  $\angle A = \angle C = 90^o$ ,  $AB = AE, BC = CD, AC = 1$ . Find the area of the pentagon.


Let  $\omega_1, \omega_2, \omega_3, \ldots, \omega_{2020!}$  be the distinct roots of  $x^{2020!} - 1$ . Suppose that  $n$  is the largest integer such that  $2^n$  divides the value   $$ \sum_{k=1}^{2020!} \frac{2^{2019!}-1}{\omega_{k}^{2020}+2}. $$   Then  $n$  can be written as  $a! + b$ , where  $a$  and  $b$  are positive integers, and  $a$  is as large as possible. Find the remainder when  $a+b$  is divided by  $1000$ .

*Proposed by vsamc*


For each positive integer  $n$ , determine the smallest possible value of the polynomial  $$ W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx. $$

Solution: \boxed{ -n }


For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened:**i)** Every country voted for exactly three problems.**ii)** Any two countries voted for different sets of problems.**iii)** Given any three countries, there was a problem none of them voted for.
Find the maximal possible number of participating countries.


We consider positive integers  $n$  having at least six positive divisors. Let the positive divisors of  $n$  be arranged in a sequence  $(d_i)_{1\le i\le k}$  with   $$ 1=d_1<d_2<\dots <d_k=n\quad (k\ge 6). $$  Find all positive integers  $n$  such that   $$ n=d_5^2+d_6^2. $$

Solution: \boxed{ n = 500 }