For the real numbers $a, b, c, d$, the following inequalities hold: $$a + b + c \le 3d, \,\,\, b + c + d \le 3a, \,\,\,c + d + a \le 3b, \,\,\,d + a + b\le 3c.$$Compare the numbers $a, b, c, d$.

Knowing that $1 < y < 2$ and $x - y + 1 = 0,$ calculate the value of the expression: $$E = \sqrt{4x^2 +4y-3} + 2\sqrt{y^2 - 6x - 2y +10}.$$

If $a, b, c$ represent the lengths of the sides of a triangle, prove that: $$\frac{a}{b-a+c}+ \frac{b}{b-a+c}+ \frac{c}{b-a+c} \ge 3$$

Let $ABC$ be an acute-angled triangle and variable $D \in [BC]$ . Let's denote by $E, F$ the feet of the perpendiculars from $D$ to $AB$, $AC$ respectively . a) Show that $$\frac{4S^2}{b^2+c^2}\le DE^2 + DF^2\le max \{h_B^2 + h_C^2 \}.$$b) Proved that, if $D_0 \in [BC]$ is the point where the minimum of the sum $DE^2 + DF^2$ is achieved, then $D_0$ is the leg of the symmetrical median of $A$ facing the bisector of angle $A$. c) Specify the position, of $D \in [BC]$ for which the maximum of the sum $DE^2 + DF^2$ is achieved. (The area of the triangle $ABC$ was denoted by $S$ and $h_b, h_c$ are the lengths of the altitudes from $B$ and $C$ respectively)

In the square $ABCD$ we consider $ E \in (AB)$, $ F \in (AD)$ and $EF \cap AC = \{P\}$. Show that: a) $\frac{1}{AE} + \frac{1}{AF} = \frac{\sqrt2}{AP}$ b) $AP^2 \le \frac{AE \cdot AF}{2}$

a) Show that the number $(2k + 1)^3 - (2k - 1)^3$, $k \in Z$, is the sum of three perfect squares. b) Represent the number $(2n + 1)^3 -2$, $n \in N^*$, as the sum of $3n- 1$ perfect squares greater than $1$.

The negative real numbers $x, y, z, t$ satisfy simultaneously equalities, $$x + y + z = t, \,\,\,\,\frac{1}{x}+ \frac{1}{y}+\frac{1}{z}= \frac{1}{t}, \\,\,\,\, x^3 + y^3 + z^3 = 1000^3$$Compute $x + y + z + t$.

Let $SABC$ be the pyramid where$ m(\angle ASB) = m(\angle BSC) = m(\angle CSA) = 90^o$. Show that, whatever the point $M \in AS$ is and whatever the point $N \in BC$ is, holds the relation $$\frac{1}{MN^2} \le \frac{1}{SB^2} + \frac{1}{SC^2}.$$

In the rectangular parallelepiped $ABCDA'B'C'D'$, the points $E$ and $F$ are the centers of the faces $ABCD$ and $ADD' A'$, respectively, and the planes $(BCF)$ and $(B'C'E)$ are perpendicular. Let $A'M \perp B'A$, $M \in B'A$ and $BN \perp B'C$, $N \in B'C$. Denote $n = \frac{C'D}{BN}$. a) Show that $n \ge \sqrt2$. . b) Express and in terms of $n$, the ratio between the volume of the tetrahedron $BB'M N$ and the volume of the parallelepiped $ABCDA'B'C'D'$.

Let be two natural primes $ 1\le q \le p. $ Prove that $ \left( \sqrt{p^2+q} +p\right)^2 $ is irrational and its fractional part surpasses $ 3/4. $

Let $ A,B $ be two points in a plane and let two numbers $ a,b\in (0,1) . $ For each point $ M $ that is not on the line $ AB $ consider $ P $ on the segment $ AM $ and $ N $ on $ BM $ (both excluding the extremities) such that $ BN=b\cdot BM $ and $ AP=a\cdot AM. $ Find the locus of the points $ M $ for which $ AN=BP. $

Let be a natural number $ n\ge 2 $ and an expression of $ n $ variables $$ E\left( x_1,x_2,...,x_n\right) =x_1^2+x_2^2+\cdots +x_n^2-x_1x_2-x_2x_3-\cdots -x_{n-1}x_n -x_nx_1. $$Determine $ \sup_{x_1,...,x_n\in [0,1]} E\left( x_1,x_2,...,x_n\right) $ and the specific values at which this supremum is attained.

Let $ I $ be the center of the incircle of a triangle $ ABC. $ Shw that, if for any point $ M $ on the segment $ AB $ (extremities excluded) there exist two points $ N,P $ on $ BC, $ respectively, $ AC $ (both excluding the extremities) such that the center of mass of $ MNP $ coincides with $ I, $ then $ ABC $ is equilateral.

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence having $ x_1=3 $ and defined as $ x_{n+1} =\left\lfloor \sqrt 2x_n\right\rfloor , $ for every natural number $ n. $ Find all values $ m $ for which the terms $ x_m,x_{m+1},x_{m+2} $ are in arithmetic progression, where $ \lfloor\rfloor $ denotes the integer part.

Demonstrate that if $ z_1,z_2\in\mathbb{C}^* $ satisfy the relation: $$ z_1\cdot 2^{\big| z_1\big|} +z_2\cdot 2^{\big| z_2\big|} =\left( z_1+z_2\right)\cdot 2^{\big| z_1 +z_2\big|} , $$then $ z_1^6=z_2^6 $

Let be a tetahedron $ ABCD, $ and $ E $ be the projection of $ D $ on the plane formed by $ ABC. $ If $ \mathcal{A}_{\mathcal{R}} $ denotes the area of the region $ \mathcal{R}, $ show that the following affirmations are equivalent: a) $ C=E\vee CE\parallel AB $ b) $ M\in\overline{CD}\implies\mathcal{A}_{ABM}^2=\frac{CM^2}{CD^2}\cdot\mathcal{A}_{ABD}^2 +\left( 1-\frac{CM^2}{CD^2}\right)\cdot\mathcal{A}_{ABC}^2 $

Let $ f $ be a polynom of degree $ 3 $ and having rational coefficients. Prove that, if there exist two distinct nonzero rational numbers $ a,b $ and two roots $ x,y $ of $ f $ such that $ ax+by $ is rational, then all roots of $ f $ are rational.

Let $ \mathcal{M} =\left\{ A\in M_2\left( \mathbb{C}\right)\big| \det\left( A-zI_2\right) =0\implies |z| < 1\right\} . $ Prove that: $$ X,Y\in\mathcal{M}\wedge X\cdot Y=Y\cdot X\implies X\cdot Y\in\mathcal{M} . $$

Study the convergence of a sequence $ \left( x_n\right)_{n\ge 0} $ for which $ x_0\in\mathbb{R}\setminus\mathbb{Q} , $ and $ x_{n+1}\in \left\{ \frac{x_n+1}{x_n} , \frac{x_n+2}{2x_n-1}\right\} , $ for all $ n\ge 1. $

A function $ f:\mathbb{R}^2\longrightarrow\mathbb{R} $ is olympic if, any finite number of pairwise distinct elements of $ \mathbb{R}^2 $ at which the function takes the same value represent in the plane the vertices of a convex polygon. Prove that if $ p $ if a complex polynom of degree at least $ 1, $ then the function $ \mathbb{R}^2\ni (x,y)\mapsto |p(x+iy)| $ is olympic if and only if the roots of $ p $ are all equal.

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function that satisfies the conditions: $ \text{(i)}\quad \lim_{x\to\infty} (f\circ f) (x) =\infty =-\lim_{x\to -\infty} (f\circ f) (x) $ $ \text{(ii)}\quad f $ has Darboux’s property a) Prove that the limits of $ f $ at $ \pm\infty $ exist. b) Is possible for the limits from a) to be finite?

Let $ a\in (1,\infty) $ and a countinuous function $ f:[0,\infty)\longrightarrow\mathbb{R} $ having the property: $$ \lim_{x\to \infty} xf(x)\in\mathbb{R} . $$ a) Show that the integral $ \int_1^{\infty} \frac{f(x)}{x}dx $ and the limit $ \lim_{t\to\infty} t\int_{1}^a f\left( x^t \right) dx $ both exist, are finite and equal. b) Calculate $ \lim_{t\to \infty} t\int_1^a \frac{dx}{1+x^t} . $

For any partition $ P $ of $ [0,1] $ , consider the set $$ \mathcal{A}(P)=\left\{ f:[0,1]\longrightarrow\mathbb{R}\left| \exists f’\bigg|_{[0,1]}\right.\wedge\int_0^1 |f(x)|dx =1\wedge \left( y\in P\implies f (y ) =0\right)\right\} . $$Prove that there exists a partition $ P_0 $ of $ [0,1] $ such that $$ g\in \mathcal{A}\left( P_0\right)\implies \sup_{x\in [0,1]} \big| g’(x)\big| >4\cdot \# P. $$ Here, $ \# D $ denotes the natural number $ d $ such that $ 0=x_0<x_1<\cdots <x_d=1 $ is a partition $ D $ of $ [0,1] . $

We say that the abelian group $ G $ has property (P) if, for any commutative group $ H, $ any $ H’\le H $ and any momorphism $ \mu’:H\longrightarrow G, $ there exists a morphism $ \mu :H\longrightarrow G $ such that $ \mu\bigg|_{H’} =\mu’ . $ Show that: a) the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property (P). b) the group $ \left( \mathbb{Q}, +\right) $ has property (P).

Prove that a nontrivial finite ring is not a skew field if and only if the equation $ x^n+y^n=z^n $ has nontrivial solutions in this ring for any natural number $ n. $