Let $a$ and $b$ be positive integers. Consider the set of all non-negative integers $n$ for which the number $\left(a+\frac12\right)^n +\left(b+\frac12\right)^n$ is an integer. Show that the set is finite.

Let \( f(x) = \frac{100^x}{100^x + 10} \). Determine the value of: \[ f\left( \frac{1}{2024} \right) - f\left( \frac{2}{2024} \right) + f\left( \frac{3}{2024} \right) - f\left( \frac{4}{2024} \right) + \ldots - f\left( \frac{2022}{2024} \right) + f\left( \frac{2023}{2024} \right) \]

Prove that the number $2^9 +2^{99}$ is divisible by $100$.

Find the locus of the points $ M $ that are situated on the plane where a rhombus $ ABCD $ lies, and satisfy: $$ MA\cdot MC+MB\cdot MD=AB^2 $$ [i]Ovidiu Pop[/i]

Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?

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 ?

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

Given are the odd integers $m> 1$, $k$, and a prime $p$ such that $p> mk +1$. Prove that $p^{2}\mid {\binom{k}{k}}^{m}+{\binom{k+1}{k}}^{m}+\cdots+{\binom{p-1}{k}}^{m}$.

A quadrilateral $ABCD$ has an incircle $\Gamma$. The points $X, Y$ are chosen so that $AX-CX=AB-BC$, $BX-DX=BC-CD$, $CY-AY=AD-DC$ and $DY-BY=AB-AD$. Given that the center of $\Gamma$ lies on $XY$, show that $AC, BD, XY$ are concurrent.

Points $X$ and $Y$ on the unit circle centered at $O = (0,0)$ are at $(-1,0)$ and $(0,-1)$ respectively. Points $P$ and $Q$ are on the unit circle such that $\angle P XO = \angle QY O = 30^o$. Let $Z$ be the intersection of line $X P$ and line $Y Q$. The area bounded by segment $Z P$, segment $ZQ$, and arc $PQ$ can be expressed as $a\pi -b$ where $a$ and $b$ are rational numbers. Find $\frac{1}{ab}$ .

Find all ordered pairs of positive integers $(m,n)$ for which there exists a set $C=\{c_1,\ldots,c_k\}$ ($k\ge1$) of colors and an assignment of colors to each of the $mn$ unit squares of a $m\times n$ grid such that for every color $c_i\in C$ and unit square $S$ of color $c_i$, exactly two direct (non-diagonal) neighbors of $S$ have color $c_i$. [i]David Yang.[/i]

Let $x, y, z$ be nonzero real numbers with $$\frac{x + y}{z}=\frac{y + z}{x}=\frac{z + x}{y}.$$ Determine all possible values of $$\frac{(x + y)(y + z)(z + x)}{xyz}.$$ [i](Walther Janous)[/i]

Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.

Consider $64$ distinct natural numbers, at most equal to $2012$. Show that it is possible to choose four of them, denoted as $a,b,c,d$ such that $ a+b-c-d$ to be a multiple of $2013$

Given the triangle $ABC$ with $T$ is its [i]Fermat–Torricelli[/i] point. Let $(N_a)$ be the circumcircle of $\triangle TBC$. Choose a point $X$ on $(N_a)$ such that $TX$ is perpendicular to $BC$. The segment $BC$ intersects $(TN_aX)$ at $D$. Similar definition of points $Y, Z, E, F$. The reflection lines of the [i]Euler[/i] line of $\triangle ABC$ wrt $BC, CA, AB$ intersect $XD, YE, ZF$ at $P, Q, R$, respectively. Prove that: $AP$ is perpendicular to $QR$ if and only if $AB = AC$ or $2BC^2 = AB^2 + AC^2$.

Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards. Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells. [*]Prove that every minimal blocking set containing at most $3m^2$ cells.

There are $2025$ positive integers $a_1, a_2, \cdots, a_{2025}$ are placed around a circle. For any $k = 1, 2, \cdots, 2025$, $a_k \mid a_{k-1} + a_{k+1}$ where indices are considered modulo $n$. Prove that there exists a positive integer $N$ such that satisfies the following condition. [list] [*] [b](Condition)[/b] For any positive integer $n > N$, when $a_1 = n^n$, $a_1, a_2, \cdots, a_{2025}$ are all multiples of $n$. [/list]

Let $ABC$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $BCO$. The altitude of $ABC$ from $A$ meets $w$ at a point $P$. The line $PK$ intersects the circumcircle of $ABC$ at points $E$ and $F$. Prove that one of the segments $EP$ and $FP$ is equal to the segment $PA$.

Find all $f:\mathbb{Z}\to\mathbb{Z}$ such that \[f\left(\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\right)+f(x)=f(f(y))+\left\lfloor\frac{f(x)+f(y)}{2}\right\rfloor\] holds for all $x,y\in\mathbb{Z}$. [i]Proposed by usjl[/i]

Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.

From a set of integers $\{1,...,100\}$, $k$ integers were deleted. Is it always possible to choose $k$ distinct integers from the remaining set such that their sum is $100$ if [b](a) $k=9$?[/b] [b](b) $k=8$?[/b]

Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation $$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$ [b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational. [b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?

Cut a square into eight acute-angled triangles.

Find $\lim_{t\rightarrow 0}\frac{1}{t^3}\int_0^{t^2} e^{-x}\sin \frac{x}{t}\ dx\ (t\neq 0).$ [i]2010 Kumamoto University entrance exam/Medicine[/i]

For a fixed natural number $m \geq 2$, prove that [b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\] [b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.