Which of the following is the same as \[ \frac{2\minus{}4\plus{}6\minus{}8\plus{}10\minus{}12\plus{}14}{3\minus{}6\plus{}9\minus{}12\plus{}15\minus{}18\plus{}21}? \]$ \textbf{(A)}\ \minus{}1 \qquad \textbf{(B)}\ \minus{}\frac23 \qquad \textbf{(C)}\ \frac23 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ \frac{14}{3}$

Prove that there is a positive integer $M$ for which the following statement holds: For all prime numbers $p$, there is an integer $n$ for which $\sqrt{p} \le n \le M\sqrt{p}$ and $p \mod n \le \frac{n}{2020}$ . Note: Here, $p \mod n$ denotes the unique integer $r \in {0, 1, ..., n - 1}$ for which $n|p -r$. In other words, $p \mod n$ is the residue of $p$ upon division by $n$.

Prove that the product of three consecutive positive integers is never a perfect square.

[b]p1.[/b] Solve the equation $3^x + 9^x = 27^x$. [b]p2.[/b] An equilateral triangle in inscribed in a circle of area $1$ m$^2$. Then the second circle is inscribed in the triangle. Find the radius of the second circle. [b]p3.[/b] Solve the inequality: $2\sqrt{x^2 - 5x + 4} + 3\sqrt{x^2 + 2x - 3} \le 5\sqrt{6 - x - x^2}$ [b]p4.[/b] Peter and John played a game. Peter wrote on a blackboard all integers from $1$ to $18$ and offered John to choose $8$ different integers from this list. To win the game John had to choose 8 integers such that among them the difference between any two is either less than $7$ or greater than $11$. Can John win the game? Justify your answer. [b]p5.[/b] Prove that given $100$ different positive integers such that none of them is a multiple of $100$, it is always possible to choose several of them such that the last two digits of their sum are zeros. [b]p6.[/b] Given $100$ different squares such that the sum of their areas equals $1/2$ m$^2$ , is it possible to place them on a square board with area $1$ m$^2$ without overlays? Justify your answer. PS. You should use hide for answers.

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. [i]Proposed by Amine Natik, Morocco[/i]

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

Let $ A,B\in\mathcal{M} \left( \mathbb{R} \right) $ that satisfy $ AB=O_3. $ Prove that: [b]a)[/b] The function $ f:\mathbb{C}\longrightarrow\mathbb{C} $ defined as $ f(x)=\det \left( A^2+B^2+xBA \right) $ is a polynomial one, of degree at most $ 2. $ [b]b)[/b] $ \det\left( A^2+B^2 \right)\ge 0. $

Find the number of sequences with $2022$ natural numbers $n_1, n_2, n_3, \ldots, n_{2022}$, such that in every sequence: $\bullet$ $n_{i+1}\geq n_i$ $\bullet$ There is at least one number $i$, such that $n_i=2022$ $\bullet$ For every $(i, j)$ $n_1+n_2+\ldots+n_{2022}-n_i-n_j$ is divisible to both $n_i$ and $n_j$

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that for any real $x, y$ holds equality $$f(xf(y)) + f(xy) = 2f(x)y$$ [i]Proposed by Arseniy Nikolaev[/i]

I found this inequality in "Topics in Inequalities" (I 85) For all positive reals $x,y,z$ with $xyz=1$ prove: \[ \frac{x^9+y^9}{x^6+x^3y^3+y^6}+\frac{y^9+z^9}{y^6+y^3z^3+z^6}+\frac{z^9+x^9}{z^6+z^3x^3+x^6}\geq 2 \]

[u]Set 4[/u] [b]G10.[/b] Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The minimum possible area of the resulting shape is $A$. Find the integer closest to $100A$. [b]G11.[/b] The $10$-digit number $\underline{1A2B3C5D6E}$ is a multiple of $99$. Find $A + B + C + D + E$. [b]G12.[/b] Let $A, B, C, D$ be four points satisfying $AB = 10$ and $AC = BC = AD = BD = CD = 6$. If $V$ is the volume of tetrahedron $ABCD$, then find $V^2$. [u]Set 5[/u] [b]G13.[/b] Nate the giant is running a $5000$ meter long race. His first step is $4$ meters, his next step is $6$ meters, and in general, each step is $2$ meters longer than the previous one. Given that his $n$th step will get him across the finish line, find $n$. [b]G14.[/b] In square $ABCD$ with side length $2$, there exists a point $E$ such that $DA = DE$. Let line $BE$ intersect side $AD$ at $F$ such that $BE = EF$. The area of $ABE$ can be expressed in the form $a -\sqrt{b}$ where $a$ is a positive integer and $b$ is a square-free integer. Find $a + b$. [b]G15.[/b] Patrick the Beetle is located at $1$ on the number line. He then makes an infinite sequence of moves where each move is either moving $1$, $2$, or $3$ units to the right. The probability that he does reach $6$ at some point in his sequence of moves is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [u]Set 6[/u] [b]G16.[/b] Find the smallest positive integer $c$ greater than $1$ for which there do not exist integers $0 \le x, y \le9$ that satisfy $2x + 3y = c$. [b]G17.[/b] Jaeyong is on the point $(0, 0)$ on the coordinate plane. If Jaeyong is on point $(x, y)$, he can either walk to $(x + 2, y)$, $(x + 1, y + 1)$, or $(x, y + 2)$. Call a walk to $(x + 1, y + 1)$ an Brilliant walk. If Jaeyong cannot have two Brilliant walks in a row, how many ways can he walk to the point $(10, 10)$? [b]G18.[/b] Deja vu? Let $ABCD$ be a square with side length $1$. It is folded along a line $\ell$ that divides the square into two pieces with equal area. The maximum possible area of the resulting shape is $B$. Find the integer closest to $100B$. PS. You should use hide for answers. Sets 1-3 have been posted [url=https://artofproblemsolving.com/community/c3h3131303p28367061]here [/url] and 7-9 [url=https://artofproblemsolving.com/community/c3h3131308p28367095]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$. What integer was removed ?

Angle $A$ of the acute-angled triangle $ABC$ equals $60^o$ . Prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$, passes through the circumcircle 's centre. (V . Pogrebnyak , year 12 student , Vinnitsa,)

We are given $8$ different weights and a balance without a scale. (a) Find the smallest number of weighings necessary to find the heaviest weight. (b) How many weighting is further necessary to find the second heaviest weight?

Find, with proof, the point $P$ in the interior of an acute-angled triangle $ABC$ for which $BL^2+CM^2+AN^2$ is a minimum, where $L,M,N$ are the feet of the perpendiculars from $P$ to $BC,CA,AB$ respectively. [i]Proposed by United Kingdom.[/i]

Let $ABC$ be a triangle, the circle having $BC$ as diameter cuts $AB,AC$ at $F,E$ respectively. Let $P$ a point on this circle. Let $C',B$' be the projections of $P$ upon the sides $AB,AC$ respectively. Let $H$ be the orthocenter of the triangle $AB'C'$. Show that $\angle EHF = 90^o$.

For any positive integer $n$, we define $$S_n=\sum_{k=1}^n \frac{2^k}{k^2}.$$ Prove that there are no polynomials $P,Q$ with real coefficients such that for any positive integer $n$, we have $\frac{S_{n+1}}{S_n}=\frac{P(n)}{Q(n)}$.

Determine the number of ways to partition the $n^2$ squares of an $n\times n$ grid into $n$ connected pieces of sizes $1$, $3$, $5$, $\dots$, $2n-1$ so that each piece is symmetric across the diagonal connecting the bottom right to the top left corner of the grid. A connected piece is a set of squares that any two of them are connected by a sequence of adjacent squares in the set. Two squares are adjacent if and only if they share an edge.

Find all positive integers $x$ and $y$ such that $x^{x+y}=y^{3x}$.

There are $n$ integers $a$ such that $0\leq a<91$ and $a$ is a solution to $x^3+8x^2-x+83\equiv0\pmod{91}$. What is $n$?

Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$.

Determine all functions $f:\mathbb{N}\to \mathbb{N}$ such that for all $a, b \in \mathbb{N}$ the following conditions hold: $(i)$ $f(f(a)+b) \mid b^a-1$; $(ii)$ $f(f(a))\geq f(a)-1$.

Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with \[\frac{x + y}{2} \quad \text{and} \quad 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}.\] After $2017$ minutes the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.

Consider the isosceles right triangle$ ABC, \angle A = 90^o$, and point $D \in (AB)$ such that $AD = \frac13 AB$. In the half-plane determined by the line $AB$ and point $C$ , consider a point $E$ such that $\angle BDE = 60^o$ and $\angle DBE = 75^o$. Lines $BC$ and $DE$ intersect at point $G$, and the line passing through point $G$ parallel to the line $AC$ intersects the line $BE$ at point $H$. Prove that the triangle $CEH$ is equilateral.

If $a, b, c$ and $d$ are complex non-zero numbers such that $$2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|.$$ Prove that $$\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .$$