Given a real number $\alpha$, find all pairs $(f,g)$ of functions $f,g :\mathbb{R} \to \mathbb{R}$ such that $$xf(x+y)+\alpha \cdot yf(x-y)=g(x)+g(y) \;\;\;\;\;\;\;\;\;\;\; ,\forall x,y \in \mathbb{R}.$$

Each day John's mother sends him to the store with $\$1$ to buy widgets and gadgets, each of which cost a whole number of cents. On the first day John comes back with $4$ widgets, $5$ gadgets, and $35$ cents in change. On the second day, John comes back with $5$ widgets, $4$ gadgets, and $39$ cents in change. On the third day, John comes back with only $c$ cents in change. He hands his mother the change, telling her that he had tripped coming home and broken all the widgets and gadgets. His mother, thinking for a moment, begins yelling at him for lying, as she noticed that there was no way he could have received exactly $c$ cents in change given the price of widgets and gadgets. What is the sum of the digits of the least possible value of $c$? $\textbf{(A) }10\qquad\textbf{(B) }13\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }\text{impossible to determine}$

The triangle $ABC$ satisfies $0\le AB\le 1\le BC\le 2\le CA\le 3$. What is the maximum area it can have?

The real numbers $a, b, c, d$ satisfy $a > b > c > d$ and $$a + b + c + d = 2004 \,\,\, and \,\,\, a^2 - b^2 + c^2 - d^2 = 2004.$$ Answer, with proof, to the following questions: a) What is the smallest possible value of $a$? b) What is the number of possible values of $a$?

There are $2n$ points on the plane, no three of which lie on the same line. Some segments are drawn between them so that they do not intersect at internal points and any segment with ends among the given points intersects some of the drawn segments at an internal point. Is it true that it is always possible to choose $n$ drawn segments having no common ends?

Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

For $n>1$ consider an $n\times n$ chessboard and place identical pieces at the centers of different squares. [list=i] [*] Show that no matter how $2n$ identical pieces are placed on the board, that one can always find $4$ pieces among them that are the vertices of a parallelogram. [*] Show that there is a way to place $(2n-1)$ identical chess pieces so that no $4$ of them are the vertices of a parallelogram. [/list]

Let $ a,b,c$ and $ d$ be any four real numbers, not all equal to zero. Prove that the roots of the polynomial $ f(x) \equal{} x^{6} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ can't all be real.

Find the area enclosed by the graph of $|x|+|2y|=12$. [i]Proposed by Nathan Ramesh

For the function $$ g(a) = \underbrace{\max}_{x\in R} \left\{ \cos x + \cos \left(x + \frac{\pi}{6} \right)+ \cos \left(x + \frac{\pi}{4} \right) + cos(x + a) \right\},$$ let $b \in R$ be the input that maximizes $g$. If $\cos^2 b = \frac{m+\sqrt{n}+\sqrt{p}-\sqrt{q}}{24}$ for positive integers $m, n, p, q$, find $m + n + p + q$.

Four unit squares form a $2 \times 2$ grid. Each of the $12$ unit line segments forming the sides of the squares is colored either red or blue in such way that each square has $2$ red sides and blue sides. One example is shown below (red is solid, blue is dashed). Find the number of such colorings. [asy] size(4cm); defaultpen(linewidth(1.2)); draw((0, 0) -- (2, 0) -- (2, 1)); draw((0, 1) -- (1, 1) -- (1, 2) -- (2,2)); draw((0, 0) -- (0, 1), dotted); draw((1, 0) -- (1, 1) -- (2, 1) -- (2, 2), dotted); draw((0, 1) -- (0, 2) -- (1, 2), dotted); [/asy]

Let $f,g$ are defined in $(a,b)$ such that $f(x),g(x)\in\mathcal{C}^2$ and non-decreasing in an interval $(a,b)$ . Also suppose $f^{\prime \prime}(x)=g(x),g^{\prime \prime}(x)=f(x)$. Also it is given that $f(x)g(x)$ is linear in $(a,b)$. Show that $f\equiv 0 \text{ and } g\equiv 0$ in $(a,b)$.

Let $P$ be a point inside a regular tetrahedron ABCD with edge length $1$. Show that $$d(P,AB)+d(P,AC)+d(P,AD)+d(P,BC)+d(P,BD)+d(P,CD) \ge \frac{3}{2} \sqrt2$$ , with equality only when $P$ is the centroid of $ABCD$. Here $d(P,XY)$ denotes the distance from point $P$ to line $XY$.

Original in Hungarian; translated with Google translate; polished by myself. Let $f: [0, \infty) \to [0, \infty)$ be a function that is linear between adjacent integers, and for $n = 0, 1, \dots$ satisfies $$f(n) = \begin{cases} 0, & \textrm{if }2\mid n,\\4^l + 1, & \textrm{if }2 \nmid n, 4^{l - 1} \leq n < 4^l(l = 1, 2, \dots).\end{cases}$$ Let $f^1(x) = f(x)$, and $f^k(x) = f(f^{k - 1}(x))$ for all integers $k \geq 2$. Determine the values of $\liminf\nolimits_{k\to\infty}f^k(x)$ and $\limsup\nolimits_{k\to\infty}f^k(x)$ for almost all $x \in [0, \infty)$ under Lebesgue measure. (Not sure whether the last sentence translates correctly; the original: Határozzuk meg Lebesgue majdnem minden $x\in [0, \infty)$-re a $\liminf\nolimits_{k\to\infty}f^k(x)$ és $\limsup\nolimits_{k\to\infty}f^k(x)$ értékét.)

Given a finite set $X$, let $f$ be a rule such that $f$ maps every [i]even-element-subset[/i] $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions: (I) there exists an [i]even-element-subset[/i] $D$ of $X$ such that $f(D)>1990$; (II) for any two disjoint [i]even-element-subsets [/i]$A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds. Prove that there exist two subsets $P,Q$ of $X$ satisfying: (1) $P\cap Q=\emptyset$, $P\cup Q=X$; (2) for any [i]non-even-element-subset [/i]$S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$; (3) for any [i]even-element-subset[/i] $T$ of $Q$, we have $f(T)\le 1990$.

Points $A_1$ and $C_1$ are on $BC$ and $AB$ of acute-angled triangle $ABC$ . $AA_1$ and $CC_1$ intersect in $K$. Circumcircles of $AA_1B,CC_1B$ intersect in $P$ - incenter of $AKC$. Prove, that $P$ - orthocenter of $ABC$

On a standard die one of the dots is removed at random with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? $ \textbf{(A)}\ \frac {5}{11} \qquad \textbf{(B)} \ \frac {10}{21} \qquad \textbf{(C)}\ \frac {1}{2} \qquad \textbf{(D)} \ \frac {11}{21} \qquad \textbf{(E)}\ \frac {6}{11}$

[b]p1[/b]. Is it possible to find $n$ positive numbers such that their sum is equal to $1$ and the sum of their squares is less than $\frac{1}{10}$? [b]p2.[/b] In the country of Sepulia, there are several towns with airports. Each town has a certain number of scheduled, round-trip connecting flights with other towns. Prove that there are two towns that have connecting flights with the same number of towns. [b]p3.[/b] A $4 \times 4$ magic square is a $4 \times 4$ table filled with numbers $1, 2, 3,..., 16$ - with each number appearing exactly once - in such a way that the sum of the numbers in each row, in each column, and in each diagonal is the same. Is it possible to complete $\begin{bmatrix} 2 & 3 & * & * \\ 4 & * & * & *\\ * & * & * & *\\ * & * & * & * \end{bmatrix}$ to a magic square? (That is, can you replace the stars with remaining numbers $1, 5, 6,..., 16$, to obtain a magic square?) [b]p4.[/b] Is it possible to label the edges of a cube with the numbers $1, 2, 3, ... , 12$ in such a way that the sum of the numbers labelling the three edges coming into a vertex is the same for all vertices? [b]p5.[/b] (Bonus) Several ants are crawling along a circle with equal constant velocities (not necessarily in the same direction). If two ants collide, both immediately reverse direction and crawl with the same velocity. Prove that, no matter how many ants and what their initial positions are, they will, at some time, all simultaneously return to the initial positions. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCDEF$ be a convex hexagon with $AB=DE$, $BC=EF$, $CD=FA$, and $\angle A-\angle D = \angle C -\angle F = \angle E -\angle B$. Prove that the diagonals $AD$, $BE$, and $CF$ are concurrent.

Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

Find all functions $f : Z \to Z$ such that $f(-1) = f(1)$ and $f(x)+ f(y) = f(x+2xy)+ f(y-2xy)$ for all $x,y \in Z$

A graph consists of a set of vertices, some of which are connected by (undirected) edges. A [i]star[/i] of a graph is a set of edges with a common endpoint. A [i]matching[/i] of a graph is a set of edges such that no two have a common endpoint. Show that if the number of edges of a graph $G$ is larger than $2(k-1)^2$, then $G$ contains a matching of size $k$ or a star of size $k$.

Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).

At a party attended by $n$ married couples, each person talks to everyone else at the party except his or her spouse. The conversations involve sets of persons or cliques $C_1, C_2, \cdots, C_k$ with the following property: no couple are members of the same clique, but for every other pair of persons there is exactly one clique to which both members belong. Prove that if $n \geq 4$, then $k \geq 2n$. [i]Proposed by USA.[/i]

Let $a>k$ be natural numbers and $r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n$ be sequences of natural numbers such that: $(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)$ Prove that these sequences are equal.