Let $0 \leq \alpha , \beta , \gamma \leq \frac{\pi}{2}$, such that $\cos ^{2} \alpha + \cos ^{2} \beta + \cos ^{2} \gamma = 1$. Prove that $2 \leq (1 + \cos ^{2} \alpha ) ^{2} \sin^{4} \alpha + (1 + \cos ^{2} \beta ) ^{2} \sin ^{4} \beta + (1 + \cos ^{2} \gamma ) ^{2} \sin ^{4} \gamma \leq (1 + \cos ^{2} \alpha )(1 + \cos ^{2} \beta)(1 + \cos ^{2} \gamma ).$

Find $4$ consecutive positive integers whose product is $1680$.

Let $m_n$ be the smallest value of the function ${{f}_{n}}\left( x \right)=\sum\limits_{k=0}^{2n}{{{x}^{k}}}$ Show that $m_n \to \frac{1}{2}$, as $n \to \infty.$

Find the largest real $m$, such that for all positive real $a, b, c$ with sum $1$, the inequality $\sqrt{\frac{ab} {ab+c}}+\sqrt{\frac{bc} {bc+a}}+\sqrt{\frac{ca} {ca+b}} \geq m$ is satisfied.

Let $ f(x)$ be a polynomial with integer coefficients and let $ p$ be a prime. Denote by $ z_1,...,z_{p\minus{}1}$ the $ (p\minus{}1)$th complex roots of unity. Prove that $ f(z_1)\cdots f(z_{p\minus{}1})\equiv f(1)\cdots f(p\minus{}1) \pmod{p}$.

Find all function $f:[0,\infty )\to\mathbb{R}$ such that $f$ is monotonic and \[ [f(x)+f(y)]^2=f(x^2-y^2)+f(2xy) \] for all $x\geq y\geq 0$

Positive integers $x, y, z$ satisfy $(x + yi)^2 - 46i = z$. What is $x + y + z$?

Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.

Find all real solutions of the system of equations $$x^2-y^2=2(xz+yz+x+y),$$$$y^2-z^2=2(yx+zx+y+z),$$$$z^2-x^2=2(zy+xy+z+x).$$

Let $a$, $b$, and $c$ be the $3$ roots of $x^3-x+1=0$. Find $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}.$

Let $x,y,z$ be positive numbers with the sum $1$. Prove that $$\frac x{y^2+z}+\frac y{z^2+x}+\frac z{x^2+y}\ge\frac94.$$

Let $a,b,c$ be distinct real numbers and $P(x)$ a polynomial with real coefficients. Suppose that the remainders of $P(x)$ upon division by $(x-a), (x-b)$ and $(x-c)$ are $a,b$ and $c$, respectively. Find the polynomial that is obtained as the remainder of $P(x)$ upon division by $(x-a)(x-b)(x-c)$.

Let f be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$. If f has exactly $2$ integer roots, how many such polynomials $f$ exist?

There are real numbers $x, y$ such that $x \ne 0$, $y \ne 0$, $xy + 1 \ne 0$ and $x + y \ne 0$. Suppose the numbers $x + \frac{1}{x} + y + \frac{1}{y}$ and $x^3+\frac{1}{x^3} + y^3 + \frac{1}{y^3}$ are rational. Prove that then the number $x^2+\frac{1}{x^2} + y^2 + \frac{1}{y^2}$ is also rational.

Let $n$ be a positive integer. Prove that there exist polynomials$f(x)$and $g(x$) with integer coefficients such that \[f(x)\left(x + 1 \right)^{2^n}+ g(x) \left(x^{2^n}+ 1 \right) = 2.\]

Let be a natural number $ n $ and $ n $ real numbers $ a_1,a_2,\ldots ,a_n $ such that $$ a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} ,\quad\forall m\in\{ 1,2,\ldots ,n \} . $$ Prove that $ a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} . $

There is a unique degree-$10$ monic polynomial with integer coefficients $f(x)$ such that $$f \left( \sum^9_{j=0}\sqrt[10]{2021^j}\right)= 0.$$ Find the remainder when $f(1)$ is divided by $1000$.

Let $P_1(x) = ax^2 - bx - c$, $P_2(x) = bx^2 - cx - a$, $P_3(x) = cx^2 - ax - b$ be three quadratic polynomials. Suppose there exists a real number $\alpha$ such that $P_1(\alpha) = P_2(\alpha) = P_3(\alpha)$. Prove that $a = b = c$.

Vessel $A$ has liquids $X$ and $Y$ in the ratio $X:Y=8:7$. Vessel $B$ holds a mixture of $X$ and $Y$ in the ratio $X:Y=5:9$. What ratio should you mix the liquids in both vessels if you need the mixture to be $X:Y=1:1$? $\textbf{(A)}~4:3$ $\textbf{(B)}~30:7$ $\textbf{(C)}~17:25$ $\textbf{(D)}~7:30$

Prove that for any function $f(x)$, continuous or otherwise, $$f(f(x)) = x^2 - 1996$$ cannot hold for all real numbers $x$. (S Bogatiy, M Smurov,)

[b]p4[/b] Define the sequence ${a_n}$ as follows: 1) $a_1 = -1$, and 2) for all $n \ge 2$, $a_n = 1 + 2 + . . . + n - (n + 1)$. For example, $a_3 = 1+2+3-4 = 2$. Find the largest possible value of $k$ such that $a_k+a_{k+1} = a_{k+2}$. [b]p5[/b] The taxicab distance between two points $(a, b)$ and $(c, d)$ on the coordinate plane is $|c-a|+|d-b|$. Given that the taxicab distance between points $A$ and $B$ is $8$ and that the length of $AB$ is $k$, find the minimum possible value of $k^2$. [b]p6[/b] For any two-digit positive integer $\overline{AB}$, let $f(\overline{AB}) = \overline{AB}-A\cdot B$, or in other words, the result of subtracting the product of its digits from the integer itself. For example, $f(\overline{72}) = 72-7\cdot 2 = 58$. Find the maximum possible $n$ such that there exist distinct two-digit integers$ \overline{XY}$ and $\overline{WZ}$ such that $f(\overline{XY} ) = f(\overline{WZ}) = n$.

Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions: [b](i)[/b] $ \forall m,n \geq 1$ we have $ a_{m \plus{} n} \leq 2 \left(a_m \plus{} a_n \right)$ [b](ii)[/b] $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k \plus{} 1)^{2008}}$ such that for each sequence element we have the inequality $ a_n \leq Q.$

Suppose that $ a$, $ b$, and $ c$ are positive real numbers such that $ a^{\log_3 7} \equal{} 27$, $ b^{\log_7 11} \equal{} 49$, and $ c^{\log_{11} 25} \equal{} \sqrt {11}$. Find \[ a^{(\log_3 7)^2} \plus{} b^{(\log_7 11)^2} \plus{} c^{(\log_{11} 25)^2}. \]

A square $ R$ of sidelength $ 250$ lies inside a square $ Q$ of sidelength $ 500$. Prove that: One can always find two points $ A$ and $ B$ on the perimeter of $ Q$ such that the segment $ AB$ has no common point with the square $ R$, and the length of this segment $ AB$ is greater than $ 521$.

[color=darkred]Find all functions $f:\mathbb{R}\to\mathbb{R}$ with the following property: for any open bounded interval $I$, the set $f(I)$ is an open interval having the same length with $I$ .[/color]