Let $n \in \mathbb{N}, n \ge 2$ and the positive real numbers $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ such that $a_1+a_2+…+a_n=b_1+b_2+…+b_n=S.$ $\textbf{a)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}.$ $\textbf{b)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.$

Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

Show that \[ \int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n} \] for all positive integers $n$.

Let $S$ be a sequence $n_1, n_2,..., n_{1995}$ of positive integers such that $n_1 +...+ n_{1995 }=m < 3990$. Prove that for each integer $q$ with $1 \le q \le m$, there is a sequence $n_{i_1} , n_{i_2} , ... , n_{i_k}$ , where $1 \le i_1 < i_2 < ...< i_k \le 1995$, $n_{i_1} + ...+ n_{i_k} = q$ and $k$ depends on $q$.

A father and two sons went to visit their grandmother, who Raya lives $33$ km from the city. My father has a motor roller, the speed of which $25$ km/h, and with a passenger - $20$ km/h (with two passengers on a scooter It’s impossible to move). Each of the brothers walks along the road at a speed of $5$ km/h. Prove that all three can get to grandma's in $3$ hours

Prove that every positive rational number can be expressed uniquely as a finite sum of the form $$a_1+\frac{a_2}{2!}+\frac{a_3}{3!}+\dots+\frac{a_n}{n!},$$ where $a_n$ are integers such that $0 \leq a_n \leq n-1$ for all $n > 1$.

Let $f : R \to R$ be a function satisfying the inequality $|f(x + y) -f(x) - f(y)| < 1$ for all reals $x, y$. Show that $\left| f\left( \frac{x}{2008 }\right) - \frac{f(x)}{2008} \right|  < 1$ for all real numbers $x$.

a) Factorize $A= x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2$ b) Prove that there are no integers $x,y,z$ such that $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=2000 $

Without using tables, find the exact value of the product: \[P = \prod^7_{k=1} \cos \left(\frac{k \pi}{15} \right).\]

For the function $f : Z^2_{\ge0} \to Z_{\ge 0}$ it is known that $$f(0, j) = f(i, 0) = 1, \,\,\,\,\, \forall i, j \in N_0$$ $$f(i, j) = if (i, j - 1) + jf(i - 1, j),\,\,\,\,\, \forall i, j \in N$$ Prove that for every natural number $n$ the following inequality holds: $$\sum_{0\le i+j\le n+1} f(i, j) \le 2 \left(\sum^n_{k=0}\frac{1}{k!}\right)\left(\sum^n_{p=1}p!\right)+ 3$$

Tyler has an infinite geometric series with sum $10$. He increases the first term of his sequence by $4$ and swiftly changes the subsequent terms so that the common ratio remains the same, creating a new geometric series with sum $15$. Compute the common ratio of Tyler’s series.

Which of the following rationals is greater , $\frac{1995^{1994} + 1}{1995^{1995} + 1}$ or $\frac{1995^{1995} + 1}{ 1995^{1996} +1}$ ?

Find all pairs of positive integers $(x, y)$ such that $(xy+1)(xy+x+2)$ be a perfect square .

Let $-1 < x < 1.$ Show that \[ \sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}. \] Deduce that \[ \csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8. \]

[b]p4.[/b] For how many three-digit multiples of $11$ in the form $\underline{abc}$ does the quadratic $ax^2 + bx + c$ have real roots? [b]p5.[/b] William draws a triangle $\vartriangle ABC$ with $AB =\sqrt3$, $BC = 1$, and $AC = 2$ on a piece of paper and cuts out $\vartriangle ABC$. Let the angle bisector of $\angle ABC$ meet $AC$ at point $D$. He folds $\vartriangle ABD$ over $BD$. Denote the new location of point $A$ as $A'$. After William folds $\vartriangle A'CD$ over $CD$, what area of the resulting figure is covered by three layers of paper? [b]p6.[/b] Compute $(1)(2)(3) + (2)(3)(4) + ... + (18)(19)(20)$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the points of intersection of the graphs $y = \cos x$ and $x = 100 \cos (100y)$ for which both coordinates are positive. Let $a$ be the sum of their $x$-coordinates and $b$ be the sum of their $y$-coordinates. Determine the value of $\frac{a}{b}$.

We call a set $ S$ on the real line $ \mathbb{R}$ [i]superinvariant[/i] if for any stretching $ A$ of the set by the transformation taking $ x$ to $ A(x) \equal{} x_0 \plus{} a(x \minus{} x_0), a > 0$ there exists a translation $ B,$ $ B(x) \equal{} x\plus{}b,$ such that the images of $ S$ under $ A$ and $ B$ agree; i.e., for any $ x \in S$ there is a $ y \in S$ such that $ A(x) \equal{} B(y)$ and for any $ t \in S$ there is a $ u \in S$ such that $ B(t) \equal{} A(u).$ Determine all [i]superinvariant[/i] sets.

The natural numbers $ a_1 $, $ a_2 $, $ \dots $, $ a_n $ satisfy the condition $ 1 / a_1 + 1 / a_2 + \ldots + 1 / a_n = 1 $. Prove that all these numbers do not exceed $$ n ^ {2 ^ n} $$

Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as \begin{align*} (x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z) \end{align*} with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.

Let $a_0=2$ and for $n\geq 1$, $a_n=\frac{\sqrt3 a_{n-1}+1}{\sqrt3-a_{n-1}}$. Find the value of $a_{2002}$ in the form $p+q\sqrt3$ where $p$ and $q$ are rational numbers

Show that the equation $x^2 + y^2 + z^2 = ( x-y)(y-z)(z-x)$ has infintely many solutions in integers $x,y,z$.

Let $x,y{}$ and $z{}$ be positive real numbers satisfying $x+y+z=1$. Prove that [list=a] [*]\[1-\frac{x^2-yz}{x^2+x}=\frac{(1-y)(1-z)}{x^2+x};\] [*]\[\frac{x^2-yz}{x^2+x}+\frac{y^2-zx}{y^2+y}+\frac{z^2-xy}{z^2+z}\leqslant 0.\] [/list]

[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]

Determine all pairs of [i]distinct[/i] real numbers $(x, y)$ such that both of the following are true: [list] [*]$x^{100} - y^{100} = 2^{99} (x-y)$ [*]$x^{200} - y^{200} = 2^{199} (x-y)$ [/list]

Let $a$, $b$, $c$ be the real numbers. It is true, that $a + b$, $b + c$ and $c + a$ are three consecutive integers, in a certain order, and the smallest of them is odd. Prove that the numbers $a$, $b$, $c$ are also consecutive integers in a certain order.