Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$

Outside the square $ABCD$, the rhombus $BCMN$ is constructed with angle $BCM$ obtuse . Let $P$ be the intersection point of the lines $BM$ and $AN$ . Prove that $DM \perp CP$ and the triangle $DPM$ is right isosceles .

Find all positive integers $n$ so that $$17^n +9^{n^2} = 23^n +3^{n^2} .$$

Outside the square $ABCD$ is constructed the right isosceles triangle $ABD$ with hypotenuse $[AB]$. Let $N$ be the midpoint of the side $[AD]$ and ${M} = CE \cap AB$, ${P} = CN \cap AB$ , ${F} = PE \cap MN$. On the line $FP$ the point $Q$ is considered such that the $[CE$ is the bisector of the angle $QCB$. Prove that $MQ \perp CF$.

Let $a,b,c\in \left( 0,\infty \right)$.Prove the inequality $\frac{a-\sqrt{bc}}{a+2\left( b+c \right)}+\frac{b-\sqrt{ca}}{b+2\left( c+a \right)}+\frac{c-\sqrt{ab}}{c+2\left( a+b \right)}\ge 0.$

Let $ABCDA'B'C'D'$ be a cube with side $AB = a$. Consider points $E \in (AB)$ and $F \in (BC)$ such that $AE + CF = EF$. a) Determine the measure the angle formed by the planes $(D'DE)$ and $(D'DF)$. b) Calculate the distance from $D'$ to the line $EF$.

Find the smallest integer $n$ for which the set $A = \{n, n +1, n +2,...,2n\}$ contains five elements $a<b<c<d<e$ so that $$\frac{a}{c}=\frac{b}{d}=\frac{c}{e}$$

Prove that three discs of radius $1$ cannot cover entirely a square surface of side $2$, but they can cover more than $99.75\%$ of it.

Find x, y, z $\in Z$ $x^2+y^2+z^2=2^n(x+y+z)$ $n\in N$

Let $ a $ be an odd natural that is not a perfect square, and $ m,n\in\mathbb{N} . $ Then a) $ \left\{ m\left( a+\sqrt a \right) \right\}\neq\left\{ n\left( a-\sqrt a \right) \right\} $ b) $ \left[ m\left( a+\sqrt a \right) \right]\neq\left[ n\left( a-\sqrt a \right) \right] $ Here, $ \{\},[] $ denotes the fractionary, respectively the integer part.

Let $ P,Q $ be the midpoints of the diagonals $ BD, $ respectively, $ AC, $ of the quadrilateral $ ABCD, $ and points $ M,N,R,S $ on the segments $ BC,CD,PQ, $ respectively $ AC, $ except their extremities, such that $$ \frac{BM}{MC}=\frac{DN}{NC}=\frac{PR}{RQ}=\frac{AS}{SC} . $$Show that the center of mass of the triangle $ AMN $ is situated on the segment $ RS. $

Let $ ABCD $ be a quadrilateral inscribed in a circle of diameter $ AC. $ Fix points $ E,F $ of segments $ CD, $ respectively, $ BC $ such that $ AE $ is perpendicular to $ DF $ and $ AF $ is perpendicular to $ BE. $ Show that $ AB=AD. $

Let be a natural number $ n. $ Calculate $$ \sum_{k=1}^{n^2}\#\left\{ d\in\mathbb{N}| 1\le d\le k\le d^2\le n^2\wedge k\equiv 0\pmod d \right\} . $$ Here, $ \# $ means cardinal.

Let be a function $ f:\mathbb{N}\longrightarrow\mathbb{N} $ satisfying $ \text{(i)} f(1)=1 $ $ \text{(ii)} f(p)=1+f(p-1), $ for any prime $ p $ $ \text{(iii)} f(p_1p_2\cdots p_u)=f(p_1)+f(p_2)+\cdots f(p_u), $ for any natural number $ u $ and any primes $ p_1,p_2,\ldots ,p_u. $ Show that $ 2^{f(n)}\le n^3\le 3^{f(n)}, $ for any natural $ n\ge 2. $

Let $ n $ be a natural number, and $ A $ the set of the first $ n $ natural numbers. Find the number of nondecreasing functions $ f:A\longrightarrow A $ that have the property $$ x,y\in A\implies |f(x)-f(y)|\le |x-y|. $$

Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then: P. $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$ Q. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$ Prove that $ P \Longleftrightarrow Q$

Find all continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy: $ \text{(i)}\text{id}+f $ is nondecreasing $ \text{(ii)} $ There is a natural number $ m $ such that $ \text{id}+f+f^2\cdots +f^m $ is nonincreasing. Here, $ \text{id} $ represents the identity function, and ^ denotes functional power.

Find all derivable functions that have real domain and codomain, and are equal to their second functional power.

Let $A,B\in M_n(C)$ be two square matrices satisfying $A^2+B^2 = 2AB$. 1.Prove that $\det(AB-BA)=0$. 2.If $rank(A-B)=1$, then prove that $AB=BA$.

Let $ A\in\mathcal{M}_4\left(\mathbb{R}\right) $ be an invertible matrix whose trace is equal to the trace of its adjugate, which is nonzero. Show that $ A^2+I $ is singular if and only if there exists a nonzero matrix in $ \mathcal{M}_4\left( \mathbb{R} \right) $ that anti-commutes with it.

For a ring $ A, $ and an element $ a $ of it, define $ s_a,d_a:A\longrightarrow A, s_a(x)=ax,d_a=xa.$ a) Prove that if $ A $ is finite, then $ s_a $ is injective if and only if $ d_a $ is injective. b) Give example of a ring which has an element $ b $ for which $ s_b $ is injective and $ d_b $ is not, or, conversely, $ s_b $ is not injective, but $ d_b $ is.

Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $ Show that $ f $ and $ g $ are the same function.

Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties: $ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $ $ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $ a) Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $ b) Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $

Let be a finite group $ G $ that has an element $ a\neq 1 $ for which exists a prime number $ p $ such that $ x^{1+p}=a^{-1}xa, $ for all $ x\in G. $ a) Prove that the order of $ G $ is a power of $ p. $ b) Show that $ H:=\{x\in G|\text{ord} (x)=p\}\le G $ and $ \text{ord}^2(H)>\text{ord}(G). $