Sphere $S$ touches a plane. Let $A,B,C,D$ be four points on the plane such that no three of them are collinear. Consider the point $A'$ such that $S$ in tangent to the faces of tetrahedron $A'BCD$. Points $B',C',D'$ are defined similarly. Prove that $A',B',C',D'$ are coplanar and the plane $A'B'C'D'$ touches $S$. [i]Proposed by Alexey Zaslavsky (Russia)[/i]

In how many ways can $47$ be written as the sum of two primes? $\text{(A)}\ 0 \qquad \text{(B)}\ 1 \qquad \text{(C)}\ 2 \qquad \text{(D)}\ 3 \qquad \text{(E)}\ \text{more than 3}$

A basketball player made $ 5$ baskets during a game. Each basket was worth either $ 2$ or $ 3$ points. How many different numbers could represent the total points scored by the player? $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2$ (mod $n$). Compute the number of positive integers k such that $k | f(k)$.

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

Bamal, Halvan, and Zuca are playing [i]The Game[/i]. To start, they‘re placed at random distinct vertices on regular hexagon $ABCDEF$. Two or more players collide when they‘re on the same vertex. When this happens, all the colliding players lose and the game ends. Every second, Bamal and Halvan teleport to a random vertex adjacent to their current position (each with probability $\dfrac{1}{2}$), and Zuca teleports to a random vertex adjacent to his current position, or to the vertex directly opposite him (each with probability $\dfrac{1}{3}$). What is the probability that when [i]The Game[/i] ends Zuca hasn‘t lost? [i]Proposed by Edwin Zhao[/i] [hide=Solution][i]Solution.[/i] $\boxed{\dfrac{29}{90}}$ Color the vertices alternating black and white. By a parity argument if someone is on a different color than the other two they will always win. Zuca will be on opposite parity from the others with probability $\dfrac{3}{10}$. They will all be on the same parity with probability $\dfrac{1}{10}$. At this point there are $2 \cdot 2 \cdot 3$ possible moves. $3$ of these will lead to the same arrangement, so we disregard those. The other $9$ moves are all equally likely to end the game. Examining these, we see that Zuca will win in exactly $2$ cases (when Bamal and Halvan collide and Zuca goes to a neighboring vertex). Combining all of this, the answer is $$\dfrac{3}{10}+\dfrac{2}{9} \cdot \dfrac{1}{10}=\boxed{\dfrac{29}{90}}$$ [/hide]

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 $f (n) = 19091997$.