For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$$$a^3+b^3+c^3+d^3=0$$Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero

In convex quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at point $O$ at angle $90^{\circ}$. Let $K$, $L$, $M$ and $N$ be orthogonal projections of point $O$ to sides $AB$, $BC$, $CD$ and $DA$ of quadrilateral $ABCD$. Prove that $KLMN$ is cyclic

If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$

In table of dimensions $2n \times 2n$ there are positive integers not greater than $10$, such that numbers lying in unit squares with common vertex are coprime. Prove that there exist at least one number which occurs in table at least $\frac{2n^2}{3}$ times

Find all real numbers $(x,y)$ satisfying the following: $$x+\frac{3x-y}{x^2+y^2}=3$$$$y-\frac{x+3y}{x^2+y^2}=0$$

It is given acute triangle $ABC$ with orthocenter at point $H$. Prove that $$AH \cdot h_a+BH \cdot h_b+CH \cdot h_c=\frac{a^2+b^2+c^2}{2}$$where $a$, $b$ and $c$ are sides of a triangle, and $h_a$, $h_b$ and $h_c$ altitudes of $ABC$

It is given set with $n^2$ elements $(n \geq 2)$ and family $\mathbb{F}$ of subsets of set $A$, such that every one of them has $n$ elements. Assume that every two sets from $\mathbb{F}$ have at most one common element. Prove that $i)$ Family $\mathbb{F}$ has at most $n^2+n$ elements $ii)$ Upper bound can be reached for $n=3$

Prove the inequality $$ \frac{y^2-x^2}{2x^2+1}+\frac{z^2-y^2}{2y^2+1}+\frac{x^2-z^2}{2z^2+1} \geq 0$$where $x$, $y$ and $z$ are real numbers

Angle bisector from vertex $A$ of acute triangle $ABC$ intersects side $BC$ in point $D$, and circumcircle of $ABC$ in point $E$ (different from $A$). Let $F$ and $G$ be foots of perpendiculars from point $D$ to sides $AB$ and $AC$. Prove that area of quadrilateral $AEFG$ is equal to the area of triangle $ABC$

Let $n$ be an odd positive integer bigger than $1$. Prove that $3^n+1$ is not divisible with $n$

In plane there are $n$ noncollinear points $A_1$, $A_2$,...,$A_n$. Prove that there exist a line which passes through exactly two of these points

It is given positive real number $a$ such that: $$\left\{\frac{1}{a}\right\}=\{a^2\}$$$$ 2<a^2<3$$Find the value of $$a^{12}-\frac{144}{a}$$

Let $AA_1$, $BB_1$ and $CC_1$ be altitudes of triangle $ABC$ and let $A_1A_2$, $B_1B_2$ and $C_1C_2$ be diameters of Euler circle of triangle $ABC$. Prove that lines $AA_2$, $BB_2$ and $CC_2$ are concurrent