Let $a$ and $b$ be positive numbers. Prove the inequality $$\sqrt[3]{\frac ab}+\sqrt[3]{\frac ba}\le\sqrt[3]{2(a+b)\left(\frac1a+\frac1b\right)}.$$

The triangle $ABC$ is given. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$. Similarly, $Q_1$ and $Q_2$ are points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ intersect at the point $R{}$, and the circles $P_1P_2R$ and $Q_1Q_2R$ intersect a second time at the point $S{}$ lying inside the triangle $P_1Q_1R$. Let $M{}$ be the midpoint of the segment $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.

For real numbers $ x_{i}>1,1\leq i\leq n,n\geq 2,$ such that: $ \frac{x_{i}^2}{x_{i}\minus{}1}\geq S\equal{}\displaystyle\sum^n_{j\equal{}1}x_{j},$ for all $ i\equal{}1,2\dots, n$ find, with proof, $ \sup S.$

You live in an economy where all coins are of value $1/k$ for some positive integer $k$ (i.e. $1, 1/2, 1/3, \dots$). You just recently bought a coin exchanging machine, called the [i] Cape Town Machine [/i]. For any integer $n > 1$, this machine can take in $n$ of your coins of the same value, and return a coin of value equal to the sum of values of those coins (provided the coin returned is part of the economy). Given that the product of coins values that you have is $2015^{-1000}$, what is the maximum numbers of times you can use the machine over all possible starting sets of coins? [i] Proposed by Yang Liu [/i]

Let $n$ be a positive integer. On a blackboard, Bobo writes a list of $n$ non-negative integers. He then performs a sequence of moves, each of which is as follows: -for each $i = 1, . . . , n$, he computes the number $a_i$ of integers currently on the board that are at most $i$, -he erases all integers on the board, -he writes on the board the numbers $a_1, a_2,\ldots , a_n$. For instance, if $n = 5$ and the numbers initially on the board are $0, 7, 2, 6, 2$, after the first move the numbers on the board will be $1, 3, 3, 3, 3$, after the second they will be $1, 1, 5, 5, 5$, and so on. (a) Show that, whatever $n$ and whatever the initial configuration, the numbers on the board will eventually not change any more. (b) As a function of $n$, determine the minimum integer $k$ such that, whatever the initial configuration, moves from the $k$-th onwards will not change the numbers written on the board.

Find all triples of complex numbers $(x, y, z)$ for which $$(x + y)^3 + (y + z)^3 + (z + x)^3 - 3(x + y)(y + z)(z + x) = x^2(y + z) + y^2(z + x ) + z^2(x + y) = 0$$

Let $ a,\ b$ are positive constants. Determin the value of a positive number $ m$ such that the areas of four parts of the region bounded by two parabolas $ y\equal{}ax^2\minus{}b,\ y\equal{}\minus{}ax^2\plus{}b$ and the line $ y\equal{}mx$ have equal area.

Let $A$ be an even number but not divisible by $10$. The last two digits of $A^{20}$ are: (A): $46$, (B): $56$, (C): $66$, (D): $76$, (E): None of the above.

The radius $R$ of a cylindrical box is $8$ inches, the height $H$ is $3$ inches. The volume $V = \pi R^2H$ is to be increased by the same fixed positive amount when $R$ is increased by $x$ inches as when $H$ is increased by $x$ inches. This condition is satisfied by: $ \textbf{(A)}\ \text{no real value of} \text{ } x\qquad$ $\textbf{(B)}\ \text{one integral value of} \text{ } x\qquad$ $\textbf{(C)}\ \text{one rational, but not integral, value of} \text{ } x\qquad$ $\textbf{(D)}\ \text{one irrational value of} \text{ } x\qquad$ $\textbf{(E)}\ \text{two real values of} \text{ } x $

Let $n$ be odd natural number and $x_1,x_2,\cdots,x_n$ be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum. ( $X=\{\mid x_i-x_j \mid | i<j\}$ )

Is it possible to write positive integers from $1$ to $2025$ in the cells of a \( 45 \times 45 \) grid such that each number is used exactly once, and at the same time, each written number is either greater than all the numbers located in its side-adjacent cells or smaller than all the numbers located in its side-adjacent cells? [i]Proposed by Anton Trygub[/i]

Point $P$ lies inside triangle $ABC$. Let the projections of $P$ onto sides $BC$,$CA$,$AB$ be $D$, $E$, $F$ respectively. Let the projections from $A$ to the lines $BP$ and $CP$ be $M$ and $N$ respectively. Prove that $ME$, $NF$ and $BC$ are concurrent.

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

The numbers $2,4,\ldots,2^{100}$ are written on a board. At a move, one may erase the numbers $a,b$ from the board and replace them with $ab/(a+b).$ Prove that the last numer on the board will be greater than 1. [i]From the folklore[/i]

Find all monic polynomials $P,Q$ which are non-constant, have real coefficients and they satisfy $2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})$ and $P(1)=1$ for all real $x$.

A "labyrinth" is an $8 \times 8$ chessboard with walls between some neighboring squares. If a rook can traverse the entire board without jumping over the walls, the labyrinth is "good" ; otherwise it is "bad" . Are there more good labyrinths or bad labyrinths? (A Shapovalov)

Determine all pairs of $(a,b)$ of non negative integers such that: $$\dfrac{a+b}{2} - \sqrt{ab}~=~1$$

The real numbers $a$, $b$, and $c$ are such that the quadratic trinomials $ax^2 + bx + c$ and $cx^2 + bx + a$ each have two strictly positive real roots. Show that the sum of all these roots is at least $4$.

Given a positive integer $ n$, for all positive integers $ a_1, a_2, \cdots, a_n$ that satisfy $ a_1 \equal{} 1$, $ a_{i \plus{} 1} \leq a_i \plus{} 1$, find $ \displaystyle \sum_{i \equal{} 1}^{n} a_1a_2 \cdots a_i$.

Determine on which side is the steering wheel disposed in the car depicted in the figure. [img]https://4.bp.blogspot.com/-s2rjZw-d4UY/XMg5BXCE9SI/AAAAAAAAKHc/WOpvqjWw7lAciDEiNj_TX7io6sfItSPnQCK4BGAYYCw/s320/Sharygin%2Bfinal%2B2007%2B8.1.png[/img]

Let $n \geq 1$ be an integer, and $a_0, a_1, \dots, a_{n-1}$ be real numbers such that \[ 1 \geq a_{n-1} \geq a_{n-2} \geq \dots \geq a_1 \geq a_0 \geq 0. \] We assume that $\lambda$ is a real root of the polynomial \[ x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0. \] Prove that $|\lambda| \leq 1$.

Two equal circles intersect at points $A$ and $B$. $P$ is the point of one of the circles that is different from $A$ and $B, X$ and $Y$ are the second intersection points of the lines of $PA, PB$ with the other circle. Prove that the line passing through $P$ and perpendicular to $AB$ divides one of the arcs $XY$ in half.

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

In a regular polygon with $134$ sides, $67$ diagonals are drawn so that exactly one diagonal emerges from each vertex. We call the [i]length[/i] of a diagonal the number of sides of the polygon included between the vertices of the diagonal and which is less than or equal to $67$. If we order the [i]lengths [/i] of the diagonals in ascending order, we obtain a succession of $67$ numbers $(d_1,d_2,...,d_{67})$. It will be possible to draw diagonals such that a) $(d_1,d_2,...,d_{67})=\underbrace{2 ... 2}_{6},\underbrace{3 ... 3}_{61}$ ? b) $(d_1,d_2,...,d_{67}) =\underbrace{3 ... 3}_{8},\underbrace{6 ... 6}_{55}.\underbrace{8 ... 8}_{4} $ ?

Three natural numbers $ a,b,c $ with $ \gcd (a,b) =1 $ define in the Diophantine plane a line $ d: ax+by-c=0. $ Prove that: [b]a)[/b] the distance between any two points from $ d $ is at least $ \sqrt{a^2+b^2} . $ [b]b)[/b] the restriction of $ d $ to the first quadrant of the Diophantine plane is a finite line having at most $ 1+\frac{c}{ab} $ elements.