(a) On a long pavement, a sequence of $999$ integers is written in chalk. The numbers need not be in increasing order and need not be distinct. Merlijn encircles $500$ of the numbers with red chalk. From left to right, the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$. Next, Jeroen encircles $500$ of the numbers with blue chalk. From left to right, the numbers circled in blue are precisely the numbers $500, 499, 498, ...,2,1$. Prove that the middle number in the sequence of $999$ numbers is circled both in red and in blue. (b) Merlijn and Jeroen cross the street and find another sequence of $999$ integers on the pavement. Again Merlijn circles $500$ of the numbers with red chalk. Again the numbers circled in red are precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Now Jeroen circles $500$ of the numbers, not necessarily the same as Merlijn, with green chalk. The numbers circled in green are also precisely the numbers $1, 2, 3, ...,499, 500$ from left to right. Prove: there is a number that is circled both in red and in green that is not the middle number of the sequence of $999$ numbers.

Find the number of ordered triples $(a,b,c)$ of integers satisfying $0\le a,b,c \le 1000$ for which \[a^3+b^3+c^3\equiv 3abc+1\pmod{1001}.\] [i]Proposed by James Lin[/i]

Alice and Bob determine a number with $2018$ digits in the decimal system by choosing digits from left to right. Alice starts and then they each choose a digit in turn. They have to observe the rule that each digit must differ from the previously chosen digit modulo $3$. Since Bob will make the last move, he bets that he can make sure that the final number is divisible by $3$. Can Alice avoid that? [i](Proposed by Richard Henner)[/i]

Let $M$ be the midpoint of the cathetus $AC$ of a right-angled triangle $ABC$ $(\angle C=90^{\circ})$. The perpendicular from $M$ to the bisector of angle $ABC$ meets $AB$ at point $N$. Prove that the circumcircle of triangle $ANM$ touches the bisector of angle $ABC$. Proposed by:D.Shvetsov

Find real $a,b$ such that polynomial $P(x)=x^{n+1}+ax+b$ to be divisible by $(x-1)^2$. Then find the quotient $P(x):(x-1)^2 , n\in \mathbb{N}^*$

Triangle $BCF$ has a right angle at $B$. Let $A$ be the point on line $CF$ such that $FA=FB$ and $F$ lies between $A$ and $C$. Point $D$ is chosen so that $DA=DC$ and $AC$ is the bisector of $\angle{DAB}$. Point $E$ is chosen so that $EA=ED$ and $AD$ is the bisector of $\angle{EAC}$. Let $M$ be the midpoint of $CF$. Let $X$ be the point such that $AMXE$ is a parallelogram. Prove that $BD,FX$ and $ME$ are concurrent.

The diagram below shows two concentric circles whose areas are $7$ and $53$ and a pair of perpendicular lines where one line contains diameters of both circles and the other is tangent to the smaller circle. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/3/b/87cbb97a799686cf5dbec9dcd79b6b03e1995c.png[/img]

In triangle $\vartriangle ABC$, $E$ and $F$ are the feet of the altitudes from $B$ to $\overline{AC}$ and $C$ to $\overline{AB}$, respectively. Line $\overleftrightarrow{BC}$ and the line through $A$ tangent to the circumcircle of $ABC$ intersect at $X$. Let $Y$ be the intersection of line $\overleftrightarrow{EF}$ and the line through $A$ parallel to $\overline{BC}$. If $XB = 4$, $BC = 8$, and $EF = 4\sqrt3$, compute $XY$.

Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ satisfying (for all $x,y \in \mathbb{R}$): $f(x+y)^2 - f(2x^2) = f(y-x)f(y+x) + 2x\cdot f(y)$.

Find all positive integers that are equal to $700$ times the sum of its digits.

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? $\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ \text{Infinity}$