**p1.** Is it possible to place six points in the plane and connect them by nonintersecting segments so that each point will be connected with exactly
a) Three other points?
b) Four other points?**p2.** Martian bank notes can have denomination of  $1, 3, 5, 25$  marts. Is it possible to change a note of  $25$  marts to exactly  $10$  notes of smaller denomination?**p3.** What is bigger: $99 \cdot 99  \cdot .... \cdot 99$  (product of  $20$  factors) or $9999  \cdot 9999  \cdot ...  \cdot 9999$  (product of  $10$  factors)?**p4.** How many whole numbers, from  $1$  to  $2006$ , are divisible neither by  $5$  nor by  $7$ ?

PS. You should use hide for answers.

Solution: \boxed{ 1376 }


Find the positive integer  $n$  such that  $\frac12 \cdot \frac34 + \frac56 \cdot \frac78 +   \frac{9}{10}\cdot \frac{11}{12 }= \frac{n}{1200}$  .


If  $X=1+x+x^2+x^3+\cdots$  and  $Y=1+y+y^2+y^3+\cdots$ , what is  $1+xy+x^2y^2+x^3y^3+\cdots$  in terms of  $X$  and  $Y$  only?


The Sequence  $\{a_{n}\}_{n \geqslant 0}$   is defined by  $a_{0}=1, a_{1}=-4$  and  $a_{n+2}=-4a_{n+1}-7a_{n}$  , for  $n \geqslant 0$ . Find the number of positive integer divisors of  $a^2_{50}-a_{49}a_{51}$ .


If  $x$  and  $y$  are positive integers such that  $(x-4)(x-10)=2^y$ , then Find maximum value of  $x+y$


In an alphabet of  $n$  letters, is  $syllable$  is any ordered pair of two (not necessarily distinct) letters. Some syllables are considered  $indecent$ . A  $word$  is any sequence, finite or infinite, of letters, that does not contain indecent syllables. Find the least possible number of  indecent syllables for which infinite words do not exist.


Let  $Q(x)=a_0+a_1x+\dots+a_nx^n$  be a polynomial with integer coefficients, and  $0\le a_i<3$  for all  $0\le i\le n$ .

Given that  $Q(\sqrt{3})=20+17\sqrt{3}$ , compute  $Q(2)$ .


Find out how many positive integers  $n$  not larger than  $2009$  exist such that the last digit of  $n^{20}$  is  $1$ .


The area of parallelogram  $ABCD$  is  $51\sqrt{55}$  and  $\angle{DAC}$  is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?


Johny's father tells him: "I am twice as old as you will be seven years from the time I was thrice as old as you were". What is Johny's age?


Let  $n$  be the product of the first  $10$  primes, and let   $$ S=\sum_{xy\mid n} \varphi(x) \cdot y, $$  where  $\varphi(x)$  denotes the number of positive integers less than or equal to  $x$  that are relatively prime to  $x$ , and the sum is taken over ordered pairs  $(x, y)$  of positive integers for which  $xy$  divides  $n$ . Compute  $\tfrac{S}{n}.$


Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is  $\tfrac{1}{3}$ . The probability that a man has none of the three risk factors given that he does not have risk factor A is  $\tfrac{p}{q}$ , where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .


In quadrilateral  $ABCD$ ,  $AB \parallel CD$  and  $BC \perp AB$ . Lines  $AC$  and  $BD$  intersect at  $E$ . If  $AB = 20$ ,  $BC = 2016$ , and  $CD = 16$ , find the area of  $\triangle BCE$ .

*Proposed by Harrison Wang*


Alice and Bob live on the same road. At time  $t$ , they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at  $3:19\text{pm}$ , and Bob arrived at Alice's house at  $3:29\text{pm}$ . Charlie, who was driving by, noted that Alice and Bob passed each other at  $3:11\text{pm}$ . Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day.

*Proposed by Kevin You*


Find all pairs of prime numbers  $p, q$  such that the number  $pq + p - 6$  is also prime.

Solution: \boxed{ (2, 3) }


Graphistan has  $2011$  cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer  $k$  such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than  $k.$

Solution: \boxed{ 1005 }


Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point with probability  $\frac{9}{10}$  . Each time Aileen successfully bats the birdie over the net, her opponent, independent of all previous hits, returns the birdie with probability  $\frac{3}{4}$  . Each time Aileen bats the birdie, independent of all previous hits, she returns the birdie with probability  $\frac{m}{n}$  , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .


There are three eight-digit positive integers which are equal to the sum of the eighth powers of their digits. Given that two of the numbers are  $24678051$  and  $88593477$ , compute the third number.

*Proposed by Vincent Huang*


Let  $ABC$  be an acute-angled, not equilateral triangle, where vertex  $A$  lies on the perpendicular bisector of the segment  $HO$ , joining the orthocentre  $H$  to the circumcentre  $O$ . Determine all possible values for the measure of angle  $A$ .

(U.S.A. - 1989 IMO Shortlist)

Solution: \boxed{ 60^\circ }


A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was  $7/9$ . How many players took part at the tournament?


What is the maximum possible value for the sum of the squares of the roots of  $x^4+ax^3+bx^2+cx+d$  where  $a$ ,  $b$ ,  $c$ , and  $d$  are  $2$ ,  $0$ ,  $1$ , and  $7$  in some order?


Katrine has a bag containing 4 buttons with distinct letters M, P, F, G on them (one letter per button).  She picks buttons randomly, one at a time, without replacement, until she picks the button with letter G.  What is the probability that she has at least three picks and her third pick is the button with letter M?


Congruent circles  $\Gamma_1$  and  $\Gamma_2$  have radius  $2012,$  and the center of  $\Gamma_1$  lies on  $\Gamma_2.$  Suppose that  $\Gamma_1$  and  $\Gamma_2$   intersect at  $A$  and  $B$ . The line through  $A$  perpendicular to  $AB$  meets  $\Gamma_1$  and  $\Gamma_2$  again at  $C$  and  $D$ , respectively. Find the length of  $CD$ .

*Author: Ray Li*


The remainder when  $x^{100} -x^{99} +... -x +1$  is divided by  $x^2 -1$  can be written in the form  $ax +b$ . Find  $2a +b$ .

*Proposed by Calvin Garces*


Let  $R=(8,6)$ . The lines whose equations are  $8y=15x$  and  $10y=3x$  contain points  $P$  and  $Q$ , respectively, such that  $R$  is the midpoint of  $\overline{PQ}$ . The length of  $PQ$  equals  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .


Let  $E$  be the set of all continuously differentiable real valued functions  $f$  on  $[0,1]$  such that  $f(0)=0$  and  $f(1)=1$ . Define  $$ J(f)=\int^1_0(1+x^2)f'(x)^2\text dx. $$  a) Show that  $J$  achieves its minimum value at some element of  $E$ .
b) Calculate  $\min_{f\in E}J(f)$ .

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


Take the number   $2^{2004}$  and calculate the sum   $S$  of all its digits. Then the sum of all the digits of   $S$  is calculated to obtain   $R$ . Next, the sum of all the digits of  $R$ is calculated   and so on until a single digit number is reached. Find it.  (For example if we take    $2^7=128$ , we find that    $S=11,R=2$ . So in this case of    $2^7$  the searched digit will be  $2$ ).

Solution: \boxed{ 1 }


There are positive integers  $m$  and  $n$  such that  $m^2 -n = 32$  and   $\sqrt[5]{m +\sqrt{n}}+  \sqrt[5]{m -\sqrt{n}}$  is a real root of the polynomial  $x^5 - 10x^3 + 20x - 40$ . Find  $m + n$ .


Find the smallest positive integer  $k$  such that there is exactly one prime number of the form  $kx + 60$  for the integers  $0 \le x \le 10$ .


Compute the number of positive integer divisors of  $2121$  with a units digit of  $1$ .

*2021 CCA Math Bonanza Individual Round #1*


Let  $f(x)$  be a continuous function satisfying  $f(x)=1+k\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} f(t)\sin (x-t)dt\ (k:constant\ number)$

Find the value of  $k$  for which  $\int_0^{\pi} f(x)dx$  is maximized.

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


Find the sum of all prime numbers  $p$  which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct)  $a$ ,  $b$  and  $c$ .


$12$  friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are  $B_1, B_2, ..., B_{12}$ . Find the largest possible value of the sum  $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$  .


Several pairs of positive integers  $(m ,n )$  satisfy the condition  $19m + 90 + 8n = 1998$ .  Of these,  $(100, 1 )$   is the pair with the smallest value for  $n$ . Find the pair with the smallest value for  $m$ .

Solution: \boxed{ (4, 229) }


Natural numbers  $k, l,p$  and  $q$  are such that if  $a$  and  $b$  are roots of  $x^2 - kx + l = 0$  then  $a +\frac1b$  and  $b + \frac1a$  are the roots of  $x^2 -px + q = 0$ . What is the sum of all possible values of  $q$ ?


Find with all integers  $n$  when   $|n^3 - 4n^2 + 3n - 35|$   and  $|n^2 + 4n + 8|$   are prime numbers.

Solution: \boxed{n = 5}


Solve the equation
\[
  \sin 9^\circ \sin 21^\circ \sin(102^\circ + x^\circ) = \sin 30^\circ \sin 42^\circ \sin x^\circ
\]
for  $x$  where  $0 < x < 90$ .

Solution: \boxed{ x = 9^\circ }


Compute the only element of the set \[\{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \backslash \{-2, 2\}\right\}.\]

*Proposed by Michael Tang*


For  $p \in \mathbb{R}$ , let  $(a_n)_{n \ge 1}$  be the sequence defined by
\[ a_n=\frac{1}{n^p} \int_0^n |\sin( \pi x)|^x \mathrm dx. \]
Determine all possible values of  $p$  for which the series  $\sum_{n=1}^\infty a_n$  converges.

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


If  $k, l, m$  are positive integers with  $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}<1$ , ﬁnd the maximum possible value of  $\frac{1}{k}+\frac{1}{l}+\frac{1}{m}$ .


Consider a sequence  $F_0=2$ ,  $F_1=3$  that has the property  $F_{n+1}F_{n-1}-F_n^2=(-1)^n\cdot2$ . If each term of the sequence can be written in the form  $a\cdot r_1^n+b\cdot r_2^n$ , what is the positive difference between  $r_1$  and  $r_2$ ?


Let  $ABCD$  be a cyclic quadrilateral with  $AB = 1$ ,  $BC = 2$ ,  $CD = 3$ ,  $DA = 4$ . Find the square of the area of quadrilateral  $ABCD$ .


In the acute triangle  $ABC$ , the sum of the distances from the vertices  $B$  and  $C$  to of the orthocenter  $H$  is equal to  $4r,$  where  $r$  is the radius of the circle inscribed in this triangle. Find the perimeter of triangle  $ABC$  if it is known that  $BC=a$ .

(Gryhoriy Filippovskyi)

Solution: \boxed{ 3a }


Let  $a_1, a_2, \ldots$  be a sequence of real numbers such that  $a_1=4$  and  $a_2=7$  such that for all integers  $n$ ,  $\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}$  forms an arithmetic progression, and  $a_{2n}, a_{2n+1}, a_{2n+2}$  forms an arithmetic progression. Find, with proof, the prime factorization of  $a_{2019}$ .

Solution: \boxed{4 \cdot 7^{1009}}


Let  $x_1 \dots, x_{42}$ , be real numbers such that  $5x_{i+1}-x_i-3x_ix_{i+1}=1$  for each  $1 \le i \le 42$ , with  $x_1=x_{43}$ . Find all the product of all possible values for  $x_1 + x_2 + \dots + x_{42}$ .

*Proposed by Michael Ma*


A sequence of distinct circles  $\omega_1, \omega_2, \cdots$  is inscribed in the parabola  $y=x^2$  so that  $\omega_n$  and  $\omega_{n+1}$  are tangent for all  $n$ .  If  $\omega_1$  has diameter  $1$  and touches the parabola at  $(0,0)$ , find the diameter of  $\omega_{1998}$ .


Determine the maximum value of  $m^{2}+n^{2}$ , where  $m$  and  $n$  are integers satisfying  $m,n\in \{1,2,...,1981\}$  and  $(n^{2}-mn-m^{2})^{2}=1.$


Given the isosceles triangle  $ABC$  with  $| AB | = | AC | = 10$  and  $| BC | = 15$ . Let points  $P$  in  $BC$  and  $Q$  in  $AC$  chosen such that  $| AQ | = | QP | = | P C |$ . Calculate the ratio of areas of the triangles   $(PQA): (ABC)$ .


Upon cutting a certain rectangle in half, you obtain two rectangles that are scaled down versions of the original. What is the ratio of the longer side length to the shorter side length?


Find an integer  $n$ , where  $100 \leq n \leq 1997$ , such that
\[ \frac{2^n+2}{n}  \]
is also an integer.


A farmer buys a batch of trees, which he wishes to plant in a square grid. For example, if he had  $25$  trees, then he could plant them as shown below.

[asy]
size(3cm,0);
dot((0,0));
dot((0,1));
dot((0,2));
dot((0,3));
dot((0,4));
dot((1,0));
dot((1,1));
dot((1,2));
dot((1,3));
dot((1,4));
dot((2,0));
dot((2,1));
dot((2,2));
dot((2,3));
dot((2,4));
dot((3,0));
dot((3,1));
dot((3,2));
dot((3,3));
dot((3,4));
dot((4,0));
dot((4,1));
dot((4,2));
dot((4,3));
dot((4,4));
[/asy]

However, the farmer finds that he cannot plant his trees in a square grid. If he had  $20$  more trees, or if he had  $39$  fewer trees, then he could plant his trees in a square grid. How many trees did the farmer buy?


In a square  $ABCD$  of diagonals  $AC$  and  $BD$ , we call  $O$  at the center of the square. A square  $PQRS$  is constructed with sides parallel to those of  $ABCD$  with  $P$  in segment  $AO, Q$  in segment  $BO, R$  in segment  $CO, S$  in segment  $DO$ .  If area of  $ABCD$  equals two times the  area of  $PQRS$ ,  and  $M$  is the midpoint of the  $AB$  side, calculate the measure of the angle  $\angle AMP$ .


If  $x, y, z \in \mathbb{R}$  are solutions to the system of equations  $$ \begin{cases}
x - y + z - 1 = 0
xy + 2z^2 - 6z + 1 = 0
\end{cases} $$  what is the greatest value of  $(x - 1)^2 + (y + 1)^2$ ?


Find the least positive integer  $k$  such that for any  $a \in [0, 1]$  and any positive integer  $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]

Solution: \boxed{ k = 4 }


Consider the sequence  $ \left( x_n \right)_{n\ge 1} $  having  $ x_1>1 $  and satisfying the equation  $$  x_1+x_2+\cdots +x_{n+1} =x_1x_2\cdots x_{n+1} ,\quad\forall n\in\mathbb{N} .  $$  Show that this sequence is convergent and find its limit.


Solve the system of equations in real numbers:
\[
\begin{cases*}
(x - 1)(y - 1)(z - 1) = xyz - 1,
(x - 2)(y - 2)(z - 2) = xyz - 2.
\end{cases*}
\]
*Vladimir Bragin*

Solution: \boxed{x = y = z = 1}


The kid cut out of grid paper with the side of the cell  $1$  rectangle along the grid  lines and calculated its area and perimeter. Carlson snatched his scissors and cut out of this rectangle along the lines of the grid a square adjacent to the boundary of the rectangle.
- My rectangle ... - kid sobbed. - There is something strange about this figure!
- Nonsense, do not mention it -  Carlson said - waving his hand carelessly. - Here you see, in this figure the perimeter is the same as the area of the rectangle was, and the area is the same as was the perimeter!
What size square did Carlson cut out?


A farmer wants to build a rectangular region, using a river as one side and some fencing as the other three sides. He has 1200 feet of fence which he can arrange to different dimensions. He creates the rectangular region with length  $ L$  and width  $ W$  to enclose the greatest area. Find  $ L\plus{}W$ .


Find all integers  $n>1$  such that any prime divisor of  $n^6-1$  is a divisor of  $(n^3-1)(n^2-1)$ .

Solution: \boxed{n = 2}


Let  $n$  be a positive integer and let  $P$  be the set of monic polynomials of degree  $n$  with complex coefficients. Find the value of
\[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \]

Solution: \boxed{ 1 }


Let  $[ABCD]$  be a convex quadrilateral with area  $2014$ , and let  $P$  be a point on  $[AB]$  and  $Q$  a point on  $[AD]$  such that triangles  $[ABQ]$  and  $[ADP]$  have area  $1$ . Let  $R$  be the intersection of  $[AC]$  and  $[PQ]$ . Determine  $\frac{\overline{RC}}{\overline{RA}}$ .


Let  $(p_1, p_2, \dots) = (2, 3, \dots)$  be the list of all prime numbers, and  $(c_1, c_2, \dots) = (4, 6, \dots)$  be the list of all composite numbers, both in increasing order. Compute the sum of all positive integers  $n$  such that  $|p_n - c_n| < 3$ .

*Proposed by Brandon Wang*


Let  $A_1A_2A_3A_4A_5A_6A_7A_8$   be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of  $A_1A_2A_3A_4A_5A_6A_7A_8$  and such convex quadrilaterals will be called "sub quadrilaterals".Find the smallest  $n$  satisfying:
We can color n "button" such that for all  $i,k \in\{1,2,3,4,5,6,7,8\},i\neq k,s(i,k)$  are the same where  $s(i,k)$  denote the number of the  "sub quadrilaterals" has  $A_i,A_k$  be the vertices and the intersection of two its diagonals is "button".


The sequence of digits
\[
1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 \dots
\]
is obtained by writing the positive integers in order. If the  $10^n$ -th digit in this sequence occurs in the part of the sequence in which the  $m$ -digit numbers are placed, define  $f(n)$  to be  $m$ . For example,  $f(2)=2$  because the 100th digit enters the sequence in the placement of the two-digit integer 55. Find, with proof,  $f(1987)$ .

Solution: \boxed{ f(1987) = 1984 }


Let set  $A=\{1,2,\ldots,n\} ,$  and  $X,Y$  be two subsets (not necessarily distinct) of  $A.$  Define that  $\textup{max} X$  and  $\textup{min} Y$  represent the greatest element of  $X$  and the least element of  $Y,$  respectively. Determine the number of two-tuples  $(X,Y)$  which satisfies  $\textup{max} X>\textup{min} Y.$

Solution: \boxed{2^{2n} - (n+1)2^n}


Let  $ABC$  be a triangle. Take  $n$  point lying on the side  $AB$  (different from  $A$  and  $B$ ) and connect all of them by straight lines to the vertex  $C$ . Similarly, take  $n$  points on the side  $AC$  and connect them to  $B$ . Into how many regions is the triangle  $ABC$  partitioned by these lines?

Further, take  $n$  points on the side  $BC$  also and join them with  $A$ . Assume that no three straight lines meet at a point other than  $A,B$  and  $C$ . Into how many regions is the triangle  $ABC$  partitioned now?

Solution: \boxed{3n^2 + 3n + 1}


Let  $\triangle ABC$  be an equilateral triangle and let  $P$  be a point on  $\left[AB\right]$ .
 $Q$  is the point on  $BC$  such that  $PQ$  is perpendicular to  $AB$ .  $R$  is the point on  $AC$  such that  $QR$  is perpendicular to  $BC$ . And  $S$  is the point on  $AB$  such that  $RS$  is perpendicular to  $AC$ .
 $Q'$  is the point on  $BC$  such that  $PQ'$  is perpendicular to  $BC$ .  $R'$  is the point on  $AC$  such that  $Q'R'$  is perpendicular to  $AC$ . And  $S'$  is the point on  $AB$  such that  $R'S'$  is perpendicular to  $AB$ .
Determine  $\frac{|PB|}{|AB|}$  if  $S=S'$ .

Solution: \boxed{ \frac{2}{7} }


Find the smallest  $n$  such that every subset of  $\{1, 2, 3, . . . , 2004 \}$  with  $n$  elements contains at least two elements that are relatively prime.


Find all functions  $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$  that satisfy the following two conditions:
[list] $\bullet\ f(n)$  is a perfect square for all  $n\in\mathbb{Z}_{>0}$  $\bullet\ f(m+n)=f(m)+f(n)+2mn$  for all  $m,n\in\mathbb{Z}_{>0}$ .[/list]

Solution: \boxed{f(n) = n^2}


Call a natural number  $n$  *good* if for any natural divisor  $a$  of  $n$ , we have that  $a+1$  is also divisor of  $n+1$ . Find all good natural numbers.

*S. Berlov*

Solution: \boxed{ 1 }


In the coordinate plane, denote by  $ S(a)$  the area of the region bounded by the line passing through the point  $ (1,\ 2)$  with the slope  $ a$  and the parabola  $ y\equal{}x^2$ . When  $ a$  varies in the range of  $ 0\leq a\leq 6$ , find the value of  $ a$  such that  $ S(a)$  is minimized.

Solution: \boxed{ a = 2 }


The set  $X$  has  $1983$  members. There exists a family of subsets  $\{S_1, S_2, \ldots , S_k \}$  such that:**(i)** the union of any three of these subsets is the entire set  $X$ , while**(ii)** the union of any two of them contains at most  $1979$  members. What is the largest possible value of  $k ?$


In trapezoid  $ABCD$ , the diagonals intersect at  $E$ , the area of  $\triangle ABE$  is 72 and the area of  $\triangle CDE$  is 50. What is the area of trapezoid  $ABCD$ ?


Two three-letter strings,  $aaa$  and  $bbb$ , are transmitted electronically. Each string is sent letter by letter. Due to faulty equipment, each of the six letters has a 1/3 chance of being received incorrectly, as an  $a$  when it should have been a  $b$ , or as a  $b$  when it should be an  $a$ . However, whether a given letter is received correctly or incorrectly is independent of the reception of any other letter. Let  $S_a$  be the three-letter string received when  $aaa$  is transmitted and let  $S_b$  be the three-letter string received when  $bbb$  is transmitted. Let  $p$  be the probability that  $S_a$  comes before  $S_b$  in alphabetical order. When  $p$  is written as a fraction in lowest terms, what is its numerator?

Solution: \boxed{ 532 }


Let  $ a,\ b$  be postive real numbers. For a real number  $ t$ , denote by  $d(t)$  the distance between the origin and the line  $ (ae^t)x \plus{} (be^{ \minus{} t})y \equal{} 1$ .

Let  $ a,\ b$  vary with  $ ab \equal{} 1$ , find the minimum value of  $ \int_0^1 \frac {1}{d(t)^2}\ dt$ .

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


In the acute-angled triangle  $ABC$ , the altitudes  $BP$  and  $CQ$  were drawn, and the point  $T$  is the intersection point  of the altitudes of  $\Delta PAQ$ . It turned out that  $\angle CTB = 90 {} ^ \circ$ . Find the measure of  $\angle BAC$ .

(Mikhail Plotnikov)


The four-digit base ten number  $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$  has all nonzero digits and is a multiple of  $99$ . Additionally, the two-digit base ten number  $\underline{a}\;\underline{b}$  is a divisor of  $150$ , and the two-digit base ten number  $\underline{c}\;\underline{d}$  is a divisor of  $168$ . Find the remainder when the sum of all possible values of the number  $\underline{a}\;\underline{b}\;\underline{c}\;\underline{d}$  is divided by  $1000$ .

*Proposed by **treemath***


A cubic block with dimensions  $n$  by  $n$  by  $n$  is made up of a collection of  $1$  by  $1$  by  $1$  unit cubes. What is the smallest value of  $n$  so that if the outer layer of unit cubes are removed from the block, more than half the original unit cubes will still remain?


You are given a positive integer  $k$  and not necessarily distinct positive integers  $a_1, a_2 , a_3 , \ldots,
a_k$ . It turned out that for any coloring of all positive integers from  $1$  to  $2021$  in one of the  $k$  colors so that there are exactly  $a_1$  numbers of the first color,  $a_2$  numbers of the second color,  $\ldots$ , and  $a_k$  numbers of the  $k$ -th color, there is always a number  $x \in \{1, 2, \ldots, 2021\}$ , such that the total number of numbers colored in the same color as  $x$  is exactly  $x$ . What are the possible values of  $k$ ?

*Proposed by Arsenii Nikolaiev*


Let  $p$  be a prime and let  $f(x) = ax^2 + bx + c$  be a quadratic polynomial with integer coefficients such that  $0 < a, b, c \le p$ . Suppose  $f(x)$  is divisible by  $p$  whenever  $x$  is a positive integer. Find all possible values of  $a + b + c$ .


In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners  $ABCD$ , he starts at  $A$  and walks to  $C$ . Then, he walks to the midpoint of side  $AD$ , say  $A_1$ . Then, he walks to the midpoint of side  $CD$  say  $C_1$ , and then the midpoint of  $A_1D$  which is  $A_2$ . He continues in this fashion, indefinitely. The total length of his path if  $AB=5$  and  $BC=12$  is of the form  $a + b\sqrt{c}$ . Find  $\displaystyle\frac{abc}{4}$ .


Consider non-negative real numbers  $a, b, c$  satisfying the condition  $a^2 + b^2 + c^2 = 2$  . Find the maximum value of the following expression   $$ P=\frac{\sqrt{b^2+c^2}}{3-a}+\frac{\sqrt{c^2+a^2}}{3-b}+a+b-2022c $$


Triangle  $ABC$  has  $AB=25$ ,  $AC=29$ , and  $BC=36$ .  Additionally,  $\Omega$  and  $\omega$  are the circumcircle and incircle of  $\triangle ABC$ .  Point  $D$  is situated on  $\Omega$  such that  $AD$  is a diameter of  $\Omega$ , and line  $AD$  intersects  $\omega$  in two distinct points  $X$  and  $Y$ .  Compute  $XY^2$ .

*Proposed by David Altizio*


Find all  $f: R\rightarrow R$  such that
(i) The set  $\{\frac{f(x)}{x}| x\in R-\{0\}\}$  is finite
(ii)  $f(x-1-f(x)) = f(x)-1-x$  for all  $x$

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


Let  $x,y$  and  $z$  be positive real numbers such that  $xy+z^2=8$ . Determine the smallest possible value of the expression   $$ \frac{x+y}{z}+\frac{y+z}{x^2}+\frac{z+x}{y^2}. $$


Let ABCD be a unit square. Draw a quadrant of a circle with A as centre and B;D
as end points of the arc. Similarly, draw a quadrant of a circle with B as centre and
A;C as end points of the arc. Inscribe a circle ? touching the arc AC internally, the
arc BD internally and also touching the side AB. Find the radius of the circle ?.


Let  $ABCD$  be a convex quadrilateral with  $AC \perp BD$ , and let  $P$  be the intersection of  $AC$  and  $BD$ . Suppose that the distance from  $P$  to  $AB$  is  $99$ , the distance from  $P$  to  $BC$  is  $63$ , and the distance from  $P$  to  $CD$  is  $77$ . What is the distance from  $P$  to  $AD$ ?


Let  $ABCD$  be a square with side length  $1$ . How many points  $P$  inside the square (not on its sides) have the property that the square can be cut into  $10$  triangles of equal area such that all of them have  $P$  as a vertex?

*Proposed by Josef Tkadlec - Czech Republic*


Establish necessary and sufficient conditions on the constant  $k$  for the existence of a continuous real valued function  $f(x)$  satisfying   $$ f(f(x))=kx^9 $$   for all real  $x$ .

Solution: \boxed{ k \geq 0 }


Find all positive integers  $k$  for which number  $3^k+5^k$  is a power of some integer with exponent greater than  $1$ .

Solution: \boxed{ k = 1 }


Patchouli is taking an exam with  $k > 1$  parts, numbered Part  $1, 2, \dots, k$ . It is known that for  $i = 1, 2, \dots, k$ , Part  $i$  contains  $i$  multiple choice questions, each of which has  $(i+1)$  answer choices. It is known that if she guesses randomly on every single question, the probability that she gets exactly one question correct is equal to  $2018$  times the probability that she gets no questions correct. Compute the number of questions that are on the exam.

*Proposed by Yannick Yao*


 $25$  little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants to give to each of the  $24$  remaining donkeys a pot of one of six colours of the rainbow (except red), so that at least one pot of each colour is given to some donkey (but successive donkeys can receive pots of the same colour). Which of the two friends has more ways to get his plan implemented, and how many times more?

*Eeyore is a character in the Winnie-the-Pooh books by A. A. Milne. He is generally depicted as a pessimistic, gloomy, depressed, old grey stuffed donkey, who is a friend of the title character, Winnie-the-Pooh. His name is an onomatopoeic representation of the braying sound made by a normal donkey. Of course, Winnie-the-Pooh is a fictional anthropomorphic bear.*

*Proposed by F. Petrov*


Determine the least possible value of the natural number  $n$  such that  $n!$  ends in exactly  $1987$  zeros.


Let  $\triangle ABC$  be a triangle with  $AB=3$  and  $AC=5$ . Select points  $D, E,$  and  $F$  on  $\overline{BC}$  in that order such that  $\overline{AD}\perp \overline{BC}$ ,  $\angle BAE=\angle CAE$ , and  $\overline{BF}=\overline{CF}$ . If  $E$  is the midpoint of segment  $\overline{DF}$ , what is  $BC^2$ ?

Let  $T = TNYWR$ , and let  $T = 10X + Y$  for an integer  $X$  and a digit  $Y$ . Suppose that  $a$  and  $b$  are real numbers satisfying  $a+\frac1b=Y$  and  $\frac{b}a=X$ . Compute  $(ab)^4+\frac1{(ab)^4}$ .


Let  $ (a_n)^{\infty}_{n\equal{}1}$  be a sequence of integers with  $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$  For all quadruple  $ (i,j,k,l)$  of indices such that  $ 1 \leq i < j \leq k < l$  and  $ i \plus{} l \equal{} j \plus{} k$  we have the inequality  $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$  Determine the least possible value of  $ a_{2008}.$


Let  $C$  be a cube.  Let  $P$ ,  $Q$ , and  $R$  be random vertices of  $C$ , chosen uniformly and independently from the set of vertices of  $C$ .  (Note that  $P$ ,  $Q$ , and  $R$  might be equal.)  Compute the probability that some face of  $C$  contains  $P$ ,  $Q$ , and  $R$ .


Find the sum of all integers  $x$ ,  $x \ge 3$ , such that  $201020112012_x$  (that is,  $201020112012$  interpreted as a base  $x$  number) is divisible by  $x-1$


If  $S$  is a sequence of positive integers let  $p(S)$  be the product of the members of  $S$ . Let  $m(S)$  be the arithmetic mean of  $p(T)$  for all non-empty subsets  $T$  of  $S$ . Suppose that  $S'$  is formed from  $S$  by appending an additional positive integer. If  $m(S) = 13$  and  $m(S') = 49$ , find  $S'$ .

Solution: \boxed{\{1, 1, 7, 22\}}


The fraction  $\frac1{10}$  can be expressed as the sum of two unit fraction in many ways, for example,  $\frac1{30}+\frac1{15}$  and  $\frac1{60}+\frac1{12}$ .

Find the number of ways that  $\frac1{2007}$  can be expressed as the sum of two distinct positive unit fractions.


Find all pairs of positive integers  $(a,b)$  satisfying the following conditions:

- $a$  divides  $b^4+1$ ,
- $b$  divides  $a^4+1$ ,
- $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$ .

*Yang Liu*

Solution: \boxed{(2, 1)}


In a rectangle  $ABCD, E$  is the midpoint of  $AB, F$  is a point on  $AC$  such that  $BF$  is perpendicular to  $AC$ , and  $FE$  perpendicular to  $BD$ . Suppose  $BC = 8\sqrt3$ . Find  $AB$ .


Let  $P\left(X\right)=X^5+3X^4-4X^3-X^2-3X+4$ . Determine the number of monic polynomials  $Q\left(x\right)$  with integer coefficients such that  $\frac{P\left(X\right)}{Q\left(X\right)}$  is a polynomial with integer coefficients. Note: a monic polynomial is one with leading coefficient  $1$  (so  $x^3-4x+5$  is one but not  $5x^3-4x^2+1$  or  $x^2+3x^3$ ).

*2016 CCA Math Bonanza Individual #9*


For every positive integer  $n$ , determine the biggest positive integer  $k$  so that  $2^k |\ 3^n+1$

Solution: \boxed{ k = 2 }


There are  $1000$  students in a school. Every student has exactly  $4$  friends. A group of three students  $ \left \{A,B,C \right \}$  is said to be a *friendly triplet* if any two students in the group are friends. Determine the maximal possible number of friendly triplets.

*Proposed by Nikola Velov*


Suppose that the angles of  $\triangle ABC$  satisfy  $\cos(3A) + \cos(3B) + \cos(3C) = 1$ . Two sides of the triangle have lengths  $10$  and  $13$ . There is a positive integer  $m$  so that the maximum possible length for the remaining side of  $\triangle ABC$  is  $\sqrt{m}$ . Find  $m$ .


The CMU Kiltie Band is attempting to crash a helicopter via grappling hook.  The helicopter starts parallel (angle  $0$  degrees) to the ground.  Each time the band members pull the hook, they tilt the helicopter forward by either  $x$  or  $x+1$  degrees, with equal probability, if the helicopter is currently at an angle  $x$  degrees with the ground.  Causing the helicopter to tilt to  $90$  degrees or beyond will crash the helicopter.  Find the expected number of times the band must pull the hook in order to crash the helicopter.

*Proposed by Justin Hsieh*


If   $$ \sum_{k=1}^{40} \left( \sqrt{1 + \frac{1}{k^{2}} + \frac{1}{(k + 1)^{2}}}\right) = a + \frac {b}{c} $$  where  $a, b, c \in \mathbb{N}, b < c, gcd(b,c) =1 ,$  then what is the value of  $a+ b ?$


The points  $A$ ,  $B$ ,  $C$ ,  $D$ , and  $E$  lie in one plane and have the following properties:  $AB = 12, BC = 50, CD = 38,  AD = 100, BE = 30, CE = 40$ .

Find the length of the segment  $ED$ .


Let  $a$ ,  $b$ , and  $c$  be positive real numbers such that  $a^2+b^2+c^2=989$  and  $(a+b)^2+(b+c)^2+(c+a)^2=2013$ . Find  $a+b+c$ .


Find all real numbers  $ x$  such that  $ 4x^5 \minus{} 7$  and  $ 4x^{13} \minus{} 7$  are both perfect squares.

Solution: \boxed{ x = 2 }


Find the area of the  $MNRK$  trapezoid with the lateral side   $RK = 3$  if the distances from the vertices  $M$  and  $N$  to the line  $RK$  are  $5$  and  $7$ , respectively.


If  $a$  ,  $b$  are integers and  $s=a^3+b^3-60ab(a+b)\geq 2012$  , find the least possible value of  $s$ .


Find all integers n greater than or equal to  $3$  such that  $\sqrt{\frac{n^2 - 5}{n + 1}}$  is a rational number.


There are  $2022$  distinct integer points on the plane. Let  $I$  be the number of pairs among these points with exactly  $1$  unit apart. Find the maximum possible value of  $I$ .
(*Note. An integer point is a point with integer coordinates.*)

*Proposed by CSJL.*


Determine, with proof, the number of ordered triples  $\left(A_{1}, A_{2}, A_{3}\right)$  of sets which have the property that

(i)  $A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},$  and
(ii)  $A_{1} \cap A_{2} \cap A_{3}=\emptyset.$ Express your answer in the form  $2^{a} 3^{b} 5^{c} 7^{d},$  where  $a, b, c, d$  are nonnegative integers.

Solution: \boxed{2^{10} \cdot 3^{10}}


Suzie flips a fair coin 6 times. The probability that Suzie flips 3 heads in a row but not 4 heads in a row is given by  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m + n$ .

Solution: \boxed{ 19 }


Let  $a_1$ ,  $a_2, \dots, a_{2015}$  be a sequence of positive integers in  $[1,100]$ .
Call a nonempty contiguous subsequence of this sequence *good* if the product of the integers in it leaves a remainder of  $1$  when divided by  $101$ .
In other words, it is a pair of integers  $(x, y)$  such that  $1 \le x \le y \le 2015$  and \[a_xa_{x+1}\dots a_{y-1}a_y \equiv 1 \pmod{101}. \]Find the minimum possible number of good subsequences across all possible  $(a_i)$ .

*Proposed by Yang Liu*


The Matini company released a special album with the flags of the  $ 12$  countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is  $50\%$ .

Solution: \boxed{ 3 }


Points  $X$  and  $Y$  are the midpoints of arcs  $AB$  and  $BC$  of the circumscribed circle of triangle  $ABC$ . Point  $T$  lies on side  $AC$ . It turned out that the bisectors of the angles  $ATB$  and  $BTC$  pass through points  $X$  and  $Y$  respectively. What angle  $B$  can be in  triangle  $ABC$ ?

Solution: \boxed{ 90^\circ }


Find the maximum of  $x+y$  given that  $x$  and  $y$  are positive real numbers that satisfy \[x^3+y^3+(x+y)^3+36xy=3456.\]


Denote by  $d(n)$  be the biggest prime divisor of  $|n|>1$ . Find all polynomials with integer coefficients satisfy;  $$ P(n+d(n))=n+d(P(n))  $$  for the all  $|n|>1$  integers such that  $P(n)>1$  and  $d(P(n))$  can be defined.

Solution: \boxed{ P(n) = n }


Consider the set  $$ \mathcal{S}=\{(a, b, c, d, e): 0<a<b<c<d<e<100\} $$  where  $a, b, c, d, e$  are integers. If  $D$  is the average value of the fourth element of such a tuple in the set, taken over all the elements of  $\mathcal{S}$ , find the largest integer less than or equal to  $D$ .


Find the maximum value of the positive real number  $k$  such that the inequality  $$ \frac{1}{kab+c^2} +\frac{1} {kbc+a^2} +\frac{1} {kca+b^2}  \geq \frac{k+3}{a^2+b^2+c^2}   $$  holds for all positive real numbers  $a,b,c$  such that  $a^2+b^2+c^2=2(ab+bc+ca).$

Solution: \boxed{ k = 2 }


Given an  $m\times n$  grid with unit squares coloured either black or white, a black square in the grid is *stranded*if there is some square to its left in the same row that is white and there is some square above it in the same column that is white.
Find a closed formula for the number of  $2\times n$  grids with no stranded black square.
Note that  $n$  is any natural number and the formula must be in terms of  $n$  with no other variables.

Solution: \boxed{2 \cdot 3^n - 2^n}


Let  $\alpha$  be a real number. Determine all polynomials  $P$  with real coefficients such that   $$ P(2x+\alpha)\leq (x^{20}+x^{19})P(x) $$   holds for all real numbers  $x$ .

*Proposed by Walther Janous, Austria*

Solution: \boxed{ P(x) \equiv 0 }


What is the greatest number of positive integers lesser than or equal to 2016 we can choose such that it doesn't have two of them differing by 1,2, or 6?


Consider the set  $A = \{1, 2, 3, ..., 2008\}$ . We say that a set is of *type*  $r, r \in \{0, 1, 2\}$ , if that set is a nonempty subset of  $A$  and the sum of its elements gives the remainder  $r$  when divided by  $3$ . Denote by  $X_r, r \in \{0, 1, 2\}$  the class of sets of type  $r$ . Determine which of the classes  $X_r, r \in \{0, 1, 2\}$ ,  is the largest.

Solution: \boxed{ X_1 }


If  $x$ ,  $y$ ,  $z$  are nonnegative integers satisfying the equation below, then compute  $x+y+z$ .
\[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\]

*Proposed by Jeffrey Shi*


Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice?


A rectangle,  $HOMF$ , has sides  $HO=11$  and  $OM=5$ . A triangle  $\Delta ABC$  has  $H$  as orthocentre,   $O$  as circumcentre,  $M$  be the midpoint of  $BC$ ,  $F$  is the feet of altitude from  $A$ . What is the length of  $BC$  ?

[asy]
unitsize(0.3 cm);
pair F, H, M, O;
F = (0,0);
H = (0,5);
O = (11,5);
M = (11,0);
draw(H--O--M--F--cycle);
label(" $F$ ", F, SW);
label(" $H$ ", H, NW);
label(" $M$ ", M, SE);
label(" $O$ ", O, NE);
[/asy]


Let  $c$  be a complex number.  Suppose there exist distinct complex numbers  $r$ ,  $s$ , and  $t$  such that for every complex number  $z$ , we have
\[
  (z - r)(z - s)(z - t) = (z - cr)(z - cs)(z - ct).
\]
Compute the number of distinct possible values of  $c$ .


For all  $x,y,z$  positive real numbers, find the all  $c$  positive real numbers that providing  $$ \frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2 $$

Solution: \boxed{ c = 1 }


How many positive integers less than  $2019$  are divisible by either  $18$  or  $21$ , but not both?


Find all triples  $(p, q, r)$  of prime numbers such that all of the following numbers are
integers $\frac{p^2 + 2q}{q+r}, \frac{q^2+9r}{r+p}, \frac{r^2+3p}{p+q}$ *Proposed by Tajikistan*

Solution: \boxed{ (2, 3, 7) }


Laura is putting together the following list:  $a_0, a_1, a_2, a_3, a_4, ..., a_n$ , where  $a_0 = 3$  and  $a_1 = 4$ .
She knows that the following equality holds for any value of  $n$  integer greater than or equal to  $1$ :  $$ a_n^2-2a_{n-1}a_{n+1} =(-2)^n. $$  Laura calculates the value of  $a_4$ . What value does it get?


According to Moor's Law, the number of shoes in Moor's room doubles every year.  In 2013, Moor's room starts out having exactly one pair of shoes.  If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes?  Note that left and right shoes are distinct, and Moor must always wear one of each.


Reduce   $$ \frac{2}{\sqrt{4-3\sqrt[4]{5} + 2\sqrt[4]{25}-\sqrt[4]{125}}} $$   to its lowest form.

Then generalize this result and show that it holds for any positive  $n$ .

Solution: \boxed{ 1 + \sqrt[4]{5} }


If you increase a number  $X$  by  $20\%,$  you get  $Y.$  By what percent must you decrease  $Y$  to get  $X?$


Numbers  $\frac{49}{1}, \frac{49}{2}, ... , \frac{49}{97}$  are writen on a blackboard. Each time, we can replace two numbers (like  $a, b$ ) with  $2ab-a-b+1$ . After  $96$  times doing that prenominate action, one number will be left on the board. Find all the possible values fot that number.


Board with dimesions  $2018 \times 2018$  is divided in unit cells  $1 \times 1$ . In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If  $W$  is number of remaining white chips, and  $B$  number of remaining black chips on board and  $A=min\{W,B\}$ , determine maximum of  $A$


Five positive integers from  $1$  to  $15$  are chosen without replacement.  What is the probability that their sum is divisible by  $3$ ?

Solution: \boxed{ \frac{1}{3} }


Find all odd natural numbers  $n$  such that  $d(n)$  is the largest divisor of the number  $n$  different from  $n$ .
( $d(n)$  is the number of divisors of the number n including  $1$  and  $n$  ).


In the blackboard there are drawn  $25$  points as shown in the figure.
Gastón must choose  $4$  points that are vertices of a square.
In how many different ways can he make this choice?  $$ \begin{matrix}\bullet & \bullet & \bullet & \bullet & \bullet \bullet & \bullet & \bullet & \bullet & \bullet \bullet & \bullet & \bullet & \bullet & \bullet \bullet & \bullet & \bullet & \bullet & \bullet \bullet & \bullet & \bullet & \bullet & \bullet \end{matrix} $$


Let  $a,x,y$  be positive integer such that  $a>100,x>100,y>100$  and  $y^2-1=a^2(x^2-1)$   . Find the minimum value of  $\frac{a}{x}$ .


A  $40$  feet high screen is put on a vertical wall  $10$  feet above your eye-level. How far should you stand to maximize the angle subtended by the screen (from top to bottom) at your eye?


Let  $n$  be a positive integer, and let  $x=\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+2}+\sqrt{n}}$  and  $y=\frac{\sqrt{n+2}+\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}$ .

It is given that  $14x^2+26xy+14y^2=2014$ . Find the value of  $n$ .


In trapezoid  $ABCD$ ,  $AD \parallel BC$  and  $\angle ABC + \angle CDA = 270^{\circ}$ .  Compute  $AB^2$  given that  $AB \cdot \tan(\angle BCD) = 20$  and  $CD = 13$ .

*Proposed by Lewis Chen*


Given that  $\log_{10}2 \approx 0.30103$ , ﬁnd the smallest positive integer  $n$  such that the decimal representation of  $2^{10n}$  does not begin with the digit  $1$ .


Lisa writes a positive whole number in the decimal system on the blackboard and now makes in each turn the following:
The last digit is deleted from the number on the board and then the remaining shorter number (or 0 if the number was one digit) becomes four times the number deleted number added. The number on the board is now replaced by the result of this calculation.
Lisa repeats this until she gets a number for the first time was on the board.
(a) Show that the sequence of moves always ends.
(b) If Lisa begins with the number  $53^{2022} - 1$ , what is the last number on the board?

Example: If Lisa starts with the number  $2022$ , she gets  $202 + 4\cdot 2 = 210$  in the first move and overall the result   $$ 2022 \to 210 \to 21 \to 6 \to 24 \to 18 \to 33 \to 15 \to 21 $$  .
Since Lisa gets  $21$  for the second time, the turn order ends.

*(Stephan Pfannerer)*


Let  $n$  be a positive integer. At most how many distinct unit vectors can be selected in  $\mathbb{R}^n$  such that from any three of them, at least two are orthogonal?


Solution: \boxed{ 500 < a_1 }


Farmer Boso has a busy farm with lots of animals. He tends to  $5b$  cows,  $5a +7$  chickens, and  $b^{a-5}$  insects. Note that each insect has  $6$  legs. The number of cows is equal to the number of insects. The total number of legs present amongst his animals can be expressed as  $\overline{LLL }+1$ , where  $L$  stands for a digit. Find  $L$ .


Each of the  $n$  students writes one of the numbers  $1,2$  or  $3$  on each of the  $29$  boards. If any two students wrote different numbers on at least one of the boards and any three students wrote the same number on at least one of the boards, what is the maximum possible value of  $n$ ?


Suppose that  $10$  points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.


Let  $x, y$  and  $z$  be positive real numbers such that  $x \geq y+z$ .
Proof that  $$ \frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7 $$  When does equality occur?

(Walther Janous)


Let  $A$  be a real  $n\times n$  matrix satisfying  $$ A+A^{\text T}=I, $$  where  $A^{\text T}$  denotes the transpose of  $A$  and  $I$  the  $n\times n$  identity matrix. Show that  $\det A>0$ .

Solution: \boxed{ \det A > 0 }


Andrew writes down all of the prime numbers less than  $50$ . How many times does he write the digit  $2$ ?


For all positive integers  $n$ , let  $f(n)$  return the smallest positive integer  $k$  for which  $\tfrac{n}{k}$  is not an integer.  For example,  $f(6) = 4$  because  $1$ ,  $2$ , and  $3$  all divide  $6$  but  $4$  does not.  Determine the largest possible value of  $f(n)$  as  $n$  ranges over the set  $\{1,2,\ldots, 3000\}$ .


Three non-collinear points  $A_1, A_2, A_3$  are given in a plane. For   $n = 4, 5, 6, \ldots$ ,  $A_n$  be the centroid of the triangle  $A_{n-3}A_{n-2}A_{n-1}$ .

[list]
a) Show that there is exactly one point  $S$ , which lies in the interior of the triangle  $A_{n-3}A_{n-2}A_{n-1}$  for all  $n\ge 4$ .
b) Let  $T$  be the intersection of the line  $A_1A_2$  with  $SA_3$ . Determine the two ratios,  $A_1T : TA_2$  and  $TS : SA_3$ .
[/list]


Determine all such pairs pf  positive integers  $(a, b)$  such that   $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot  lcm(a -1, b)$ , where  $lcm (a, b)$  denotes the smallest common multiple, and  $gcd (a, b)$  denotes the greatest common divisor of numbers  $a, b$ .

Solution: \boxed{(6, 15)}


In triangle  $ABC$ ,  $AB = 52$ ,  $BC = 34$  and  $CA = 50$ . We split  $BC$  into  $n$  equal segments by placing  $n-1$  new points. Among these points are the feet of the altitude, median and angle bisector from  $A$ . What is the smallest possible value of  $n$ ?


Let  $ n$  and  $ k$  be given natural numbers, and let  $ A$  be a set such that \[ |A| \leq \frac{n(n+1)}{k+1}.\] For  $ i=1,2,...,n+1$ , let  $ A_i$  be sets of size  $ n$  such that \[ |A_i \cap A_j| \leq k \;(i \not=j)\ ,\] \[ A=  \bigcup_{i=1}^{n+1} A_i.\] Determine the cardinality of  $ A$ .

*K. Corradi*


A mathematician is walking through a library with twenty-six shelves, one for each letter of the alphabet. As he walks, the mathematician will take at most one book off each shelf. He likes symmetry, so if the letter of a shelf has at least one line of symmetry (e.g., M works, L does not), he will pick a book with probability  $\frac12$ . Otherwise he has a  $\frac14$  probability of taking a book. What is the expected number of books that the mathematician will take?


Let  $f(n)$  denote the largest odd factor of  $n$ , including possibly  $n$ . Determine the value of
\[\frac{f(1)}{1} + \frac{f(2)}{2} + \frac{f(3)}{3} + \cdots + \frac{f(2048)}{2048},\]
rounded to the nearest integer.


*Nocycleland* is a country with  $500$  cities and  $2013$  two-way roads, each one of them connecting two cities. A city  $A$  *neighbors*  $B$  if there is one road that connects them, and a city  $A$  *quasi-neighbors*  $B$  if there is a city  $C$  such that  $A$  neighbors  $C$  and  $C$  neighbors  $B$ .
It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities  $A$ ,  $B$ ,  $C$  and  $D$  such that  $A$  neighbors  $B$ ,  $B$  neighbors  $C$ ,  $C$   neighbors  $D$ , and  $D$  neighbors  $A$ .
Show that there is at least one city that quasi-neighbors at least  $57$  other cities.


In Prime Land, there are seven major cities, labelled  $C_0$ ,  $C_1$ , \dots,  $C_6$ . For convenience, we let  $C_{n+7} = C_n$  for each  $n=0,1,\dots,6$ ; i.e. we take the indices modulo  $7$ .  Al initially starts at city  $C_0$ .

Each minute for ten minutes, Al flips a fair coin.  If the coin land heads, and he is at city  $C_k$ , he moves to city  $C_{2k}$ ; otherwise he moves to city  $C_{2k+1}$ .  If the probability that Al is back at city  $C_0$  after  $10$  moves is  $\tfrac{m}{1024}$ , find  $m$ .

*Proposed by Ray Li*


Let  $G$  be the set of points  $(x, y)$  such that  $x$  and  $y$  are positive integers less than or equal to 20.  Say that a ray in the coordinate plane is *ocular* if it starts at  $(0, 0)$  and passes through at least one point in  $G$ .  Let  $A$  be the set of angle measures of acute angles formed by two distinct ocular rays.  Determine
\[
  \min_{a \in A} \tan a.
\]


Find the maximal possible value of  $n$  such that there exist points  $P_1,P_2,P_3,\ldots,P_n$  in the plane and real numbers  $r_1,r_2,\ldots,r_n$  such that the distance between any two different points  $P_i$  and  $P_j$  is  $r_i+r_j$ .

Solution: \boxed{ n = 4 }


A bag contains  $8$  green candies and  $4$  red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers  $m$  and  $n$  so that  $\frac{m}{n}$  is the probability that you do not eat a green candy after you eat a red candy. Find  $m+n$ .


Suppose  $a$  and  $b$  are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs  $(a, b)$ .

Solution: \boxed{(2, 2), (-2, -2)}


Find the largest natural number  $ n$  for which there exist different sets  $ S_1,S_2,\ldots,S_n$  such that:

 $ 1^\circ$   $ |S_i\cup S_j|\leq 2004$  for each two  $ 1\leq i,j\le n$  and
 $ 2^\circ$   $ S_i\cup S_j\cup S_k\equal{}\{1,2,\ldots,2008\}$  for each three integers  $ 1\le i<j<k\le n$ .


Determine all possible values of the expression  $xy+yz+zx$  with real numbers  $x, y, z$  satisfying the conditions  $x^2-yz = y^2-zx = z^2-xy = 2$ .


Find the least real number  $K$  such that for all real numbers  $x$  and  $y$ , we have  $(1 + 20 x^2)(1 + 19 y^2) \ge K xy$ .

Solution: \boxed{ K = 8\sqrt{95} }


There are  $2n$  different numbers in a row. By one move we can interchange any two numbers or interchange any  $3$  numbers cyclically (choose  $a,b,c$  and place  $a$  instead of  $b$ ,  $b$  instead of  $c$ ,  $c$  instead of  $a$ ). What is the minimal number of moves that is always sufficient to arrange the numbers in increasing order ?

Solution: \boxed{n}


For a given positive integer  $m$ , the series   $$ \sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)} $$  evaluates to  $\frac{a}{bm^2}$ , where  $a$  and  $b$  are positive integers. Compute  $a+b$ .


Tickets for the football game are \ $10 for students and \$ 15 for non-students. If 3000 fans attend and pay \$36250, how many students went?


For all positive reals  $ a$ ,  $ b$ , and  $ c$ , what is the value of positive constant  $ k$  satisfies the following inequality?
 $ \frac{a}{c\plus{}kb}\plus{}\frac{b}{a\plus{}kc}\plus{}\frac{c}{b\plus{}ka}\geq\frac{1}{2007}$  .


Find all nonnegative integers  $a, b, c$  such that  $$ \sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}. $$

Solution: \boxed{ (0, 0, 2014) }


The 40 unit squares of the 9 9-table (see below) are labeled. The horizontal or vertical row of 9 unit squares is good if it has more labeled unit squares than unlabeled ones. How many good (horizontal and vertical) rows totally could have the table?


Find the smallest positive integer  $n$  with the following property:

For any integer  $m$  with  $0 < m < 2004$ , there exists an integer  $k$  such that

\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]


Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?

SolutionWith probability  $1$  the number of corners covered is  $0$ ,  $1$ , or  $2$  for example by the diameter of a square being  $\sqrt{2}$  so it suffices to compute the probability that the square covers  $2$  corners. This is due to the fact that density implies the mean number of captured corners is  $1$ . For the lattice with offset angle  $\theta \in \left[0,\frac{\pi}{2}\right]$  consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint  $\left(\frac{1}{2},0 \right)$ . The locus of such midpoint locations so that the square captures the  $2$  points  $(0,0),(1,0)$ , is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as  $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$
[asy]
draw((0,0)--(80,40));
draw((0,0)--(-40,80));
draw((80,40)--(40,120));
draw((-40,80)--(40,120));
draw((80,40)--(-20,40));
draw((-40,80)--(60,80));
draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5)));
draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5)));
draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5))));
draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5))));
draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40)));
draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80)));
[/asy]

Solution: \boxed{2 - \frac{6}{\pi}}


Define polynomials  $f_{n}(x)$  for  $n \geq 0$  by  $f_{0}(x)=1, f_{n}(0)=0$  for  $n \geq 1,$  and  $$ \frac{d}{d x} f_{n+1}(x)=(n+1) f_{n}(x+1) $$  for  $n \geq 0 .$  Find, with proof, the explicit factorization of  $f_{100}(1)$  into powers of distinct primes.


Colin has  $900$  Choco Pies. He realizes that for some integer values of  $n \le 900$ , if he eats n pies a day, he will be able to eat the same number of pies every day until he runs out. How many possible values of  $n$  are there?


At lunch, the seven members of the Kubik family sit down to eat lunch together at a round table.  In how many distinct ways can the family sit at the table if Alexis refuses to sit next to Joshua?  (Two arrangements are not considered distinct if one is a rotation of the other.)


For every positive integer  $n$ , determine the greatest possible value of the quotient  $$ \frac{1-x^{n}-(1-x)^{n}}{x(1-x)^n+(1-x)x^n} $$  where  $0 < x < 1$ .

Solution: \boxed{ 2^n - 2 }


In the  $xy$ -plane, for  $a>1$  denote by  $S(a)$  the area of the figure bounded by the curve  $y=(a-x)\ln x$  and the  $x$ -axis.

Find the value of integer  $n$  for which  $\lim_{a\rightarrow \infty} \frac{S(a)}{a^n\ln a}$  is non-zero real number.

Solution: \boxed{ n = 2 }


Given the sequence  $\{y_n\}_{n=1}^{\infty}$  defined by  $y_1=y_2=1$  and
\[y_{n+2} = (4k-5)y_{n+1}-y_n+4-2k, \qquad n\ge1\]
find all integers  $k$  such that every term of the sequence is a perfect square.

Solution: \boxed{k = 3}


Let  $ S \equal{} \{1,2,\ldots,2008\}$ . For any nonempty subset  $ A\in S$ , define  $ m(A)$  to be the median of  $ A$  (when  $ A$  has an even number of elements,  $ m(A)$  is the average of the middle two elements). Determine the average of  $ m(A)$ , when  $ A$  is taken over all nonempty subsets of  $ S$ .


 $ABCD$  - quadrilateral inscribed in circle, and  $AB=BC,AD=3DC$  . Point  $R$  is on the  $BD$  and  $DR=2RB$ . Point  $Q$  is on  $AR$  and  $\angle ADQ = \angle BDQ$ .  Also  $\angle ABQ + \angle CBD = \angle QBD$  .  $AB$  intersect line  $DQ$  in point  $P$ .
Find  $\angle APD$


Nick is taking a  $10$  question test where each answer is either true or false with equal probability. Nick forgot to study, so he guesses randomly on each of the  $10$  problems. What is the probability that Nick answers exactly half of the questions correctly?


The Evil League of Evil is plotting to poison the city's water supply. They plan to set out from their headquarters at  $(5, 1)$  and put poison in two pipes, one along the line  $y=x$  and one along the line  $x=7$ . However, they need to get the job done quickly before Captain Hammer catches them. What's the shortest distance they can travel to visit both pipes and then return to their headquarters?


Find all integers  $n$  for which both  $4n + 1$  and  $9n + 1$  are perfect squares.


Find  $ f(x): (0,\plus{}\infty) \to \mathbb R$  such that \[ f(x)\cdot f(y) \plus{} f(\frac{2008}{x})\cdot f(\frac{2008}{y})\equal{}2f(x\cdot y)\] and  $ f(2008)\equal{}1$  for  $ \forall x \in (0,\plus{}\infty)$ .

Solution: \boxed{f(x) = 1}


Find the smallest positive integer  $n$  for which one can select  $n$  distinct real numbers such that each of them is equal to the sum of some two other selected numbers.

*Proposed by Anton Trygub*


Determine the smallest natural number  $a\geq 2$  for which there exists a prime number  $p$  and a natural number  $b\geq 2$  such that
\[\frac{a^p - a}{p}=b^2.\]


Suppose  $P(x)$  is a degree  $n$  monic polynomial with integer coefficients such that  $2013$  divides  $P(r)$  for exactly  $1000$  values of  $r$  between  $1$  and  $2013$  inclusive.  Find the minimum value of  $n$ .


Let  $ABC$  be an acute-angled triangle and  $\Gamma$  be a circle with  $AB$  as diameter intersecting  $BC$  and  $CA$  at  $F ( \not= B)$  and  $E (\not= A)$  respectively. Tangents are drawn at  $E$  and  $F$  to  $\Gamma$  intersect at  $P$ . Show that the  ratio of the circumcentre of triangle  $ABC$  to that if  $EFP$  is a rational number.


The number  $S$  is the result of the following sum:  $1 + 10 + 19 + 28 + 37 +...+ 10^{2013}$ If one writes down the number  $S$ , how often does the digit ` $5$ ' occur in the result?


Find all functions  $ f: \mathbb{N} \to \mathbb{Z} $  satisfying   $$  n \mid f\left(m\right) \Longleftrightarrow m \mid \sum\limits_{d \mid n}{f\left(d\right)}  $$   holds for all positive integers  $ m,n $

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


One rainy afternoon you write the number  $1$  once, the number  $2$  twice, the number  $3$  three times, and so forth until you have written the number  $99$  ninety-nine times. What is the  $2005$  th digit that you write?


**polyhedral**

we call a  $12$ -gon in plane good whenever:
first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points  $(0,1)$ , $(1,0)$ , $(-1,0)$  and  $(0,-1)$ .
find the faces of the <u>massivest</u> polyhedral that it's image on every three plane  $xy$ , $yz$  and  $zx$  is a good  $12$ -gon.
(it's obvios that centers of these three  $12$ -gons are the origin of coordinates for three dimensions.)

time allowed for this question is 1 hour.


Let  $n\ge 1$   be a fixed integer. Calculate the distance  $\inf_{p,f}\, \max_{0\le x\le 1} |f(x)-p(x)|$   , where  $p$   runs over polynomials of degree less than  $n$   with real coefficients and  $f$   runs over functions  $f(x)= \sum_{k=n}^{\infty} c_k x^k$    defined on the closed interval  $[0,1]$  , where  $c_k \ge 0$   and  $\sum_{k=n}^{\infty} c_k=1$ .

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


Plane  $A$  passes through the points  $(1, 0, 0)$ ,  $(0, 1, 0)$ , and  $(0, 0, 1)$ . Plane  $B$  is parallel to plane  $A$ , but passes through the point  $(1, 0, 1)$ . Find the distance between planes  $A$  and  $B$ .


Let  $S=\{p/q| q\leq 2009, p/q <1257/2009, p,q \in \mathbb{N} \}$ . If the maximum element of  $S$  is  $p_0/q_0$  in reduced form, find  $p_0+q_0$ .


The real numbers x and у satisfy the equations    $$ \begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2  \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt2  \end{cases} $$
Find the numerical value of the ratio  $y/x$ .

Solution: \boxed{\frac{y}{x} = 6}


Let  $ABCD$  be a square with side length  $2$ .  Let  $M$  and  $N$  be the midpoints of  $\overline{BC}$  and  $\overline{CD}$  respectively, and let  $X$  and  $Y$  be the feet of the perpendiculars from  $A$  to  $\overline{MD}$  and  $\overline{NB}$ , also respectively.  The square of the length of segment  $\overline{XY}$  can be written in the form  $\tfrac pq$  where  $p$  and  $q$  are positive relatively prime integers.  What is  $100p+q$ ?

*Proposed by David Altizio*


Let  $a$  be number of  $n$  digits ( $ n > 1$ ). A number  $b$  of   $2n$  digits is obtained by writing two copies of  $a$  one after the other. If  $\frac{b}{a^2}$  is an integer  $k$ , find the possible values values of  $k$ .

Solution: \boxed{ k = 7 }


On each side of an equilateral triangle with side length  $n$  units, where  $n$  is an integer,  $1 \leq n \leq 100$ , consider  $n-1$  points that divide the side into  $n$  equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of  $n$  for which it is possible to turn all coins tail up after a finite number of moves.


Let  $S$  is a finite set with  $n$  elements. We divided  $AS$  to  $m$  disjoint parts such that if  $A$ ,  $B$ ,  $A \cup B$  are in the same part, then  $A=B.$  Find the minimum value of  $m$ .


How many distinct sets of  $5$  distinct positive integers  $A$  satisfy the property that for any positive integer  $x\le 29$ , a subset of  $A$  sums to  $x$ ?


Let  $S_n = n^2 + 20n + 12$ ,  $n$  a positive integer. What is the sum of all possible values of  $n$  for which  $S_n$  is a perfect square?


Determine all functions  $f: \mathbb{R} \to \mathbb{R}$  such that   $$  f(x^3) + f(y)^3 + f(z)^3 = 3xyz  $$  for all real numbers  $x$ ,  $y$  and  $z$  with  $x+y+z=0$ .

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


Let  $n\ge 3$ . Suppose  $a_1, a_2, ... , a_n$  are  $n$   distinct in pairs real numbers.
In terms of  $n$ , find the smallest possible number of different assumed values by the following  $n$  numbers:  $$ a_1 + a_2, a_2 + a_3,..., a_{n- 1} + a_n, a_n + a_1 $$


(My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the *flux* of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and the flux of 12333332 is 1, while the flux of 9182736450 is 8. What is the average value of the flux of the positive integers from 1 to 999, inclusive?


Find all ordered triplets  $(p,q,r)$  of positive integers such that  $p$  and  $q$  are two (not necessarily distinct) primes,  $r$  is even, and
\[p^3+q^2=4r^2+45r+103.\]

Solution: \boxed{ (7, 2, 4) }


Suppose that  $c\in\left(\frac{1}{2},1\right)$ . Find the least  $M$  such that for every integer  $n\ge 2$  and real numbers  $0<a_1\le a_2\le\ldots \le a_n$ , if  $\frac{1}{n}\sum_{k=1}^{n}ka_{k}=c\sum_{k=1}^{n}a_{k}$ , then we always have that  $\sum_{k=1}^{n}a_{k}\le M\sum_{k=1}^{m}a_{k}$  where  $m=[cn]$


Find the number of integers  $n$  with  $1 \le n \le 2017$  so that  $(n-2)(n-0)(n-1)(n-7)$  is an integer
multiple of  $1001$ .


Define an ordered quadruple of integers  $(a, b, c, d)$  as interesting if  $1 \le a<b<c<d \le 10$ , and  $a+d>b+c$ . How many ordered quadruples are there?


In the triangle  $ABC$  , the angle bisector  $AD$  divides the side  $BC$  into the ratio  $BD: DC = 2: 1$ . In what ratio, does the median  $CE$  divide this bisector?


The sweeties shop called "Olympiad" sells boxes of  $6,9$  or  $20$  chocolates. Groups of students from a school that is near the shop collect money to buy a chocolate for each student; to make this they buy a box and than give to everybody a chocolate. Like this students can create groups of  $15=6+9$  students,  $38=2*9+20$  students, etc. The seller has promised to the students that he can satisfy any group of students, and if he will need to open a new box of chocolate for any group (like groups of  $4,7$  or  $10$  students) than he will give all the chocolates for free to this group. Can there be constructed the biggest group that profits free chocolates, and if so, how many students are there in this group?


Let  $S_0=0$  and let  $S_k$  equal  $a_1+2a_2+\ldots+ka_k$  for  $k\geq 1$ .  Define  $a_i$  to be  $1$  if  $S_{i-1}<i$  and  $-1$  if  $S_{i-1}\geq i$ .  What is the largest  $k\leq 2010$  such that  $S_k=0$ ?


Given a set of points in space, a *jump* consists of taking two points,  $P$  and  $Q,$  and replacing  $P$  with the reflection of  $P$  over  $Q$ . Find the smallest number  $n$  such that for any set of  $n$  lattice points in  $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.

*Author: Anderson Wang*

Solution: \boxed{ 1025 }


Call a pair of integers  $(a,b)$  *primitive* if there exists a positive integer  $\ell$  such that  $(a+bi)^\ell$  is real. Find the smallest positive integer  $n$  such that less than  $1\%$  of the pairs  $(a, b)$  with  $0 \le a, b \le n$  are primitive.

*Proposed by Mehtaab Sawhney*


What is the largest positive integer  $n$  for which there are no *positive* integers  $a,b$  such that  $8a+11b=n$ ?

*2019 CCA Math Bonanza Lightning Round #2.2*


In a cycling competition with  $14$  stages, one each day, and  $100$  participants, a competitor was characterized by finishing  $93^{\text{rd}}$  each day.What is the best place he could have finished in the overall standings? (Overall standings take into account the total cycling time over all stages.)


On a faded piece of paper it is possible to read the following:
\[(x^2 + x + a)(x^{15}- \cdots ) = x^{17} + x^{13} + x^5 - 90x^4 + x - 90.\]
Some parts have got lost, partly the constant term of the first factor of the left side, partly the majority of the summands of the second factor. It would be possible to restore the polynomial forming the other factor, but we restrict ourselves to asking the following question: What is the value of the constant term  $a$ ? We assume that all polynomials in the statement have only integer coefficients.


Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer  $ n < 60$  (both Kelly and Jason know that  $ n < 60$ , but that they don't know what the value of  $ n$  is). The computer tells Kelly the unit digit of  $ n$ , and it tells Jason the number of divisors of  $ n$ . Then, Kelly and Jason have the following dialogue:

Kelly: I don't know what  $ n$  is, and I'm sure that you don't know either. However, I know that  $ n$  is divisible by at least two different primes.

Jason: Oh, then I know what the value of  $ n$  is.

Kelly: Now I also know what  $ n$  is.

Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of  $ n$ ?


Triangle  $ABC$  has  $AB=9$  and  $BC: AC=40: 41$ . What's the largest area that this triangle can have?


Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distinguishable arrangements of the 8 coins.


The integers from  $1$  through  $9$  inclusive, are placed in the squares of a  $3 \times 3$  grid. Each square contains a different integer. The product of the integers in the first and second rows are  $60$  and  $96$  respectively. Find the sum of the integers in the third row.

*Proposed by bissue*


Let  $a$  and  $b$  be positive whole numbers such that  $\frac{4.5}{11}<\frac{a}{b}<\frac{5}{11}$ .
Find the fraction  $\frac{a}{b}$  for which the sum  $a+b$  is as small as possible.
Justify your answer


Find all reals  $ k$  such that
\[ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \plus{} 1\geq k(a \plus{} b \plus{} c \plus{} d)
\]
holds for all  $ a,b,c,d\geq \minus{} 1$ .

*Edited by orl.*


A function  $f$  defined on integers such that $f (n) =n + 3$  if  $n$  is odd $f (n) = \frac{n}{2}$  if  $n$  is even

If  $k$  is an odd integer, determine the values for which  $f (f (f (k))) = k$ .

Solution: \boxed{ k = 1 }


Let  $n$  be a positive integer. Each number  $1, 2, ..., 1000$  has been colored with one of  $n$  colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number  $n$  for which it is possible.

Solution: \boxed{ n = 10 }


Let  $b$  and  $c$  be real numbers not both equal to  $1$  such that  $1,b,c$  is an arithmetic progression and  $1,c,b$  is a geometric progression. What is  $100(b-c)$ ?

*Proposed by Noah Kravitz*


How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart?


$\triangle ABC$  has side lengths  $AB=20$ ,  $BC=15$ , and  $CA=7$ . Let the altitudes of  $\triangle ABC$  be  $AD$ ,  $BE$ , and  $CF$ . What is the distance between the orthocenter (intersection of the altitudes) of  $\triangle ABC$  and the incenter of  $\triangle DEF$ ?


We define the sets  $A_1,A_2,...,A_{160}$  such that  $\left|A_{i} \right|=i$  for all  $i=1,2,...,160$ . With the elements of these sets we create new sets  $M_1,M_2,...M_n$  by the following procedure: in the first step we choose some of the sets  $A_1,A_2,...,A_{160}$  and we remove from each of them the same number of elements. These elements that we removed are the elements of  $M_1$ . In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define  $M_2$ . We continue similarly until there are no more elements in  $A_1,A_2,...,A_{160}$ , thus defining the sets  $M_1,M_2,...,M_n$ . Find the minimum value of  $n$ .


Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An *operation* consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.

Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.

*Proposed by Anthony Wang*

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


Determine the greatest number, who is the product of some positive integers, and the sum of these numbers is  $1976.$

Solution: \boxed{ 2 \cdot 3^{658} }


Let  $ f(x) \equal{} e^{(p \plus{} 1)x} \minus{} e^x$  for real number  $ p > 0$ . Answer the following questions.

(1) Find the value of  $ x \equal{} s_p$  for which  $ f(x)$  is minimal and draw the graph of  $ y \equal{} f(x)$ .

(2) Let  $ g(t) \equal{} \int_t^{t \plus{} 1} f(x)e^{t \minus{} x}\ dx$ . Find the value of  $ t \equal{} t_p$  for which  $ g(t)$  is minimal.

(3) Use the fact  $ 1 \plus{} \frac {p}{2}\leq \frac {e^p \minus{} 1}{p}\leq 1 \plus{} \frac {p}{2} \plus{} p^2\ (0 < p\leq 1)$  to find the limit  $ \lim_{p\rightarrow \plus{}0} (t_p \minus{} s_p)$ .

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


Determine all natural numbers n for which the number  $A = n^4 + 4n^3 +5n^2 + 6n$  is a perfect square of a natural number.

Solution: \boxed{ n = 1 }


About pentagon  $ABCDE$  is known that angle  $A$  and angle  $C$  are right and that the sides  $| AB | = 4$ ,  $| BC | = 5$ ,  $| CD | = 10$ ,  $| DE | = 6$ . Furthermore, the point  $C'$  that appears by mirroring  $C$  in the line  $BD$ , lies on the line segment  $AE$ . Find angle  $E$ .


Two lines intersect inside a unit square, splitting it into four regions. Find the maximum product of the areas of the four regions.

[i]Proposed by Nathan Ramesh


Let  $\triangle ABC$  be a triangle with a right angle  $\angle ABC$ .  Let  $D$  be the midpoint of  $\overline{BC}$ , let  $E$  be the midpoint of  $\overline{AC}$ , and let  $F$  be the midpoint of  $\overline{AB}$ .  Let  $G$  be the midpoint of  $\overline{EC}$ .  One of the angles of  $\triangle DFG$  is a right angle.  What is the least possible value of  $\frac{BC}{AG}$ ?


Rachel has  $3$  children, all of which are at least  $2$  years old. The ages of the children are all pairwise relatively prime, but Rachel’s age is a multiple of each of her children’s ages. What is Rachel’s minimum possible age?


A set  $A$  of positive integers is called *uniform* if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer  $n>1$  such that there exist a uniform set  $A$  with  $n$  elements.


Consider the polynomial  $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ , with degree less than or equal to 2.

When  $ f$  varies with subject to the constrain  $ f(0) \equal{} 0,\ f(2) \equal{} 2$ , find the minimum value of  $ S\equal{}\int_0^2 |f'(x)|\ dx$ .


Let  $n$  be a positive integer. Find the number of permutations  $a_1$ ,  $a_2$ ,  $\dots a_n$  of the
sequence  $1$ ,  $2$ ,  $\dots$  ,  $n$  satisfying  $$ a_1 \le 2a_2\le 3a_3 \le \dots \le na_n $$  .

Proposed by United Kingdom


Find the smallest positive integer  $n$  with the property that in the set  $\{70, 71, 72,... 70 + n\}$  you can choose two different numbers whose product is the square of an integer.


A rectangular table  $ 9$  rows  $ \times$   $ 2008$  columns is fulfilled with numbers  $ 1$ ,  $ 2$ , ..., $ 2008$  in a such way that each number appears exactly  $ 9$  times in table and difference between any two numbers from same column is not greater than  $ 3$ . What is maximum value of minimum sum in column (with minimal sum)?


The two diagonals of a quadrilateral have lengths  $12$  and  $9$ , and the two diagonals are perpendicular to each other. Find the area of the quadrilateral.


Diagonals of trapezium   $ABCD$  are mutually perpendicular and the midline of the trapezium is  $5$ . Find the length of the segment that connects the midpoints of the bases of the trapezium.


You know that the Jones family has five children, and the Smith family has three children.  Of the eight children you know that there are five girls and three boys.  Let  $\dfrac{m}{n}$   be the probability that at least one of the families has only girls for children.  Given that  $m$  and  $n$  are relatively prime positive integers,  find  $m+ n$ .


For each  $k$  , find the least  $n$  in terms of  $k$  st the following holds:
There exists  $n$  real numbers  $a_1 , a_2 ,\cdot \cdot \cdot , a_n$  st for each  $i$  :  $$ 0 < a_{i+1} - a_{i} < a_i - a_{i-1} $$   And , there exists  $k$  pairs  $(i,j)$  st  $a_i - a_j = 1$ .

Solution: \boxed{ 2k }


What is the least positive integer  $k$  such that, in every convex  $101$ -gon, the sum of any  $k$  diagonals is greater than or equal to the sum of the remaining diagonals?

Solution: \boxed{ k = 4900 }


What is the largest positive integer  $n$  for which there is a unique integer  $k$  such that  $\frac{8}{15} < \frac{n}{n + k} < \frac{7}{13}$ ?


Let  $A_1$ ,  $\ldots$ ,  $A_{2022}$  be the vertices of a regular  $2022$ -gon in the plane. Alice and Bob play a game. Alice secretly chooses a line and colors all points in the plane on one side of the line blue, and all points on the other side of the line red. Points on the line are colored blue, so every point in the plane is either red or blue. (Bob cannot see the colors of the points.)

In each round, Bob chooses a point in the plane (not necessarily among  $A_1, \ldots, A_{2022}$ ) and Alice responds truthfully with the color of that point. What is the smallest number  $Q$  for which Bob has a strategy to always determine the colors of points  $A_1, \ldots, A_{2022}$  in  $Q$  rounds?


In trapezoid  $ABCD$ ,  $AB < CD$ ,  $AB\perp BC$ ,  $AB\parallel CD$ , and the diagonals  $AC$ ,  $BD$  are perpendicular at point  $P$ . There is a point  $Q$  on ray  $CA$  past  $A$  such that  $QD\perp DC$ . If
\[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then  $\frac{BP} {AP}-\frac{AP}{BP}$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ . Compute  $m+n$ .

*Ray Li.*

Solution: \boxed{ 47 + 14 = 61 }


Determine the smallest possible value of the expression   $$ \frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a} $$   where  $a,b,c \in \mathbb{R}$  satisfy  $a+b+c = -1$  and  $abc \leqslant -3$


Let  $l$  be a line passing the origin on the coordinate plane and has a positive slope. Consider circles  $C_1,\ C_2$  determined by the condition (i), (ii), (iii) as below.

(i) The circles  $C_1,\ C_2$  are contained in the domain determined by the inequality  $x\geq 0,\ y\geq 0.$ (ii) The circles  $C_1,\ C_2$  touch the line  $l$  at same point.

(iii) The circle  $C_1$  touches the  $x$ -axis at the point  $(1,\ 0)$  and the circle  $C_2$  touches the  $y$ -axis.

Let  $r_1,\ r_2$  be the radii of the circles  $C_1,\ C_2$ , respectively. Find the equation of the line  $l$  such that  $8r_1+9r_2$  is minimized and the minimum value.


Find all pairs  $(a,b)$  of different positive integers that satisfy the equation  $W(a)=W(b)$ , where  $W(x)=x^{4}-3x^{3}+5x^{2}-9x$ .

Solution: \boxed{ (1, 2) }


A jeweler can get an alloy that is  $40\%$  gold for  $200$  dollars per ounce, an alloy that is  $60\%$  gold for  $300$  dollar per ounce, and an alloy that is  $90\%$  gold for  $400$  dollars per ounce. The jeweler will purchase some of these gold alloy products, melt them down, and combine them to get an alloy that is  $50\%$  gold. Find the minimum number of dollars the jeweler will need to spend for each ounce of the alloy she makes.


The operation  $*$  is defined by  $a*b=a+b+ab$ , where  $a$  and  $b$  are real numbers. Find the value of \[\frac{1}{2}*\bigg(\frac{1}{3}*\Big(\cdots*\big(\frac{1}{9}*(\frac{1}{10}*\frac{1}{11})\big)\Big)\bigg).\]

*2017 CCA Math Bonanza Team Round #3*


A quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$


Let  $a$  be a real number greater than  $1$  such that  $\frac{20a}{a^2+1} = \sqrt{2}$ . Find  $\frac{14a}{a^2 - 1}$ .


A board  $64$  inches long and  $4$  inches high is inclined so that the long side of the board makes a  $30$  degree angle with the ground. The distance from the ground to the highest point on the board 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. What is  $a+b+c$ ?

*Author: Ray Li*

ClarificationThe problem is intended to be a two-dimensional problem. The board's dimensions are 64 by 4. The long side of the board makes a 30 degree angle with the ground. One corner of the board is touching the ground.


Let  $ABCD$  be a square with side length  $6$ . Circles  $X, Y$  , and  $Z$  are congruent circles with centers inside the square such that  $X$  is tangent to both sides  $\overline{AB}$  and  $\overline{AD}$ ,  $Y$  is tangent to both sides  $\overline{AB}$  and  $\overline{BC}$ , and  $Z$  is tangent to side  $\overline{CD}$  and both circles  $X$  and  $Y$  . The radius of the circle  $X$  can be written  $m -\sqrt{n}$ , where  $m$  and  $n$  are positive integers. Find  $m + n$ .


Let  $n$  be a positive integer. Compute the number of words  $w$  that satisfy the following three properties.

1.  $w$  consists of  $n$  letters from the alphabet  $\{a,b,c,d\}.$ 2.  $w$  contains an even number of  $a$ 's

3.  $w$  contains an even number of  $b$ 's.

For example, for  $n=2$  there are  $6$  such words:  $aa, bb, cc, dd, cd, dc.$

Solution: \boxed{4^{n-1} + 2^{n-1}}


Circle  $\Gamma$  with radius  $1$  is centered at point  $A$  on the circumference of circle  $\omega$  with radius  $7$ . Suppose that point  $P$  lies on  $\omega$  with  $AP=4$ . Determine the product of the distances from  $P$  to the two intersections of  $\omega$  and  $\Gamma$ .

*2018 CCA Math Bonanza Team Round #6*


We say positive integer  $n$  is  $\emph{metallic}$  if there is no prime of the form  $m^2-n$ . What is the sum of the three smallest metallic integers?

*Proposed by Lewis Chen*


Big Welk writes the letters of the alphabet in order, and starts again at  $A$  each time he gets to  $Z.$  What is the  $4^3$ -rd letter that he writes down?

Solution: \boxed{L}


What is the largest number  $N$  for which there exist  $N$  consecutive positive integers such that the sum of the digits in the  $k$ -th integer is divisible by  $k$  for  $1 \le  k \le  N$  ?

(S Tokarev)

Solution: \boxed{ N = 21 }


There are  $15$  (not necessarily distinct) integers chosen uniformly at random from the range from  $0$  to  $999$ , inclusive. Yang then computes the sum of their units digits, while Michael computes the last three digits of their sum. The probability of them getting the same result is  $\frac mn$  for relatively prime positive integers  $m,n$ . Find  $100m+n$ *Proposed by Yannick Yao*


Let   $f: \mathbb{R}\to\mathbb{R}$  is a function such that  $f( \cot x ) = \cos 2x+\sin 2x$  for all  $0 < x < \pi$ . Define  $g(x) = f(x) f(1-x)$  for  $-1 \leq x \leq 1$ . Find the maximum and minimum values of  $g$  on the closed interval  $[-1, 1].$

Solution: \boxed{1 \text{ and } -1}


Find all positive integer pairs  $(a,b)$  for which the set of positive integers can be partitioned into sets  $H_1$  and  $H_2$  such that neither  $a$  nor  $b$  can be represented as the difference of two numbers in  $H_i$  for  $i=1,2$ .

Solution: \boxed{ (a, b) }


Find the number of ordered pairs  $(a, b)$  of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\]


Let  $ABC$  be a triangle with incenter  $I$  and  $AB = 1400$ ,  $AC = 1800$ ,  $BC = 2014$ .  The circle centered at  $I$  passing through  $A$  intersects line  $BC$  at two points  $X$  and  $Y$ . Compute the length  $XY$ .

*Proposed by Evan Chen*


Let  $f(x) = 1 + 2x + 3x^2 + 4x^3 + 5x^4$  and let  $\zeta = e^{2\pi i/5} = \cos \frac{2\pi}{5} + i \sin \frac{2\pi}{5}$ . Find the value of the following expression:   $$ f(\zeta)f(\zeta^2)f(\zeta^3)f(\zeta^4). $$


A semicircle has diameter  $XY$ . A square  $PQRS$  with side length 12 is inscribed in the semicircle with  $P$  and  $S$  on the diameter. Square  $STUV$  has  $T$  on  $RS$ ,  $U$  on the semicircle, and  $V$  on  $XY$ . What is the area of  $STUV$ ?


Al travels for  $20$  miles per hour rolling down a hill in his chair for two hours, then four miles per hour climbing a hill for six hours. What is his average speed, in miles per hour?


Find all positive integers  $A$  which can be represented in the form: \[ A = \left ( m - \dfrac 1n \right) \left( n - \dfrac 1p \right) \left( p - \dfrac 1m \right) \]
where  $m\geq n\geq p \geq 1$  are integer numbers.

*Ioan Bogdan*


Let  $A, B, C$  be unique collinear points $ AB = BC =\frac13$ . Let  $P$  be a point that lies on the circle centered at  $B$  with radius  $\frac13$  and the circle centered at  $C$  with radius  $\frac13$  . Find the measure of angle  $\angle PAC$  in degrees.


Find all functions  $f:\mathbb{R}\to\mathbb{R}$  for which
\[ x(f(x+1)-f(x)) = f(x), \]
for all  $x\in\mathbb{R}$  and
\[ | f(x) - f(y) | \leq |x-y| , \]
for all  $x,y\in\mathbb{R}$ .

*Mihai Piticari*

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


Let  $f(x) =x^4+14x^3+52x^2+56x+16.$   Let  $z_1, z_2, z_3, z_4$  be the four roots of  $f$ .  Find the smallest possible value of  $|z_az_b+z_cz_d|$  where  $\{a,b,c,d\}=\{1,2,3,4\}$ .


Let  $f(x,y)$  be a function defined for all pairs of nonnegative integers  $(x, y),$  such that  $f(0,k)=f(k,0)=2^k$  and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers  $a, b.$  Determine the number of positive integers  $n\leq2016$  for which there exist two nonnegative integers  $a, b$  such that  $f(a,b)=n$ .

*Proposed by Michael Ren*


Let  $n > 0$  be an integer. We are given a balance and  $n$  weights of weight  $2^0, 2^1, \cdots, 2^{n-1}$ . We are to place each of the  $n$  weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.
Determine the number of ways in which this can be done.

*Proposed by Morteza Saghafian, Iran*

Solution: \boxed{ (2n-1)!! }


Let  $S = {1, 2, \cdots, 100}.$   $X$  is a subset of  $S$  such that no two distinct elements in  $X$  multiply to an element in  $X.$  Find the maximum number of elements of  $X$ .

*2022 CCA Math Bonanza Individual Round #3*


In an acute triangle  $ABC$ , an arbitrary point  $P$  is chosen on the altitude  $AH$ . The points  $E$  and  $F$  are the midpoints of  $AC$  and  $AB$ , respectively. The perpendiculars from  $E$  on  $CP$  and from  $F$  on  $BP$  intersect at the point  $K$ . Show that  $KB = KC$ .

Solution: \boxed{ KB = KC }


$25.$  Let  $C$  be the answer to Problem  $27.$  What is the  $C$ -th smallest positive integer with exactly four positive factors? $26.$  Let  $A$  be the answer to Problem  $25.$  Determine the absolute value of the difference between the two positive integer roots of the quadratic equation  $x^2-Ax+48=0$   $27.$  Let  $B$  be the answer to Problem  $26.$  Compute the smallest integer greater than  $\frac{B}{\pi}$

Solution: \boxed{ 8 }


Let  $\dots, a_{-1}, a_0, a_1, a_2, \dots$  be a sequence of positive integers satisfying the folloring relations:  $a_n = 0$  for  $n < 0$ ,  $a_0 = 1$ , and for  $n \ge 1$ ,
\[a_n = a_{n - 1} + 2(n - 1)a_{n - 2} + 9(n - 1)(n - 2)a_{n - 3} + 8(n - 1)(n - 2)(n - 3)a_{n - 4}.\]
Compute
\[\sum_{n \ge 0} \frac{10^n a_n}{n!}.\]


A function  $f$  is defined on the complex numbers by  $f(z)=(a+bi)z,$  where  $a$  and  $b$  are positive numbers.  This function has the property that the image of each point in the complex plane is equidistant from that point and the origin.  Given that  $|a+bi|=8$  and that  $b^2=m/n,$  where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n.$


Find all prime numbers such that the square of the prime number can be written as the sum of cubes of two positive integers.

Solution: \boxed{ p = 3 }


Source: 2018 Canadian Open Math Challenge Part B Problem 3
-----

The *doubling sum* function is defined by
\[D(a,n)=\overbrace{a+2a+4a+8a+...}^{\text{n terms}}.\]
For example, we have
\[D(5,3)=5+10+20=35\]
and
\[D(11,5)=11+22+44+88+176=341.\]
Determine the smallest positive integer  $n$  such that for every integer  $i$  between  $1$  and  $6$ , inclusive, there exists a positive integer  $a_i$  such that  $D(a_i,i)=n.$


Let  $a,b,c$  be the roots of  $x^3-9x^2+11x-1=0$ , and let  $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$ .  Find  $s^4-18s^2-8s$ .


Suppose  $(a_n)$ ,  $(b_n)$ ,  $(c_n)$  are arithmetic progressions. Given that  $a_1+b_1+c_1 = 0$  and  $a_2+b_2+c_2 = 1$ , compute  $a_{2014}+b_{2014}+c_{2014}$ .

*Proposed by Evan Chen*


Let  $S_1$  and  $S_2$  be sets of points on the coordinate plane  $\mathbb{R}^2$  defined as follows
\[S_1={(x,y)\in \mathbb{R}^2:|x+|x||+|y+|y||\le 2}\]
\[S_2={(x,y)\in \mathbb{R}^2:|x-|x||+|y-|y||\le 2}\]
Find the area of the intersection of  $S_1$  and  $S_2$


Find the smallest prime  $p$  such that the digits of  $p$  (in base 10) add up to a prime number greater than  $10$ .


Let  $ABC$  be a triangle. Let  $E$  be a point on the segment  $BC$  such that  $BE = 2EC$ . Let  $F$  be the mid-point of  $AC$ . Let  $BF$  intersect  $AE$  in  $Q$ . Determine  $BQ:QF$ .


Let  $n \ge 4$  be a natural number. Let  $A_1A_2 \cdots A_n$  be a regular polygon and  $X = \{ 1,2,3....,n \} $ . A subset  $\{ i_1, i_2,\cdots, i_k \} $  of  $X$ , with  $k \ge 3$  and  $i_1 < i_2 < \cdots < i_k$ , is called a good subset if the angles of the polygon  $A_{i_1}A_{i_2}\cdots A_{i_k}$  , when arranged in the increasing order, are in an arithmetic progression. If  $n$  is a prime, show that a proper good subset of  $X$  contains exactly four elements.

Solution: \boxed{k = 4}


Let  $q$  be a positive rational number. Two ants are initially at the same point  $X$  in the plane. In the  $n$ -th minute  $(n = 1,2,...)$  each of them chooses whether to walk due north, east, south or west and then walks the distance of  $q^n$  metres. After a whole number of minutes, they are at the same point in the plane (not necessarily  $X$ ), but have not taken exactly the same route within that time. Determine all possible values of  $q$ .

Proposed by Jeremy King, UK

Solution: \boxed{ q = 1 }


A point  $P$  in the interior of a convex polyhedron in Euclidean space is called a *pivot point* of the polyhedron if every line through  $P$  contains exactly  $0$  or  $2$  vertices of the polyhedron. Determine, with proof, the maximum number of pivot points that a polyhedron can contain.


**p1.** Meghan Trainor is all about those base-systems. If she weighs  $451$  pounds in some base, and  $127$  pounds in a base that is twice as big, how much does she weigh in base  $10$ ?**p2.** Taylor Swift made the song  $15$  in  $2008$ , and the song  $22$  in  $2012$ . If she had continued at the same rate, in what year would her song  $1989$  have come out?**p3.** Sir Mix-A-Lot likes big butts and cannot lie. He is with his three friends, who all always lie. If each of the following statements was said by a different person, who is Sir Mix-A-Lot?
A: B and D both like big butts.
B: C and D either both like big butts, or neither of them do.
C: B does not like big butts.
D: A or C (or both) like big butts.**p4.** Mark Ronson is going to give some uptown funk-tions to you. Bruno Mars is at  $1$  degree Fahrenheit right now (that is, time, in seconds,  $t = 0$ ), and his heat h(t) is given by  $h(t) = 2 \cdot h(t - 1) + t$ . If he will explode when he is over  $1,000,000$  degrees Fahrenheit, at what time t will Bruno Mars explode?**p5.** Jay-Z, who is a emcee, took  $4$  AMC contests, for a total of  $100$  problems. He got  $99$  problems correct and  $1$  problem wrong. His answer,  $z$ , to the problem he missed, was the square root of the correct  $4$ -digit answer,  $j$ . If  $z$  is also  $3$  times the sum of the digits of  $j$ , then what is the ordered tuple  $(j, z)$ ?

PS. You had better use hide for answers.

Solution: \boxed{(2916, 54)}


$ABCD$  is a square with centre  $O$ . Two congruent isosceles triangle  $BCJ$  and  $CDK$  with base  $BC$  and  $CD$  respectively are constructed outside the square. let  $M$  be the midpoint of  $CJ$ . Show that  $OM$  and  $BK$  are perpendicular to each other.

Solution: \boxed{OM \perp BK}


The numbers from 1 to  $n^2$   are randomly arranged in the cells of a  $n \times n$  square ( $n \geq 2$ ). For any pair of numbers situated on the same row or on the same column the ratio of the greater number to the smaller number is calculated.  Let us call the **characteristic** of the arrangement the smallest of these  $n^2\left(n-1\right)$  fractions. What is the highest possible value of the characteristic ?


The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids which are not parallelograms have vertices in this set of  $16$  points and have area of at least six square units?
[asy]
import graph; size(7cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
dot((-2,4),linewidth(6pt) + dotstyle);
dot((-1,4),linewidth(6pt) + dotstyle);
dot((0,4),linewidth(6pt) + dotstyle);
dot((1,4),linewidth(6pt) + dotstyle);
dot((2,4),linewidth(6pt) + dotstyle);
dot((3,4),linewidth(6pt) + dotstyle);
dot((4,4),linewidth(6pt) + dotstyle);
dot((-3,2),linewidth(6pt) + dotstyle);
dot((-2,2),linewidth(6pt) + dotstyle);
dot((-1,2),linewidth(6pt) + dotstyle);
dot((0,2),linewidth(6pt) + dotstyle);
dot((1,2),linewidth(6pt) + dotstyle);
dot((2,2),linewidth(6pt) + dotstyle);
dot((3,2),linewidth(6pt) + dotstyle);
dot((4,2),linewidth(6pt) + dotstyle);
dot((5,2),linewidth(6pt) + dotstyle); [/asy]


In the game of Avalon, there are  $10$  players,  $4$  of which are bad. A *quest* is a subset of those players. A quest fails if it contains at least one bad player. A randomly chosen quest of  $3$  players happens to fail. What is the probability that there is exactly one bad player in the failed quest?

*2018 CCA Math Bonanza Team Round #3*

Solution: \boxed{ \frac{3}{5} }


Let  $n$  points with integer coordinates be given in the  $xy$ -plane. What is the minimum value of  $n$  which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?


Prair has a box with some combination of red and green balls.  If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors!  How many possible values between  $200$  and  $1000$  are there for the total number of balls in the box?

*Proposed by Andrew Wu*


In triangle  $ABC$ ,  $D$  is a point on  $AB$  between  $A$  and  $B$ ,  $E$  is a point on  $AC$  between  $A$  and  $C$ , and  $F$  is a point on  $BC$  between  $B$  and  $C$  such that  $AF$ ,  $BE$ , and  $CD$  all meet inside  $\triangle ABC$  at a point  $G$ . Given that the area of  $\triangle ABC$  is  $15$ , the area of  $\triangle ABE$  is  $5$ , and the area of  $\triangle ACD$  is  $10$ , compute the area of  $\triangle ABF$ .


Farmer James has three types of cows on his farm. A cow with zero legs is called a  $\textit{ground beef}$ , a cow with one leg is called a  $\textit{steak}$ , and a cow with two legs is called a  $\textit{lean beef}$ . Farmer James counts a total of  $20$  cows and  $18$  legs on his farm. How many more  $\textit{ground beef}$ s than  $\textit{lean beef}$ s does Farmer James have?

*Proposed by James Lin*

Solution: \boxed{ 2 }


Given that the answer to this problem can be expressed as  $a\cdot b\cdot c$ , where  $a$ ,  $b$ , and  $c$  are pairwise relatively prime positive integers with  $b=10$ , compute  $1000a+100b+10c$ .

*Proposed by Ankit Bisain*


Find out the maximum possible area of the triangle  $ABC$  whose medians have lengths satisfying inequalities  $m_a \le 2, m_b \le 3, m_c \le  4$ .


Find all differentiable functions  $f:\mathbb{R}\to\mathbb{R}$  such that
\[f'(x)=\frac{f(x+n)-f(x)}n\]
for all real numbers  $x$  and all positive integers  $n.$

Solution: \boxed{ f(x) = ax + b }


Given a natural number  $k$ , find the smallest natural number  $C$  such that  $$ \frac C{n+k+1}\binom{2n}{n+k} $$  is an integer for every integer  $n\ge k$ .


Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be  $\frac 32$  times of the original number.


Given positive integer  $ m \geq 17$ ,  $ 2m$  contestants participate in a circular competition. In each round, we devide the  $ 2m$  contestants into  $ m$  groups, and the two contestants in one group play against each other. The groups are re-divided in the next round. The contestants compete for  $ 2m\minus{}1$  rounds so that each contestant has played a game with all the  $ 2m\minus{}1$  players. Find the least possible positive integer  $ n$ , so that there exists a valid competition and after  $ n$  rounds, for any  $ 4$  contestants, non of them has played with the others or there have been at least  $ 2$  games played within those  $ 4$ .


For every rational number  $m>0$  we consider the function  $f_m:\mathbb{R}\rightarrow\mathbb{R},f_m(x)=\frac{1}{m}x+m$ . Denote by  $G_m$  the graph of the function  $f_m$ . Let  $p,q,r$  be positive rational numbers.

a) Show that if  $p$  and  $q$  are distinct then  $G_p\cap G_q$  is non-empty.

b) Show that if  $G_p\cap G_q$  is a point with integer coordinates, then  $p$  and  $q$  are integer numbers.

c) Show that if  $p,q,r$  are consecutive natural numbers, then the area of the triangle determined by intersections of  $G_p,G_q$  and  $G_r$  is equal to  $1$ .

Solution: \boxed{ 1 }


The sum of all of the interior angles of seven polygons is  $180\times17$ . Find the total number of sides of the polygons.


It is known that  $\frac{7}{13} + \sin \phi = \cos \phi$  for some real  $\phi$ . What is sin  $2\phi$ ?


Calvin was asked to evaluate  $37 + 31 \times a$  for some number  $a$ . Unfortunately, his paper was tilted 45 degrees, so he mistook multiplication for addition (and vice versa) and evaluated  $37 \times 31 + a$  instead. Fortunately, Calvin still arrived at the correct answer while still following the order of operations. For what value of  $a$  could this have happened?

*Ray Li.*


Find the largest positive integer  $k$  such that  $\phi ( \sigma ( 2^k)) = 2^k$ . ( $\phi(n)$  denotes the number of positive integers that are smaller than  $n$  and relatively prime to  $n$ , and  $\sigma(n)$  denotes the sum of divisors of  $n$ ). As a hint, you are given that  $641|2^{32}+1$ .


Let  $2561$  given points on a circle be colored either red or green. In each step, all points are recolored simultaneously in the following way: if both direct neighbors of a point  $P$  have the same color as  $P$ , then the color of  $P$  remains unchanged, otherwise  $P$  obtains the other color. Starting with the initial coloring  $F_1$ , we obtain the colorings  $F_2, F_3,\dots$  after several recoloring steps. Determine the smallest number  $n$  such that, for any initial coloring  $F_1$ , we must have  $F_n = F_{n+2}$ .

Solution: \boxed{ n = 1280 }