Find all integer solutions of the equation $xy+5x-3y=27$.

A wobbly number is a positive integer whose digits are alternately zero and non-zero with the last digit non-zero (for example, 201). Find all positive integers which do not divide any wobbly number.

On every square of a chessboard, there are as many grains as shown on the picture. Starting from an arbitrary square, a knight starts a journey over the chessboard. After every move it eats up all the grains from the square it arrived to, but when it leaves, the same number of grains is put back on the square. After some time the knight returns to its initial square. Prove that the total number of grains the knight has eaten up during the journey is divisible by $3$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC8xL2IwOGZlODYxMDg1MWMwMWUwMjFkOGJkMWQ2MjA4YzIzZmQ5YTc5LnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0yOCBhdCA3LjIzLjA3IEFNLnBuZw==[/img]

$\boxed{A6}$The sequence $a_0,a_1,...$ is defined by the initial conditions $a_0=1,a_1=6$ and the recursion $a_{n+1}=4a_n-a_{n-1}+2$ for $n>1.$Prove that $a_{2^k-1}$ has at least three prime factors for every positive integer $k>3.$

Consider a real number $a$ that satisfies $a=(a-1)^3$. Prove that there exists an integer $N$ that satisfies $$|a^{2021}-N|<2^{-1000}.$$ [i] (Vlad Robu) [/i]

Let $x$ be a non-zero real numbers such that $x^4+\frac{1}{x^4}$ and $x^5+\frac{1}{x^5}$ are both rational numbers. Prove that $x+\frac{1}{x}$ is a rational number.

Two branches of the hyperbola $xy=1$ are $C_1,C_2$ ($C_1$ in Quadrant I, $C_2$ in Quadrant III). Three apexes of regular triangle $PQR$ are on the hyperbola. [b](a)[/b] $P,Q,R$ cannot be on the same branch. [b](b)[/b] $P(-1,-1)$ is a point on $C_2$, if $Q,R$ are on $C_1$, find their coordinates.

Nashan randomly chooses $6$ positive integers $a, b, c, d, e, f$. Find the probability that $2^a+2^b+2^c+2^d+2^e+2^f$ is divisible by $5$. [i]Proposed by Bradley Guo[/i]

Let $ABC $be a right triangle with $\angle C = 90^o$ and let $D$ be the foot of the altitude from $C$. Let $E$ be the centroid of triangle $ACD$ and let $F$ be the centroid of triangle $BCD$. The point $P$ satisfies $\angle CEP = 90^o$ and $|CP| = |AP|$, while point $Q$ satisfies $\angle CFQ = 90^o$ and $|CQ| = |BQ|$. Prove that $PQ$ passes through the centroid of triangle $ABC$.

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $x,y>0$ and $xy\ge1.$ Prove that $x^3+y^3+4xy\ge x^2+y^2+x+y+2.$ Let $x,y>0$ and $xy\ge1.$ Prove that $2(x^3+y^3+xy+x+y)\ge5(x^2+y^2).$

Three non-collinear lattice points $A,B,C$ lie on the plane $1+3x+5y+7z=0$. The minimal possible area of triangle $ABC$ can be expressed as $\frac{\sqrt{m}}{n}$ where $m,n$ are positive integers such that there does not exists a prime $p$ dividing $n$ with $p^2$ dividing $m$. Compute $100m+n$. [i]Proposed by Yannick Yao[/i]

There are $52$ people in a room. What is the largest value of $n$ such that the statement "At least $n$ people in this room have birthdays falling in the same month" is always true? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 12 $

[u]Round 1[/u] [b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!) [b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away from where they started? [b]p3.[/b] At Barrios’s absurdly priced fish and chip shop, one fish is worth $\$13$, one chip is worth $\$5$. What is the largest integer dollar amount of money a customer can enter with, and not be able to spend it all on fish and chips? [u]Round 2[/u] [b]p4.[/b] If there are $15$ points in $4$-dimensional space, what is the maximum number of hyperplanes that these points determine? [b]p5.[/b] Consider all possible values of $\frac{z_1 - z_2}{z_2 - z_3} \cdot \frac{z_1 - z_4}{z_2 - z_4}$ for any distinct complex numbers $z_1$, $z_2$, $z_3$, and $z_4$. How many complex numbers cannot be achieved? [b]p6.[/b] For each positive integer $n$, let $S(n)$ denote the number of positive integers $k \le n$ such that $gcd(k, n) = gcd(k + 1, n) = 1$. Find $S(2015)$. [u]Round 3 [/u] [b]p7.[/b] Let $P_1$, $P_2$,$...$, $P_{2015}$ be $2015$ distinct points in the plane. For any $i, j \in \{1, 2, ...., 2015\}$, connect $P_i$ and $P_j$ with a line segment if and only if $gcd(i - j, 2015) = 1$. Define a clique to be a set of points such that any two points in the clique are connected with a line segment. Let $\omega$ be the unique positive integer such that there exists a clique with $\omega$ elements and such that there does not exist a clique with $\omega + 1$ elements. Find $\omega$. [b]p8.[/b] A Chinese restaurant has many boxes of food. The manager notices that $\bullet$ He can divide the boxes into groups of $M$ where $M$ is $19$, $20$, or $21$. $\bullet$ There are exactly $3$ integers $x$ less than $16$ such that grouping the boxes into groups of $x$ leaves $3$ boxes left over. Find the smallest possible number of boxes of food. [b]p9.[/b] If $f(x) = x|x| + 2$, then compute $\sum^{1000}_{k=-1000} f^{-1}(f(k) + f(-k) + f^{-1}(k))$. [u]Round 4 [/u] [b]p10.[/b] Let $ABC$ be a triangle with $AB = 13$, $BC = 20$, $CA = 21$. Let $ABDE$, $BCFG$, and $CAHI$ be squares built on sides $AB$, $BC$, and $CA$, respectively such that these squares are outside of $ABC$. Find the area of $DEHIFG$. [b]p11.[/b] What is the sum of all of the distinct prime factors of $7783 = 6^5 + 6 + 1$? [b]p12.[/b] Consider polyhedron $ABCDE$, where $ABCD$ is a regular tetrahedron and $BCDE$ is a regular tetrahedron. An ant starts at point $A$. Every time the ant moves, it walks from its current point to an adjacent point. The ant has an equal probability of moving to each adjacent point. After $6$ moves, what is the probability the ant is back at point $A$? PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2782011p24434676]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $p>3$ be a prime number. Prove that the product of all primitive roots between 1 and $p-1$ is congruent 1 modulo $p$.

A triangle $ABC$ is inscribed in a circle $(C)$ .Let $G$ the centroid of $\triangle ABC$ . We draw the altitudes $AD,BE,CF$ of the given triangle .Rays $AG$ and $GD$ meet (C) at $M$ and $N$.Prove that points $ F,E,M,N $ are concyclic.

On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $

Let $ a_1, a_2, ..., a_{300}$ be nonnegative real numbers, with $ \sum_{i\equal{}1}^{300} a_i \equal{} 1$. Find the maximum possible value of $ \sum_{i \neq j, i|j} a_ia_j$.

Let $HOPKINS$ be an irregular convex heptagon (i.e., its angles and side lengths are all distinct, with the angles all having measure less than $180^{\circ}$) with area $1876$ such that all of its side lengths are greater than $5$, $OP=20$, and $KI=22$. Arcs with radius $2$ are drawn inside $HOPKINS$ with their centers at each of the vertices and their endpoints on the sides, creating circular sectors. Find the area of the region inside $HOPKINS$ but outside the sectors.

Let $G$ be a simple connected graph. Each edge has two phases, which is either blue or red. Each vertex are switches that change the colour of every edge that connects the vertex. All edges are initially red. Find all ordered pairs $(n,k)$, $n\ge 3$, such that: a) For all graph $G$ with $n$ vertex and $k$ edges, it is always possible to perform a series of switching process so that all edges are eventually blue. b) There exist a graph $G$ with $n$ vertex and $k$ edges and it is possible to perform a series of switching process so that all edges are eventually blue.

Let $P$ and $Q$ be the midpoints of the sides $BC$ and $CD$, respectively in a rectangle $ABCD$. Let $K$ and $M$ be the intersections of the line $PD$ with the lines $QB$ and $QA$, respectively, and let $N$ be the intersection of the lines $PA$ and $QB$. Let $X$, $Y$ and $Z$ be the midpoints of the segments $AN$, $KN$ and $AM$, respectively. Let $\ell_1$ be the line passing through $X$ and perpendicular to $MK$, $\ell_2$ be the line passing through $Y$ and perpendicular to $AM$ and $\ell_3$ the line passing through $Z$ and perpendicular to $KN$. Prove that the lines $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent.

Suppose $n\geq 3$ is an integer. There are $n$ grids on a circle. We put a stone in each grid. Find all positive integer $n$, such that we can perform the following operation $n-2$ times, and then there exists a grid with $n-1$ stones in it: $\bullet$ Pick a grid $A$ with at least one stone in it. And pick a positive integer $k\leq n-1$. Take all stones in the $k$-th grid after $A$ in anticlockwise direction. And put then in the $k$-th grid after $A$ in clockwise direction.

Distinct lines $\ell$ and $m$ lie in the $xy$-plane. They intersect at the origin. Point $P(-1, 4)$ is reflected about line $\ell$ to point $P'$, and then $P'$ is reflected about line $m$ to point $P''$. The equation of line $\ell$ is $5x - y = 0$, and the coordinates of $P''$ are $(4,1)$. What is the equation of line $m?$ $(\textbf{A})\: 5x+2y=0\qquad(\textbf{B}) \: 3x+2y=0\qquad(\textbf{C}) \: x-3y=0$ $(\textbf{D}) \: 2x-3y=0\qquad(\textbf{E}) \: 5x-3y=0$

Let non-constant polynomial $f(x)$ with real coefficients is given with the following property: for any positive integer $n$ and $k$, the value of expression $$\frac{f(n + 1)f(n + 2)... f(n + k)}{ f(1)f(2) ... f(k)} \in Z$$ Prove that $f(x)$ is divisible by $x$

In a triangle $ABC$, let $K$ be a point on the median $BM$ such that $CM = CK$. It turned out that $\angle CBM = 2\angle ABM$. Show that $BC = KM$.

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.