Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

Find the smallest positive integer $k$ such that \[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\] for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.) [i]Alex Zhu.[/i] [hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]

Are there exist some positive integers $n$ and $k$, such that the first decimals of $2^n$ (from left to the right) represent the number $5^k$ while the first decimals of $5^n$ represent the number $2^k$ ? (5)

Find the perimeter of the convex polygon whose coordinates of the vertices are the set of pairs of the integer solutions of the equation $x^2+xy = x + 2y + 9$.

Two circunmferences $ \Gamma_1$ $ \Gamma_2$ intersect at $ A$ and $ B$ $ r_1$ is the tangent from $ A$ to $ \Gamma_1$ and $ r_2$ is the tangent from $ B$ to $ \Gamma_2$ $ r_1 \cap r_2\equal{}C$ $ T\equal{} r_1 \cap \Gamma_2$ ($ T \neq A$) We consider a point $ X$ in $ \Gamma_1$ which is distinct from $ A$ and $ B$. $ XA \cap \Gamma_2 \equal{}Y$ ($ Y \neq A$) $ YB \cap XC\equal{}Z$ Prove that $ TZ \parallel XY$

We have six positive numbers $a_1, a_2, \ldots , a_6$ such that $a_1a_2\ldots a_6 =1$. Prove that: $$ \frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.$$

The smaller root of the equation $ \left(x-\frac{3}{4}\right)\left(x-\frac{3}{4}\right)+\left(x-\frac{3}{4}\right)\left(x-\frac{1}{2}\right) =0$ is: ${{ \textbf{(A)}\ -\frac{3}{4}\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{5}{8}\qquad\textbf{(D)}\ \frac{3}{4} }\qquad\textbf{(E)}\ 1 } $

Let $F_1,F_2,F_3,...$ be the [i]Fibonacci sequence[/i], the sequence of positive integers with $F_1 =F_2 =1$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \ge 1$. A [i]Fibonacci number[/i] is by definition a number appearing in this sequence. Let $P_1,P_2,P_3,...$ be the sequence consisting of all the integers that are products of two Fibonacci numbers (not necessarily distinct) in increasing order. The first few terms are $1,2,3,4,5,6,8,9,10,13,...$ since, for example $3 = 1 \cdot 3, 4 = 2 \cdot 2$, and $10 = 2 \cdot 5$. Consider the sequence $D_n$ of [i]successive [/i] differences of the $P_n$ sequence, where $D_n = P_{n+1}-P_n$ for $n \ge 1$. The first few terms of D_n are $1,1,1,1,1,2,1,1,3, ...$ . Prove that every number in $D_n$ is a [i]Fibonacci number[/i].

A cylindrical container of revolution is partially filled with a liquid whose density we ignore. Placing it with the axis inclined $30^o$ with respect to the vertical, we observe that when removing liquid so that the level falls $1$ cm, the weight of the contents decreases $40$ g. How much will the weight of that content decrease for each centimeter that lower the level if the axis makes an angle of $45^o$ with the vertical? It is supposed that the horizontal surface of the liquid does not touch any of the bases of the container.

Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by $m_{ij}=x$ if $i=j$, $m_{ij}=a$ if $i \not= j$ and $i+j$ is even, $m_{ij}=b$ if $i \not= j$ and $i+j$ is odd. For example $ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a & b& x & b\\ b & a & b & x \end{vmatrix}$. Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ . P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)

Let $(x_n)_{n\geq 1}$ be the sequence defined by $x_1\in (0,1)$ and $x_{n+1}=x_n-\frac{x_n^2}{\sqrt{n}}$ for all $n\geq 1$. Find the values of $\alpha\in\mathbb{R}$ for which the series $\sum_{n=1}^{\infty}x_n^{\alpha}$ is convergent.

For $ p\equal{}1,2,\ldots,10$ let $ S_p$ be the sum of the first $ 40$ terms of the arithmetic progression whose first term is $ p$ and whose common difference is $ 2p\minus{}1$; then $ S_1\plus{}S_2\plus{}\cdots\plus{}S_{10}$ is $ \textbf{(A)}\ 80000 \qquad \textbf{(B)}\ 80200 \qquad \textbf{(C)}\ 80400 \qquad \textbf{(D)}\ 80600 \qquad \textbf{(E)}\ 80800$

In $\triangle ABC$, point $D$ lies on side $AC$ such that $\angle ABD=\angle C$. Point $E$ lies on side $AB$ such that $BE=DE$. $M$ is the midpoint of segment $CD$. Point $H$ is the foot of the perpendicular from $A$ to $DE$. Given $AH=2-\sqrt{3}$ and $AB=1$, find the size of $\angle AME$.

Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has team $A$ played? $\textbf{(A) } 21 \qquad \textbf{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63$

Find all function $f : \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ . $$f(x+f(y))+f(x+y)=2x+f(y)+f(f(y))$$ . [hide=Original]$$f(x+f(y))+f(x+y)=2x+f(y)+y$$[/hide]

Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$? $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Let $ABCDEFG$ be a regular heptagon with center $O$. Let $M$ be the centroid of $\triangle ABD$. Prove that $\cos^2(\angle GOM)$ is rational and determine its value. [i]Evan o'Dorney[/i]

Find all functions $f:\mathbb{Z}\rightarrow \mathbb{Z}$ such that for all integers $x$, $y$, $$f(x-f(y))=f(f(y))+f(x-2y)$$ [i]Proposed by Ivan Chan Kai Chin[/i]

Six mathematician sit around a round table. Each of them has a number and they do the following transformation: Each time, two mathematician sitting next to each other is chosen, they will add $1$ to their own number. Is it possible to make all the six numbers equal if the initial numbers are a) 6,5,4,3,2,1 b) 7,5,3,2,1,4

For some values of c, the equation $x^c + y^c = z^c$ can be illustrated geometrically. For example, the case $c = 2$ can be illustrated by a right-angled triangle. By this we mean that, x, y, z is a solution of the equation $x^2 + y^2 = z^2$ if and only if there exists a right-angled triangle with catheters $x$ and $y$ and hypotenuse $z$. In this problem we will look at the cases $c = -\frac{1}{2}$ and $c = - 1$. a) Let $x, y$ and $z$ be the radii of three circles intersecting each other and a line, as shown, in the figure. Show that, $x^{-\frac{1}{2}}+ y^{-\frac{1}{2}} = z^{-\frac{1}{2}}$ [img]https://cdn.artofproblemsolving.com/attachments/5/7/5315e33e1750a3a49ae11e1b5527311117ce70.png[/img] b) Draw a geometric figure that illustrates the case in a similar way, $c = - 1$. The figure must be able to be constructed with a compass and a ruler. Describe such a construction and prove that, in the figure, lines $x, y$ and $z$ satisfy $x^{-1}+ y^{-1} = z^{-1}$. (All positive solutions of this equation should be possible values for $x, y$, and $z$ on such a figure, but you don't have to prove that.)

Let $N$ be a positive integer, and consider an $N \times N$ grid. A [i]right-down path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell below the previous cell in the sequence. A [i]right-up path[/i] is a sequence of grid cells such that each cell is either one cell to the right of or one cell above the previous cell in the sequence. Prove that the cells of the $N \times N$ grid cannot be partitioned into less than $N$ right-down or right-up paths. For example, the following partition of the $5 \times 5$ grid uses $5$ paths. [asy] size(4cm); draw((5,-1)--(0,-1)--(0,-2)--(5,-2)--(5,-3)--(0,-3)--(0,-4)--(5,-4),gray+linewidth(0.5)+miterjoin); draw((1,-5)--(1,0)--(2,0)--(2,-5)--(3,-5)--(3,0)--(4,0)--(4,-5),gray+linewidth(0.5)+miterjoin); draw((0,0)--(5,0)--(5,-5)--(0,-5)--cycle,black+linewidth(2.5)+miterjoin); draw((0,-1)--(3,-1)--(3,-2)--(1,-2)--(1,-4)--(4,-4)--(4,-3)--(2,-3)--(2,-2),black+linewidth(2.5)+miterjoin); draw((3,0)--(3,-1),black+linewidth(2.5)+miterjoin); draw((1,-4)--(1,-5),black+linewidth(2.5)+miterjoin); draw((4,-3)--(4,-1)--(5,-1),black+linewidth(2.5)+miterjoin); [/asy] [i]Proposed by Zixiang Zhou, Canada[/i]

Points $D,E,F$ are midpoints of the sides $AB,BC,CA$ of triangle $ABC$. Angle bisectors of the angles $BDC$ and $ADC$ intersect the lines $BC$ and $AC$ respectively at the points $M$ and $N$, and the line $MN$ intersects the line $CD$ at the point $O$. Let the lines $EO$ and $FO$ intersect respectively the lines $AC$ and $BC$ at the points $P$ and $Q$. Prove that $CD=PQ$. [i](Plamen Koshlukov)[/i]

Find all functions $f:\mathbb R \rightarrow \mathbb R$, such that $2xyf(x^2-y^2)=(x^2-y^2)f(x)f(2y)$

The expression $\frac{2021}{2020} - \frac{2020}{2021}$ is equal to the fraction $\frac{p}{q}$, where $p$ and $q$ are positive integers whose greatest common divisor is $1$. What is $p$? $\textbf{(A) }1\qquad\textbf{(B) }9\qquad\textbf{(C) }2020\qquad\textbf{(D) }2021\qquad\textbf{(E) }4041$

Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.