Prove that \[ \frac{1}{a^2+2} + \frac{1}{b^2+2} + \frac{1}{c^2+2} \le \frac{1}{6ab+c^2} + \frac{1}{6bc+a^2} + \frac{1}{6ca+b^2} \] for all positive real numbers $a$, $b$ and $c$ satisfying $a^2+b^2+c^2=1$.

Let $P$ be a fourth degree polynomial, with derivative $P^\prime$, such that $P(1)=P(3)=P(5)=P^\prime (7)=0$. Find the real number $x\neq 1,3,5$ such that $P(x)=0$.

Let $f: \mathbb{R}\rightarrow\mathbb{R}$ be a differentiable function with continuous derivative, that satisfies $f\big(x+f'(x)\big)=f(x)$. Let's call this property $(P)$. a) Show that if $f$ is a function with property $(P)$, then there exists a real $x$ such that $f'(x)=0$. b) Give an example of a non-constant function $f$ with property $(P)$. c) Show that if $f$ has property $(P)$ and the equation $f'(x)=0$ has at least two solutions, then $f$ is a constant function.

Prove that $\sin\sqrt{x}<\sqrt{\sin x}$ for every real $x$ such that $0<x<\frac{\pi}{2}$.

Let $f(x)=\frac{P(x)}{Q(x)}$, where $P(x), Q(x)$ are two non-constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right)$ for infinitely many values of $x$. a) Show that $\text{deg}(P)<\text{deg}(Q)$. b) Show that $P'(1)=2Q'(1)-\text{deg}(Q)\cdot Q(1)$. Here, $P'(x)$ denotes the derivative of $P(x)$ as usual.

Let $ f : \mathbf {R} \to \mathbf {R} $ be a smooth function such that $ f'(x)^2 = f(x) f''(x) $ for all $x$. Suppose $f(0)=1$ and $f^{(4)} (0) = 9$. Find all possible values of $f'(0)$.

Find the slopes of all lines passing through the origin and tangent to the curve $y^2=x^3+39x-35$.

Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.

The parabola $P$ has focus $(0,0)$ and goes through the points $(4,3)$ and $(-4,-3)$. For how many points $(x,y)\in P$ with integer coefficients is it true that $|4x+3y|\leq 1000$? $\textbf{(A) }38\qquad \textbf{(B) }40\qquad \textbf{(C) }42\qquad \textbf{(D) }44\qquad \textbf{(E) }46\qquad$

2 curves $ y \equal{} x^3 \minus{} x$ and $ y \equal{} x^2 \minus{} a$ pass through the point $ P$ and have a common tangent line at $ P$. Find the area of the region bounded by these curves.

For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$ (1) Find $\lim_{t\rightarrow 0} F(t).$ (2) Find the range of $t$ such that $F(t)\geq 1.$

Let $a,b,c \in \left[ \frac 12, 1 \right]$. Prove that \[ 2 \leq \frac{ a+b}{1+c} + \frac{ b+c}{1+a} + \frac{ c+a}{1+b} \leq 3 . \] [i]selected by Mircea Lascu[/i]

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that for all $x \in \mathbb R$ and for all $n \in \mathbb N^{\ast}$ we have \[ n^2 \int_{x}^{x + \frac{1}{n}} f(t) \, dt = n f(x) + \frac12 . \] [i]Mihai Piticari[/i]

Do either (1) or (2): (1) Show that any solution $f(t)$ of the functional equation $$f(x+y)f(x-y)=f(x)^{2} +f(y)^{2} -1$$ for $x,y\in \mathbb{R}$ satisfies $$f''(t)= \pm c^{2} f(t)$$ for a constant $c$, assuming the existence and continuity of the second derivative. Deduce that $f(t)$ is one of the functions $$ \pm \cos ct, \;\;\; \pm \cosh ct.$$ (2) Let $(a_{i})_{i=1,...,n}$ and $(b_{i})_{i=1,...,n}$ be real numbers. Define an $(n+1)\times (n+1)$-matrix $A=(c_{ij})$ by $$ c_{i1}=1, \; \; c_{1j}= x^{j-1} \; \text{for} \; j\leq n,\; \; c_{1n+1}=p(x), \;\; c_{ij}=a_{i-1}^{j-1} \; \text{for}\; i>1, j\leq n,\;\; c_{in+1}=b_{i-1}\; \text{for}\; i>1.$$ The polynomial $p(x)$ is defined by the equation $\det A=0$. Let $f$ be a polynomial and replace $(b_{i})$ with $(f(b_{i}))$. Then $\det A=0$ defines another polynomial $q(x)$. Prove that $f(p(x))-q(x)$ is a multiple of $$\prod_{i=1}^{n} (x-a_{i}).$$

Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that $a_1 + ... + a_n = b_1 + ... + b_n = 1$. Find the minimal value of $ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.

In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.

Consider an arc of a planar curve; let the radius of curvature at any point of the arc be a differentiable function of the arc length and its derivative be everywhere different from zero; moreover, let the total curvature be less than $ \frac{\pi}{2}$. Let $ P_1,P_2,P_3,P_4,P_5$ and $ P_6$ be any points on this arc, subject to the only condition that the radius of curvature at $ P_k$ is greater than at $ P_j$ if $ j<k$. Prove that the radius of the circle passing through the points $ P_1,P_3$ and $ P_5$ is less than the radius of the circle through $ P_2,P_4$ and $ P_6$

Prove that if $|x| < 1$, then \[ \frac{x}{(1-x)^2}+\frac{x^2}{(1+x^2)^2} + \frac{x^3}{(1-x^3)^2}+\cdots=\frac{x}{1-x}+\frac{2x^2}{1+x^2}+\frac{3x^3}{1-x^3}+\cdots\]

Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.

If $ a\geq b\geq c\geq d > 0$ such that $ abcd\equal{}1$, then prove that \[ \frac 1{1\plus{}a} \plus{} \frac 1{1\plus{}b} \plus{} \frac 1{1\plus{}c} \geq \frac {3}{1\plus{}\sqrt[3]{abc}}.\]

Prove that for any real number $ x$ the following inequality is true: $ \max\{|\sin x|, |\sin(x\plus{}2010)|\}>\dfrac1{\sqrt{17}}$

For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$: \[p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1\] a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points) \[(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)\] b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points) \[\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}\] c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that: \[|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1\] Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)

Let \( p \) be a monic polynomial with all distinct real roots. Show that there exists \( K \) such that \[ (p(x)^2)'' \leq K(p'(x))^2. \]

Let $f$ be a function defined on $(0, \infty )$ as follows: \[f(x)=x+\frac1x\] Let $h$ be a function defined for all $x \in (0,1)$ as \[h(x)=\frac{x^4}{(1-x)^6}\] Suppose that $g(x)=f(h(x))$ for all $x \in (0,1)$. (a) Show that $h$ is a strictly increasing function. (b) Show that there exists a real number $x_0 \in (0,1)$ such that $g$ is strictly decreasing in the interval $(0,x_0]$ and strictly increasing in the interval $[x_0,1)$.

Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that $$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$ Prove that $$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$