In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. [i]Proposed by Sammy Charney[/i] [b]Due to the problem having infinitely many solutions, all teams who inputted answers received points.[/b]

A convex polygon $P$ lies on a flat wooden table. You are allowed to drive some nails into the table. The nails must not go through $P$, but they may touch its boundary. We say that a set of nails blocks $P$ if the nails make it impossible to move $P$ without lifting it off the table. What is the minimum number of nails that suffices to block any convex polygon $P$? (N. Beluhov, S. Gerdgikov)

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 $E$

On a board there are $n\geq2$ distinct nonnegative integers such that the sum of each two distinct numbers is a power of $2$. What are the possible values of $n$?

Let $a_1(x), a_2(x)$, and $a_3(x)$ be three polynomials with integer coefficients such that every polynomial with integer coefficients can be written in the form $p_1(x)a_1(x) + p_2(x)a_2(x) + p_3(x)a_3(x)$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients. Show that every polynomial is of the form $p_1(x)a_1(x)^2 + p_2(x)a_2(x)^2 + p_3(x)a_3(x)^2$ for some polynomials $p_1(x), p_2(x), p_3(x)$ with integer coefficients.

In the beginning, there is a circle with three points on it. The points are colored (clockwise): Green, blue, red. Jonathan may perform the following actions, as many times as he wants, in any order: [list] [*] Choose two adjacent points with [u]different[/u] colors, and add a point between them with one of the two colors only. [*] Choose two adjacent points with [u]the same[/u] color, and add a point between them with any of the three colors. [*] Choose three adjacent points, at least two of them having the same color, and delete the middle point. [/list] Can Jonathan reach a state where only three points remain on the circle, colored (clockwise): Blue, green, red?

The sequence of natural numbers $1, 5, 6, 25, 26, 30, 31,...$ is made up of powers of $5$ with natural exponents or sums of powers of $5$ with different natural exponents, written in ascending order. Determine the term of the string written in position $167$.

Let $a$ be a positive integer. The sequence $\{x_n\}_{n\geq 1}$ is defined by $x_1=1$, $x_2=a$ and $x_{n+2} = ax_{n+1} + x_n$ for all $n\geq 1$. Prove that $(y,x)$ is a solution of the equation \[ |y^2 - axy - x^2 | = 1 \] if and only if there exists a rank $k$ such that $(y,x)=(x_{k+1},x_k)$. [i]Serban Buzeteanu[/i]

For any reals $x, y$, show the following inequality: $$\sqrt{(x+4)^2 + (y+2)^2} + \sqrt{(x-5)^2 + (y+4)^2} \le \sqrt{(x-2)^2 + (y-6)^2} + \sqrt{(x-5)^2 + (y-6)^2} + 20$$ [i](Proposed by Bogdan Rublov)[/i]

Find all real numbers$p, q$ for which the polynomial equation $P(x) = x^4 - \frac{8p^2}{q}x^3 + 4qx^2 - 3px + p^2 = 0$ has four positive roots.

LeBron James and Carmelo Anthony play a game of one-on-one basketball where the first player to $3$ points or more wins. LeBron James has a $20\%$ chance of making a $3$-point shot; Carmelo has a $10\%$ chance of making a $3$-pointer. LeBron has a $40\%$ chance of making a $2$-point shot from anywhere inside the $3$-point line (excluding dunks, which are also worth $2$ points); Carmelo has a $52\%$ chance of making a $ 2$-point shot from anywhere inside the 3-point line (excluding dunks). LeBron has a $90\%$ chance of dunking on Carmelo; Carmelo has a $95\%$ chance of dunking on LeBron. If each player has $3$ possessions to try to win, LeBron James goes first, and both players follow a rational strategy to try to win, what is the probability that Carmelo Anthony wins the game?

Let $ABCD$ be a parallelogram and $\omega$ be the circumcircle of the triangle $ABD$. Let $E ,F$ be the intersections of $\omega$ with lines $BC ,CD$ respectively . Prove that the circumcenter of the triangle $CEF$ lies on $\omega$.

Point \( H \) is the orthocenter of the acute triangle \( ABC \), and \( AD \) is its altitude. Tangents are drawn from points \( B \) and \( C \) to the circle with center \( A \) and radius \( AD \), which do not coincide with the line \( BC \). These tangents intersect at point \( P \). Prove that the radius of the incircle of \( \triangle BCP \) is equal to \( HD \). [i]Proposed by Danylo Khilko[/i]

Solve system of equation $$8(x^3+y^3+z^3)=73$$ $$2(x^2+y^2+z^2)=3(xy+yz+zx)$$ $$xyz=1$$ in set $\mathbb{R}^3$

A subset of the real numbers has the property that for any two distinct elements of it such as $x$ and $y$, we have $(x+y-1)^2 = xy+1$. What is the maximum number of elements in this set? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$

Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 \equal{} 1,$ and $ a_{n \plus{} 1} \equal{} a^4_n \minus{} a^3_n \plus{} 2a^2_n \plus{} 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$

Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]

$z$ is a complex number that $\left|2z+\frac{1}{z}\right|=1$, then the range value of $\arg(z)$ is________.

Let $ABC$ be a triangle with circumcircle $(O)$. Let $M,N$ be the midpoint of arc $AB,AC$ which does not contain $C,B$ and let $M',N'$ be the point of tangency of incircle of $\triangle ABC$ with $AB,AC$. Suppose that $X,Y$ are foot of perpendicular of $A$ to $MM',NN'$. If $I$ is the incenter of $\triangle ABC$ then prove that quadrilateral $AXIY$ is cyclic if and only if $b+c=2a$.

How many triangles formed by three vertices of a regular $17$-gon are obtuse? $\text{(A) }156\qquad\text{(B) }204\qquad\text{(C) }357\qquad\text{(D) }476\qquad\text{(E) }524$

Triangles $ABC$ and $A_1B_1C_1$ are similar and differently oriented. On the segment $AA_1$, a point $A'$ is taken such that $AA' / A_1A'= BC / B_1C_1$. We similarly construct $B'$ and $C'$. Prove that $A', B',C'$ lie on one straight line.

A polynomial $f(x)$ with real coefficients of degree greater than $1$ is given. Prove that there are infinitely many positive integers which cannot be represented in the form \[f(n+1)+f(n+2)+\cdots+f(n+k)\] where $n$ and $k$ are positive integers.

$C_1:\frac{x^2}{a^2}+y^2=1(a>0),C_2:y^2=2(x+m)$, one intersection of $C_1$ and $C_2$ is $P$, and $P$ is above the $x$-axis. [b](a)[/b] Find the range value of $m$ (express with $a$). [b](b)[/b] $O(0,0),A(-a,0)$. If $0<a<\frac{1}{2}$, find the maximum value of $S_{\triangle OAP}$.

How many $8$-digit numbers in base $4$ formed of the digits $1,2, 3$ are divisible by $3$?

Let $\phi(n)$ denote $\textit{Euler's phi function}$, the number of integers $1\leq i\leq n$ that are relatively prime to $n$. (For example, $\phi(6)=2$ and $\phi(10)=4$.) Let \[S=\sum_{d|2008}\phi(d),\] in which $d$ ranges through all positive divisors of $2008$, including $1$ and $2008$. Find the remainder when $S$ is divided by $1000$.