Consider the set $$M = \left\{\frac{a}{\overline{ba}}+\frac{b}{\overline{ab}} \, | a,b\in\{1,2,3,4,5,6,7,8,9\} \right\}.$$a) Show that the set $M$ contains no integer. b) Find the smallest and the largest element of $M$

Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$. a) Prove that $AF \perp FC$. b) Determine the measure of the angle $AFB$.

In the square $ABCD$ denote by $M$ the midpoint of the side $[AB]$, with $P$ the projection of point $B$ on the line $CM$ and with $N$ the midpoint of the segment $[CP]$, Bisector of the angle $DAN$ intersects the line $DP$ at point $Q$. Show that the quadrilateral $BMQN$ is a parallelogram.

Find all prime numbers with $n \ge 3$ digits, having the property: for every $k \in \{1, 2, . . . , n -2\}$, deleting any $k$ of its digits leaves a prime number.

Prove the following: a) If $ABCA'B'C'$ is a right prism and $M \in (BC), N \in (CA), P \in (AB)$ such that $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent, then the prism $ABCA'B'C'$ is regular. b) If $ABCA'B'C'$ is a regular prism and $\frac{AA'}{AB}=\frac{\sqrt6}{4}$ , then there are $M \in (BC), N \in (CA), P \in (AB)$ so that the lines $A'M, B'N$ and $C'P$ are perpendicular each other and concurrent.

Show that for every integer $n \ge 3$ there exists positive integers $x_1, x_2, . . . , x_n$, pairwise different, so that $\{2, n\} \subset \{x_1, x_2, . . . , x_n\}$ and $$\frac{1}{x_1}+\frac{1}{x_2}+.. +\frac{1}{x_n}= 1.$$

Let $n \in N, n\ge 2$, and $a_1, a_2, ..., a_n, b_1, b_2, ..., b_n$ be real positive numbers such that $$\frac{a_1}{b_1} \le \frac{a_2}{b_2} \le ... \le\frac{a_n}{b_n}.$$Find the largest real $c$ so that $$(a_1-b_1c)x_1+(a_2-b_2c)x_2+...+(a_n-b_nc)x_n \ge 0,$$for every $x_1, x_2,..., x_n > 0$, with $x_1\le x_2\le ...\le x_n$.

Let $a, b, c, d \in [0, 1]$. Prove that $$\frac{a}{1 + b}+\frac{b}{1 + c}+\frac{c}{1 + d}+\frac{d}{1 + a}+ abcd \le 3.$$

Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.

Let be a square $ ABCD, $ a point $ E $ on $ AB, $ a point $ N $ on $ CD, $ points $ F,M $ on $ BC, $ name $ P $ the intersection of $ AN $ with $ DE, $ and name $ Q $ the intersection of $ AM $ with $ EF. $ If the triangles $ AMN $ and $ DEF $ are equilateral, prove that $ PQ=FM. $

Let be two natural numbers $ n $ and $ a. $ a) Prove that there exists an $ n\text{-tuplet} $ of natural numbers $ \left( a_1,a_2,\ldots ,a_n\right) $ that satisfy the following equality. $$ 1+\frac{1}{a} =\prod_{i=1}^n \left( 1+\frac{1}{a_i} \right) $$b) Show that there exist only finitely such $ n\text{-tuplets} . $

Let be two natural numbers $ b>a>0 $ and a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property. $$ f\left( x^2+ay\right)\ge f\left( x^2+by\right) ,\quad\forall x,y\in\mathbb{R} $$ a) Show that $ f(s)\le f(0)\le f(t) , $ for any real numbers $ s<0<t. $ b) Prove that $ f $ is constant on the interval $ (0,\infty ) . $ c) Give an example of a non-monotone such function.

Solve in the set of real numbers the equation $ a^{[ x ]} +\log_a\{ x \} =x , $ where $ a $ is a real number from the interval $ (0,1). $ $ [] $ and $ \{\} $ denote the floor, respectively, the fractional part.

A function $ f:\mathbb{Q}_{>0}\longrightarrow\mathbb{Q} $ has the following property: $$ f(xy)=f(x)+f(y),\quad x,y\in\mathbb{Q}_{>0} $$ a) Demonstrate that there are no injective functions with this property. b) Do exist surjective functions having this property?

$ \sin\frac{\pi }{4n}\ge \frac{\sqrt 2 }{2n} ,\quad \forall n\in\mathbb{N} $

Find the number of functions $ A\stackrel{f}{\longrightarrow } A $ for which there exist two functions $ A\stackrel{g}{\longrightarrow } B\stackrel{h}{\longrightarrow } A $ having the properties that $ g\circ h =\text{id.} $ and $ h\circ g=f, $ where $ B $ and $ A $ are two finite sets.

Let be a surjective function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that has the property that if the sequence $ \left( f\left( x_n \right) \right)_{n\ge 1} $ is convergent, then the sequence $ \left( x_n \right)_{n\ge 1} $ is convergent. Prove that it is continuous.

Let be two natural numbers $ n\ge 2, k, $ and $ k\quad n\times n $ symmetric real matrices $ A_1,A_2,\ldots ,A_k. $ Then, the following relations are equivalent: $ 1)\quad \left| \sum_{i=1}^k A_i^2 \right| =0 $ $ 2)\quad \left| \sum_{i=1}^k A_iB_i \right| =0,\quad\forall B_1,B_2,\ldots ,B_k\in \mathcal{M}_n\left( \mathbb{R} \right) $ $ || $ denotes the determinant.

Let be a natural number $ n\ge 2 $ and two $ n\times n $ complex matrices $ A,B $ that satisfy $ (AB)^3=O_n. $ Does this imply that $ (BA)^3=O_n ? $

Let be a function $ f $ of class $ \mathcal{C}^1[a,b] $ whose derivative is positive. Prove that there exists a real number $ c\in (a,b) $ such that $$ f(f(b))-f(f(a))=(f'(c))^2(b-a) . $$

a) Let be a continuous function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R}_{>0} . $ Show that there exists a natural number $ n_0 $ and a sequence of positive real numbers $ \left( x_n \right)_{n>n_0} $ that satisfy the following relation. $$ n\int_0^{x_n} f(t)dt=1,\quad n_0<\forall n\in\mathbb{N} $$ b) Prove that the sequence $ \left( nx_n \right)_{n> n_0} $ is convergent and find its limit.

Let be a natural number $ n $ and $ 2n $ real numbers $ b_1,b_2,\ldots ,b_n,a_1<a_2<\cdots <a_n. $ Show that a) if $ b_1,b_2,\ldots ,b_n>0, $ then there exists a polynomial $ f\in\mathbb{R}[X] $ irreducible in $ \mathbb{R}[X] $ such that $$ f\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$b) there exists a polynom $ g\in\mathbb{R} [X] $ of degree at least $ 1 $ which has only real roots and such that $$ g\left( a_i \right) =b_i,\quad\forall i\in\{ 1,2,\ldots ,n \} . $$

Let $G$ be a finite group with the following property: If $f$ is an automorphism of $G$, then there exists $m\in\mathbb{N^\star}$, so that $f(x)=x^{m} $ for all $x\in G$. Prove that G is commutative. Marian Andronache

A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $ a) Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $ b) Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.