Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any $(x, y)$ real numbers we have $f(xf(y)+f(x))+f(y^2)=f(x)+yf(x+y)$

As $a$, $b$ and $c$ range over [i]all[/i] real numbers, let $m$ be the smallest possible value of $$2\left(a+b+c\right)^2+\left(ab-4\right)^2+\left(bc-4\right)^2+\left(ca-4\right)^2$$ and $n$ be the number of ordered triplets $\left(a,b,c\right)$ such that the above quantity is minimized. Compute $m+n$. [i]2016 CCA Math Bonanza Team #8[/i]

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\]for all real numbers $x$ and $y$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

Find the continuous function $f(x)$ such that the following equation holds for any real number $x$. \[\int_0^x \sin t \cdot f(x-t)dt=f(x)-\sin x.\] [i]1977 Keio University entrance exam/Medicine[/i]

From eight edges and eight diagonal of surfaces of a cube, choose $k$ lines. If any two lines of them are skew lines, then the maximum value of $k$ is________.

Let $p$ and $n$ be positive integers. Suppose that the numbers $c_{hk}$ ($h=1,2,\ldots,n$ ; $k=1,2,\ldots,ph$) satisfy $0 \leq c_{hk} \leq 1.$ Prove that $$ \left( \sum \frac{ c_{hk} }{h} \right)^2 \leq 2p \sum c_{hk} ,$$ where each summation is over all admissible ordered pairs $(h,k).$

Is there any function $ f : \mathbb{R}\to\mathbb{R}$ satisfying following conditions: $1) f(0) = 1$ $2) f(x+f(y)) = f(x+y) + 1$, for all $x,y \to\mathbb{R} $ $3)$ there exist rational, but not integer $x_0$, such $f(x_0)$ is integer

The circle, which is tangent to the circumcircle of isosceles triangle $ABC$ ($AB=AC$), is tangent $AB$ and $AC$ at $P$ and $Q$, respectively. Prove that the midpoint $I$ of the segment $PQ$ is the center of the excircle (which is tangent to $BC$) of the triangle .

The incircle of a cyclic quadrilateral $ABCD$ tangents the four sides at $E$, $F$, $G$, $H$ in counterclockwise order. Let $I$ be the incenter and $O$ be the circumcenter of $ABCD$. Show that the line connecting the centers of $\odot(OEG)$ and $\odot(OFH)$ is perpendicular to $OI$.

(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$. (b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.

Let $a_1 <a_2 < \ldots$ be an increasing sequence of positive integers. Let the series $$\sum_{i=1}^{\infty} \frac{1}{a_i }$$ be convergent. For any real number $x$, let $k(x)$ be the number of the $a_i$ which do not exceed $x$. Show that $\lim_{x\to \infty} \frac{k(x)}{x}=0.$

Problem: A mathematical moron is given the values b; c; A for a triangle ABC and is required to find a. He does this by using the cosine rule $ a^2 = b^2 + c^2 - 2bccosA$ and misapplying the low of the logarithm to this to get $ log a^2 = log b^2 + log c^2 - log(2bc cos A) $ He proceeds to evaluate the right-hand side correctly, takes the anti-logarithms and gets the correct answer. What can be said about the triangle ABC?

If $a,b,c\in (0,1)$ satisfy $a+b+c=2$ , prove that $\frac{abc}{(1-a)(1-b)(1-c)}\ge 8$

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

In a lattice, connected the elements $ a \wedge b$ and $ a \vee b$ by an edge whenever $ a$ and $ b$ are incomparable. Prove that in the obtained graph every connected component is a sublattice. [i]M. Ajtai[/i]

Given a positive integer $k$, a pigeon and a seagull play a game on an $n\times n$ board. The pigeon goes first, and they take turns doing the operations. The pigeon will choose $m$ grids and lay an egg in each grid he chooses. The seagull will choose a $k\times k$ grids and eat all the eggs inside them. If at any point every grid in the $n\times n $ board has an egg in it, then the pigeon wins. Else, the seagull wins. For every integer $n\geq k$, find all $m$ such that the pigeon wins. [i]Proposed by amano_hina[/i]

Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$. Let $C$ be the circumcenter of the triangle $AOB$. Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$. $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$, cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$). Prove that $OP\perp QR$. [i]Proposed by Mehdi E'tesami Fard, Ali Khezeli[/i]

Let $ a$, $ b$, $ c$ and $ n$ be positive integers such that $ a^n$ is divisible by $ b$, such that $ b^n$ is divisible by $ c$, and such that $ c^n$ is divisible by $ a$. Prove that $ \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1}$ is divisible by $ abc$. An even broader [i]generalization[/i], though not part of the QEDMO problem and not quite number theory either: If $ u$ and $ n$ are positive integers, and $ a_1$, $ a_2$, ..., $ a_u$ are integers such that $ a_i^n$ is divisible by $ a_{i \plus{} 1}$ for every $ i$ such that $ 1\leq i\leq u$ (we set $ a_{u \plus{} 1} \equal{} a_1$ here), then show that $ \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1}$ is divisible by $ a_1a_2...a_u$.

Find the number of ordered pairs of real numbers $ (a,b)$ such that $ (a \plus{} bi)^{2002} \equal{} a \minus{} bi$. $ \textbf{(A)}\ 1001\qquad \textbf{(B)}\ 1002\qquad \textbf{(C)}\ 2001\qquad \textbf{(D)}\ 2002\qquad \textbf{(E)}\ 2004$

(a) Prove that $\sqrt{n+1}-\sqrt n<\frac1{2\sqrt n}<\sqrt n-\sqrt{n-1}$ for all $n\in\mathbb N$. (b) Prove that the integer part of the sum $1+\frac1{\sqrt2}+\frac1{\sqrt3}+\ldots+\frac1{\sqrt{m^2}}$, where $m\in\mathbb N$, is either $2m-2$ or $2m-1$.

A hunter and an invisible rabbit play a game on an infinite square grid. First the hunter fixes a colouring of the cells with finitely many colours. The rabbit then secretly chooses a cell to start in. Every minute, the rabbit reports the colour of its current cell to the hunter, and then secretly moves to an adjacent cell that it has not visited before (two cells are adjacent if they share an edge). The hunter wins if after some finite time either:[list][*]the rabbit cannot move; or [*]the hunter can determine the cell in which the rabbit started.[/list]Decide whether there exists a winning strategy for the hunter. [i]Proposed by Aron Thomas[/i]

Prove that the Polynomial $ f(x) \equal{} x^{4} \plus{} 26x^{3} \plus{} 56x^{2} \plus{} 78x \plus{} 1989$ can't be expressed as a product $ f(x) \equal{} p(x)q(x)$ , where $ p(x)$ and $ q(x)$ are both polynomial with integral coefficients and with degree at least $ 1$.

Let the $ABCD$ be a quadrilateral without parallel sides, inscribed in a circle. Let $P$ and $Q$ be the intersection points between the lines containing the quadrilateral opposite sides. Show that the bisectors to the angles at $P$ and $Q$ are parallel to the bisectors of the angles at the intersection point of the diagonals of the quadrilateral.

Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that \[f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.\]