Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$

Let $a$, $b$ and $c$ be positive real numbers such that $abc=2015$. Prove that $$\frac{a+b}{a^2+b^2}+\frac{b+c}{b^2+c^2}+\frac{c+a}{c^2+a^2} \leq \frac{\sqrt{a}+\sqrt{b}+\sqrt{c}}{\sqrt{2015}}$$

In parallelogram $ABCD$ holds $AB=BD$. Let $K$ be a point on $AB$, different from $A$, such that $KD=AD$. Let $M$ be a point symmetric to $C$ with respect to $K$, and $N$ be a point symmetric to point $B$ with respect to $A$. Prove that $DM=DN$

Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put on to the right side of scale so scale is in balance. Find the minimal number of weights in the box

Solve the inequation: $$5\mid x\mid \leq x(3x+2-2\sqrt{8-2x-x^2})$$

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $$\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a} \leq \frac{a^2+b^2+c^2}{2}$$

Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$

On competition there were $67$ students. They were solving $6$ problems. Student who solves $k$th problem gets $k$ points, while student who solves incorrectly $k$th problem gets $-k$ points. $a)$ Prove that there exist two students with exactly the same answers to problems $b)$ Prove that there exist at least $4$ students with same number of points

For real numbers $x$, $y$ and $z$, solve the system of equations: $$x^3+y^3=3y+3z+4$$$$y^3+z^3=3z+3x+4$$$$x^3+z^3=3x+3y+4$$

Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$is prime

Let $F$ be an intersection point of altitude $CD$ and internal angle bisector $AE$ of right angled triangle $ABC$, $\angle ACB = 90^{\circ}$. Let $G$ be an intersection point of lines $ED$ and $BF$. Prove that area of quadrilateral $CEFG$ is equal to area of triangle $BDG$

There are $10001$ students at an university. Some students join together to form several clubs (a student may belong to different clubs). Some clubs join together to form several societies (a club may belong to different societies). There are a total of $k$ societies. Suppose that the following conditions hold: i.) Each pair of students are in exactly one club. ii.) For each student and each society, the student is in exactly one club of the society. iii.) Each club has an odd number of students. In addition, a club with ${2m+1}$ students ($m$ is a positive integer) is in exactly $m$ societies. Find all possible values of $k$. Proposed by Guihua Gong, Puerto Rico

Let $a$, $b$, $c$ and $d$ be real numbers such that $a+b+c+d=8$. Prove the inequality: $$\frac{a}{\sqrt[3]{8+b-d}}+\frac{b}{\sqrt[3]{8+c-a}}+\frac{c}{\sqrt[3]{8+d-b}}+\frac{d}{\sqrt[3]{8+a-c}} \geq 4$$

For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$

Let $O$ and $I$ be circumcenter and incenter of triangle $ABC$. Let incircle of $ABC$ touches sides $BC$, $CA$ and $AB$ in points $D$, $E$ and $F$, respectively. Lines $FD$ and $CA$ intersect in point $P$, and lines $DE$ and $AB$ intersect in point $Q$. Furthermore, let $M$ and $N$ be midpoints of $PE$ and $QF$. Prove that $OI \perp MN$

It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that: $a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled $b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal