A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.

Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome? [i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i]

For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine whether there exists a positive integer triple \( a, b, c \) such that there are exactly $1012$ positive integers \( K \) not greater than $2024$ that satisfies the following: the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \] holds for some positive integers $x,y,z$.

Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.

Let $p$ be an odd prime. Let $A(n)$ be the number of subsets of $\{1,2,...,n\}$ such that the sum of elements of the subset is a multiple of $p$. Prove that if $2^{p-1}-1$ is not a multiple of $p^2$, there exists infinitely many positive integer $m$ for any integer $k$ that satisfies the following. (The sum of elements of the empty set is 0.) $$\frac{A(m)-k}{p}\in\mathbb{Z}$$

Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]

Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that $74$ are on the football team; $26$ are on the baseball team; $17$ are on both the football and lacrosse teams; $18$ are on both the baseball and football teams; and $13$ are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.

Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.

Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?

Is rational or irrational,the number $$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

Let $ABC$ be a non-isosceles and acute triangle. $X$ is a point on arc $BC$ not containing $A$ such that $BA-CA = CX-BX$. The incircle of $\triangle ABC$ touches $AC$ and $AB$ at $E$ and $F$ respectively. The $X$-excircle of $\triangle XBC$ touches $XC$ and $XB$ at $Y$ and $Z$ respectively. Let $T$ be such that $TA$ and $TX$ bisects $\angle BAC$ and $\angle BXC$ respectively. Prove that $T$ lies on the radical axis of circles $(BFZ)$ and $(CEY)$. [i](Proposed by Chuah Jia Herng)[/i]

Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

Let $ABC$ be a triangle inscribed in the ellipse $\frac{x^2}{4} +\frac{y^2}{9}= 1$. If its centroid is the origin $(0,0)$, find its area.

Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.

Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$ .

Right $ \triangle ABC$ has $ AB \equal{} 3$, $ BC \equal{} 4$, and $ AC \equal{} 5$. Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$, $ W$ on $ \overline{AB}$, and $ Z$ on $ \overline{BC}$. What is the side length of the square? [asy]size(200);defaultpen(fontsize(10pt)+linewidth(.8pt)); real s = (60/37); pair A = (0,0); pair C = (5,0); pair B = dir(60)*3; pair W = waypoint(B--A,(1/3)); pair X = foot(W,A,C); pair Y = (X.x + s, X.y); pair Z = (W.x + s, W.y); label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,SE); label("$W$",W,NW); label("$X$",X,S); label("$Y$",Y,S); label("$Z$",Z,NE); draw(A--B--C--cycle); draw(X--W--Z--Y);[/asy] $ \textbf{(A)}\ \frac {3}{2}\qquad \textbf{(B)}\ \frac {60}{37}\qquad \textbf{(C)}\ \frac {12}{7}\qquad \textbf{(D)}\ \frac {23}{13}\qquad \textbf{(E)}\ 2$

Fine all positive integers $m,n\geq 2$, such that (1) $m+1$ is a prime number of type $4k-1$; (2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that \[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]

We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$

The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the triangle $ABC$ respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the triangles $DEF$, $DEA$, $DBF$, $CEF$, respectively. Prove that $$d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$$ When the equality takes place?

Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$, \[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\] find $\lim_{n \to \infty} \frac{x_n}{y_n}$.

In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.