Solve the equation \[ 3^x \minus{} 5^y \equal{} z^2.\] in positive integers. [i]Greece[/i]

There are $n$ points in the plane, some of which are connected by segments. Some of the segments are colored in white, while the others are colored black in such a way that there exist a completely white as well as a completely black closed broken line of segments, each of them passing through every one of the $n$ points exactly once. It is known that the segments $AB$ and $BC$ are white. Prove that it is possible to recolor the segments in red and blue in such a way that $AB$ and $BC$ are recolored as red, [hide=not all of which segments are recolored red]meaning that recoloring every white as red and every black as blue is not acceptable[/hide], and that there exist a completely red as well as a completely blue closed broken line of segments, each of them passing through every one of the $n$ points exactly once.

In every cell of a board $101 \times 101$ is written a positive integer. For any choice of $101$ cells from different rows and columns, their sum is divisible by $101$. Show that the number of ways to choose a cell from each row of the board, so that the total sum of the numbers in the chosen cells is divisible by $101$, is divisible by $101$.

Let $n$ be a positive integer such that the sum of all its positive divisors (inclusive of $n$) equals to $2n + 1$. Prove that $n$ is an odd perfect square. related: https://artofproblemsolving.com/community/c6h515011 https://artofproblemsolving.com/community/c6h108341 (Putnam 1976) https://artofproblemsolving.com/community/c6h368488 https://artofproblemsolving.com/community/c6h445330 https://artofproblemsolving.com/community/c6h378928

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$. Prove that $\angle OBC=\angle ODC$.

Given an isosceles triangle $ABC$ with $AB = AC$, suppose $D$ is the midpoint of the $AC$. The circumcircle of the $DBC$ triangle intersects the altitude from $A$ at point $E$ inside the triangle $ABC$, and the circumcircle of the triangle $AEB$ cuts the side $BD$ at point $F$. If $CF$ cuts $AE$ at point $G$, prove that $AE = EG$.

\[\int_{-10}^{10}|4-|3-|2-|1-|x|||||\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

A circle with diameter $23$ is cut by a chord $AC$. Two different circles can be inscribed between the large circle and $AC$. Find the sum of the two radii.

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f\left(x^2\right) - yf(y) = f(x + y) (f(x) - y) $$ for all real numbers $x$ and $y$.

A man buys a house for $10000$ dollars and rents it. He puts $ 12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325$ dollars a year taxes and realizes $ 5\frac{1}{2}\%$ on his investment. The monthly rent is: $\textbf{(A)}\ 64.82\text{ dollars} \qquad \textbf{(B)}\ 83.33\text{ dollars} \qquad \textbf{(C)}\ 72.08\text{ dollars} \qquad \textbf{(D)}\ 45.83\text{ dollars} \qquad \textbf{(E)}\ 177.08\text{ dollars}$

A circle with diameter $\overline{PQ}$ of length 10 is internally tangent at $P$ to a circle of radius 20. Square $ABCD$ is constructed with $A$ and $B$ on the larger circle, $\overline{CD}$ tangent at $Q$ to the smaller circle, and the smaller circle outside $ABCD$. The length of $\overline{AB}$ can be written in the form $m + \sqrt{n}$, where $m$ and $n$ are integers. Find $m + n$.

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

For all rational $x$ satisfying $0 \leq x < 1$, the functions $f$ is defined by \[f(x)=\begin{cases}\frac{f(2x)}{4},&\mbox{for }0 \leq x < \frac 12,\\ \frac 34+ \frac{f(2x - 1)}{4}, & \mbox{for } \frac 12 \leq x < 1.\end{cases}\] Given that $x = 0.b_1b_2b_3 \cdots $ is the binary representation of $x$, find, with proof, $f(x)$.

$x_{1},x_{2},\dots,x_{n}$ are real number such that for each $i$, the set $\{x_{1},x_{2},\dots,x_{n}\}\backslash \{x_{i}\}$ could be partitioned into two sets that sum of elements of first set is equal to the sum of the elements of the other. Prove that all of $x_{i}$'s are zero. [hide="Hint"]It is a number theory problem.[/hide]

Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and $$\int_{a}^{b} x^n f(x) \, dx =0$$ for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$

Let $ABC$ and $PQR$ be two triangles such that [list] [b](a)[/b] $P$ is the mid-point of $BC$ and $A$ is the midpoint of $QR$. [b](b)[/b] $QR$ bisects $\angle BAC$ and $BC$ bisects $\angle QPR$ [/list] Prove that $AB+AC=PQ+PR$.

Let $b$ be a real number randomly sepected from the interval $[-17,17]$. Then, $m$ and $n$ are two relatively prime positive integers such that $m/n$ is the probability that the equation \[x^4+25b^2=(4b^2-10b)x^2\] has $\textit{at least}$ two distinct real solutions. Find the value of $m+n$.

Let $p$ be a prime number and let $A$ be a subgroup of the multiplicative group $\mathbb F^*_p$ of the finite field $\mathbb F_p$ with $p$ elements. Prove that if the order of $A$ is a multiple of $6$, then there exist $x,y,z\in A$ satisfying $x+y=z$.

$ABCDEF$ is a regular hexagon. Let $R$ be the overlap between $\vartriangle ACE$ and $\vartriangle BDF$. What is the area of $R$ divided by the area of $ABCDEF$?

Consider $k$ positive integers $a_1,a_2,\ldots,a_k$ satisfying $1\le a_1<a_2<\ldots<a_k\le n$ and $\operatorname{lcm}(a_i,a_j)\le n$ for any $i,j$. Prove that $$k\le2\lfloor\sqrt n\rfloor.$$

In an acute triangle $ ABC$ segments $ BE$ and $ CF$ are altitudes. Two circles passing through the point $ A$ and $ F$ and tangent to the line $ BC$ at the points $ P$ and $ Q$ so that $ B$ lies between $ C$ and $ Q$. Prove that lines $ PE$ and $ QF$ intersect on the circumcircle of triangle $ AEF$. [i]Proposed by Davood Vakili, Iran[/i]

Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that \[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]

Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.

Let $ABCD$ be a cyclic quadrilateral such that the circles with diameters $AB$ and $CD$ touch at $S$. If $M, N$ are the midpoints of $AB, CD$, prove that the perpendicular through $M$ to $MN$ meets $CS$ on the circumcircle of $ABCD$.