Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$

Let $ABC$ be a triangle, and let $r, r_1, r_2, r_3$ denoted its inradius and the exradii opposite the vertices $A,B,C$, respectively. Suppose $a>r_1, b>r_2, c>r_3$. Prove that (a) triangle $ABC$ is acute, (b) $a+b+c>r+r_1+r_2+r_3$.

Does the equation $$z(y-x)(x+y)=x^3$$ have finitely many solutions in the set of positive integers? [i]Proposed by Nikola Velov[/i]

Find all positive integers n such that $n^5 + n + 1$ is prime.

Find all functions $f:\mathbb R\to\mathbb R$ such that \[f(x^2)+2xf(y)=yf(x)+xf(x+y).\] [i](Proposed by Yeoh Yi Shuen)[/i]

\[\int_0^1 \sqrt{x}\log(x)\,\mathrm dx\] [i]Proposed by Connor Gordon[/i]

A square was split into several rectangles so that the centers of rectangles form a convex polygon. a) Is it true for sure that each rectangle adjoins to a side of the square? b) Can the number of rectangles equal 23 ? Alexandr Shapovalov

8.8, 9.8, 11.8 a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges. b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges. c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas. ([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])

In a chess tournament participated $ 100 $ players. Every player played one game with every other player. For a win $1$ point is given, for loss $ 0 $ and for a draw both players get $0,5$ points. Ion got more points than every other player. Mihai lost only one game, but got less points than every other player. Find all possible values of the difference between the points accumulated by Ion and the points accumulated by Mihai.

In table [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9hLzBjNjFlZWFjM2ZlOTQzMTk2YTdkMzQ2MjJiYzYyMWFlN2Y0ZGZlLnBuZw==&rn=dGFibGljYWEucG5n[/img] $10$ numbers are circled, in every row and every column exactly one. Prove that among them, there are at least two equal

Tadeo draws the rectangle with the largest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$ and the rectangle with the smallest perimeter that can be divided into $2015$ squares of sidelength $1$ $cm$. What is the difference between the perimeters of the rectangles Tadeo drew?

How many rational solutions for $x$ are there to the equation $x^4+(2-p)x^3+(2-2p)x^2+(1-2p)x-p=0$ if $p$ is a prime number?

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Let $ f:[1,\infty )\longrightarrow (0,\infty ) $ be a continuous function satisfying the following properties: $ \text{(i)}\exists\lim_{x\to\infty } \frac{f(x)}{x}\in\overline{\mathbb{R}} $ $ \text{(ii)}\exists\lim_{x\to\infty } \frac{1}{x}\int_1^x f(t)dt\in\mathbb{R}. $ [b]a)[/b] Show that $ \lim_{x\to\infty } \frac{f(x)}{x}=0. $ [b]b)[/b] Prove that $ \lim_{x\to\infty } \frac{1}{x^2}\int_1^x f^2(t)dt=0. $

Find all positive integers $n$ such that $n^3$ is the product of all divisors of $n$.

On a plane $\Pi$ is given a straight line $\ell$ and on the line $\ell$ are given two different points $A_1, A_2$. We consider on the plane $\Pi$, outside the line $\ell$, two different points $A_3, A_4$. Examine if it is possible to put points $A_3$ and $A_4$ on such positions such the four points $A_1, A_2, A_3, A_4$ form the maximal number of possible isosceles triangles, in the following cases: (a) when the points $A_3, A_4$ belong to dierent semi-planes with respect to $\ell$; (b) when the points $A_3, A_4$ belong to the same semi-planes with respect to $\ell$. Give all possible cases and explain how is possible to construct in each case the points $A_3$ and $A_4$.

Alex and Bobby take turns playing the following game on an initially white row of $5000$ cells, with Alex starting first. On her turn, Alex must color two adjacent white cells black. On his turn, Bobby must color either one or three consecutive white cells black. No player can make a move after which there will be a white cell with no adjacent white cell. The game ends when one player cannot make a move (in which case that player loses), or when the entire row is colored black (in which case Alex wins). Who has a winning strategy?

[b]p1.[/b] Given a group of $n$ people. An $A$-list celebrity is one that is known by everybody else (that is, $n - 1$ of them) but does not know anybody. A $B$-list celebrity is one that is known by exactly $n - 2$ people but knows at most one person. (a) What is the maximum number of $A$-list celebrities? You must prove that this number is attainable. (b) What is the maximum number of $B$-list celebrities? You must prove that this number is attainable. [b]p2.[/b] A polynomial $p(x)$ has a remainder of $2$, $-13$ and $5$ respectively when divided by $x+1$, $x-4$ and $x-2$. What is the remainder when $p(x)$ is divided by $(x + 1)(x - 4)(x - 2)$? [b]p3.[/b] (a) Let $x$ and y be positive integers satisfying $x^2 + y = 4p$ and $y^2 + x = 2p$, where $p$ is an odd prime number. Prove: $x + y = p + 1$. (b) Find all values of $x, y$ and $p$ that satisfy the conditions of part (a). You will need to prove that you have found all such solutions. [b]p4.[/b] Let function $f(x, y, z)$ be defined as following: $$f(x, y, z) = \cos^2(x - y) + \cos^2(y - z) + \cos^2(z - x), x, y, z \in R.$$ Find the minimum value and prove the result. [b]p5.[/b] In the diagram below, $ABC$ is a triangle with side lengths $a = 5$, $b = 12$,$ c = 13$. Let $P$ and $Q$ be points on $AB$ and $AC$, respectively, chosen so that the segment $PQ$ bisects the area of $\vartriangle ABC$. Find the minimum possible value for the length $PQ$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/91a09dd3d831b299b844b07cd695ddf51cb12b.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url]. Thanks to gauss202 for sending the problems.

Let be given $a\in\{0,1,2, 3,..., 100\}.$ Find all $n \in\{1,2, 3,..., 2013\}$ such that $C_n^{2013} > C_a^{2013}$ , where $C_k^m=\frac{m!}{k!(m -k)!}$.

Let $ABC$ be a right triangle with right angle at $ B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and $I$ the incenter of the triangle, show that $ \angle IMB=15^o$.

Let $ S$ be the set of points $ (a,b)$ in the coordinate plane, where each of $ a$ and $ b$ may be $ \minus{} 1$, $ 0$, or $ 1$. How many distinct lines pass through at least two members of $ S$? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 20 \qquad \textbf{(C)}\ 24 \qquad \textbf{(D)}\ 27\qquad \textbf{(E)}\ 36$

Let $X$ and $Y$ be two sets of points in the plane and $M$ be a set of segments connecting points from $X$ and $Y$ . Let $k$ be a natural number. Prove that the segments from $M$ can be painted using $k$ colors in such a way that for any point $x \in X \cup Y$ and two colors $\alpha$ and $\beta$ $(\alpha \neq \beta)$, the difference between the number of $\alpha$-colored segments and the number of $\beta$-colored segments originating in $X$ is less than or equal to $1$.

Given regular hexagon $ABCDEF$, compute the probability that a randomly chosen point inside the hexagon is inside triangle $PQR$, where $P$ is the midpoint of $AB$, $Q$ is the midpoint of $CD$, and $R$ is the midpoint of $EF$.

The diagram shows some squares whose sides intersect other squares at the midpoints of their sides. The shaded region has total area $7$. Find the area of the largest square. [img]https://cdn.artofproblemsolving.com/attachments/3/a/c3317eefe9b0193ca15f36599be3f6c22bb099.png[/img]

Let $q$ be a positive rational number. Two ants are initially at the same point $X$ in the plane. In the $n$-th minute $(n = 1,2,...)$ each of them chooses whether to walk due north, east, south or west and then walks the distance of $q^n$ metres. After a whole number of minutes, they are at the same point in the plane (not necessarily $X$), but have not taken exactly the same route within that time. Determine all possible values of $q$. Proposed by Jeremy King, UK