Solve in $ \mathbb{R} $ the equation $ 4^{x^2-x}=\log_2 x+\sqrt{x-1} +14. $ [i]Marin Tolosi[/i]

Let $a_1, a_2,..., a_9$ be integers. Prove that if $19$ divides $a_1^9+a_2^9+...+a_9^9$ then $19$ divides the product $a_1a_2...a_9$.

Let $n \geq 2$ be an integer such that $6^n + 11^n$ is divisible by $n$. Prove that $n^{100} + 6^n + 11^n$ is divisible by $17n$ and not divisible by $289n$.

Vertices $A$, $B$ and $C$ of a triangle are connected with points $A'$ , $B'$ and $C'$ lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments $AA'$, $BB'$ and $CC'$ lie in a straight line? (Folklore)

Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.

Is it possible to pair up the numbers $0, 1, 2, 3,... , 61$ in such a way that when we sum each pair, the product of the $31$ numbers we get is a perfect f ifth power?

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy \begin{tabular}{c@{\,}c@{\,}c@{\,}c} & $C$ & $C$ & $A$ \\ + & $B$ & $2$ & $B$ \\\hline & $A$ & $8$ & $8$ \\ \end{tabular} what is $A \cdot B \cdot C$? [i]2021 CCA Math Bonanza Individual Round #5[/i]

Two players play a game involving an $n \times n$ grid of chocolate. Each turn, a player may either eat a piece of chocolate (of any size), or split an existing piece of chocolate into two rectangles along a grid-line. The player who moves last loses. For how many positive integers $n$ less than $1000$ does the second player win? (Splitting a piece of chocolate refers to taking an $a \times b$ piece, and breaking it into an $(a-c) \times b$ and a $c \times b$ piece, or an $a \times (b-d)$ and an $a \times d$ piece.) [i]Proposed by Lewis Chen[/i]

Define $L_n$ as the least common multiple of all the integers from $1$ to $n$ inclusive. There is a unique integer $h$ such that $\frac{1}{1}+\frac{1}{2}+\frac{1}{3} \ldots +\frac{1}{17}=\frac{h}{L_{17}}$. What is the remainder when $h$ is divided by $17?$ $\textbf{(A) } 1 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 9$

The number of distinct pairs of integers $(x,y)$ such that \[0 < x < y\quad \text{and}\quad \sqrt{1984} = \sqrt{x} + \sqrt{y}\] is $\textbf{(A) }0\qquad \textbf{(B) }1\qquad \textbf{(C) }2\qquad \textbf{(D) }3\qquad \textbf{(E) }7$

For two complex numbers $z_1,z_2$ satisfy that $|z_1|=|z_1+z_2|=3,|z_1-z_2|=3\sqrt3$, then $\log_3|(z_1\overline{z_2})^{2000}+(\overline{z_1}z_2)^{2000}|=$________.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

If 100 dice are rolled, what is the probability that the sum of the numbers rolled is even? [i]Lightning 1.1[/i]

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

Two circles are intersecting in points $P$ and $Q$. Construct two points $A$ and $B$ on these circles so that $P\in AB$ and the product $AP.PB$ is maximal.

We call a $6\times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.

To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$

Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$ \[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\ &w(a,b)&-2&-1&0&1&2\\ \hline &-2&-1&-2&2&-2&-1\\ &-1&-2&4&-4&4&-2\\ a&0&2&-4&12&-4&2\\ &1&-2&4&-4&4&-2\\ &2&-1&-2&2&-2&-1\\ \hline\end{array}\] For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)

For the real numbers $x$ and $y$, $[\sqrt{x}] = 10$ and $[\sqrt{y}] =14$. How large is $\left[\sqrt{[ \sqrt{x+y} ]}\right]$ ? (Note: the square roots are the positive values ​​and $[x]$ is the largest integer less than or equal to x.)

Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.

Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality $(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$