Let the class of functions $f_n$ be defined such that $f_1(x)=|x^3-x^2|$ and $f_{k+1}(x)=|f_k(x)-x^3|$ for all $k\ge1$. Denote by $S_n$ the sum of all $y$-values of $f_n(x)$'s "sharp" points in the First Quadrant. (A "sharp" point is a point for which the derivative is not defined.) Find the ratio of odd to even terms, $$\lim_{k\to\infty}\frac{S_{2k+1}}{S_{2k}}$$

Let the decimal representations of numbers $A$ and $B$ be given as: $A = 0.a_1a_2\cdots a_k > 0$, $B = 0.b_1b_2\cdots b_k > 0$ (where $a_k, b_k$ can be 0), and let $S$ be the count of numbers $0.c_1c_2\cdots c_k$ such that $0.c_1c_2\cdots c_k < A$ and $0.c_kc_{k-1}\cdots c_1 < B$ ($c_k, c_1$ can also be 0). (Here, $0.c_1c_2\cdots c_r (c_r \neq 0)$ is considered the same as $0.c_1c_2\cdots c_r0\cdots0$). Prove: $\left| S - 10^k AB \right| \leq 9k.$

A set $S$ of unit cells of an $n\times n$ array, $n\geq 2$, is said [i]full[/i] if each row and each column of the array contain at least one element of $S$, but which has this property no more when any of its elements is removed. A full set having maximum cardinality is said [i]fat[/i], while a full set of minimum cardinality is said [i]meagre[/i]. i) Determine the cardinality $m(n)$ of the meagre sets, describe all meagre sets and give their count. ii) Determine the cardinality $M(n)$ of the fat sets, describe all fat sets and give their count. [i](Dan Schwarz)[/i]

There are 10 cards, labeled from 1 to 10. Three cards denoted by $ a,\ b,\ c\ (a > b > c)$ are drawn from the cards at the same time. Find the probability such that $ \int_0^a (x^2 \minus{} 2bx \plus{} 3c)\ dx \equal{} 0$.

Let $ ABC$ be an acute triangle. $ H$ is the orthocenter and $ M$ is the middle of the side $ BC$. A line passing through $ H$ and perpendicular to $ HM$ intersect the segment $ AB$ and $ AC$ in $ P$ and $ Q$. Prove that $ MP \equal{} MQ$

Let $ABC$ be an acute triangle, and its altitudes $AX,BY,CZ$ concurrent at $H$. Construct circles $(K_a), (K_b), (K_c)$ circumscribing the triangles $AY Z, BZX, CXY$ . Construct a circle $(K)$ that is internally tangent to all the three circles $(Ka), (K_b), (K_c)$. Prove that $(K)$ is tangent to the circumcircle $(O)$ of the triangle $ABC$. Đỗ Thanh Sơn

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

Let $ABC$ be a triangle with $AB=2\sqrt6, BC=5, CA=\sqrt{26}$, midpoint $M$ of $BC$, circumcircle $\Omega$, and orthocenter $H$. Let $BH$ intersect $AC$ at $E$ and $CH$ intersect $AB$ at $F$. Let $R$ be the midpoint of $EF$ and let $N$ be the midpoint of $AH$. Let $AR$ intersect the circumcircle of $AHM$ again at $L$. Let the circumcircle of $ANL$ intersect $\Omega$ and the circumcircle of $BNC$ at $J$ and $O$, respectively. Let circles $AHM$ and $JMO$ intersect again at $U$, and let $AU$ intersect the circumcircle of $AHC$ again at $V \neq A$. The square of the length of $CV$ can be expressed in the form $\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Michael Ren[/i]

Alan, Beth, Carlos, and Diana were discussing their possible grades in mathematics class this grading period. Alan said, "If I get an A, then Beth will get an A." Beth said, "If I get an A, then Carlos will get an A." Carlos said, "If I get an A, then Diana will get an A." All of these statements were true, but only two of the students received an A. Which two received A's? \[ \textbf{(A)} \text{Alan, Beth} \qquad \textbf{(B)} \text{Beth, Carlos} \qquad \textbf{(C)} \text{Carlos, Diana} \qquad \textbf{(D)} \text{Alan, Diana} \qquad \textbf{(E)} \text{Beth, Diana} \]

Ana plays a game on a $100\times 100$ chessboard. Initially, there is a white pawn on each square of the bottom row and a black pawn on each square of the top row, and no other pawns anywhere else.\\ Each white pawn moves toward the top row and each black pawn moves toward the bottom row in one of the following ways: [list] [*] it moves to the square directly in front of it if there is no other pawn on it; [*] it [b]captures[/b] a pawn on one of the diagonally adjacent squares in the row immediately in front of it if there is a pawn of the opposite color on it. [/list] (We say a pawn $P$ [b]captures[/b] a pawn $Q$ of the opposite color if we remove $Q$ from the board and move $P$ to the square that $Q$ was previously on.)\\ \\ Ana can move any pawn (not necessarily alternating between black and white) according to those rules. What is the smallest number of pawns that can remain on the board after no more moves can be made? [i]Proposed by José Alejandro Reyes González, Mexico[/i]

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Suppose that the circumradius $R$ and the inradius $r$ of a triangle $ABC$ satisfy $R = 2r$. Prove that the triangle is equilateral.

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

The Fibonacci numbers $F_n$ are defined as $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for all $n> 2$. Let $A$ be the minimum area of a (possibly degenerate) convex polygon with $2020$ sides, whose side lengths are the first $2020$ Fibonacci numbers $F_1$, $F_2$, $...$ , $F_{2020}$ (in any order). A degenerate convex polygon is a polygon where all angles are $\le 180^o$. If $A$ can be expressed in the form $$\frac{\sqrt{(F_a-b)^2-c}}{d}$$ , where $a, b, c$ and $d$ are positive integers, compute the minimal possible value of $a + b + c + d$.

Find all functions $f\colon R \to R$ such that \[f\left(x^{2}+yf(x)\right) = f(x)^{2}+xf(y)\] for all reals $x,y$.

Find all real numbers $x$ which satisfy $\frac{n}{3n+1}\leq x\leq \frac{4n+1}{2n-1}$, for all $n\in\mathbb{N}^*$. [i]Gheorghe Boroica[/i]

Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that $$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

The sequence $(a_{n})$ is defined by $a_1=1$ and $a_n=n(a_1+a_2+\cdots+a_{n-1})$ , $\forall n>1$. [b](a)[/b] Prove that for every even $n$, $a_{n}$ is divisible by $n!$. [b](b)[/b] Find all odd numbers $n$ for the which $a_{n}$ is divisible by $n!$.

Three lines are drawn in a plane. Which of the following could NOT be the total number of points of intersections? (A) $0$ (B) $1$ (C) $2$ (D) $3$ (E) They all could.

Sides $\overline{AB}$ and $\overline{AC}$ of equilateral triangle $ABC$ are tangent to a circle at points $B$ and $C$ respectively. What fraction of the area of $\triangle ABC$ lies outside the circle? $ \textbf{(A) }\dfrac{4\sqrt{3}\pi}{27}-\frac{1}{3}\qquad \textbf{(B) } \frac{\sqrt{3}}{2}-\frac{\pi}{8}\qquad \textbf{(C) } \frac{1}{2} \qquad \textbf{(D) }\sqrt{3}-\frac{2\sqrt{3}\pi}{9}\qquad \textbf{(E) } \frac{4}{3}-\dfrac{4\sqrt{3}\pi}{27}$

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

Let $x$ be an integer of $n$ digits, all equal to $ 1$. Show that if $x$ is prime, then $n$ is also prime.

For positive numbers $ a $, $ b $ and $ c $ satisfying the equality $ a + b + c = 1 $, prove the inequality $$ \frac {a ^ 7 + b ^ 7} {a ^ 5 + b ^ 5} + \frac {b ^ 7 + c ^ 7} {b ^ 5 + c ^ 5} + \frac {c ^ 7 + a ^ 7} {c ^ 5 + a ^ 5} \geq \frac {1} {3}. $$