An  $n \times n$  complex matrix  $A$  is called \emph{t-normal} if $AA^t = A^t A$  where  $A^t$  is the transpose of  $A$ .  For each  $n$ ,
determine the maximum dimension of a linear space of complex  $n
\times n$  matrices consisting of t-normal matrices.

Proposed by Shachar Carmeli, Weizmann Institute of Science

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


A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by  $\overline{AB}$ , has length 31. Find the sum of the lengths of the three diagonals that can be drawn from  $A$ .


Let  $A$  be the set of positive integers that are the product of two consecutive integers.  Let  $B$  the set of positive integers that are the product of three consecutive integers.  Find the sum of the two smallest elements of  $A\cap B$ .


Let  $\mathcal{P}$  be the set of all polynomials  $p(x)=x^4+2x^2+mx+n$ , where  $m$  and  $n$  range over the positive reals. There exists a unique  $p(x) \in \mathcal{P}$  such that  $p(x)$  has a real root,  $m$  is minimized, and  $p(1)=99$ . Find  $n$ .

*Proposed by **AOPS12142015***


A rectangular prism has three distinct faces of area  $24$ ,  $30$ , and  $32$ . The diagonals of each distinct face of the prism form sides of a triangle. What is the triangle’s area?


In each cell of the table  $4 \times 4$ , in which the lines are labeled with numbers  $1,2,3,4$ , and columns with  letters  $a,b,c,d$ ,  one number is written:  $0$  or  $1$  . Such a table is called *valid* if there are exactly two units in each of its rows and in each column. Determine the number of *valid* tables.


Find the unique 3 digit number  $N=\underline{A}$   $\underline{B}$   $\underline{C},$  whose digits  $(A, B, C)$  are all nonzero, with the property that the product  $P=\underline{A}$   $\underline{B}$   $\underline{C}$   $\times$   $\underline{A}$   $\underline{B}$   $\times$   $\underline{A}$  is divisible by  $1000$ .

*Proposed by Kyle Lee*


There are  $24$  participants attended a meeting. Each two of them shook hands once or not. A total of  $216$  handshakes occured in the meeting. For any two participants who have shaken hands, at most  $10$  among the rest  $22$  participants have shaken hands with exactly one of these two persons. Define a *friend circle* to be a group of  $3$  participants in which each person has shaken hands with the other two. Find the minimum possible value of friend circles.


A positive integer  $N$  has  $20$  digits when written in base  $9$  and  $13$  digits when written in base  $27$ . How many digits does  $N$  have when written in base  $3$ ?

*Proposed by Aaron Lin*


How many  $6$ -digit positive integers have their digits in nondecreasing order from left to right? Note that  $0$  cannot be a leading digit.


Find all natural numbers  $n$  for which  $n + 195$  and  $n - 274$  are  perfect cubes.


A natural number  $n$  is called *perfect* if it is equal to the sum of all its natural divisors other than  $n$ . For example, the number  $6$  is perfect because  $6 = 1 + 2 + 3$ . Find all even perfect numbers that can be given as the sum of two cubes positive integers.

Solution: \boxed{ 28 }


Find the biggest positive integer  $n$  such that  $n$  is  $167$  times the amount of it's positive divisors.


Among all polynomials  $P(x)$  with integer coefficients for which  $P(-10) = 145$  and  $P(9) = 164$ , compute the smallest possible value of  $|P(0)|.$


Points  $E$  and  $C$  are chosen on a semicircle with diameter  $AB$  and center  $O$  such that  $OE \perp AB$  and the intersection point  $D$  of  $AC$  and  $OE$  is inside the semicircle.  Find all values of  $\angle{CAB}$  for which the quadrilateral  $OBCD$  is tangent.


Find all the integers  $x, y$  and  $z$  greater than or equal to  $0$  such that  $2^x + 9 \cdot  7^y = z^3$

Solution: \boxed{ (0, 1, 4) }


Let  $M$  be a set consisting of  $n$  points in the plane, satisfying:
i) there exist  $7$  points in  $M$  which constitute the vertices of a convex heptagon;
ii) if for any  $5$  points in  $M$  which constitute the vertices of a convex pentagon, then there is a point in  $M$  which lies in the interior of the pentagon.
Find the minimum value of  $n$ .

*Leng Gangsong*


12. What is the sum of all possible  $\left(\begin{array}{l}i  j\end{array}\right)$  subject to the restrictions that  $i \geq 10, j \geq 0$ , and  $i+j \leq 20$  ? Count different  $i, j$  that yield the same value separately - for example, count both  $\left(\begin{array}{c}10  1\end{array}\right)$  and  $\left(\begin{array}{c}10  9\end{array}\right)$ .


Let  $AXYZB$  be a convex pentagon inscribed in a semicircle with diameter  $AB$ .  Suppose that  $AZ-AX=6$ ,  $BX-BZ=9$ ,  $AY=12$ , and  $BY=5$ .  Find the greatest integer not exceeding the perimeter of quadrilateral  $OXYZ$ , where  $O$  is the midpoint of  $AB$ .

*Proposed by Evan Chen*


Convex pentagon  $ABCDE$  has side lengths  $AB=5$ ,  $BC=CD=DE=6$ , and  $EA=7$ . Moreover, the pentagon has an inscribed circle (a circle tangent to each side of the pentagon). Find the area of  $ABCDE$ .


In a country consisting of  $2015$  cities, between any two cities there is exactly one direct round flight operated by some air company. Find the minimal possible number of air companies if direct flights between any three cities are operated by three different air companies.


For each natural number  $n\geq 2$ , solve the following system of equations in the integers  $x_1, x_2, ..., x_n$ :  $$ (n^2-n)x_i+\left(\prod_{j\neq i}x_j\right)S=n^3-n^2,\qquad \forall 1\le i\le n $$  where  $$ S=x_1^2+x_2^2+\dots+x_n^2. $$

Solution: \boxed{ x_i = 1 }


Let  $n$  be a positive integer.  For each partition of the set  $\{1,2,\dots,3n\}$  into arithmetic  progressions,  we  consider  the  sum  $S$  of  the  respective  common differences  of  these  arithmetic  progressions.   What  is  the  maximal  value that  $S$  can attain?

(An *arithmetic progression* is a set of the form  $\{a,a+d,\dots,a+kd\}$ , where  $a,d,k$  are positive integers, and  $k\geqslant 2$ ; thus an arithmetic progression has at least three elements, and successive elements have difference  $d$ , called the *common difference* of the arithmetic progression.)


While taking the SAT, you become distracted by your own answer sheet.  Because you are not bound to the College Board's limiting rules, you realize that there are actually  $32$  ways to mark your answer for each question, because you could fight the system and bubble in multiple letters at once: for example, you could mark  $AB$ , or  $AC$ , or  $ABD$ , or even  $ABCDE$ , or nothing at all!

You begin to wonder how many ways you could mark off the 10 questions you haven't yet answered.  To increase the challenge, you wonder how many ways you could mark off the rest of your answer sheet without ever marking the same letter twice in a row.  (For example, if  $ABD$  is marked for one question,  $AC$  cannot be marked for the next one because  $A$  would be marked twice in a row.)  If the number of ways to do this can be expressed in the form  $2^m p^n$ , where  $m,n > 1$  are integers and  $p$  is a prime, compute  $100m+n+p$ .

*Proposed by Alexander Dai*


We have a group of  $n$  kids. For each pair of kids, at least one has sent a message to
the other one. For each kid  $A$ , among the kids to whom  $A$  has sent a message, exactly $25 \% $  have sent a message to  $A$ . How many possible two-digit values of  $n$  are there?

*Proposed by Bulgaria*


Let  $a_0 = 1$  and define the sequence  $\{a_n\}$  by \[a_{n+1} = \frac{\sqrt{3}a_n - 1}{a_n + \sqrt{3}}.\] If  $a_{2017}$  can be expressed in the form  $a+b\sqrt{c}$  in simplest radical form, compute  $a+b+c$ .

*2016 CCA Math Bonanza Lightning #3.2*


Two circles have radius  $2$  and  $3$ , and the distance between their centers is  $10$ . Let  $E$  be the intersection of their two common external tangents, and  $I$  be the intersection of their two common internal tangents. Compute  $EI$ .

(A *common external tangent* is a tangent line to two circles such that the circles are on the same side of the line, while a *common internal tangent* is a tangent line to two circles such that the circles are on opposite sides of the line).

*Proposed by Connor Gordon)*


Andile and Zandre play a game on a  $2017 \times 2017$  board. At the beginning, Andile declares some of the squares *forbidden*, meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game.

What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)


There are positive integers  $b$  and  $c$  such that the polynomial  $2x^2 + bx + c$  has two real roots which differ by  $30.$  Find the least possible value of  $b + c.$


Let  $m$  and  $n$  be positive integers where  $m$  has  $d$  digits in base ten and  $d\leq n$ . Find the sum of all the digits (in  base ten) of the product  $(10^n-1)m$ .

Solution: \boxed{ 9n }


A  $3$  by  $3$  determinant has three entries equal to  $2$ , three entries equal to  $5$ , and three entries equal to  $8$ . Find the maximum possible value of the determinant.


Let  $Z$  denote the set of points in  $\mathbb{R}^{n}$  whose coordinates are  $0$  or  $1.$  (Thus  $Z$  has  $2^{n}$  elements, which are the vertices of a unit hypercube in  $\mathbb{R}^{n}$ .) Given a vector subspace  $V$  of  $\mathbb{R}^{n},$  let  $Z(V)$  denote the number of members of  $Z$  that lie in  $V.$  Let  $k$  be given,  $0\le k\le n.$  Find the maximum, over all vector subspaces  $V\subseteq\mathbb{R}^{n}$  of dimension  $k,$  of the number of points in  $V\cap Z.$

Solution: \boxed{ 2^k }


Denote by  $P(n)$  the greatest prime divisor of  $n$ . Find all integers  $n\geq 2$  for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]

Solution: \boxed{n = 3}


Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament.
They decided on the following rules of attendance:

(a) There should always be at least one person present on each day.

(b) On no two days should the same subset attend.

(c) The members present on day  $N$  should include for each  $K<N$ ,  $(K\ge 1)$  at least one member who was present on day  $K$ .

For how many days can the parliament sit before one of the rules is broken?


Let  $n>1$  be a natural number and  $x_k{}$  be the residue of  $n^2$  modulo  $\lfloor n^2/k\rfloor+1$  for all natural  $k{}$ . Compute the sum \[\bigg\lfloor\frac{x_2}{1}\bigg\rfloor+\bigg\lfloor\frac{x_3}{2}\bigg\rfloor+\cdots+\left\lfloor\frac{x_n}{n-1}\right\rfloor.\]


Yura put  $2001$  coins of  $1$ ,  $2$  or  $3$  kopeykas in a row.  It turned out that between any two  $1$ -kopeyka coins there is at least one coin; between any two  $2$ -kopeykas coins there are at least two coins; and between any two  $3$ -kopeykas coins there are at least  $3$  coins.  How many  $3$ -koyepkas coins could Yura put?


Find the number of ordered tuples  $\left(C,A,M,B\right)$  of non-negative integers such that \[C!+C!+A!+M!=B!\]

*2019 CCA Math Bonanza Team Round #4*


As a reward for working for NIMO, Evan divides  $100$  indivisible marbles among three of his volunteers: David, Justin, and Michael. (Of course, each volunteer must get at least one marble!) However, Evan knows that, in the middle of the night, Lewis will select a positive integer  $n > 1$  and, for each volunteer, steal exactly  $\frac 1n$  of his marbles (if possible, i.e. if  $n$  divides the number of marbles). In how many ways can Evan distribute the  $100$  marbles so that Lewis is unable to steal marbles from every volunteer, regardless of which  $n$  he selects?

*Proposed by Jack Cornish*


The cells of a  $2021\times 2021$  table are filled with numbers using the following rule. The bottom left cell, which we label with coordinate  $(1, 1)$ , contains the number  $0$ . For every other cell  $C$ , we consider a route from  $(1, 1)$  to  $C$ , where at each step we can only go one cell to the right or one cell up (not diagonally). If we take the number of steps in the route and add the numbers from the cells along the route, we obtain the number in cell  $C$ . For example, the cell with coordinate  $(2, 1)$  contains  $1 = 1 + 0$ , the cell with coordinate  $(3, 1)$  contains  $3 = 2 + 0 + 1$ , and the cell with coordinate  $(3, 2)$  contains  $7 = 3 + 0 + 1 + 3$ . What is the last digit of the number in the cell  $(2021, 2021)$ ?


Let  $p$  be a prime number such that  $\frac{28^p-1}{2p^2+2p+1}$  is an integer. Find all possible values of number of divisors of  $2p^2+2p+1$ .


A rectangular wooden block has a square top and bottom, its volume is  $576$ , and the surface area of its vertical sides is  $384$ . Find the sum of the lengths of all twelve of the edges of the block.


For given integer  $n \geq 3$ , set  $S =\{p_1, p_2, \cdots, p_m\}$  consists of permutations  $p_i$  of  $(1, 2, \cdots, n)$ . Suppose that among every three distinct numbers in  $\{1, 2, \cdots, n\}$ , one of these number does not lie in between the other two numbers in every permutations  $p_i$  ( $1 \leq i \leq m$ ). (For example, in the permutation  $(1, 3, 2, 4)$ ,  $3$  lies in between  $1$  and  $4$ , and  $4$  does not lie in between  $1$  and  $2$ .) Determine the maximum value of  $m$ .

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


What is the value of  $ { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=odd} $   $ - { \sum_{1 \le i< j \le 10}(i+j)}_{i+j=even} $


Let  $a$  and  $b$  be two-digit positive integers. Find the greatest possible value of  $a+b$ , given that the greatest common factor of  $a$  and  $b$  is  $6$ .

*Proposed by Jacob Xu*

Solution*Solution*.  $\boxed{186}$ We can write our two numbers as  $6x$  and  $6y$ . Notice that  $x$  and  $y$  must be relatively prime. Since  $6x$  and  $6y$  are two digit numbers, we just need to check values of  $x$  and  $y$  from  $2$  through  $16$  such that  $x$  and  $y$  are relatively prime. We maximize the sum when  $x = 15$  and  $y = 16$ , since consecutive numbers are always relatively prime. So the sum is  $6 \cdot (15+16) = \boxed{186}$ .


For  $n \geq 1$ , let  $a_n$  be the number beginning with  $n$   $9$ 's followed by  $744$ ; eg.,  $a_4=9999744$ . Define   $$ f(n)=\text{max}\{m\in \mathbb{N} \mid2^m ~ \text{divides} ~ a_n \} $$  , for  $n\geq 1$ . Find  $f(1)+f(2)+f(3)+ \cdots + f(10)$ .


Evaluate the following expression:   $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$   The terms with minus signs are exactly the powers of two.


Let  $ABC$  be a triangle in which  $\angle ABC = 60^{\circ}$ . Let  $I$  and  $O$  be the incentre and circumcentre of  $ABC$ , respectively. Let  $M$  be the midpoint of the arc  $BC$  of the circumcircle of  $ABC$ , which does not contain the point  $A$ . Determine  $\angle BAC$  given that  $MB = OI$ .


Let  $f(n) = \sum^n_{d=1} \left\lfloor \frac{n}{d} \right\rfloor$  and  $g(n) = f(n) -f(n - 1)$ . For how many  $n$  from  $1$  to  $100$  inclusive is  $g(n)$  even?


Find a positive integrer number  $n$  such that, if yor put a number  $2$  on the left and a number  $1$  on the right, the new number is equal to  $33n$ .


In a bag there are  $1007$  black and  $1007$  white balls, which are randomly numbered  $1$  to  $2014$ . In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after  $2014$  steps?

Solution: \boxed{1007^2 = 1014049}


$\textbf{Problem C.1}$ There are two piles of coins, each containing  $2010$  pieces. Two players  $A$  and  $B$  play a game taking turns ( $A$  plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?

Solution: \boxed{ B }


In angle  $\angle AOB=60^{\circ}$  are two circle which circumscribed and tangjent to each other . If we write with  $r$  and  $R$  the radius of smaller and bigger circle respectively and if  $r=1$  find  $R$  .


Determine the minimum value of  $$ x^{2014} + 2x^{2013} + 3x^{2012} + 4x^{2011} +\ldots + 2014x + 2015 $$  where  $x$  is a real number.


Consider the set  $E$  of all positive integers  $n$  such that when divided by  $9,10,11$  respectively, the remainders(in that order) are all  $>1$  and form a non constant geometric progression. If  $N$  is the largest element of  $E$ , find the sum of digits of  $E$


A quarry wants to sell a large pile of gravel. At full price, the gravel would sell for  $3200$  dollars. But during the first week the quarry only sells  $60\%$  of the gravel at full price. The following week the quarry drops the price by  $10\%$ , and, again, it sells  $60\%$  of the remaining gravel. Each week, thereafter, the quarry reduces the price by another  $10\%$  and sells  $60\%$  of the remaining gravel. This continues until there is only a handful of gravel left. How many dollars does the quarry collect for the sale of all its gravel?


Let  $r$  be a fixed positive real number. It is known that for some positive integer  $n$  the following statement is true: for any positive real numbers  $a_1,...,a_n$  satisfying the equation  $a_1+...+a_n=r(\frac{1}{a_1}+...+\frac{1}{a_n})$  they also satisfy the equation  $\frac{1}{\sqrt{r}-a_1}+...+\frac{1}{\sqrt{r}-a_n}=\frac{1}{\sqrt{r}}$  ( $a_i\neq\sqrt{r}$ ). Find  $n$ .

Solution: \boxed{ n = 2 }


How many rooks can be placed in an  $n\times n$  chessboard such that each rook is threatened by at most  $2k$  rooks?
(15 points)
*Proposed by Mostafa Einollah zadeh*

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


A positive integer  $N$  greater than  $1$  is described as special if in its base- $8$  and base- $9$  representations, both the leading and ending digit of  $N$  are equal to  $1$ . What is the smallest special integer in decimal representation?

*Proposed by Michael Ren*


Determine, with proof, the smallest positive integer  $c$  such that for any positive integer  $n$ , the decimal representation of the number  $c^n+2014$  has digits all less than  $5$ .

*Proposed by Evan Chen*


Triangle  $ABC$  has  $AB=10$ ,  $BC=17$ , and  $CA=21$ . Point  $P$  lies on the circle with diameter  $AB$ . What is the greatest possible area of  $APC$ ?

Solution: \boxed{ \frac{189}{2} }


Determine all positive integers  $n$  so that both  $20n$  and  $5n + 275$  are perfect squares.

(A perfect square is a number which can be expressed as  $k^2$ , where  $k$  is an integer.)


Find the last two digits of  $\tbinom{200}{100}$ . Express the answer as an integer between  $0$  and  $99$ . (e.g. if the last two digits are  $05$ , just write  $5$ .)


Find all positive integers  $n$  with  $n \ge 2$  such that the polynomial \[ P(a_1, a_2, ..., a_n) = a_1^n+a_2^n + ... + a_n^n - n a_1 a_2 ... a_n \] in the  $n$  variables  $a_1$ ,  $a_2$ ,  $\dots$ ,  $a_n$  is irreducible over the real numbers, i.e. it cannot be factored as the product of two nonconstant polynomials with real coefficients.

*Proposed by Yang Liu*

Solution: \boxed{ n \ge 4 }


Find all pairs of positive integers  $(x, y)$  such that  $\frac{xy^3}{x+y}$  is the cube of a prime.

Solution: \boxed{(2, 14)}


In a tennis tournament there are participants from  $n$  different countries. Each team consists of a coach and a player whom should settle in a hotel. The rooms considered for the settlement of coaches are different from players' ones. Each player wants to be in a room whose roommates are **all** from countries which have a defense agreement with the player's country. Conversely, each coach wants to be in a room whose roommates are **all** from countries which don't have a defense agreement with the coach's country. Find the minimum number of the rooms such that we can **always** grant everyone's desire.

*proposed by Seyed Reza Hosseini and Mohammad Amin Ghiasi*

Solution: \boxed{ n + 1 }


For every positive integer  $n$ , define the number of non-empty subsets  $\mathcal N\subseteq \{1,\ldots ,n\}$  such that  $\gcd(n\in\mathcal N)=1$ . Show that  $f(n)$  is a perfect square if and only if  $n=1$ .

Solution: \boxed{ n = 1 }


Determine all intergers  $n\geq 2$  such that  $a+\sqrt{2}$  and  $a^n+\sqrt{2}$  are both rational for some real number  $a$  depending on  $n$

Solution: \boxed{ n = 2 }


Mary and Pat play the following number game. Mary picks an initial integer greater than  $2017$ . She then multiplies this number by  $2017$  and adds  $2$  to the result. Pat will add  $2019$  to this new number and it will again be Mary’s turn. Both players will continue to take alternating turns. Mary will always multiply the current number by  $2017$  and add  $2$  to the result when it is her turn. Pat will always add  $2019$  to the current number when it is his turn. Pat wins if any of the numbers obtained by either player is divisible by  $2018$ . Mary wants to prevent Pat from winning the game.
Determine, with proof, the smallest initial integer Mary could choose in order to achieve this.


Let  $ a $  ,  $ b $  ,  $ c $  ,  $ d $  ,  $ (a + b + c + 18 + d) $  ,  $ (a + b + c + 18 - d) $  ,  $ (b + c) $  , and  $ (c + d) $  be distinct prime numbers such that  $ a + b + c = 2010 $ ,  $ a $ ,  $ b $ ,  $  c $ ,  $ d \neq 3 $  , and  $ d \le 50 $ . Find the maximum value of the diﬀerence between two of these prime numbers.


Find all natural numbers   $k$  such that there exist natural numbers  $a_1,a_2,...,a_{k+1}$  with  $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider  $0$  to be a natural number.

Solution: \boxed{ k = 3 }


An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.


For  $f(x)=x^4+|x|,$  let  $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$

Find the value of   $\frac{I_1}{I_2}.$


Let   $$ N=2^5+2^{5^2}+2^{5^3}+\cdots{}+2^{5^{2015}}. $$  Wrotten in the usual decimal form, find the last two digits of the number  $N$ .


Let  $\omega_1$  be a circle of radius  $1$  that is internally tangent to a circle  $\omega_2$  of radius  $2$  at point  $A$ . Suppose  $\overline{AB}$  is a chord of  $\omega_2$  with length  $2\sqrt3$  that intersects  $\omega_1$  at point  $C\ne A$ . If the tangent line of  $\omega_1$  at  $C$  intersects  $\omega_2$  at points  $D$  and  $E$ , find  $CD^4 + CE^4$ .


Find the largest positive real  $ k$ , such that for any positive reals  $ a,b,c,d$ , there is always:
\[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

Solution: \boxed{ k = 174960 }


How many integers  $n$  with  $10 \le n \le 500$  have the property that the hundreds digit of  $17n$  and  $17n+17$  are different?

*Proposed by Evan Chen*


For each positive integer  $k$  denote  $C(k)$  to be sum of its distinct prime divisors. For example  $C(1)=0,C(2)=2,C(45)=8$ . Find all positive integers  $n$  for which  $C(2^n+1)=C(n)$ .

Solution: \boxed{ n = 3 }


Let  $n$  and  $k$  be two integers which are greater than  $1$ . Let  $a_1,a_2,\ldots,a_n,c_1,c_2,\ldots,c_m$  be non-negative real numbers such that
i)  $a_1\ge a_2\ge\ldots\ge a_n$  and  $a_1+a_2+\ldots+a_n=1$ ;
ii) For any integer  $m\in\{1,2,\ldots,n\}$ , we have that  $c_1+c_2+\ldots+c_m\le m^k$ .
Find the maximum of  $c_1a_1^k+c_2a_2^k+\ldots+c_na_n^k$ .

Solution: \boxed{ 1 }


Let  $S = \{a_1, \ldots, a_n \}$  be a finite set of positive integers of size  $n \ge 1$ , and let  $T$  be the set of all positive integers that can be expressed as sums of perfect powers (including  $1$ ) of distinct numbers in  $S$ , meaning
\[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \]
Show that there is a positive integer  $N$  (only depending on  $n$ ) such that  $T$  contains no arithmetic progression of length  $N$ .

*Yang Liu*

Solution: \boxed{ N }


A semicircle is erected over the segment  $AB$  with center  $M$ . Let  $P$  be one point  different  from  $A$  and  $B$  on the semicircle and  $Q$  the midpoint of the arc of the circle  $AP$ . The point of intersection of the straight line  $BP$  with the parallel to  $P Q$  through  $M$  is  $S$ . Show that  $PM = PS$  holds.

*(Karl Czakler)*

Solution: \boxed{ PM = PS }


One integer was removed from the set  $S=\left \{ 1,2,3,...,n \right \}$  of the integers from  $1$  to  $n$ . The arithmetic mean of the other integers of  $S$  is equal to  $\frac{163}{4}$ .
What integer was removed ?


Let  $n$  be a positive integer. Find the greatest possible integer  $m$ , in terms of  $n$ , with the following property: a table with  $m$  rows and  $n$  columns can be filled with real numbers in such a manner that for any two different rows  $\left[ {{a_1},{a_2},\ldots,{a_n}}\right]$  and  $\left[ {{b_1},{b_2},\ldots,{b_n}} \right]$  the following holds: \[\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1\]
*Poland (Tomasz Kobos)*


For a positive integer n, let  $w(n)$  denote the number of distinct prime
divisors of n. Determine the least positive integer k such that
 $2^{w(n)} \leq k \sqrt[4]{n}$
for all positive integers n.


An  $\textrm{alien}$  script has  $n$  letters  $b_1,b_2,\dots,b_n$ . For some  $k<n/2$  assume that all words formed by any of the  $k$  letters (written left to right) are meaningful. These words are called  $k$ -words. Such a  $k$ -word is considered  $\textbf{sacred}$  if:

i. no letter appears twice and,
ii. if a letter  $b_i$  appears in the word then the letters  $b_{i-1}$  and  $b_{i+1}$  do not appear. (Here  $b_{n+1} = b_1$  and  $b_0 = b_n$ ).

For example, if  $n = 7$  and  $k = 3$  then  $b_1b_3b_6, b_3b_1b_6, b_2b_4b_6$  are sacred  $3$ -words. On the other hand  $b_1b_7b_4, b_2b_2b_6$  are not sacred.
What is the total number of sacred  $k$ -words?
Use your formula to find the answer for  $n = 10$  and  $k = 4$ .


Princeton’s Math Club recently bought a stock for  $\$ 2 $ and sold it for $ \ $9$  thirteen days later. Given that the stock either increases or decreases by  $\$ 1 $ every day and never reached $ \ $0$ , in how many possible ways could the stock have changed during those thirteen days?


Let  $ ABC$  be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio  $ \angle A : \angle C$ .


$M$  is an integer set with a finite number of elements. Among any three elements of this set, it is always possible to choose two such that the sum of these two numbers is an element of  $M.$  How many elements can  $M$  have at most?


In triangle  $ ABC$ , one has marked the incenter, the foot of altitude from vertex  $ C$  and the center of the excircle tangent to side  $ AB$ . After this, the triangle was erased. Restore it.

Solution: \boxed{ \triangle ABC }


Let  $f_{0}(x)=x$ , and for each  $n\geq 0$ , let  $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$ . Find the smallest real number that is at least as large as
\[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\]
for all  $a \in [0,1]$ .


The integers  $a, b,$  and  $c$  form a strictly increasing geometric sequence. Suppose that  $abc = 216$ . What is the maximum possible value of  $a + b + c$ ?


How many ways are there to make two  $3$ -digit numbers  $m$  and  $n$  such that  $n=3m$  and each of six digits  $1$ ,  $2$ ,  $3$ ,  $6$ ,  $7$ ,  $8$  are used exactly once?


All angles of  $ABC$  are in  $(30,90)$ . Circumcenter of  $ABC$  is  $O$  and circumradius is  $R$ . Point  $K$  is projection of  $O$  to angle bisector of  $\angle B$ , point  $M$  is midpoint  $AC$ . It is known, that  $2KM=R$ . Find  $\angle B$


No math tournament exam is complete without a self referencing question. What is the product of
the smallest prime factor of the number of words in this problem times the largest prime factor of the
number of words in this problem


Let  $A$ ,  $B$ , and  $C$  be distinct points on a line with  $AB=AC=1$ . Square  $ABDE$  and equilateral triangle  $ACF$  are drawn on the same side of line  $BC$ . What is the degree measure of the acute angle formed by lines  $EC$  and  $BF$ ?

*Ray Li*


Let the sum  $\sum_{n=1}^{9} \frac{1}{n(n+1)(n+2)}$  written in its lowest terms be  $\frac{p}{q}$  . Find the value of  $q - p$ .


Find the maximum possible value of  $k$  for which there exist distinct reals  $x_1,x_2,\ldots ,x_k $  greater than  $1$  such that for all  $1 \leq i, j \leq k$ ,  $$ x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}. $$  *Proposed by Morteza Saghafian*


Write either  $1$  or  $-1$  in each of the cells of a  $(2n) \times (2n)$ -table, in such a way that there are exactly  $2n^2$  entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be  $M$ . Determine the largest possible value of  $M$ .


Let  $ABC$  be a triangle with circumcircle  $\Gamma$  and let  $D$  be the midpoint of minor arc  $BC$ . Let  $E, F$  be on  $\Gamma$  such that  $DE \bot AC$  and  $DF \bot AB$ . Lines  $BE$  and  $DF$  meet at  $G$ , and lines  $CF$  and  $DE$  meet at  $H$ . Given that  $AB = 8, AC = 10$ , and  $\angle BAC = 60^\circ$ , find the area of  $BCHG$ .

*Note: this is a modified version of Premier #2*


Consider a board on  $2013 \times 2013$  squares, what is the maximum number of chess knights that can be placed so that no  $2$  attack each other?


Let  $C$  be a unit cube and let  $p$  denote the orthogonal projection onto the plane. Find the maximum area of  $p(C)$ .


Find the sum of all values of  $a$  such that there are positive integers  $a$  and  $b$  satisfying  $(a - b)\sqrt{ab} = 2016$ .


$1989$  equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently.


Let  $ABC$  have side lengths  $3$ ,  $4$ , and  $5$ . Let  $P$  be a point inside  $ABC$ . What is the minimum sum of lengths of the altitudes from  $P$  to the side lengths of  $ABC$ ?


Determine the largest integer  $N$ , for which there exists a  $6\times N$  table  $T$  that has the following properties: $*$  Every column contains the numbers  $1,2,\ldots,6$  in some ordering. $*$  For any two columns  $i\ne j$ , there exists a row  $r$  such that  $T(r,i)= T(r,j)$ . $*$  For any two columns  $i\ne j$ , there exists a row  $s$  such that  $T(s,i)\ne T(s,j)$ .

(Proposed by Gerhard Woeginger, Austria)


problem 1 :A sequence is defined by $ x_1 = 1, x_2 = 4$  and  $ x_{n+2} = 4x_{n+1} -x_n$  for  $n \geq 1$ . Find all natural numbers  $m$  such that the number  $3x_n^2 + m$  is a perfect square for all natural numbers  $n$

Solution: \boxed{ m = 1 }


Define a "digitized number" as a ten-digit number  $a_0a_1\ldots a_9$  such that for  $k=0,1,\ldots, 9$ ,  $a_k$  is equal to the number of times the digit  $k$  occurs in the number.  Find the sum of all digitized numbers.


Let  $BCDE$  be a trapezoid with  $BE\parallel CD$ ,  $BE = 20$ ,  $BC = 2\sqrt{34}$ ,  $CD = 8$ ,  $DE = 2\sqrt{10}$ . Draw a line through  $E$  parallel to  $BD$  and a line through  $B$  perpendicular to  $BE$ , and let  $A$  be the intersection of these two lines. Let  $M$  be the intersection of diagonals  $BD$  and  $CE$ , and let  $X$  be the intersection of  $AM$  and  $BE$ . If  $BX$  can be written as  $\frac{a}{b}$ , where  $a, b$  are relatively prime positive integers, find  $a + b$


Two players play alternately on a  $ 5 \times 5$  board. The first player always enters a  $ 1$  into an empty square and the second player always enters a  $ 0$  into an empty square. When the board is full, the sum of the numbers in each of the nine  $ 3 \times 3$  squares is calculated and the first player's score is the largest such sum. What is the largest score the first player can make, regardless of the responses of the second player?


Find the minimum value of  $c$  such that for any positive integer  $n\ge 4$  and any set  $A\subseteq \{1,2,\cdots,n\}$ , if  $|A| >cn$ , there exists a function  $f:A\to\{1,-1\}$  satisfying  $$ \left| \sum_{a\in A}a\cdot f(a)\right| \le 1. $$

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


A four digit number is called *stutterer* if its first two digits are the same and its last two digits are also the same, e.g.  $3311$  and  $2222$  are stutterer numbers. Find all stutterer numbers that are square numbers.

Solution: \boxed{ 7744 }


Let  $n$  be a positive integer. For a positive integer  $m$ , we partition the set  $\{1, 2, 3,...,m\}$  into  $n$  subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of  $n$ , find the largest positive integer  $m$  for which such a partition exists.

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


The sequences  $ (a_n),(b_n)$  are defined by  $ a_1\equal{}1,b_1\equal{}2$  and \[a_{n \plus{} 1} \equal{} \frac {1 \plus{} a_n \plus{} a_nb_n}{b_n}, \quad b_{n \plus{} 1} \equal{} \frac {1 \plus{} b_n \plus{} a_nb_n}{a_n}.\]

Show that  $ a_{2008} < 5$ .

Solution: \boxed{ a_{2008} < 5 }


David has a collection of 40 rocks, 30 stones, 20 minerals and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform?

*Ray Li*


How many distinct triangles  $ABC$  are tjere, up to simplilarity, such that the magnitudes of the angles  $A, B$  and  $C$  in degrees are positive integers and satisfy  $$ \cos{A}\cos{B} + \sin{A}\sin{B}\sin{kC} = 1 $$  for some positive integer  $k$ , where  $kC$  does not exceet  $360^{\circ}$ ?


Find the total number of primes  $p<100$  such that  $\lfloor (2+\sqrt{5})^p \rfloor-2^{p+1}$  is divisible by  $p$ . Here  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ .


Let  $a_1,a_2,a_3,\dots,a_6$  be an arithmetic sequence with common difference  $3$ . Suppose that  $a_1$ ,  $a_3$ , and  $a_6$  also form a geometric sequence. Compute  $a_1$ .


Find all functions  $f : R \to  R$  satisfying the conditions:
1.  $f (x + 1) \ge  f (x) + 1$  for all  $x \in R$
2.  $f (x y) \ge f (x)f (y)$  for all  $x, y \in R$

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


Points  $A$ ,  $B$ , and  $C$  lie in that order on line  $\ell$  such that  $AB=3$  and  $BC=2$ .  Point  $H$  is such that  $CH$  is perpendicular to  $\ell$ .  Determine the length  $CH$  such that  $\angle AHB$  is as large as possible.


Determine the smallest natural number  $n =>2$  with the property:
 For every positive integers  $a_1, a_2,.  .  .  , a_n$   the product of all  differences  $a_j-a_i$ ,
   $1 <=i <j <=n$ , is divisible by 2001.


Let  $N$  be the number of (positive) divisors of  $2010^{2010}$  ending in the digit  $2$ . What is the remainder when  $N$  is divided by 2010?


A  $(2^n - 1) \times (2^n +1)$  board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangle is a power of 2. Find the minimum number of rectangles that the board may be divided into.

Solution: \boxed{ 2n }


Let  $\overline{AB}$  be a line segment with length  $10$ . Let  $P$  be a point on this segment with  $AP = 2$ . Let  $\omega_1$  and  $\omega_2$  be the circles with diameters  $\overline{AP}$  and  $\overline{P B}$ , respectively. Let  $XY$  be a line externally tangent to  $\omega_1$  and  $\omega_2$  at distinct points  $X$  and  $Y$  , respectively. Compute the area of  $\vartriangle XP Y$  .


On a chess board ( $8*8$ ) there are written the numbers  $1$  to  $64$ : on the first line, from left to right, there are the numbers  $1, 2, 3, ... , 8$ ; on the second line, from left to right, there are the numbers  $9, 10, 11, ... , 16$ ;etc. The  $\"+\"$  and  $\"-\"$  signs are put to each number such that, in each line and in each column, there are  $4$   $\"+\"$  signs and  $4$   $\"-\"$  signs. Then, the  $64$  numbers are added. Find all the possible values of this sum.


Determine the smallest natural number written in the decimal system with the product of the digits equal to  $10! = 1 \cdot 2  \cdot 3\cdot  ... \cdot9\cdot10$ .


In triangle  $\triangle ABC$ , the points  $A', B', C'$  are on sides  $BC, AC, AB$  respectively. Also,  $AA', BB', CC'$  intersect at the point  $O$ (they are concurrent at  $O$ ). Also,  $\frac {AO}{OA'}+\frac {BO}{OB'}+\frac {CO}{OC'} = 92$ . Find the value of  $\frac {AO}{OA'}\times \frac {BO}{OB'}\times \frac {CO}{OC'}$ .


Let  $S = \{1,2,3,\ldots,n\}$ . Consider a function  $f\colon S\to S$ . A subset  $D$  of  $S$  is said to be invariant if for all  $x\in D$  we have  $f(x)\in D$ . The empty set and  $S$  are also considered as invariant subsets. By  $\deg (f)$  we define the number of invariant subsets  $D$  of  $S$  for the function  $f$ .**i)** Show that there exists a function  $f\colon S\to S$  such that  $\deg (f)=2$ .**ii)** Show that for every  $1\leq k\leq n$  there exists a function  $f\colon S\to S$  such that  $\deg (f)=2^{k}$ .


John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.
John looked at the number (which was  $2002$  ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.
What was Mary's Number?


A natural number  $k$  is said  $n$ -squared if by colouring the squares of a  $2n \times k$  chessboard, in any manner, with  $n$  different colours, we can find  $4$  separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of  $n$ , the smallest natural  $k$  that is  $n$ -squared.

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


For a positive integer  $n$ , let  $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$ .

Find  $I_1+I_2+I_3+I_4$ .

*1992 University of Fukui entrance exam/Medicine*


Determine the number of solutions of the simultaneous equations  $ x^2 \plus{} y^3 \equal{} 29$  and  $ \log_3 x \cdot \log_2 y \equal{} 1.$


Let  $f(x)$  be the polynomial  $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$ .  Let  $c$  be the coefficient of  $x^{48}$  in  $f(x)$ .  When  $c$  is divided by 101, what is the remainder?  (The remainder is an integer between 0 and 100.)


Given a convex quadrilateral  $ABCD$  in which  $\angle BAC = 20^o$ ,   $\angle CAD = 60^o$ ,   $\angle ADB = 50^o$  , and  $\angle BDC = 10^o$ . Find  $\angle ACB$ .


How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]


Find all positive integers  $n$  such that the product of all positive divisors of  $n$  is  $24^{240}$ .

Solution: \boxed{ n = 24^5 }


Let  $h$  be a positive integer. The sequence  $a_n$  is defined by  $a_0 = 1$  and

\[a_{n+1} = \{\begin{array}{c} \frac{a_n}{2} \text{  if  } a_n \text{  is  even  }a_n+h \text{  otherwise  }.\end{array}\]

For example,  $h = 27$  yields  $a_1=28, a_2 = 14, a_3 = 7, a_4 = 34$  etc. For which  $h$  is there an  $n > 0$  with  $a_n = 1$ ?

Solution: \boxed{ h }


Find the smallest positive integer  $ n$  such that  $ 107n$  has the same last two digits as  $ n$ .


Let  $\star$  be an operation defined in the set of nonnegative integers with the following properties: for any nonnegative integers  $x$  and  $y$ ,
(i)  $(x + 1)\star 0 = (0\star x) + 1$ (ii)  $0\star (y + 1) = (y\star 0) + 1$ (iii)  $(x + 1)\star (y + 1) = (x\star y) + 1$ .
If  $123\star 456 = 789$ , find  $246\star 135$ .


Let  $\mathbb{Z}$  denote the set of all integers. Find all polynomials  $P(x)$  with integer coefficients that satisfy the following property:

For any infinite sequence  $a_1$ ,  $a_2$ ,  $\dotsc$  of integers in which each integer in  $\mathbb{Z}$  appears exactly once, there exist indices  $i < j$  and an integer  $k$  such that  $a_i +a_{i+1} +\dotsb +a_j = P(k)$ .

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


In a  $m\times n$  chessboard ( $m,n\ge 2$ ), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least  $2/3$  of the chessboard is covered by dominoes.

*Proposed by DVDthe1st, mzy and jjax*

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


Given a convex polygon M invariant under a  $90^\circ$  rotation, show that there exist two circles, the ratio of whose radii is  $\sqrt2$ , one containing M and the other contained in M.
*A. Khrabrov*

Solution: \boxed{ \sqrt{2} }


Let  $ABC$  be an equilateral triangle of area  $1998$  cm $^2$ . Points  $K, L, M$  divide the segments  $[AB], [BC] ,[CA]$ , respectively, in the ratio  $3:4$  . Line  $AL$  intersects the lines  $CK$  and  $BM$  respectively at the points  $P$  and  $Q$ , and the line  $BM$  intersects the line  $CK$  at point  $R$ . Find the area of the triangle  $PQR$ .

Solution: \boxed{54 \, \text{cm}^2}


There are 47 students in a classroom with seats arranged in 6 rows  $ \times$  8 columns, and the seat in the  $ i$ -th row and  $ j$ -th column is denoted by  $ (i,j).$  Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat  $ (i,j),$  if his/her new seat is  $ (m,n),$  we say that the student is moved by  $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$  and define the position value of the student as  $ a\plus{}b.$  Let  $ S$  denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of  $ S.$


Find all pairs of positive integers  $m$ ,  $n$  such that the  $(m+n)$ -digit number \[\underbrace{33\ldots3}_{m}\underbrace{66\ldots 6}_{n}\] is a perfect square.

Solution: \boxed{ (1, 1) }


Find the smallest constant  $C > 1$  such that the following statement holds: for every integer  $n \geq 2$  and sequence of non-integer positive real numbers  $a_1, a_2, \dots, a_n$  satisfying   $$ \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n} = 1, $$   it's possible to choose positive integers  $b_i$  such that
(i) for each  $i = 1, 2, \dots, n$ , either  $b_i = \lfloor a_i \rfloor$  or   $b_i = \lfloor a_i \rfloor + 1$ , and
(ii) we have   $$ 1 < \frac{1}{b_1} + \frac{1}{b_2} + \cdots + \frac{1}{b_n} \leq C. $$  (Here  $\lfloor \bullet \rfloor$  denotes the floor function, as usual.)

*Merlijn Staps*

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


An equilateral triangle with side length  $6$  has a square of side length  $6$  attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked  $A$  and  $B$  in the figure) can be written as  $m + \sqrt{n}$  where  $m$  and  $n$  are positive integers. Find  $m + n$ .

[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);

[/asy]


A regular  $n$ -gon is inscribed in a unit circle. Compute the product from a fixed vertex to all the other vertices.

Solution: \boxed{n}


A right triangle has perimeter  $2008$ , and the area of a circle inscribed in the triangle is  $100\pi^3$ .  Let  $A$  be the area of the triangle.  Compute  $\lfloor A\rfloor$ .


What is the sum of all primes  $p$  such that  $7^p - 6^p + 2$  is divisible by 43?


Gage skated 1 hr 15 min each day for 5 days and 1 hr 30 min each day for 3 days. How long would he have to skate the ninth day in order to average 85 minutes of skating each day for the entire time? $\text{(A)}\ \text{1 hr} \qquad \text{(B)}\ \text{1 hr 10 min} \qquad \text{(C)}\ \text{1 hr 20 min} \qquad \text{(D)}\ \text{1 hr 40 min} \qquad \text{(E)}\ \text{2 hr}$

Solution: \boxed{2 \text{ hr}}


Siva has the following expression, which is missing operations:  $$ \frac12 \,\, \_  \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_  \,\,\frac{1}{16} \,\, \_  \,\,\frac{1}{32}. $$   For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than  $\frac12$  can be expressed as  $\frac{a}{b}$  for relatively prime positive integers  $a$ ,  $b$ , then find  $a + b$ .


Use  $ \log_{10} 2 \equal{} 0.301,\ \log_{10} 3 \equal{} 0.477,\ \log_{10} 7 \equal{} 0.845$  to find the value of  $ \log_{10} (10!)$ .
Note that you must answer according to the rules:fractional part of  $ 0.5$  and higher is rounded up, and everything strictly less than  $ 0.5$  is rounded down,
say  $ 1.234\longrightarrow 1.23$ . Then find the minimum integer value  $ n$  such that  $ 10! < 2^{n}$ .


There are  $2017$  turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after  $2015$  seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with another turtle can be written in the form  $\tfrac{p}{q}$  where  $p$  and  $q$  are relatively prime positive integers. Find  $p + q$ .


For a finite graph  $G$ , let  $f(G)$  be the number of triangles and  $g(G)$  the number of tetrahedra formed by edges of  $G$ . Find the least constant  $c$  such that \[g(G)^3\le c\cdot f(G)^4\] for every graph  $G$ .

*Proposed by Marcin Kuczma, Poland*


A rectangular piece of paper  $ABCD$  has sides of lengths  $AB = 1$ ,  $BC = 2$ . The rectangle is folded in half such that  $AD$  coincides with  $BC$  and  $EF$  is the folding line. Then fold the paper along a line  $BM$  such that the corner  $A$  falls on line  $EF$ . How large, in degrees, is  $\angle ABM$ ?
[asy]
size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy]


The repeating decimals  $0.abab\overline{ab}$  and  $0.abcabc\overline{abc}$  satisfy
\[0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37},\]
where  $a,b$ , and  $c$  are (not necessarily distinct) digits. Find the three-digit number  $abc$ .


Let  $p$  be a prime number such that  $p\equiv 1\pmod{4}$ . Determine  $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$ , where  $\{x\}=x-[x]$ .


In a finite sequence of real numbers the sum of any seven successive terms is negative and the sum of any eleven successive terms is positive. Determine the maximum number of terms in the sequence.


**Q.** Find all polynomials  $P: \mathbb{R \times R}\to\mathbb{R\times R}$  with real coefficients, such that   $$ P(x,y) = P(x+y,x-y), \ \forall\ x,y \in \mathbb{R}. $$  *Proposed by TuZo*

Solution: \boxed{ P(x, y) = (a, b) }


Let  $X$  be a set with  $n$  elements and  $0 \le k \le n$ . Let  $a_{n,k}$  be the maximum number of permutations of the set  $X$  such that every two of them have at least  $k$  common components (where a common component of  $f$  and g is an  $x \in X$  such that  $f(x) = g(x)$ ). Let  $b_{n,k}$  be the maximum number of permutations of the set  $X$  such that every two of them have at most  $k$  common components.
(a) Show that  $a_{n,k} \cdot b_{n,k-1} \le n!$ .
(b) Let  $p$  be prime, and find the exact value of  $a_{p,2}$ .

Solution: \boxed{ (p-2)! }


Let  $m$  and  $n$  be positive integers such that  $x=m+\sqrt{n}$  is a solution to the equation  $x^2-10x+1=\sqrt{x}(x+1)$ .  Find  $m+n$ .


In the plane there are  $2020$  points, some of which are black and the rest are green. For every black point, the following applies: *There are exactly two green points that represent the distance  $2020$  from that black point.*
Find the smallest possible number of green dots.

(Walther Janous)


Considering all numbers of the form  $n = \lfloor \frac{k^3}{2012} \rfloor$ , where  $\lfloor x \rfloor$  denotes the greatest integer less than or equal to  $x$ , and  $k$  ranges from  $1$  to  $2012$ , how many of these  $n$ ’s are distinct?


A circle radius  $320$  is tangent to the inside of a circle radius  $1000$ . The smaller circle is tangent to a diameter of the larger circle at a point  $P$ . How far is the point  $P$  from the outside of the larger circle?


Find all positive integers  $d$  for which there exist polynomials  $P(x)$  and  $Q(x)$  with real coefficients such that degree of  $P$  equals  $d$  and  $$ P(x)^2+1=(x^2+1)Q(x)^2. $$

Solution: \boxed{ d }


$ABCD$  is a cyclic quadrilateral inscribed in a circle of radius  $5$ , with  $AB=6$ ,  $BC=7$ ,  $CD=8$ . Find  $AD$ .


For every positive integer  $k$ , let  $\mathbf{T}_k = (k(k+1), 0)$ , and define  $\mathcal{H}_k$  as the homothety centered at  $\mathbf{T}_k$  with ratio  $\tfrac{1}{2}$  if  $k$  is odd and  $\tfrac{2}{3}$  is  $k$  is even. Suppose  $P = (x,y)$  is a point such that   $$ (\mathcal{H}_{4} \circ \mathcal{H}_{3} \circ \mathcal{H}_2 \circ \mathcal{H}_1)(P) = (20, 20). $$   What is  $x+y$ ?
(A *homothety*  $\mathcal{H}$  with nonzero ratio  $r$  centered at a point  $P$  maps each point  $X$  to the point  $Y$  on ray  $\overrightarrow{PX}$  such that  $PY = rPX$ .)


Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37} b(a\plus{}d)\equiv b\pmod {37} c(a\plus{}d)\equiv c\pmod{37} bc\plus{}d^2\equiv d\pmod{37} ad\minus{}bc\equiv 1\pmod{37}\end{cases}\]


Triangle  $ABC$  is right angled at  $A$ . The circle with center  $A$  and radius  $AB$  cuts  $BC$  and  $AC$  internally at  $D$  and  $E$  respectively. If  $BD = 20$  and  $DC = 16$ , determine  $AC^2$ .


Let the three sides of a triangle be  $\ell, m, n$ , respectively, satisfying  $\ell>m>n$  and  $\left\{\frac{3^\ell}{10^4}\right\}=\left\{\frac{3^m}{10^4}\right\}=\left\{\frac{3^n}{10^4}\right\}$ , where  $\{x\}=x-\lfloor{x}\rfloor$  and  $\lfloor{x}\rfloor$  denotes the integral part of the number  $x$ . Find the minimum perimeter of such a triangle.


Let  $m$  be a positive integer, and let  $a_0, a_1,\ldots,a_m$  be a sequence of reals such that  $a_0=37$ ,  $a_1=72$ ,  $a_m=0$ , and \[a_{k+1}=a_{k-1}-\frac{3}{a_k}\] for  $k=1,2, \dots, m-1$ . Find  $m$ .


The isoelectric point of glycine is the pH at which it has zero charge. Its charge is  $-\frac13$  at pH  $3.55$ , while its charge is  $\frac12$  at pH  $9.6$ . Charge increases linearly with pH. What is the isoelectric point of glycine?


There are two round tables with  $n{}$  dwarves sitting at each table. Each dwarf has only two friends: his neighbours to the left and to the right. A good wizard wants to seat the dwarves at one round table so that each two neighbours are friends. His magic allows him to make any  $2n$  pairs of dwarves into pairs of friends (the dwarves in a pair may be from the same or from different tables). However, he knows that an evil sorcerer will break  $n{}$  of those new friendships. For which  $n{}$  is the good wizard able to achieve his goal no matter what the evil sorcerer does?

*Mikhail Svyatlovskiy*

Solution: \boxed{ n }


Find all functions  $f$  from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational  $a, b$ .

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


For a positive integer  $n$ , there is a school with  $2n$  people. For a set  $X$  of students in this school, if any two students in  $X$  know each other, we call  $X$  *well-formed*. If the maximum number of students in a well-formed set is no more than  $n$ , find the maximum number of well-formed set.

Here, an empty set and a set with one student is regarded as well-formed as well.


The intersection of two squares with perimeter  $8$  is a rectangle with diagonal length  $1$ . Given that the distance between the centers of the two squares is  $2$ , the  perimeter of the rectangle can be expressed as  $P$ . Find  $10P$ .


In a competition there are  $18$  teams and in each round  $18$  teams are divided into  $9$  pairs where the  $9$  matches are played coincidentally. There are  $17$  rounds, so that each pair of teams play each other exactly once. After  $n$  rounds, there always exists  $4$  teams such that there was exactly one match played between these teams in those  $n$  rounds. Find the maximum value of  $n$ .


In a room there are  $144$  people. They are joined by  $n$  other people who are each carrying  $k$  coins. When these coins are shared among all  $n + 144$  people, each person has  $2$  of these coins. Find the minimum possible value of  $2n + k$ .


Point D is from AC of triangle ABC so that 2AD=DC. Let DE be perpendicular to BC and AE
intersects BD at F. It is known that triangle BEF is equilateral. Find <ADB?


Let  $S_n$  be the sum of the reciprocals of the non-zero digits of the integers from 1 to  $10^n$  inclusive. Find the smallest positive integer  $n$  for which  $S_n$  is an integer.


For the NEMO, Kevin needs to compute the product
		\[
		9 \times 99 \times 999 \times \cdots \times 999999999.
		\]
		Kevin takes exactly  $ab$  seconds to multiply an  $a$ -digit integer by a  $b$ -digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.

*Proposed by Evan Chen*


We define the function  $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$ . Then choose any distinct  $a, b, c \in \mathbb{R}$  such that the following holds:  $f(a,b)=f(b,c)=f(c,a)$ . Over all such choices of  $a, b, c$ , what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]


A sequence of vertices  $v_1,v_2,\ldots,v_k$  in a graph, where  $v_i=v_j$  only if  $i=j$  and  $k$  can be any positive integer, is called a  $\textit{cycle}$  if  $v_1$  is attached by an edge to  $v_2$ ,  $v_2$  to  $v_3$ , and so on to  $v_k$  connected to  $v_1$ .  Rotations and reflections are distinct:  $A,B,C$  is distinct from  $A,C,B$  and  $B,C,A$ .  Supposed a simple graph  $G$  has  $2013$  vertices and  $3013$  edges.  What is the minimal number of cycles possible in  $G$ ?


Gus has to make a list of  $250$  positive integers, not necessarily distinct, such that each number is equal to the number of numbers in the list that are different from it. For example, if  $15$  is a number from the list so the list contains  $15$  numbers other than  $15$ . Determine the maximum number of distinct numbers the Gus list can contain.


At a certain grocery store, cookies may be bought in boxes of  $10$  or  $21.$  What is the minimum positive number of cookies that must be bought so that the cookies may be split evenly among  $13$  people?

*Author: Ray Li*


An infinite sheet of paper is divided into equal squares, some of which are colored red. In each  $2\times3$  rectangle, there are exactly two red squares. Now consider an arbitrary  $9\times11$  rectangle. How many red squares does it contain? (The sides of all considered rectangles go along the grid lines.)


Find all functions  $f:\mathbb{N} \to \mathbb{N}$  such that for all  $m,n\in \mathbb{N}$ :
- $f(2)=2$ ,
- $f(mn)=f(m)f(n)$ ,
- $f(n+1)>f(n)$ .

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


Define the *hotel elevator cubic*as the unique cubic polynomial  $P$  for which  $P(11) = 11$ ,  $P(12) = 12$ ,  $P(13) = 14$ ,  $P(14) = 15$ . What is  $P(15)$ ?

*Proposed by Evan Chen*


Four positive integers  $a, b, c, d$  satisfy the condition:  $a < b < c < d$ . For what smallest possible value of  $d$  could the following condition be true: the arithmetic mean of numbers  $a, b, c$  is twice smaller than the arithmetic mean of numbers  $a, b, c, d$ ?


Find all real numbers  $x$  that satisfy the equation   $$ \frac{x-2020}{1}+\frac{x-2019}{2}+\cdots+\frac{x-2000}{21}=\frac{x-1}{2020}+\frac{x-2}{2019}+\cdots+\frac{x-21}{2000}, $$   and simplify your answer(s) as much as possible. Justify your solution.

Solution: \boxed{x = 2021}


A quadratic polynomial  $p(x)$  with integer coefficients satisfies  $p(41) = 42$ . For some integers  $a, b > 41$ ,  $p(a) = 13$  and  $p(b) = 73$ . Compute the value of  $p(1)$ .

*Proposed by Aaron Lin*


The polynomial  $P$  is a quadratic with integer coefficients.  For every positive integer  $n$ , the integers  $P(n)$  and  $P(P(n))$  are relatively prime to  $n$ .  If  $P(3) = 89$ , what is the value of  $P(10)$ ?


Let  $k$  be a real number such that the product of real roots of the equation   $$ X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0 $$   is  $-2013$ . Find the sum of the squares of these real roots.


Find all natural numbers  $n> 1$  for which the following applies:
The sum of the number  $n$  and its second largest divisor is  $2013$ .

(R. Henner, Vienna)

Solution: \boxed{ n = 1342 }


A student did not notice multiplication sign between two three-digit numbers and wrote it as a six-digit number. Result was 7 times more that it should be. Find these numbers.

*(2 points)*

Solution: \boxed{(143, 143)}


Suppose that each of  $n$  people knows exactly  one piece of information and all  $n$  pieces are  different.  Every time person  $A$  phones person  $B$ ,  $A$  tells  $B$  everything he knows, while tells  $A$  nothing. What is the minimum of phone calls between pairs of people needed for everyone to know everything?

Solution: \boxed{2n - 2}


Let  $ S(n) $  be the sum of the squares of the positive integers less than and coprime to  $ n $ . For example,  $ S(5) = 1^2 + 2^2 + 3^2 + 4^2 $ , but  $ S(4) = 1^2 + 3^2 $ . Let  $ p = 2^7 - 1 = 127 $  and  $ q = 2^5 - 1 = 31 $  be primes. The quantity  $ S(pq) $  can be written in the form   $$  \frac{p^2q^2}{6}\left(a - \frac{b}{c} \right)  $$   where  $ a $ ,  $ b $ , and  $ c $  are positive integers, with  $ b $  and  $ c $  coprime and  $ b < c $ . Find  $ a $ .


When a function  $f(x)$  is differentiated  $n$  times ,the function we get id denoted  $f^n(x)$ .If  $f(x)=\dfrac {e^x}{x}$ .Find the value of
\[\lim_{n \to \infty} \dfrac {f^ {2n}(1)}{(2n)!}\]

Solution: \boxed{ 1 }


Determine the smallest integer  $k > 1$  such that there exist  $k$  distinct primes whose squares sum to a power of  $2$ .

Solution: \boxed{ k = 5 }


Let  $S$  be the sum of the base 10 logarithms of all the proper divisors of 1000000. What is the integer nearest to  $S$ ?


Two players play a game on a pile of  $n$  beans. On each player's turn, they may take exactly  $1$ ,  $4$ , or  $7$  beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many  $n$  between  $2014$  and  $2050$  (inclusive) does the second player win?


**Q11.** Let be given a sequense  $a_1=5, \; a_2=8$  and  $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$  Calculate the greatest common divisor of  $a_{2011}$  and  $a_{2012}$ .


It is given that there exists a unique triple of positive primes  $(p,q,r)$  such that  $p<q<r$  and \[\dfrac{p^3+q^3+r^3}{p+q+r} = 249.\] Find  $r$ .

Solution: \boxed{ r = 19 }


Let  $p$  prime and  $m$  a positive integer. Determine all pairs  $( p,m)$  satisfying the equation:  $ p(p+m)+p=(m+1)^3$

Solution: \boxed{(2, 1)}


$p(x)$  is the cubic  $x^3 - 3x^2 + 5x$ . If  $h$  is a real root of  $p(x) = 1$  and  $k$  is a real root of  $p(x) = 5$ , find  $h + k$ .

Solution: \boxed{ h + k = 2 }


A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass  $a$  and  $b$  and entangles them; this process destroys the omon with mass  $a$ , preserves the one with mass  $b$ , and creates a new omon whose mass is  $\frac 12 (a+b)$ . The physicist can then repeat the process with the two resulting omons, choosing which omon to destroy at every step. The physicist initially has two omons whose masses are distinct positive integers less than  $1000$ . What is the maximum possible number of times he can use his machine without producing an omon whose mass is not an integer?

*Proposed by Michael Kural*


A mustache is created by taking the set of points  $(x, y)$  in the  $xy$ -coordinate plane that satisfy  $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$  and  $-24 \le x \le 24$ .  What is the area of the mustache?


Let  $\{a_n\}_{n=1}^{\infty}$  and  $\{b_n\}_{n=1}^{\infty}$  be sequences of integers such that  $a_1 = 20$ ,  $b_1 = 15$ , and for  $n \ge 1$ , \[\left\{\begin{aligned} a_{n+1}&=a_n^2-b_n^2,  b_{n+1}&=2a_nb_n-b_n^2 \end{aligned}\right.\] Let  $G = a_{10}^2-a_{10}b_{10}+b_{10}^2$ . Determine the number of positive integer factors of  $G$ .

*Proposed by Michael Ren*


Let  $ n\geq 3 $  be an integer and let  $ x_1,x_2,\ldots,x_{n-1} $  be nonnegative integers such that

\begin{eqnarray*} \ x_1 + x_2 + \cdots + x_{n-1} &=& n  x_1 + 2x_2 + \cdots + (n-1)x_{n-1} &=& 2n-2. \end{eqnarray*}

Find the minimal value of  $ F(x_1,x_2,\ldots,x_n) = \sum_{k=1}^{n-1} k(2n-k)x_k $ .


Let  $a_1, a_2, a_3, a_4$  be integers with distinct absolute values. In the coordinate plane, let  $A_1=(a_1,a_1^2)$ ,  $A_2=(a_2,a_2^2)$ ,  $A_3=(a_3,a_3^2)$  and  $A_4=(a_4,a_4^2)$ . Assume that lines  $A_1A_2$  and  $A_3A_4$  intersect on the  $y$ -axis at an acute angle of  $\theta$ . The maximum possible value for  $\tan \theta$  can be expressed in the form  $\dfrac mn$  for relatively prime positive integers  $m$  and  $n$ . Find  $100m+n$ .

*Proposed by James Lin*


Consider the sequence  $$ 1,7,8,49,50,56,57,343\ldots $$  which consists of sums of distinct powers of $7$ , that is,  $7^0$ ,  $7^1$ ,  $7^0+7^1$ ,  $7^2$ , $\ldots$  in increasing order. At what position will  $16856$  occur in this sequence?


Compute the maximum integer value of  $k$  such that  $2^k$  divides  $3^{2n+3}+40n-27$  for any positive integer  $n$ .


Find the greatest positive real number  $M$  such that for all positive real sequence  $(a_n)$  and for all real number  $m < M$ , it is possible to find some index  $n \ge 1$  that satisfies the inequality  $a_1 + a_2 + a_3 + ...+ a_n +a_{n+1} > m a_n$ .

Solution: \boxed{ M = 4 }


Find the number of pairs of positive integers  $a$  and  $b$  such that   $a\leq 100\,000$ ,  $b\leq 100\,000$ , and  $$ \frac{a^3-b}{a^3+b}=\frac{b^2-a^2}{b^2+a^2}. $$


Suppose that  $m$  and  $n$  are relatively prime positive integers with  $A = \tfrac mn$ , where
\[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find  $m$ . In other words, find the numerator of  $A$  when  $A$  is written as a fraction in simplest form.

*Proposed by Evan Chen*


Each person stands on a whole number on the number line from  $0$  to  $2022$   . In each turn, two people are selected by a distance of at least  $2$ . These go towards each other by  $1$ . When no more such moves are possible, the process ends.
Show that this process always ends after a finite number of moves, and determine all possible configurations where people can end up standing. (whereby is for each configuration is only of interest how many people stand at each number.)

*(Birgit Vera Schmidt)*

original wordingBei jeder ganzen Zahl auf dem Zahlenstrahl von 0 bis 2022 steht zu Beginn eine Person.
In jedem Zug werden zwei Personen mit Abstand mindestens 2 ausgewählt. Diese gehen jeweils um 1 aufeinander zu. Wenn kein solcher Zug mehr möglich ist, endet der Vorgang.
Man zeige, dass dieser Vorgang immer nach endlich vielen Zügen endet, und bestimme alle möglichen Konfigurationen, wo die Personen am Ende stehen können. (Dabei ist für jede Konfiguration nur von Interesse, wie viele Personen bei jeder Zahl stehen.)

Solution: \boxed{ 1011 }


A sequence  $a_1, a_2, \ldots$  satisfies  $a_1 = \dfrac 52$  and  $a_{n + 1} = {a_n}^2 - 2$  for all  $n \ge 1.$  Let  $M$  be the integer which is closest to  $a_{2023}.$  The last digit of  $M$  equals  $$ \mathrm a. ~ 0\qquad \mathrm b.~2\qquad \mathrm c. ~4 \qquad \mathrm d. ~6 \qquad \mathrm e. ~8 $$


Which number is greater:  $$ A=\frac{2.00\ldots04}{1.00\ldots04^2+2.00\ldots04},\text{ or }B=\frac{2.00\ldots02}{1.00\ldots02^2+2.00\ldots02}, $$  where each of the numbers above contains  $1998$  zeros?

Solution: \boxed{ B }


The target below is made up of concentric circles with diameters  $4$ ,  $8$ ,  $12$ ,  $16$ , and  $20$ . The area of the dark region is  $n\pi$ . Find  $n$ .

[asy]
size(150);
defaultpen(linewidth(0.8));
int i;
for(i=5;i>=1;i=i-1)
{
if (floor(i/2)==i/2)
{
filldraw(circle(origin,4*i),white);
}
else
{
filldraw(circle(origin,4*i),red);
}
}
[/asy]


In circle  $\Omega$ , let  $\overline{AB}=65$  be the diameter and let points  $C$  and  $D$  lie on the same side of arc  $\overarc{AB}$  such that  $CD=16$ , with  $C$  closer to  $B$  and  $D$  closer to  $A$ . Moreover, let  $AD, BC, AC,$  and  $BD$  all have integer lengths. Two other circles, circles  $\omega_1$  and  $\omega_2$ , have  $\overline{AC}$  and  $\overline{BD}$  as their diameters, respectively. Let circle  $\omega_1$  intersect  $AB$  at a point  $E \neq A$  and let circle  $\omega_2$  intersect  $AB$  at a point  $F \neq B$ . Then  $EF=\frac{m}{n}$ , for relatively prime integers  $m$  and  $n$ . Find  $m+n$ .
[asy]
size(7cm);
pair A=(0,0), B=(65,0), C=(117/5,156/5), D=(125/13,300/13), E=(23.4,0), F=(9.615,0);
draw(A--B--C--D--cycle);
draw(A--C);
draw(B--D);
dot(" $A$ ", A, SW);
dot(" $B$ ", B, SE);
dot(" $C$ ", C, NE);
dot(" $D$ ", D, NW);
dot(" $E$ ", E, S);
dot(" $F$ ", F, S);
draw(circle((A + C)/2, abs(A - C)/2));
draw(circle((B + D)/2, abs(B - D)/2));
draw(circle((A + B)/2, abs(A - B)/2));
label(" $\mathcal P$ ", (A + B)/2 + abs(A - B)/2 * dir(-45), dir(-45));
label(" $\mathcal Q$ ", (A + C)/2 + abs(A - C)/2 * dir(-210), dir(-210));
label(" $\mathcal R$ ", (B + D)/2 + abs(B - D)/2 * dir(70), dir(70));
[/asy]
*Proposed by **AOPS12142015***


Let  $ABCD$  be a square, and let  $M$  be the midpoint of side  $BC$ . Points  $P$  and  $Q$  lie on segment  $AM$  such that  $\angle BPD=\angle BQD=135^\circ$ . Given that  $AP<AQ$ , compute  $\tfrac{AQ}{AP}$ .


Starting from 37, adding 5 before each previous term, forms the following sequence:
\[37,537,5537,55537,555537,...\]
How many prime numbers are there in this sequence?

Solution: \boxed{ 1 }


Let  $f(t)$  be the cubic polynomial for  $t$  such that  $\cos 3x=f(\cos x)$  holds for all real number  $x$ .
Evaluate

\[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]


Ana has  $22$  coins. She can take from her friends either  $6$  coins or  $18$  coins, or she can give  $12$  coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.


Find the least  $k$  for which the number  $2010$  can be expressed as the sum of the squares of  $k$  integers.


Let  $x_0,x_1,x_2,\dots$  be the sequence such that  $x_0=1$  and for  $n\ge 0,$ \[x_{n+1}=\ln(e^{x_n}-x_n)\]
(as usual, the function  $\ln$  is the natural logarithm). Show that the infinite series
\[x_0+x_1+x_2+\cdots\]
converges and find its sum.

Solution: \boxed{e - 1}


 $n$  balls are placed independently uniformly at random into  $n$  boxes. One box is selected at random, and is found to contain  $b$  balls. Let  $e_n$  be the expected value of  $b^4$ . Find   $$ \lim_{n \to
\infty}e_n. $$


To any triangle with side lengths  $a,b,c$  and the corresponding angles  $\alpha, \beta, \gamma$  (measured in radians), the 6-tuple  $(a,b,c,\alpha, \beta, \gamma)$  is assigned. Find the minimum possible number  $n$  of distinct terms in the 6-tuple assigned to a scalene triangle.


Rectangle  $ABCD$  has  $AB = 8$  and  $BC = 13$ . Points  $P_1$  and  $P_2$  lie on  $AB$  and  $CD$  with  $P_1P_2 \parallel BC$ . Points  $Q_1$  and  $Q_2$  lie on  $BC$  and  $DA$  with  $Q_1Q_2 \parallel AB$ . Find the area of quadrilateral  $P_1Q_1P_2Q_2$ .


Points  $C\neq D$  lie on the same side of line  $AB$  so that  $\triangle ABC$  and  $\triangle BAD$  are congruent with  $AB = 9$ ,  $BC=AD=10$ , and  $CA=DB=17$ .  The intersection of these two triangular regions has area  $\tfrac mn$ , where  $m$  and  $n$  are relatively prime positive integers.  Find  $m+n$ .


A thousand integer divisions are made:  $2018$  is divided by each of the integers from  $ 1$  to  $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger?


How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.


Define an ordered triple  $ (A, B, C)$  of sets to be minimally intersecting if  $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$  and  $ A \cap B \cap C \equal{} \emptyset$ . For example,  $ (\{1,2\},\{2,3\},\{1,3,4\})$  is a minimally intersecting triple. Let  $ N$  be the number of minimally intersecting ordered triples of sets for which each set is a subset of  $ \{1,2,3,4,5,6,7\}$ . Find the remainder when  $ N$  is divided by  $ 1000$ .

**Note**:  $ |S|$  represents the number of elements in the set  $ S$ .


Black and white checkers are placed on an  $8 \times 8$  chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?


A triangle is composed of circular cells arranged in  $5784$  rows: the first row has one cell, the second has two cells, and so on (see the picture). The cells are divided into pairs of adjacent cells (circles touching each other), so that each cell belongs to exactly one pair. A pair of adjacent cells is called **diagonal** if the two cells in it *aren't* in the same row. What is the minimum possible amount of diagonal pairs in the division?
An example division into pairs is depicted in the image.


A gambler plays the following coin-tossing game. He can bet an arbitrary positive amount of money. Then a fair coin is tossed, and the gambler wins or loses the amount he bet depending on the outcome. Our gambler, who starts playing with   $ x$  forints, where  $ 0<x<2C$ , uses the following strategy: if at a given time his capital is  $ y<C$ , he risks all of it; and if he has  $ y>C$ , he only bets  $ 2C\minus{}y$ . If he has exactly  $ 2C$  forints, he stops playing. Let  $ f(x)$  be the probability that he reaches  $ 2C$  (before going bankrupt). Determine the value of  $ f(x)$ .


In the real axis, there is bug standing at coordinate  $x=1$ . Each step, from the position  $x=a$ , the bug can jump to either  $x=a+2$  or  $x=\frac{a}{2}$ . Show that there are precisely  $F_{n+4}-(n+4)$  positions (including the initial position) that the bug can jump to by at most  $n$  steps.

Recall that  $F_n$  is the  $n^{th}$  element of the Fibonacci sequence, defined by  $F_0=F_1=1$ ,  $F_{n+1}=F_n+F_{n-1}$  for all  $n\geq 1$ .

Solution: \boxed{ F_{n+4} - (n+4) }


Let  $ABCDEF$  be a convex hexagon of area  $1$ , whose opposite sides are parallel. The lines  $AB$ ,  $CD$  and  $EF$  meet in pairs to determine the vertices of a triangle. Similarly, the lines  $BC$ ,  $DE$  and  $FA$  meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least  $3/2$ .

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


Given that
\begin{eqnarray*}&(1)& \text{x and y are both integers between 100 and 999, inclusive;}\qquad \qquad \qquad \qquad \qquad  &(2)& \text{y is the number formed by reversing the digits of x; and} &(3)& z=|x-y|. \end{eqnarray*}How many distinct values of  $z$  are possible?


Let  $n,k$  be positive integers such that  $n>k$ . There is a square-shaped plot of land, which is divided into  $n\times n$  grid so that each cell has the same size. The land needs to be plowed by  $k$  tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.


Daniel writes over a board, from top to down, a list of positive integer numbers less or equal to 10. Next to each number of Daniel's list, Martin writes the number of times exists this number into the Daniel's list making a list with the same length. If we read the Martin's list from down to top, we get the same
list of numbers that Daniel wrote from top to down. Find the greatest length of the Daniel's list can have.


Define a list of number with the following properties:
- The first number of the list is a one-digit natural number.
- Each number (since the second) is obtained by adding  $9$  to the number before in the list.
- The number  $2012$  is in that list.
Find the first number of the list.


A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid  $C$  and a frustum-shaped solid  $F$ , in such a way that the ratio between the areas of the painted surfaces of  $C$  and  $F$  and the ratio between the volumes of  $C$  and  $F$  are both equal to  $k$ . Given that  $k=m/n$ , where  $m$  and  $n$  are relatively prime positive integers, find  $m+n$ .


Bob chooses a  $4$ -digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If  $N$  is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of  $N$ . For example, if Bob’s string is  $0000$  and the stream of bits begins  $101000001 \dots$ , then  $N = 7$ .


Let be given an integer  $n\ge 2$  and a positive real number  $p$ . Find the maximum of
\[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\]
where  $x_i$  are non-negative real numbers with sum  $p$ .


How many values of  $x\in\left[ 1,3 \right]$  are there, for which  $x^2$  has the same decimal part as  $x$ ?


Let  $p=4k+1$  be a prime.  $S$  is a set of all possible residues equal or smaller then  $2k$  when  $\frac{1}{2} \binom{2k}{k} n^k$  is divided by  $p$ . Show that \[ \sum_{x \in S} x^2 =p \]

Solution: \boxed{p}


**i.)** Let  $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$  What is the remainder when the polynomial  $g(x^{12}$  is divided by the polynomial  $g(x)$ ?

**ii.)** If  $k$  is a positive number and  $f$  is a function such that, for every positive number   $x, f(x^2 + 1 )^{\sqrt{x}} = k.$  Find the value of

\[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number  $y.$

**iii.)** The function  $f$  satisfies the functional equation  $f(x) + f(y) = f(x+y) - x \cdot y - 1$  for every pair  $x,y$  of real numbers. If  $f(1) = 1,$  then find the numbers of integers  $n,$  for which  $f(n) = n.$


There's a real number written on every field of a  $n \times n$  chess board. The sum of all numbers of a "cross" (union of a line and a column) is  $\geq a$ . What's the smallest possible sum of all numbers on the board¿

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


Let  $m = 30030$  and let  $M$  be the set of its positive divisors which have exactly  $2$  prime factors. Determine the smallest positive integer  $n$  with the following property: for any choice of  $n$  numbers from  $M$ , there exist 3 numbers  $a$ ,  $b$ ,  $c$  among them satisfying  $abc=m$ .


Let  $R$  be the region in the Cartesian plane of points  $(x,y)$  satisfying  $x\geq 0$ ,  $y\geq 0$ , and  $x+y+\lfloor x\rfloor+\lfloor y\rfloor\leq 5$ .  Determine the area of  $R$ .


How many binary strings of length  $10$  do not contain the substrings  $101$  or  $010$ ?


In a chess festival that is held in a school with  $2017$  students, each pair of students played at most one match versus each other. In the end, it is seen that for any pair of students which have played a match versus each other, at least one of them has played at most  $22$  matches. What is the maximum possible number of matches in this event?


Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?


We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?


Find the greatest possible distance between any two points inside or along the perimeter of an equilateral triangle with side length  $2$ .

*Proposed by Alex Li*


Let  $ABC$  be a triangle with  $AB=13$ ,  $BC=14$ , and  $CA=15$ . Points  $P$ ,  $Q$ , and  $R$  are chosen on segments  $BC$ ,  $CA$ , and  $AB$ , respectively, such that triangles  $AQR$ ,  $BPR$ ,  $CPQ$  have the same perimeter, which is  $\frac{4}{5}$  of the perimeter of  $PQR$ . What is the perimeter of  $PQR$ ?

*2021 CCA Math Bonanza Individual Round #2*


For how many integers  $n$  with  $1 \le n \le 2012$  is the product
\[
  \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right)
\]
equal to zero?


Find  $3x^2 y^2$  if  $x$  and  $y$  are integers such that  $y^2 + 3x^2 y^2 = 30x^2 + 517$ .


Kira has  $3$  blocks with the letter  $A$ ,  $3$  blocks with the letter  $B$ , and  $3$  blocks with the letter  $C$ . She puts these  $9$  blocks in a sequence. She wants to have as many distinct distances between blocks with the same letter as possible. For example, in the sequence  $ABCAABCBC$  the blocks with the letter A have distances  $1, 3$ , and  $4$  between one another, the blocks with the letter  $B$  have distances  $2, 4$ , and  $6$  between one another, and the blocks with the letter  $C$  have distances  $2, 4$ , and  $6$  between one another. Altogether, we got distances of  $1, 2, 3, 4$ , and  $6$ ; these are  $5$  distinct distances. What is the maximum number of distinct distances that can occur?


Let  $S$  be a set of  $n$  distinct real numbers. Let  $A_{S}$  be the set of numbers that occur as averages of two distinct
elements of  $S$ . For a given  $n \geq 2$ , what is the smallest possible number of elements in  $A_{S}$ ?

Solution: \boxed{ 2n-3 }


The function  $f(x)=x^5-20x^4+ax^3+bx^2+cx+24$  has the interesting property that its roots can be arranged to form an arithmetic sequence. Determine  $f(8)$ .


A positive integer is said to be "nefelibata" if, upon taking its last digit and placing it as the first digit, keeping the order of all the remaining digits intact (for example, 312 -> 231), the resulting number is exactly double the original number. Find the smallest possible nefelibata number.


Let  $ABF$  be a right-angled triangle with  $\angle AFB = 90$ , a square  $ABCD$  is externally to the triangle. If  $FA = 6$ ,  $FB = 8$  and  $E$  is the circumcenter of the square  $ABCD$ , determine the value of  $EF$

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


The integers  $1, 2, \dots, n$  are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers.  The averages of the numbers on the five slips are  $1234$ ,  $345$ ,  $128$ ,  $19$ , and  $9.5$  in some order.  Compute  $n$ .

*Proposed by Evan Chen*


A  $39$ -tuple of real numbers  $(x_1,x_2,\ldots x_{39})$  satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of  $\cos(x_1)$  and the maximum of  $\sin(x_1)$  over all tuples  $(x_1,x_2,\ldots x_{39})$  satisfying the condition is  $\tfrac ab$  for coprime positive integers  $a$ ,  $b$  (these maxima aren't necessarily achieved using the same tuple of real numbers). Find  $a + b$ .

*Proposed by Evan Chang*


Marisa has a collection of  $2^8-1=255$  distinct nonempty subsets of  $\{1, 2, 3, 4, 5, 6, 7, 8\}$ . For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process  $2^8-2=254$  times until there is only one set left in the collection. What is the expected size of this set?


Let  $ABC$  be a right-angled triangle ( $\angle C = 90^\circ$ ) and  $D$  be the midpoint of an altitude from C. The reflections of the line  $AB$  about  $AD$  and  $BD$ , respectively, meet at point  $F$ . Find the ratio  $S_{ABF}:S_{ABC}$ .
Note:  $S_{\alpha}$  means the area of  $\alpha$ .


Let  $d$  be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than  $d$  degrees. What is the minimum possible value for  $d$ ?


For any positive integer  $n$ , and  $i=1,2$ , let  $f_{i}(n)$  denote the number of divisors of  $n$  of the form  $3 k+i$  (including  $1$  and  $n$  ). Define, for any positive integer  $n$ ,   $$ f(n)=f_{1}(n)-f_{2}(n) $$   Find the value of  $f\left(5^{2022}\right)$  and  $f\left(21^{2022}\right)$ .

Solution: \boxed{ f(5^{2022}) = 1 }


A right circular cone has base radius  $ r$  and height  $ h$ .  The cone lies on its side on a flat table.  As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered  at the point where the vertex touches the table.  The cone first returns to its original position on the table after making  $ 17$  complete rotations.  The value of  $ h/r$  can be written in the form  $ m\sqrt {n}$ , where  $ m$  and  $ n$  are positive integers and  $ n$  is not divisible by the square of any prime.  Find  $ m \plus{} n$ .


Let  $a$  and  $b$  be positive integers such that  $2a - 9b + 18ab = 2018$ . Find  $b - a$ .


Assume  $f:\mathbb N_0\to\mathbb N_0$  is a function such that  $f(1)>0$  and, for any nonnegative integers  $m$  and  $n$ ,  $$ f\left(m^2+n^2\right)=f(m)^2+f(n)^2. $$  (a) Calculate  $f(k)$  for  $0\le k\le12$ .
(b) Calculate  $f(n)$  for any natural number  $n$ .

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


On Binary Island, residents communicate using special paper. Each piece of paper is a  $1 \times n$  row of initially uncolored squares. To send a message, each square on the paper must either be colored either red or green. Unfortunately the paper on the island has become damaged, and each sheet of paper has  $10$  random consecutive squares each of which is randomly colored red or green.

Malmer and Weven would like to develop a scheme that allows them to send messages of length  $2016$  between one another. They would like to be able to send any message of length  $2016$ , and they want their scheme to work with perfect accuracy. What is the smallest value of  $n$  for which they can develop such a strategy?

*Note that when sending a message, one can see which 10 squares are colored and what colors they are. One also knows on which square the message begins, and on which square the message ends.*


Let  $a$  and  $b$  be positive integers such that all but  $2009$  positive integers are expressible in the form  $ma + nb$ , where  $m$  and  $n$  are nonnegative integers. If  $1776 $ is one of the numbers that is not expressible, find  $a + b$ .


A positive integer is equal to the sum of the squares of its four smallest positive divisors. What is the largest prime that divides this positive integer?


The side lengths of a scalene triangle are roots of the polynomial   $$ x^3-20x^2+131x-281.3. $$   Find the square of the area of the triangle.


Equilateral triangle  $ABC$  has side length  $6$ . Circles with centers at  $A$ ,  $B$ , and  $C$  are drawn such that their respective radii  $r_A$ ,  $r_B$ , and  $r_C$  form an arithmetic sequence with  $r_A<r_B<r_C$ . If the shortest distance between circles  $A$  and  $B$  is  $3.5$ , and the shortest distance between circles  $A$  and  $C$  is  $3$ , then what is the area of the shaded region? Express your answer in terms of pi.

[asy]
size(8cm);
draw((0,0)--(6,0)--6*dir(60)--cycle);
draw(circle((0,0),1));
draw(circle(6*dir(60),1.5));
draw(circle((6,0),2));
filldraw((0,0)--arc((0,0),1,0,60)--cycle, grey);
filldraw(6*dir(60)--arc(6*dir(60),1.5,240,300)--cycle, grey);
filldraw((6,0)--arc((6,0),2,120,180)--cycle, grey);
label(" $A$ ",(0,0),SW);
label(" $B$ ",6*dir(60),N);
label(" $C$ ",(6,0),SE);
[/asy]


Let  $ABCD$  be a trapezoid such that  $|AC|=8$ ,  $|BD|=6$ , and  $AD \parallel BC$ . Let  $P$  and  $S$  be the midpoints of  $[AD]$  and  $[BC]$ , respectively. If  $|PS|=5$ , find the area of the trapezoid  $ABCD$ .

Solution: \boxed{ 24 }


It is known that, for all positive integers  $k,$
\[1^{2}+2^{2}+3^{2}+\cdots+k^{2}=\frac{k(k+1)(2k+1)}{6}. \]Find the smallest positive integer  $k$  such that  $1^{2}+2^{2}+3^{2}+\cdots+k^{2}$  is a multiple of  $200.$


Find the smallest positive integer  $n$  for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit  $0$  when written in base ten.

*Proposed by Evan Chen*


You have a circular necklace with  $10$  beads on it, all of which are initially unpainted. You randomly select  $5$  of these beads. For each selected bead, you paint that selected bead and the two beads immediately next to it (this means we may paint a bead multiple times). Once you have finished painting, what is the probability that every bead is painted?


Find all primes  $p,q, r$  such that  $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$ .

Titu Andreescu, Mathematics Department, College of Texas, USA

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


A set  $A$  contains exactly  $n$  integers, each of which is greater than  $1$  and every of their prime factors is less than  $10$ . Determine the smallest  $n$  such that  $A$  must contain at least two distinct elements  $a$  and  $b$  such that  $ab$  is the square of an integer.


Toner Drum and Celery Hilton are both running for president. A total of  $129$  million people cast their vote in a random order, with exactly  $63$  million and  $66$  million voting for Toner Drum and Celery Hilton, respectively. The Combinatorial News Network displays the face of the leading candidate on the front page of their website. If the two candidates are tied, both faces are displayed. What is the probability that Toner Drum's face is never displayed on the front page?

*2017 CCA Math Bonanza Individual Round #13*


Let  $A$  be a set containing  $4k$  consecutive positive integers, where  $k \geq 1$  is an
integer. Find the smallest  $k$  for which the set A can be partitioned into two subsets
having the same number of elements, the same sum of elements, the same sum
of the squares of elements, and the same sum of the cubes of elements.

Solution: \boxed{ k = 4 }


Let  $n$  be a positive integer.  Let  $(a, b, c)$  be a random ordered triple of nonnegative integers such that  $a + b + c = n$ , chosen uniformly at random from among all such triples.  Let  $M_n$  be the expected value (average value) of the largest of  $a$ ,  $b$ , and  $c$ .  As  $n$  approaches infinity, what value does  $\frac{M_n}{n}$  approach?


Let  $p_1,p_2,p_3,p_4$  be four distinct primes, and let  $1=d_1<d_2<\ldots<d_{16}=n$  be the divisors of  $n=p_1p_2p_3p_4$ . Determine all  $n<2001$  with the property that $d_9-d_8=22$ .

Solution: \boxed{ n = 1995 }


A rectangular box with side lengths  $1$ ,  $2$ , and  $16$  is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes.

*2020 CCA Math Bonanza Lightning Round #2.2*


For a given integer  $n\ge 2$ , let  $a_0,a_1,\ldots ,a_n$  be integers satisfying  $0=a_0<a_1<\ldots <a_n=2n-1$ . Find the smallest possible number of elements in the set  $\{ a_i+a_j \mid 0\le i \le j \le n \}$ .


Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example,  $2, 4, 8,...$ ). What is the maximal possible number of different integers on the blackboard?


Find all real values of the parameter  $a$  for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these values of  $a$ .

Solution: \boxed{ a = 33 }


Find all positive integers  $n$  such that it is possible to split the numbers from  $1$  to  $2n$  in two groups  $(a_1,a_2,..,a_n)$ ,  $(b_1,b_2,...,b_n)$  in such a way that  $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$ .

*Proposed by Alef Pineda*

Solution: \boxed{ n = 1 }


Let  $S$  be the set of integers of the form  $2^x+2^y+2^z$ , where  $x,y,z$  are pairwise distinct non-negative integers.  Determine the  $100$ th smallest element of  $S$ .


Denote by  $\mathbb{N}$  the positive integers. Let  $f:\mathbb{N} \rightarrow \mathbb{N}$  be a function such that, for any  $w,x,y,z \in \mathbb{N}$ , \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that  $f(n!) \ge n!$  for every positive integer  $n$ .

*Pakawut Jiradilok*

Solution: \boxed{ f(n!) \ge n! }


Define the function  $\xi : \mathbb Z^2 \to \mathbb Z$  by  $\xi(n,k) = 1$  when  $n \le k$  and  $\xi(n,k) = -1$  when  $n > k$ , and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of  $x_1x_2 \dots x_{1000}$  in  $P$ .
(b) Show that if  $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$  then  $P(x_1,x_2,\dots,x_{1000}) = 0$ .

*Proposed by Evan Chen*


Hello all. Post your solutions below.**Also, I think it is beneficial to everyone if you all attempt to comment on each other's solutions.**

4/1/31. A group of  $100$  friends stands in a circle. Initially, one person has  $2019$  mangos, and
no one else has mangos. The friends split the mangos according to the following rules:

• sharing: to share, a friend passes two mangos to the left and one mango to the right.
• eating: the mangos must also be eaten and enjoyed. However, no friend wants to be
selfish and eat too many mangos. Every time a person eats a mango, they must also
pass another mango to the right.

A person may only share if they have at least three mangos, and they may only eat if they
have at least two mangos. The friends continue sharing and eating, until so many mangos
have been eaten that no one is able to share or eat anymore.

Show that there are exactly eight people stuck with mangos, which can no longer be
shared or eaten.


Find the maximum number  $E$  such that the following holds: there is an edge-colored graph with 60 vertices and  $E$  edges, with each edge colored either red or blue, such that in that coloring, there is no monochromatic cycles of length 3 and no monochromatic cycles of length 5.


Let  $\phi(n,m), m \neq 1$ , be the number of positive integers less than or equal to  $n$  that are coprime with  $m.$  Clearly,  $\phi(m,m) = \phi(m)$ , where  $\phi(m)$  is Euler’s phi function. Find all integers  $m$  that satisfy the following inequality:
\[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\]
for every positive integer  $n.$

Solution: \boxed{ m }


Consider three points  $P(0,-\sqrt{2}),\ Q(0,\ \sqrt{2}),\ A(a,\ \sqrt{a^2+1})\ (0\leq a\leq 1)$ .

(1) Show that the difference of two line segments  $PA-AQ$  is constant regardless of  $a$ , then find the value.

(2) Let  $B$  be the point of intersection between the half-line passing through  $A$  with the end point  $Q$  and the parabola  $y=\frac{\sqrt{2}}{8}x^2$ , and let  $C$  be the point of intersection between the perpendicular line drawn from the point  $B$  to the line  $y=2$ .
Show that the sum of the line segments  $PA+AB+BC$  is constant regardless of  $a$ , then find the value.

Solution: \boxed{ 4 + \sqrt{2} }


Vertices  $A, B, C$  of a equilateral triangle of side  $1$  are in the surface of a sphere with radius  $1$  and center  $O$ . Let  $D$  be the orthogonal projection of  $A$  on the plane  $\alpha$  determined by points  $B, C, O$ . Let  $N$  be one of the intersections of the line perpendicular to  $\alpha$  passing through  $O$  with the sphere. Find the angle  $\angle DNO$ .


Find the smallest positive integer  $n$  such that a cube with sides of length  $n$  can be divided up into exactly  $2007$  smaller cubes, each of whose sides is of integer length.

Solution: \boxed{ n = 13 }


Put  $1,2,....,2018$  (2018 numbers) in a row randomly and call this number  $A$ . Find the remainder of  $A$  divided by  $3$ .


Let  $(a_n)_{n\ge 1}$  be a sequence such that  $a_n > 1$  and  $a_{n+1}^2 \ge a_n a_{n + 2}$ , for any  $n\ge 1$ . Show that the sequence  $(x_n)_{n\ge 1}$  given by  $x_n = \log_{a_n} a_{n + 1}$  for  $n\ge 1$  is convergent and compute its limit.


Find the maximum value of real number  $k$  such that
\[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\]
holds for all non-negative real numbers  $a,\ b,\ c$  satisfying  $a+b+c=1$ .

Solution: \boxed{ k = 4 }


An integer is between  $0$  and  $999999$  (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that the sum will be  $19$ ?


 $\triangle ABC$  has area  $240$ . Points  $X, Y, Z$  lie on sides  $AB$ ,  $BC$ , and  $CA$ , respectively. Given that  $\frac{AX}{BX} = 3$ ,  $\frac{BY}{CY} = 4$ , and  $\frac{CZ}{AZ} = 5$ , find the area of  $\triangle XYZ$ .

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pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label(" $A$ ",A,N);
label(" $B$ ",B,S);
label(" $C$ ",C,E);
label(" $X$ ",X,W);
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label(" $Z$ ",Z,NE);[/asy]


There are \(100\) countries participating in an olympiad. Suppose \(n\) is a positive integers such that each of the \(100\) countries is willing to communicate in exactly \(n\) languages. If each set of \(20\) countries can communicate in exactly one common language, and no language is common to all \(100\) countries, what is the minimum possible value of \(n\)?


Let  $ABCD$  be a convex quadrilateral with positive integer side lengths,  $\angle{A} = \angle{B} = 120^{\circ}, |AD - BC| = 42,$  and  $CD = 98$ . Find the maximum possible value of  $AB$ .


A  $ 20\times20\times20$  block is cut up into 8000 non-overlapping unit cubes and a number is assigned to each. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three  $ 1\times20\times20$  slices parallel to the faces of the block contain this unit cube. Find the sume of all numbers of the cubes outside these slices.


Let  $B = (20, 14)$  and  $C = (18, 0)$  be two points in the plane. For every line  $\ell$  passing through  $B$ , we color red the foot of the perpendicular from  $C$  to  $\ell$ .  The set of red points enclose a bounded region of area  $\mathcal{A}$ . Find  $\lfloor \mathcal{A} \rfloor$  (that is, find the greatest integer not exceeding  $\mathcal A$ ).

*Proposed by Yang Liu*


Let  $a < b < c < d < e$  be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that  $e$  can take.


Let  $ \overline{AB}$  be a diameter of circle  $ \omega$ . Extend  $ \overline{AB}$  through  $ A$  to  $ C$ . Point  $ T$  lies on  $ \omega$  so that line  $ CT$  is tangent to  $ \omega$ . Point  $ P$  is the foot of the perpendicular from  $ A$  to line  $ CT$ . Suppose  $ AB \equal{} 18$ , and let  $ m$  denote the maximum possible length of segment  $ BP$ . Find  $ m^{2}$ .


Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.


In a room, each person is an painter and/or a musician.  $2$  percent of the painters are musicians, and  $5$  percent of the musicians are painters. Only one person is both an painter and a musician. How many people are in the room?

*Proposed by Evan Chang*


Let  $A$  be a subset of  $\{1, 2, \dots , 1000000\}$  such that for any  $x, y \in A$  with  $x\neq y$ , we have  $xy\notin A$ . Determine the maximum possible size of  $A$ .


Consider all  $6$ -digit numbers of the form  $abccba$  where  $b$  is odd. Determine the number of all such  $6$ -digit numbers that are divisible by  $7$ .


The sum of the (decimal) digits of a natural number  $n$  equals  $100$ , and the sum of digits of  $44n$  equals  $800$ .  Determine the sum of digits of  $3n$ .


$a$ , $b$ , $c$  are real. What is the highest value of  $a+b+c$  if  $a^2+4b^2+9c^2-2a-12b+6c+2=0$


Define  $p(n)$  to be th product of all non-zero digits of  $n$ . For instance  $p(5)=5$ ,  $p(27)=14$ ,  $p(101)=1$  and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\]


For all positive integers  $n$ , let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute  $f(2019) - f(2018)$ .  Here  $\varphi(n)$  denotes the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ .


Find all positive integers that can be represented as   $$ \frac{abc+ab+a}{abc+bc+c} $$  for some positive integers  $a, b, c$ .

*Proposed by Oleksii Masalitin*

Solution: \boxed{1 \text{ and } 2}


Let  $P(x)=x^3+ax^2+bx+c$  be a polynomial where  $a,b,c$  are integers and  $c$   is odd. Let  $p_{i}$  be the value of  $P(x)$  at  $x=i$ . Given that  $p_{1}^3+p_{2}^{3}+p_{3}^{3}=3p_{1}p_{2}p_{3}$ , find the value of  $p_{2}+2p_{1}-3p_{0}.$