Find the remainder when \[ \sum_{i=2}^{63} \frac{i^{2011}-i}{i^2-1}. \] is divided by 2016. [i]Author: Alex Zhu[/i]

Find all function $f:\mathbb{N} \longrightarrow \mathbb{N}$ such that forall positive integers $x$ and $y$, $\frac{f(x+y)-f(x)}{f(y)}$ is again a positive integer not exceeding $2022^{2022}$.

Some vertices of a regular $2024$-gon are marked such that for any regural polygon, all of whose vertices are vertices of the $2024$-gon, at least one of his vertices is marked. Find the minimal possible number of marked vertices [i]A. Voidelevich[/i]

Each interior point of an equilateral triangle with sides equal to $1$ lies in one of six circles of the same radius $ r $. Prove that $ r \geq \frac {{\sqrt 3}} {{10}} $.

Let $ABC$ be a triangle with orthocenter $H$. Let $P,Q,R$ be the reflections of $H$ with respect to sides $BC,AC,AB$, respectively. Show that $H$ is incenter of $PQR$.

In an infinite grid of regular triangles, Niels and Henrik are playing a game they made up. Every other time, Niels picks a triangle and writes $\times$ in it, and every other time, Henrik picks a triangle where he writes a $o$. If one of the players gets four in a row in some direction (see figure), he wins the game. Determine whether one of the players can force a victory. [img]https://cdn.artofproblemsolving.com/attachments/6/e/5e80f60f110a81a74268fded7fd75a71e07d3a.png[/img]

Prove that the polynomial $x^{200} y^{200} +1$ cannot be represented in the form $f(x)g(y)$, where $f$ and $g$ are polynomials of only $x$ and $y$, respectively.

Positive integers $x_1,...,x_m$ (not necessarily distinct) are written on a blackboard. It is known that each of the numbers $F_1,...,F_{2018}$ can be represented as a sum of one or more of the numbers on the blackboard. What is the smallest possible value of $m$? (Here $F_1,...,F_{2018}$ are the first $2018$ Fibonacci numbers: $F_1=F_2=1, F_{k+1}=F_k+F_{k-1}$ for $k>1$.)

A triangle $ABC$ is given. Let $M$ be the midpoint of the side $AC$ of the triangle and $Z$ the image of point $B$ along the line $BM$. The circle with center $M$ and radius $MB$ intersects the lines $BA$ and $BC$ at the points $E$ and $G$ respectively. Let $H$ be the point of intersection of $EG$ with the line $AC$, and $K$ the point of intersection of $HZ$ with the line $EB$. The perpendicular from point $K$ to the line $BH$ intersects the lines $BZ$ and $BH$ at the points $L$ and $N$, respectively. If $P$ is the second point of intersection of the circumscribed circles of the triangles $KZL$ and $BLN$, prove that, the lines $BZ, KN$ and $HP$ intersect at a common point.

Let $ABC$ be the triangle with $\angle ABC = 76^o$ and $\angle ACB = 72^o$. Points $P$ and $Q$ lie on the sides $(AB)$ and $(AC)$, respectively, such that $\angle ABQ = 22^o$ and $\angle ACP = 44^o$. Find the measure of angle $\angle APQ$.

[b]8.1[/b] Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle. [img]https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png[/img] [b]8.2[/b]. Let the integers $a$ and $b$ be represented as $x^2-5y^2$, where $x$ and $y$ are integer numbers. Prove that the number $ab$ can also be presented in this form. [b]8.3[/b] Solve the equation $x(x + d)(x + 2d)(x + 3d) = a$. [b]8.4 / 9.1[/b] Let $a+b+c=1$, $m+n+p=1 $. Prove that $$-1 \le am + bn + cp \le 1 $$ [b]8.5[/b] Inscribe a triangle with the largest area in a semicircle. [b]8.6[/b] Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius. [img]https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png[/img] [b]8.7[/b] Find the circle of smallest radius that contains a given triangle. [b]8.8 / 9.2[/b] Given a polynomial $$x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,$$ which is divisible by $ x-1$. Prove that it is divisible by $x^2-1$. [b]8.9[/b] Prove that for any prime number $p$ other than $2$ and from $5$, there is a natural number $k$ such that only ones are involved in the decimal notation of the number $pk$.. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

In triangle $ABC$, the incircle touches sides $BC,CA,AB$ at $D,E,F$ respectively. Assume there exists a point $X$ on the line $EF$ such that \[\angle{XBC} = \angle{XCB} = 45^{\circ}.\] Let $M$ be the midpoint of the arc $BC$ on the circumcircle of $ABC$ not containing $A$. Prove that the line $MD$ passes through $E$ or $F$. United Kingdom

2004 computers make up a network using several cables. If for a subset $S$ in the set of all computers, there isn't a cable that connects two computers in $S$, $S$ is called independant. One lets the arbitrary independant set consists at most 50 computers, and uses the least number of cables. (1) Let $c(L)$ be the number of cables which connects the computer $L$. Prove that for two computers $A,B$, $c(A)=c(B)$ if there is a cable which connects $A$ and $B$, $|c(A)-c(B)|\leq 1$ otherwise. (2) Determine the number of used cables.

A penny is placed in the coordinate plane $\left(0,0\right)$. The penny can be moved $1$ unit to the right, $1$ unit up, or diagonally $1$ unit to the right and $1$ unit up. How many different ways are there for the penny to get to the point $\left(5,5\right)$? $\text{(A) }8\qquad\text{(B) }25\qquad\text{(C) }99\qquad\text{(D) }260\qquad\text{(E) }351$

Given two vectors $\overrightarrow a$, $\overrightarrow b$, find the range of possible values of $\|\overrightarrow a - 2 \overrightarrow b\|$ where $\|\overrightarrow v\|$ denotes the magnitude of a vector $\overrightarrow v$. [i](Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 1)[/i]

What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$? $ \textbf{(A)}\ \sqrt3 \qquad \textbf{(B)}\ 2\sqrt2 \qquad \textbf{(C)}\ \sqrt 2 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \text{None}$

Estimate the number of $1$s in the hexadecimal representation of $2020!$. If $E$ is your estimate and $A$ is the correct answer, you will receive $\max (25 - 0.5|A - E|, 0)$ points, rounded to the nearest integer.

Let $m,k,n$ be positive integers. Determine all triples $(m,k,n)$ satisfying the following equation: $3^m5^k=n^3+125$

Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$.

For each parabola $y = x^2+ px+q$ intersecting the coordinate axes in three distinct points, consider the circle passing through these points. Prove that all these circles pass through a single point, and find this point.

The fraction $ \frac{a^{\minus{}4}\minus{}b^{\minus{}4}}{a^{\minus{}2}\minus{}b^{\minus{}2}}$ is equal to: $ \textbf{(A)}\ a^{\minus{}6}\minus{}b^{\minus{}6} \qquad \textbf{(B)}\ a^{\minus{}2}\minus{}b^{\minus{}2} \qquad \textbf{(C)}\ a^{\minus{}2}\plus{}b^{\minus{}2} \\ \textbf{(D)}\ a^2\plus{}b^2 \qquad \textbf{(E)}\ a^2\minus{}b^2$

On the circle, 99 points are marked, dividing this circle into 99 equal arcs. Petya and Vasya play the game, taking turns. Petya goes first; on his first move, he paints in red or blue any marked point. Then each player can paint on his own turn, in red or blue, any uncolored marked point adjacent to the already painted one. Vasya wins, if after painting all points there is an equilateral triangle, all three vertices of which are colored in the same color. Could Petya prevent him?

A board that has some of its squares painted black is called [i]acceptable [/i] if there are no four black squares that form a $2 \times 2$ subboard. Find the largest real number $\lambda$ such that for every positive integer $n$ the following proposition holds: mercy: if an $n \times n$ board is acceptable and has fewer than $\lambda n^2$ dark squares, then an additional square black can be painted so that the board is still acceptable.

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 $k$.

On sides $BC$, $CA$, $AB$ of a triangle $ABC$ points $K$, $L$, $M$ are chosen, respectively, and a point $P$ is inside $ABC$ is chosen so that $PL\parallel BC$, $PM\parallel CA$, $PK\parallel AB$. Determine if it is possible that each of three trapezoids $AMPL$, $BKPM$, $CLPK$ has an inscribed circle.