Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

For every nonempty set of real numbers $S{}$ denote $\sigma(S)$ the sum of its elements. Let $A{}$ be a set of $n{}$ positive integers. Show that the set of all sums $\sigma{}$ of all nonempty sets of $A{}$ can be partitioned in $n{}$ groups such that the ratio between the greatest number and the smallest number from each group is less than $2$.

$24$ students attend a mathematical circle. For any team consisting of $6$ students, the teacher considers it to be either [b]GOOD [/b] or [b]OK[/b]. For the tournament of mathematical battles, the teacher wants to partition all the students into $4$ teams of $6$ students each. May it happen that every such partition contains either $3$ [b]GOOD[/b] teams or exactly one [b]GOOD[/b] team and both options are present?

Let $M$ and $N$ be points on the side $BC$ of a triangle $ABC$ such that $BM = CN$ ($M$ lies between $B$ and $N$). Points $P$ and $Q$ lie on $AN$ and $AM$ respectively, so that $\angle PMC =\angle MAB$ and $\angle QNB = \angle NAC$. Prove that $\angle QBC = \angle PCB$.

In the array of $13$ squares shown below, $8$ squares are colored red, and the remaining $5$ squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated $90^{\circ}$ around the central square is $\tfrac{1}{n}$, where $n$ is a positive integer. Find $n$. [asy] draw((0,0)--(1,0)--(1,1)--(0,1)--(0,0)); draw((2,0)--(2,2)--(3,2)--(3,0)--(3,1)--(2,1)--(4,1)--(4,0)--(2,0)); draw((1,2)--(1,4)--(0,4)--(0,2)--(0,3)--(1,3)--(-1,3)--(-1,2)--(1,2)); draw((-1,1)--(-3,1)--(-3,0)--(-1,0)--(-2,0)--(-2,1)--(-2,-1)--(-1,-1)--(-1,1)); draw((0,-1)--(0,-3)--(1,-3)--(1,-1)--(1,-2)--(0,-2)--(2,-2)--(2,-1)--(0,-1)); size(100); [/asy]

The state income tax where Kristin lives is levied at the rate of $ p\%$ of the first $ \$28000$ of annual income plus $ (p \plus{} 2)\%$ of any amount above $ \$28000$. Kristin noticed that the state income tax she paid amounted to $ (p \plus{} 0.25)\%$ of her annual income. What was her annual income? $ \textbf{(A) }\$28000\qquad\textbf{(B) }\$32000\qquad\textbf{(C) }\$35000\qquad\textbf{(D) }\$42,000\qquad\textbf{(E) }\$56000$

Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,1)$. Denote by $s(M)=OH_1^2+OH_2^2+OH_3^2,$ $(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\},$ where $H_1,H_2,H_3,$ are the orthocenters of the triangles $MAB,~MBC$ and $MCA.$ $a)$ Prove that if $ABC$ is equilateral$,$ then $s(M)=6,(\forall) M \in\mathcal{C}\setminus \left\{A,B,C\right\},$ $b)$ Prove that if there exist three distinct points $M_1,M_2,M_3\in\mathcal{C}\setminus \left\{A,B,C\right\}$ such that $s(M_1)=$$s(M_2)$$=s(M_3),$ then $ABC$ is equilateral$.$

A circle passing through vertices $A$ and $B$ of triangle $ABC$ intersects the sides $AC$ and $BC$ again at points $P$ and $Q$, respectively. Given that the median from vertex $C$ bisect the arc $PQ$ of the circle. Prove that $ABC$ is an isosceles triangle.

After Sally takes 20 shots, she has made $55\%$ of her shots. After she takes 5 more shots, she raises her percentage to $56\%$. How many of the last 5 shots did she make? $\textbf{(A)} 1 \qquad\textbf{(B)} 2 \qquad\textbf{(C)} 3 \qquad\textbf{(D)} 4 \qquad\textbf{(E)} 5$

Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$ and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$

Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]

Let $\astrosun O_1$ and $\astrosun O_2$ intersect at $P$ and $Q$, their common external tangent touches $\astrosun O_1$ and $\astrosun O_2$ at $A$ and $B$ respectively. A circle $\Gamma$ passing through $A$ and $B$ intersects $\astrosun O_1$, $\astrosun O_2$ at $D$, $C$. Prove that $\displaystyle \frac{CP}{CQ}=\frac{DP}{DQ}$

Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system $$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\ a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$ has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.

On the sides $(AB)$ and $(AC)$ of the triangle $ABC$ are considered the points $M$ and $N$ respectively so that $ \angle ABC =\angle ANM$. Point $D$ is symmetric of point $A$ with respect to $B$, and $P$ and $Q$ are the midpoints of the segments $[MN]$ and $[CD]$, respectively. Prove that the points $A, P$ and $Q$ are collinear if and only if $AC = AB \sqrt {2}$

Let $f:\mathbb{R}\rightarrow[0,\infty )$ be a function with the property that $|f(x)-f(y)|\le |x-y|$ for every $x,y\in\mathbb{R}$. Show that: a) If $\lim_{n\rightarrow \infty} f(x+n)=\infty$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\infty$. b) If $\lim_{n\rightarrow \infty} f(x+n)=\alpha ,\alpha\in[0,\infty )$ for every $x\in\mathbb{R}$, then $\lim_{x\rightarrow\infty}=\alpha$.

We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$. a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command. B=%Error. "rank" is a bad command. C = r$, such that $A=BC$. b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.

Solve the following system of equations in rational numbers: \[ (x^2+1)^3=y+1,\\ (y^2+1)^3=z+1,\\ (z^2+1)^3=x+1.\]

An acute-angled triangle is inscribed in a circle of radius $R$. The distance between the center of the circle and the point of intersection of the medians of the triangle is $d$. Find the radius of a circle inscribed in a triangle whose vertices are the feet of the altitudes of this triangle.

We say that three positive integers $a$, $b$ and $c$ form a [i]family[/i] if the following two conditions are satisfied: [list] [*]$a + b + c = 900$. [*]There exists an integer $n$, with $n \geqslant 2$, such that $$\frac{a}{n-1}=\frac{b}{n}=\frac{c}{n+1}.$$ [/list] Determine the number of such families.

Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

$10$ teams participated in a football tournament: every two teams played each other exactly once. After the end of the tournament it turned out that all teams got different amount of points and some teams won more games, than the winner of the tournament, call them strong. What is the maximum number of teams that could be strong? (In football the winner of the match gets three points, the loser - 0 points, and if the match ends in a draw both teams get 1 point.)

$ \lim_{n\to\infty }\left( \frac{1}{e}\sum_{i=0}^n \frac{1}{i!} \right)^{n!} $

Consider two coprime integers $p{}$ and $q{}$ which are greater than $1{}$ and differ from each other by more than $1{}$. Prove that there exists a positive integer $n{}$ such that \[\text{lcm}(p+n, q+n)<\text{lcm}(p,q).\]

Let $ABC$ be an equilateral triangle such that length of its altitude is $1$. Circle with center on the same side of line $AB$ as point $C$ and radius $1$ touches side $AB$. Circle rolls on the side $AB$. While the circle is rolling, it constantly intersects sides $AC$ and $BC$. Prove that length of an arc of the circle, which lies inside the triangle, is constant

A four-element set $\{a, b, c, d\}$ of positive integers is called [i]good[/i] if there are two of them such that their product is a mutiple of the greatest common divisor of the remaining two. For example, the set $\{2, 4, 6, 8\}$ is good since the greatest common divisor of $2$ and $6$ is $2$, and it divides $4\times 8=32$. Find the greatest possible value of $n$, such that any four-element set with elements less than or equal to $n$ is good. [i]Proposed by Victor and Isaías de la Fuente[/i]