Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

In the plane are given $n^2 + 1$ points, no three of which lie on a line. Each line segment connecting a pair of these points is colored by either red or blue. A [i]path [/i] of length $k$ is a sequence of $k$ segments where the end of each segment (except for the last one) is the beginning of the next one. A path is [i]simple [/i] if it does not intersect itself. Prove that there exists a monochromatic simple path of length $n$.

we have a triangle $ ABC $ and make rectangles $ ABA_1B_2 $ , $ BCB_1C_2 $ and $ CAC_1A_2 $ out of it. then pass a line through $ A_2 $ perpendicular to $ C_1A_2 $ and pass another line through $ A_1 $ perpendicular to $ A_1B_2 $. let $ A' $ the common point of this two lines. like this we make $ B' $ and $ C' $. prove $ AA' $ , $ BB' $ and $ CC' $ intersect each other in a same point.

Two externally tangent circles with different radii are given. Their common tangents form a triangle. Find the area of this triangle in terms of the radii of the two circles.

Prove that if $0<x<\pi/2$, then $\sec^6 x+\csc^6 x+(\sec^6 x)(\csc^6 x)\geq 80$.

Find all the intervals $I$ where any element of the interval $x \in I$ satisfies $$\cos x +\sin x >1.$$ Do the same computation when $x$ satisfies $$\cos x +\vert \sin x \vert>1.$$

Let $f(x)$ be a twice continuously differentiable function on closed interval $[a,b]$ Prove that $\lim_{n \to \infty} n^2[\int_{a}^{b}f(x)dx-\frac{b-a}{n}\sum_{k=1}^{n}f(a+\frac{2k-1}{2n}(b-a))]=\frac{(b-a)^2}{24}[f'(b)-f'(a)]$

Find all functions $g:R\rightarrow R$ for which there exists a strictly increasing function $ f:R\rightarrow R $ such that $f(x+y)=f(x)g(y)+f(y)$ $\forall x,y \in R$.

A cent, a stuiver ($5$ cent coin), a dubbeltje ($10$ cent coin), a kwartje ($25$ cent coin), a gulden ($100$ cent coin) and a rijksdaalder ($250$ cent coin) are divided among four children in such a way that each of them receives at least one of the six coins. How many such distributions are there?

Solid camphor is insoluble in water but is soluble in vegetable oil. The best explanation for this behavior is that camphor is a(n) ${ \textbf{(A)}\ \text{Ionic solid} \qquad\textbf{(B)}\ \text{Metallic solid} \qquad\textbf{(C)}\ \text{Molecular solid} \qquad\textbf{(D)}\ \text{Network solid} } $

For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?

There are $10$ teams, named $T_1$ through $T_{10},$ participating in a draft in which there are $20$ players available, named $P_1$ through $P_{20}.$ Suppose each team independent of the others has uniform random preference on the $20$ players. Team $T_1$ will draft their favorite player, and then each subsequent team $T_2, \ldots , T_{10}$ draft their favorite player among the ones not already drafted. Each team drafts exactly one player. Given that $P_1$ is among the $10$ favorite players for each team, the probability that $P_1$ is drafted can be written as $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $m + n.$

Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$

Let $ \sigma_1 ,\sigma_2 $ be two permutations of order $ n $ such that $ \sigma_1 (k)=\sigma_2 (n-k+1) $ for $ k=\overline{1,n} . $ Prove that the number of inversions of $ \sigma_1 $ plus the number of inversions of $ \sigma_2 $ is $ \frac{n(n+1)}{2} . $

Calculate $ \int_0^{2\pi }\prod_{i=1}^{2002} cos^i (it) dt. $ [i]Dorin Andrica[/i]

Five children are divided into groups and in each group they take the hand forming a wheel to dance spinning. How many different wheels those children can form, if it is valid that there are groups of $1$ to $5$ children, and can there be any number of groups?

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$. Show that there is no number $n$ with $f (n) = 19091997$.

The rooms of a building are arranged in a $m\times n$ rectangular grid (as shown below for the $5\times 6$ case). Every room is connected by an open door to each adjacent room, but the only access to or from the building is by a door in the top right room. This door is locked with an elaborate system of $mn$ keys, one of which is located in every room of the building. A person is in the bottom left room and can move from there to any adjacent room. However, as soon as the person leaves a room, all the doors of that room are instantly and automatically locked. Find, with proof, all $m$ and $n$ for which it is possible for the person to collect all the keys and escape the building. [asy] unitsize(5mm); defaultpen(linewidth(.8pt)); fontsize(25pt); for(int i=0; i<=5; ++i) { for(int j=0; j<= 6; ++j) { draw((0,i)--(9,i)); draw((1.5*j,0)--(1.5*j,5)); }} dot((.75, .5)); label("$\ast$",(8.25,4.5)); dot((11, 3)); label("$\ast$",(11,1.75)); label("room with locked external door",(18,1.9)); label("starting position",(15.3,3)); [/asy]

In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.

Let $r_A,r_B,r_C$ rays from point $P$. Define circles $w_A,w_B,w_C$ with centers $X,Y,Z$ such that $w_a$ is tangent to $r_B,r_C , w_B$ is tangent to $r_A, r_C$ and $w_C$ is tangent to $r_A,r_B$. Suppose $P$ lies inside triangle $XYZ$, and let $s_A,s_B,s_C$ be the internal tangents to circles $w_B$ and $w_C$; $w_A$ and $w_C$; $w_A$ and $w_B$ that do not contain rays $r_A,r_B,r_C$ respectively. Prove that $s_A, s_B, s_C$ concur at a point $Q$, and also that $P$ and $Q$ are isotomic conjugates. [b]PS: The rays can be lines and the problem is still true.[/b]

Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$. Find the product of the radii of the two circles.

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10); draw(A--B--C--D--E--F--G--A); label("$A$", A, W); label("$B$", B, NE); label("$C$", C, S); label("$D$", D, NE); label("$E$", E, SE); label("$F$", F, N); label("$G$", G, SE); label("$5$", (11,7.5)); label("$5$", (11,-7.5)); label("$20$", (C+D)/2, N); label("$20$", (F+E)/2, S); label("$10$", (31,0)); [/asy] $ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 291\qquad\textbf{(C)}\ 294\qquad\textbf{(D)}\ 297\qquad\textbf{(E)}\ 300 $