Let $a, b$ be two real numbers such that $$\sqrt[3]{a}- \sqrt[3]{b} = 10, ,\,\,\,\,\,\, ab = \left( \frac{8 - a - b}{6}\right)^3$$ Find $a - b$.

Consider $70$-digit numbers with the property that each of the digits $1,2,3,...,7$ appear $10$ times in the decimal expansion of $n$ (and $8,9,0$ do not appear). Show that no number of this form can divide another number of this form.

On a set $G$ we are given an operation $*: G \times G \to G$, that for every pair $(x,y)$ of elements of $G$ gives back $x*y \in G$, and for every elements $x,y,z \in G$ the equation $(x*y)*z=x*(y*z)$ holds. $G$ is partitioned into three non-empty sets $A,B$ and $C$. Can it be that for every three elements $a \in A, b \in B, c \in C$ we have $a*b \in C, b*c \in A, c*a \in B$

We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours.

If $f(n)=\tfrac{1}{3}n(n1)(n+2)$, then $f(r)-f(r-1)$ equals: $\textbf{(A)}\ r(r+1) \qquad \textbf{(B)}\ (r+1)(r+2) \qquad \textbf{(C)}\ \tfrac{1}{3}r(r+1) \qquad\\ \textbf{(D)}\ \tfrac{1}{3}(r+1)(r+2) \qquad \textbf{(E)}\ \tfrac{1}{3}r(r+1)(r+2) $

Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$ for any positive integer $ n $ not less than 2 . Find the minimum of $m.$

Rectangle $ PQRS$ lies in a plane with $ PQ = RS = 2$ and $ QR = SP = 6$. The rectangle is rotated $ 90^\circ$ clockwise about $ R$, then rotated $ 90^\circ$ clockwise about the point that $ S$ moved to after the first rotation. What is the length of the path traveled by point $ P$? ${ \textbf{(A)}\ (2\sqrt3 + \sqrt5})\pi \qquad \textbf{(B)}\ 6\pi \qquad \textbf{(C)}\ (3 + \sqrt {10})\pi \qquad \textbf{(D)}\ (\sqrt3 + 2\sqrt5)\pi \\ \textbf{(E)}\ 2\sqrt {10}\pi$

We are given a map divided into $13\times 13$ fields. It is also known that at one of the fields a tank of the enemy is stationed, which we must destroy. To achieve this we need to hit it twice with shots aimed at the centre of some field. When the tank gets hit it gets moved to a neighbouring field out of precaution. At least how many shots must we fire, so that the tank gets destroyed certainly? [i]We can neither see the tank, nor get any other feedback regarding its position.[/i]

The triangle $ABC$ is given. Inside its sides $AB$ and $AC$, the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$. Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$, where $[DEF]$ denotes the content of the triangle $DEF$. (David Hruska, Josef Tkadlec)

Find the number of integer solutions of the equation $x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$

In an acute scalene triangle $ABC$, points $D,E,F$ lie on sides $BC, CA, AB$, respectively, such that $AD \perp BC, BE \perp CA, CF \perp AB$. Altitudes $AD, BE, CF$ meet at orthocenter $H$. Points $P$ and $Q$ lie on segment $EF$ such that $AP \perp EF$ and $HQ \perp EF$. Lines $DP$ and $QH$ intersect at point $R$. Compute $HQ/HR$. [i]Proposed by Zuming Feng[/i]

Let $a_1,a_2,\cdots,a_n$ be integers such that $1=a_1\le a_2\le \cdots\le a_{2019}=99$. Find the minimum $f_0$ of the expression $$f=(a_1^2+a_2^2+\cdots+a_{2019}^2)-(a_1a_3+a_2a_4+\cdots+a_{2017}a_{2019}),$$ and determine the number of sequences $(a_1,a_2,\cdots,a_n)$ such that $f=f_0$.

A rubber band is wrapped around two pipes as shown. One has radius $3$ inches and the other has radius $9$ inches. The length of the band can be expressed as $a\pi+b\sqrt{c}$ where $a$, $b$, $c$ are integers and $c$ is square free. What is $a+b+c$? [asy] size(4cm); draw(circle((0,0),3)); draw(circle((12,0),9)); draw(3*dir(120)--(12,0)+9*dir(120)); draw(3*dir(240)--(12,0)+9*dir(240)); [/asy]

Let $O$ be the clrcumcenter of triangle $ABC$. Points $X$ and $Y$ on side $BC$ are such that $AX = BX$ and $AY = CY$. Prove that the circumcircle of triangle $AXY$ passes through the circumceuters of triangles $AOB$ and $AOC$.

Show that for every odd integer $N>5$ there exist vectors $\bf u,v,w$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $N\bf =|u|=|v|=|w|.$ [i]Based on problem 2 of the 2018 Kürschák contest[/i]

Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that \[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\] [i]Proposed by Dzianis Pirshtuk, Belarus[/i]

Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$. [i]Proposed by fattypiggy123[/i]

For which real numbers $c$ is there a straight line that intersects the curve \[ y = x^4 + 9x^3 + cx^2 + 9x + 4\] in four distinct points?

How many pairs of real numbers $ \left(x,y\right)$ are there such that $ x^{4} \minus{} 2^{ \minus{} y^{2} } x^{2} \minus{} \left\| x^{2} \right\| \plus{} 1 \equal{} 0$, where $ \left\| a\right\|$ denotes the greatest integer not exceeding $ a$. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{Infinitely many}$

In the following table, each question mark is to be replaced by "Possible" or "Not Possible" to indicate whether a nonvertical line with the given slope can contain the given number of lattice points (points both of whose coordinates are integers). How many of the $12$ entries will be "Possible"? \begin{tabular}{|c|c|c|c|c|} \cline{2-5} \multicolumn{1}{c|}{} & \textbf{zero} & \textbf{exactly one} & \textbf{exactly two} & \textbf{more than two}\\ \hline \textbf{zero slope} & ? & ? & ? & ?\\ \hline \textbf{nonzero rational slope} & ? & ? & ? & ?\\ \hline \textbf{irrational slope} & ? & ? & ? & ?\\ \hline \end{tabular} $ \textbf{(A) }4 \qquad \textbf{(B) }5 \qquad \textbf{(C) }6 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9 \qquad $

A group of people is divided over two busses in such a way that there are as many seats in total as people. The chance that two friends are seated on the same bus is $\frac{1}{2}$. a) Show that the number of people in the group is a square. b) Show that the number of seats on each bus is a triangular number.

Let $ABC$ be a triangle, and let $\ell$ be the line passing through the circumcenter of $ABC$ and parallel to the bisector of the angle $\angle A$. Prove that the line $\ell$ passes through the orthocenter of $ABC$ if and only if $AB = AC$ or $\angle BAC = 120^o$

Let $(a_n)_{n=1}^{\infty}$ and $(b_n)_{n=1}^{\infty}$ be two sequences of positive numbers. Show that the following statements are equivalent: [list=1] [*]There is a sequence $(c_n)_{n=1}^{\infty}$ of positive numbers such that $\sum_{n=1}^{\infty}{\frac{a_n}{c_n}}$ and $\sum_{n=1}^{\infty}{\frac{c_n}{b_n}}$ both converge;[/*] [*]$\sum_{n=1}^{\infty}{\sqrt{\frac{a_n}{b_n}}}$ converges.[/*] [/list] [i]Proposed by Tomáš Bárta, Charles University, Prague[/i]

Prove that if $x$ is such that $$x +\frac{1}{x}= 2\cos \alpha $$ then, for all $n = 0, 1, 2, . . . ,$ $$x^n ++\frac{1}{x^n}= 2\cos n \alpha .$$

Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$. Evaluate the following definite integral. \[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]