Determine all triplet of non-negative integers $ (x,y,z) $ satisfy $$ 2^x3^y+1=7^z $$

Prove that there are infinitely many twin primes if and only if there are infinitely many integers that cannot be written in any of the following forms: \[6uv+u+v, \;\; 6uv+u-v, \;\; 6uv-u+v, \;\; 6uv-u-v,\] for some positive integers $u$ and $v$.

Numbers $7^2$,$8^2,\ldots,2023^2$,$2024^2$ are written on the board. Is it possible to add to one of them $7$, to some other one $8$, $\ldots$, to the remaining $2024$ such that all numbers became prime [i]M. Zorka[/i]

Inside the trihedral angle $ OABC $, whose plane angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ is chosen equidistant from the faces of this angle. Through point $ S $ a plane is drawn that intersects the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove that the sum $$ \frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}$$ has a constant value, i.e. independent of the position of the plane $ MNP $.

Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.

Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.

Find all pairs of polynomials $P(x),Q(x)$ with integer coefficients such that $P(Q(x)) = (x - 1)(x - 2)...(x - 9)$ for all real numbers $x$

Two numbers are such that their difference, their sum, and their product are to one another as $1:7:24$. The product of the two numbers is: $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 24\qquad\textbf{(D)}\ 48\qquad\textbf{(E)}\ 96 $

In a school there are $n$ classes and $n$ students. The students are enrolled in classes, such that no two of them have exactly the same classes. Prove that we can close a class in a such way that there still are no two of them which have exactly the same classes.

We are given an equilateral triangle ABC with the length of its side equal to $1$. There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$. All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called [i]nodes[/i]. We assign a real number for each [i]node[/i] such that the following conditions are satisfied: (I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively; (II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal. 1) Find the minimum distance between the [i]node[/i] with the maximal number to the [i]node[/i] with the minimal number; 2) Denote by $S$ the sum of the numbers of all [i]nodes[/i], find $S$.

Prove that there is no function $f:\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $f(x)^2 \geq f(x+y)(f(x)+y)$ for all $x,y \in \mathbb{R}^{+}$.

What is the largest possible number of acute angles in an $n$-gon which is not selfintersecting (no two non-adjacent edges interesect)?

Let $a_{1}, \ldots, a_{n}$ be an infinite sequence of strictly positive integers, so that $a_{k} < a_{k+1}$ for any $k.$ Prove that there exists an infinity of terms $ a_{m},$ which can be written like $a_m = x \cdot a_p + y \cdot a_q$ with $x,y$ strictly positive integers and $p \neq q.$

The bisector of angle $A$ of triangle $ABC$ ($AB > AC$) meets its circumcircle at point $P$. The perpendicular to $AC$ from $C$ meets the bisector of angle $A$ at point $K$. A cừcle with center $P$ and radius $PK$ meets the minor arc $PA$ of the circumcircle at point $D$. Prove that the quadrilateral $ABDC$ is circumscribed.

For how many integers $0\leq n < 2013$, is $n^4+2n^3-20n^2+2n-21$ divisible by $2013$? $ \textbf{(A)}\ 6 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 16 \qquad\textbf{(E)}\ \text{None of above} $

There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class? $\textbf{(A) }3 : 4\qquad\textbf{(B) }4 : 3\qquad\textbf{(C) }3 : 2\qquad\textbf{(D) }7 : 4\qquad \textbf{(E) }2 : 1$

Suppose that each of the diagonals $AD,BE,CF$ divides the hexagon $ABCDEF$ into two parts of the same area and perimeter. Does the hexagon necessarily have a center of symmetry?

Find all prime numbers $p$ such that the number $$3^p+4^p+5^p+9^p-98$$ has at most $6$ positive divisors.

Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)? (A Andzans)

Morteza places a function $[0,1]\to [0,1]$ (that is a function with domain [0,1] and values from [0,1]) in each cell of an $n \times n$ board. Pavel wants to place a function $[0,1]\to [0,1]$ to the left of each row and below each column (i.e. to place $2n$ functions in total) so that the following condition holds for any cell in this board: If $h$ is the function in this cell, $f$ is the function below its column, and $g$ is the function to the left of its row, then $h(x) = f(g(x))$ for all $x \in [0, 1]$. Prove that Pavel can always fulfil his plan.

How many triangles are in this figure? (Some triangles may overlap other triangles.) [asy] draw((0,0)--(42,0)--(14,21)--cycle); draw((14,21)--(18,0)--(30,9));[/asy] $ \text{(A)}\ 9\qquad\text{(B)}\ 8\qquad\text{(C)}\ 7\qquad\text{(D)}\ 6\qquad\text{(E)}\ 5 $

Let $x$, $y$, and $z$ be nonnegative integers that are less than or equal to 100. Suppose that $x + y + z$, $xy + z$, $x + yz$, and $xyz$ are (in some order) four consecutive terms of an arithmetic sequence. Compute the number of such ordered triples $(x, y, z)$.

[b]p1.[/b] Vivek has three letters to send out. Unfortunately, he forgets which letter is which after sealing the envelopes and before putting on the addresses. He puts the addresses on at random sends out the letters anyways. What are the chances that none of the three recipients get their intended letter? [b]p2.[/b] David is a horrible bowler. Luckily, Logan and Christy let him use bumpers. The bowling lane is $2$ meters wide, and David's ball travels a total distance of $24$ meters. How many times did David's bowling ball hit the bumpers, if he threw it from the middle of the lane at a $60^o$ degree angle to the horizontal? [b]p3.[/b] Find $\gcd \,(212106, 106212)$. [b]p4.[/b] Michael has two fair dice, one six-sided (with sides marked $1$ through $6$) and one eight-sided (with sides marked $1-8$). Michael play a game with Alex: Alex calls out a number, and then Michael rolls the dice. If the sum of the dice is equal to Alex's number, Michael gives Alex the amount of the sum. Otherwise Alex wins nothing. What number should Alex call to maximize his expected gain of money? [b]p5.[/b] Suppose that $x$ is a real number with $\log_5 \sin x + \log_5 \cos x = -1$. Find $$|\sin^2 x \cos x + \cos^2 x \sin x|.$$ [b]p6.[/b] What is the volume of the largest sphere that FIts inside a regular tetrahedron of side length $6$? [b]p7.[/b] An ant is wandering on the edges of a cube. At every second, the ant randomly chooses one of the three edges incident at one vertex and walks along that edge, arriving at the other vertex at the end of the second. What is the probability that the ant is at its starting vertex after exactly $6$ seconds? [b]p8.[/b] Determine the smallest positive integer $k$ such that there exist $m, n$ non-negative integers with $m > 1$ satisfying $$k = 2^{2m+1} - n^2.$$ [b]p9.[/b] For $A,B \subset Z$ with $A,B \ne \emptyset$, define $A + B = \{a + b|a \in A, b \in B\}$. Determine the least $n$ such that there exist sets $A,B$ with $|A| = |B| = n$ and $A + B = \{0, 1, 2,..., 2012\}$. [b]p10.[/b] For positive integers $n \ge 1$, let $\tau (n)$ and $\sigma (n)$ be, respectively, the number of and sum of the positive integer divisors of $n$ (including $1$ and $n$). For example, $\tau (1) = \sigma (1) = 1$ and $\tau (6) = 4$, $\sigma (6) = 12$. Find the number of positive integers $n \le 100$ such that $$\sigma (n) \le (\sqrt{n} - 1)^2 +\tau (n)\sqrt{n}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence such that inequality $$a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0$$ holds for all $n \geq 3$. Prove that for all $n\geq2$ the inequality $$a_n \geq a_{n-1}+a_{n-2}+\dots+a_1$$ holds as well.

In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting $32$ triangular regions can be colored red and blue so that any two regions sharing an edge have different colors.