Find all pairs of real numbers $(x,y)$ which satisfy the system $$\begin{cases} x-y = 7 \\ \sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7\end{cases}$$

$ABC$ is an acute-angled triangle. $D$ is a variable point in space such that all faces of the tetrahedron $ABCD$ are acute-angled. $P$ is the foot of the perpendicular from $D$ to the plane $ABC$. Find the locus of $P$ as $D$ varies.

Hexagon $ABCDEF$ is inscribed in a circle. If $\measuredangle ACE = 35^{\circ}$ and $\measuredangle CEA = 55^{\circ}$, then compute the sum of the degree measures of $\angle ABC$ and $\angle EFA$. [i]Proposed by Isabella Grabski[/i]

Let $S$ be the set of all $10-$term arithmetic progressions that include the numbers $4$ and $10.$ For example, $(-2,1, 4,7,10,13, 16,19,22,25)$ and $(10,8\tfrac{1}{2}, 7,5\tfrac{1}{2}, 4, 2\tfrac{1}{2},1,\tfrac{1}{2},-2, -3\tfrac{1}{2})$ are both members of $S.$ Find the sum of all values of $a_{10}$ for each $(a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8, a_9, a_{10}) \in S,$ that is, $\sum_{a_1, a_2, a_3, ... , a_{10} \in S} a_{10}.$

Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.

Bob the bomb-defuser has stumbled upon an active bomb. He opens it up, and finds the red and green wires conveniently located for him to cut. Being a seasoned member of the bomb-squad, Bob quickly determines that it is the green wire that he should cut, and puts his wirecutters on the green wire. But just before he starts to cut, the bomb starts to count down, ticking every second. Each time the bomb ticks, starting at time $t=15$ seconds, Bob panics and has a certain chance to move his wirecutters to the other wire. However, he is a rational man even when panicking, and has a $\frac{1}{2t^2}$ chance of switching wires at time t, regardless of which wire he is about to cut. When the bomb ticks at $t=1$, Bob cuts whatever wire his wirecutters are on, without switching wires. What is the probability that Bob cuts the green wire?

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$.