Find all pairs $(n,r)$ with $n$ a positive integer and $r$ a real such that $2x^2+2x+1$ divides $(x+1)^n - r$.

Circles $o_1$ and $o_2$ tangent to some line at points $A$ and $B$, respectively, intersect at points $X$ and $Y$ ($X$ is closer to the line $AB$). Line $AX$ intersects $o_2$ at point $P\neq X$. Tangent to $o_2$ at point $P$ intersects line $AB$ at point $Q$. Prove that $\sphericalangle XYB = \sphericalangle BYQ$.

Consider the non-decreasing sequence of positive integers \[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5,... \] in which the $n^{\text{th}}$ positive integer appears $n$ times. The remainder when the $1993^{\text{rd}}$ term is divided by $5$ is $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose?

The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.)

Show that the sum $S=5+5^2+5^3+…+5^{2016}$ is divisible by $31$

The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd? $\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$

The rules for the movement of a king on a chessboard are as follows: The king can legally move to any of the (up to $8$) squares adjacent diagonally or on a side. Andrew places a king on an ordinary $8 \times 8$ chessboard. He then makes $64$ total moves with the king such that the king visits every square on the board, never crosses its own path, and winds up at its original position (where Andrew first placed it). Along the way, Andrew counts the number of times the king moves diagonally (from one square to another that shares no side). Call that number $M$. Find the maximum possible value of $M$.

For positive real numbers $a,b,c,d$ such that $$ \dfrac{a^2}{b+c+d} + \dfrac{b^2}{a+c+d} + \dfrac{c^2}{a+b+d} = \dfrac{3d^2}{a+b+c} $$ prove that $$ \dfrac{3}{a}+ \dfrac{3}{b} + \dfrac{3}{c}+ \dfrac{3}{d} \geq \dfrac{16}{a+b+2d} + \dfrac{16}{b+c+2d} + \dfrac{16}{a+c+2d}. $$ Proposed by [i]Mojtaba Zare[/i]

Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero.

[b]p1.[/b] a) There are barrels weighing $1, 2, 3, 4, ..., 19, 20$ pounds. Is it possible to distribute them equally (by weight) into three trucks? b) The same question for barrels weighing $1, 2, 3, 4, ..., 9, 10$ pounds. [b]p2.[/b] There are apples and pears in the basket. If you add the same number of apples there as there are now pears (in pieces), then the percentage of apples will be twice as large as what you get if you add as many pears to the basket as there are now apples. What percentage of apples are in the basket now? [b]p3.[/b] What is the smallest number of integers from $1000$ to $1500$ that must be marked so that any number $x$ from $1000$ to $1500$ differs from one of the marked numbers by no more than $10\% $of the value of $x$? [b]p4.[/b] Draw a perpendicular from a given point to a given straight line, having a compass and a short ruler (the length of the ruler is significantly less than the distance from the point to the straight line; the compass reaches from the point to the straight line “with a margin”). [b]p5.[/b] There is a triangle on the chessboard (left figure). It is allowed to roll it around the sides (in this case, the triangle is symmetrically reflected relative to the side around which it is rolled). Can he, after a few steps, take the position shown in right figure? [img]https://cdn.artofproblemsolving.com/attachments/f/5/eeb96c92f30b837e7ed2cdf7cf77b0fbb8ceda.png[/img] [b]p6.[/b] The natural number $a$ is less than the natural number $b$. In this case, the sum of the digits of number $a$ is $100$ less than the sum of the digits of number $b$. Prove that between the numbers $ a$ and $b$ there is a number whose sum of digits is $43$ more than the sum of the digits of $a$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer $k<5,$ no collection of $k$ pairs made by the child contains the shoes from exactly $k$ of the adults is $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $A,B,C$ be matrices with complex elements such that $[A,B]=C, [B,C]=A$ and $[C,A]=B$, where $[X,Y]$ denotes the $XY-YX$ commutator of the matrices. Prove that $e^{4 \pi A}$ is the identity matrix.

Let $a$ and $b$ be positive real numbers with $a^2 + b^2 =\frac12$. Prove that $$\frac{1}{1 - a}+\frac{1}{1-b}\ge 4.$$ When does equality hold? [i](Walther Janous)[/i]

Let a binary operation $*$ on ordered pairs of integers be defined by $(a,b)*(c,d)=(a-c,b+d)$. Then, if $(3,2)*(0,0)$ and $(x,y)*(3,2)$ represent idential pairs, $x$ equals: $\textbf{(A) }-3\qquad \textbf{(B) }0\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }6$

On a circle, $n \geq 3$ points are marked. Each marked point is coloured red, green or blue. In one step, one can erase two neighbouring marked points of different colours and mark a new point between the locations of the erased points with the third colour. In a final state, all marked points have the same colour which is called the colour of the final state. Find all $n$ for which there exists an initial state of $n$ marked points with one missing colour, from which one can reach a final state of any of the three colours by applying a suitable sequence of steps.

Let $ABCD$ be an inscribed quadrilateral. Let the circles with diameters $AB$ and $CD$ intersect at two points $X_1$ and $Y_1$, the circles with diameters $BC$ and $AD$ intersect at two points $X_2$ and $Y_2$, the circles with diameters $AC$ and $BD$ intersect at two points $X_3$ and $Y_3$. Prove that the lines $X_1Y_1, X_2Y_2$ and $X_3Y_3$ are concurrent. Maxim Didin

Prove that if the numbers $ p $, $ q $, $ r $ satisfy the equality $$ p+q + r=1$$ $$ \frac{1}{p} + \frac{1}{q} + \frac{1}{r} = 0$$ then for any numbers $ a $, $ b $, $ c $ equality holds $$a^2 + b^2 + c^2 = (pa + qb + rc)^2 + (qa + rb + pc)^2 + (ra + pb + qc)^2.$$

Let $f:[0,1]\to\mathbb R$ be a twice continuously differentiable increasing function. Define the sequences given by $L_n=\frac1n\sum_{k=0}^{n-1}f\left(\frac kn\right)$ and $U_n=\frac1n\sum_{k=0}^nf\left(\frac kn\right)$, $n\ge1$. 1. The interval $[L_n,U_n]$ is divided into three equal segments. Prove that, for large enough $n$, the number $I=\int^1_0f(x)\text dx$ belongs to the middle one of these three segments.

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]

Csaba stands in the middle of a $15$ m $\times 15$ m room at a workplace where everyone strictly adheres to $1,5$ m social distancing. At least how many people are there other than Csaba in the room if Csaba cannot reach any wall without the others moving? [i]The people are viewed as points.[/i]

Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$.

Given a circle with center $O$ and radius equal to $1$. $AB$ and $AC$ are the tangents to this circle from point $A$. Point $M$ on the circle is such that the areas of quadrilaterals $OBMC$ and $ABMC$ are equal. Find $MA$.

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

A convex quadrilateral is determined by the points of intersection of the curves $x^4+y^4=100$ and $xy=4$; determine its area.