Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be a cyclic octagon. Let $B_i$ by the intersection of $A_iA_{i+1}$ and $A_{i+3}A_{i+4}$. (Take $A_9 = A_1$, $A_{10} = A_2$, etc.) Prove that $B_1, B_2, \ldots , B_8$ lie on a conic. [i]David Yang.[/i]

Show that there exists a positive real $C$ such that for any naturals $H,N$ satisfying $H \geq 3, N \geq e^{CH}$, for any subset of $\{1,2,\ldots,N\}$ with size $\lceil \frac{CHN}{\ln N} \rceil$, one can find $H$ naturals in it such that the greatest common divisor of any two elements is the greatest common divisor of all $H$ elements.

Let $K_1$ be an open disk in the complex plane whose boundary passes through the points -1 and +1, and let $K_2$ be the mirror image of $K_1$ across the real axis. Also, let $D_1 = K_1 \cap K_2$ , and let $D_2$ be the outside of $D_1$ . Suppose that the function $u_1( z )$ is harmonic on $D_1$ and continuous on its closure, $u_2(z)$ harmonic on $D_2$ (including $\infty$) and continuous on its closure, and $u_1(z) = u_2(z)$ at the common boundary of the domains $D_1$ and $D_2$ . Prove that if $u_1( x )\geq 0$ for all $-1 < x <1$, then $u_2 ( x )\geq 0$ for all $x>1$ and $x<-1$.

Triangle $ABC$ has an incenter $I$ l its incircle touches the side $BC$ at $T$. The line through $T$ parallel to $IA$ meets the incircle again at $S$ and the tangent to the incircle at $S$ meets $AB , AC$ at points $C' , B'$ respectively. Prove that triangle $AB'C'$ is similar to triangle $ABC$.

In the pattern below, the cat (denoted as a large circle in the figures below) moves clockwise through the four squares and the mouse (denoted as a dot in the figures below) moves counterclockwise through the eight exterior segments of the four squares. [asy]defaultpen(linewidth(0.8)); size(350); path p=unitsquare; int i; for(i=0; i<5; i=i+1) { draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); } path cat=Circle((0.5,0.5), 0.3); draw(shift(0,1)*cat^^shift(4,1)*cat^^shift(7,0)*cat^^shift(9,0)*cat^^shift(12,1)*cat); dot((1.5,0)^^(5,0.5)^^(8,1.5)^^(10.5,2)^^(12.5,2)); label("1", (1,2), N); label("2", (4,2), N); label("3", (7,2), N); label("4", (10,2), N); label("5", (13,2), N); [/asy] If the pattern is continued, where would the cat and mouse be after the 247th move? $\textbf{(A)}$ [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(1,0)*cat); dot((0,0.5)); [/asy] $\textbf{(B)}$ [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(1,1)*cat); dot((0,0.5)); [/asy] $\textbf{(C)}$ [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(1,0)*cat); dot((0,1.5)); [/asy] $\textbf{(D)}$ [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(0,0)*cat); dot((0,1.5)); [/asy] $\textbf{(E)}$ [asy]defaultpen(linewidth(0.8)); size(60); path p=unitsquare; int i=0; draw(shift(3i,0)*(p^^shift(1,0)*p^^shift(0,1)*p^^shift(1,1)*p)); path cat=Circle((0.5,0.5), 0.3); draw(shift(0,1)*cat); dot((1.5,0)); [/asy]

Each lattice point with nonnegative coordinates is labeled with a nonnegative integer in such away that the point $(0, 0)$ is labeled by $0$, and for every $x, y \geq 0$, the set of numbers labeled on thepoints $(x, y), (x, y + 1),$ and $(x + 1, y)$ is $\{n, n + 1, n + 2\}$ for some nonnegative integer $n$. Determine,with proof, all possible labels for the point $(2000, 2024)$.

$ A$ is a set of points on the plane, $ L$ is a line on the same plane. If $ L$ passes through one of the points in $ A$, then we call that $ L$ passes through $ A$. (1) Prove that we can divide all the rational points into $ 100$ pairwisely non-intersecting point sets with infinity elements. If for any line on the plane, there are two rational points on it, then it passes through all the $ 100$ sets. (2) Find the biggest integer $ r$, so that if we divide all the rational points on the plane into $ 100$ pairwisely non-intersecting point sets with infinity elements with any method, then there is at least one line that passes through $ r$ sets of the $ 100$ point sets.

Let $p$ be a polynomial with integer coefficients such that $p(15)=6$, $p(22)=1196$, and $p(35)=26$. Find an integer $n$ such that $p(n)=n+82$.

For a particular peculiar pair of dice, the probabilities of rolling 1, 2, 3, 4, 5 and 6 on each die are in the ratio $ 1: 2: 3: 4: 5: 6$. What is the probability of rolling a total of 7 on the two dice? $ \textbf{(A) } \frac 4{63} \qquad \textbf{(B) } \frac 18 \qquad \textbf{(C) } \frac 8{63} \qquad \textbf{(D) } \frac 16 \qquad \textbf{(E) } \frac 27$

Given a prime number $p{}$ and integers $x{}$ and $y$, find the remainder of the sum $x^0y^{p-1}+x^1y^{p-2}+\ldots+x^{p-2}y^1+x^{p-1}y^0$ upon division by $p{}$.

Let $p(x)=x^{3}+a_{1}x^{2}+a_{2}x+a_{3}$ have rational coefficients and have roots $r_{1}$, $r_{2}$, and $r_{3}$. If $r_{1}-r_{2}$ is rational, must $r_{1}$, $r_{2}$, and $r_{3}$ be rational?

Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\tfrac nd$ where $n$ and $d$ are integers with $1\leq d\leq 5$. What is the probability that \[(\cos(a\pi)+i\sin(b\pi))^4\] is a real number? $\textbf{(A) }\dfrac3{50}\qquad\textbf{(B) }\dfrac4{25}\qquad\textbf{(C) }\dfrac{41}{200}\qquad\textbf{(D) }\dfrac6{25}\qquad\textbf{(E) }\dfrac{13}{50}$

The number selection game is the following single-player game. Originally, on the table there were positive integers $1, 2,...,22$ (All positive integers not exceeding $22$ appear exactly once). In each move, the player chooses the three numbers $a, b, c$ that are on the table, then the selected numbers $a, b, c$ disappear but a new number $a + b + c$ appears; At the same time, the player's score is added $(a + b)(b+c)(c + a)$. The initial score was $0$. The game ends after $10$ moves (when there are only two numbers left on the board). Call $M, m$ respectively the highest and the lowest possible score of a game. Determine the value of $\dfrac{M}{m}$.

Let the major axis of an ellipse be $AB$, let $O$ be its center, and let $F$ be one of its foci. $P$ is a point on the ellipse, and $CD$ a chord through $O$, such that $CD$ is parallel to the tangent of the ellipse at $P$. $PF$ and $CD$ intersect at $Q$. Compare the lengths of $PQ$ and $OA$.

For a positive real number $a$, let $C$ be the cube with vertices at $(\pm a, \pm a, \pm a)$ and let $T$ be the tetrahedron with vertices at $(2a,2a,2a),(2a, -2a, -2a),(-2a, 2a, -2a),(-2a, -2a, -2a)$. If the intersection of $T$ and $C$ has volume $ka^3$ for some $k$, find $k$.

If facilities for division are not available, it is sometimes convenient in determining the decimal expansion of $1/a$, $a>0$, to use the iteration $$x_{k+1}=x_k(2-ax_k), \quad \quad k=0,1,2,\dots ,$$ where $x_0$ is a selected “starting” value. Find the limitations, if any, on the starting values $x_0$, in order that the above iteration converges to the desired value $1/a$.

Determine all positive real numbers $a$ such that there exists a positive integer $n$ and partition $A_1$, $A_2$, ..., $A_n$ of infinity sets of the set of the integers satisfying the following condition: for every set $A_i$, the positive difference of any pair of elements in $A_i$ is at least $a^i$.

Prove that polynomial $x^8+98x^4+1$ can be factorized in $Z[X]$.

Let $p_k$ be the $k$-th prime number. Find the remainder when $\sum_{k=2}^{2550}p_k^{p_k^4-1}$ is divided by $2550$.

The target is a triangle divided by three families of parallel lines into $100$ equal regular triangles with single sides. A sniper shoots at a target. He aims at triangle and hits either it or one of the sides adjacent to it. He sees the results of his shooting and can choose when stop shooting. What is the greatest number of triangles he can with a guarantee of hitting five times?

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

Emmy goes to buy radishes at the market. Radishes are sold in bundles of $3$ for $\$5$and bundles of $5$ for $\$7$. What is the least number of dollars Emmy needs to buy exactly $100$ radishes?

On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1,\, x_2,\, x_3$. a. Prove that the following quantity is a constant: $$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$ b. Prove the following inequality: $$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$

Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list. a) Prove that we can always find a good list with 12 elements, whatever the lists are; b) Prove that students can make lists such that no shorter list is good.

Consider the solid $Q$ obtained by attaching unit cubes $Q_1...Q_6$ at the six faces of a unit cube $Q$. Prove or disprove that the space can be filled up with such solids so that no two of them have a common interior point.