Found problems: 85335
Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at least one of those ten players.
Let \(m\) be a positive integer. Find, in terms of \(m\), all polynomials \(P(x)\) with integer coefficients such that for every integer \(n\), there exists an integer \(k\) such that \(P(k)=n^m\).
[i]Proposed by Raymond Feng[/i]
An [i]up-right path[/i] from $(a, b) \in \mathbb{R}^2$ to $(c, d) \in \mathbb{R}^2$ is a finite sequence $(x_1, y_z), \dots, (x_k, y_k)$ of points in $ \mathbb{R}^2 $ such that $(a, b)= (x_1, y_1), (c, d) = (x_k, y_k)$, and for each $1 \le i < k$ we have that either $(x_{i+1}, y_{y+1}) = (x_i+1, y_i)$ or $(x_{i+1}, y_{i+1}) = (x_i, y_i + 1)$. Two up-right paths are said to intersect if they share any point.
Find the number of pairs $(A, B)$ where $A$ is an up-right path from $(0, 0)$ to $(4, 4)$, $B$ is an up-right path from $(2, 0)$ to $(6, 4)$, and $A$ and $B$ do not intersect.
An international society has its members from six different countries. The list of members contain $1978$ names, numbered $1, 2, \dots, 1978$. Prove that there is at least one member whose number is the sum of the numbers of two members from his own country, or twice as large as the number of one member from his own country.
Sheldon was really annoying Leonard. So to keep him quiet, Leonard decided to do something. He gave Sheldon the following grid
$\begin{tabular}{|c|c|c|c|c|c|}
\hline
1 & 1 & 1 & 1 & 1 & 0\\
\hline
1 & 1 & 1 & 1 & 0 & 0\\
\hline
1 & 1 & 1 & 0 & 0 & 0\\
\hline
1 & 1 & 0 & 0 & 0 & 1\\
\hline
1 & 0 & 0 & 0 & 1 & 0\\
\hline
0 & 0 & 0 & 1 & 0 & 0\\
\hline
\end{tabular}$
and asked him to transform it to the new grid below
$\begin{tabular}{|c|c|c|c|c|c|}
\hline
1 & 2 & 18 &24 &28 &30\\
\hline
21 & 3 & 4 &16 &22 &26\\
\hline
23 &19 & 5 & 6 &14 &20\\
\hline
32 &25 &17 & 7 & 8 &12\\
\hline
33 &34 &27 &15 & 9 &10\\
\hline
35 &31 &36 &29 &13 &11\\
\hline
\end{tabular}$
by only applying the following algorithm:
$\bullet$ At each step, Sheldon must choose either two rows or two columns.
$\bullet$ For two columns $c_1, c_2$, if $a,b$ are entries in $c_1, c_2$ respectively, then we say that $a$ and $b$ are corresponding if they belong to the same row. Similarly we define corresponding entries of two rows. So for Sheldon's choice, if two corresponding entries have the same parity, he should do nothing to them, but if they have different parities, he should add 1 to both of them.
Leonard hoped this would keep Sheldon occupied for some time, but Sheldon immediately said, "But this is impossible!". Was Sheldon right? Justify.
A fair coin is to be tossed $10$ times. Let $i/j$, in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j$.
The positive real numbers $a, b, c$ satisfy the equation $a+b+c=1$.
Prove the identity: $\sqrt{\frac{(a+bc)(b+ca)}{c+ab}}+\sqrt{\frac{(b+ca)(c+ab)}{a+bc}}+\sqrt{\frac{(c+ab)(a+bc)}{b+ca}} = 2$
How many ways can we pick four $3$-element subsets of $\{1, 2, ..., 6\}$ so that each pair of subsets share exactly one element?
Prove that the inequality \[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\] holds for all positive reals $a$, $b$, $c$.
Show that the infinite arithmetic progression $\{1,4,7,10 \ldots\}$ has infinitely many 3 -term sub sequences in harmonic progression such that for any two such triples $\{a_1, a_2 , a_3 \}$ and $\{b_1, b_2 ,b_3\}$ in harmonic progression , one has $$\frac{a_1} {b_1} \ne \frac {a_2}{b_2}$$.
Define $\triangle ABC$ with incenter $I$ and $AB=5$, $BC=12$, $CA=13$. A circle $\omega$ centered at $I$ intersects $ABC$ at $6$ points. The green marked angles sum to $180^\circ.$ Find $\omega$'s area divided by $\pi.$
Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.
Let $AB$ and $A_1B_1$ be two skew segments, $O$ and $O_1$ their respective midpoints. Prove that $OO_1$ is shorter than a half sum of $AA_1$ and $BB_1$.
Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \]
Suppose $\mathcal{T}=A_0A_1A_2A_3$ is a tetrahedron with $\angle A_1A_0A_3 = \angle A_2A_0A_1 = \angle A_3A_0A_2 = 90^\circ$, $A_0A_1=5, A_0A_2=12$ and $A_0A_3=9$. A cube $A_0B_0C_0D_0E_0F_0G_0H_0$ with side length $s$ is inscribed inside $\mathcal{T}$ with $B_0\in \overline{A_0A_1}, D_0 \in \overline{A_0A_2}, E_0 \in \overline{A_0A_3}$, and $G_0\in \triangle A_1A_2A_3$; what is $s$?
Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$
Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.
For a positive integer $a$, define the sequence ($x_n$) by $x_1 = x_2 = 1$ and $x_{n+2 }= (a^4 +4a^2 +2)x_{n+1} -x_n -2a^2$ , for n $\ge 1$. Show that $x_n$ is a perfect square and that for $n > 2$ its square root equals the first entry in the matrix $\begin{pmatrix}
a^2+1 & a \\
a & 1
\end{pmatrix}^{n-2}$
An $n$-stick is a connected figure consisting of $n$ matches of length $1$ which are placed horizontally or vertically and no two touch each other at points other than their ends. Two shapes that can be transformed into each other by moving, rotating or flipping are considered the same.
An $n$-mino is a shape which is built by connecting $n$ squares of side length 1 on their sides such that there's a path on the squares between each two squares of the $n$-mino.
Let $S_n$ be the number of $n$-sticks and $M_n$ the number of $n$-minos, e.g. $S_3=5$ And $M_3=2$.
(a) Prove that for any natural $n$, $S_n \geq M_{n+1}$.
(b) Prove that for large enough $n$ we have $(2.4)^n \leq S_n \leq (16)^n$.
A [b]grid segment[/b] is a segment on the plane of length 1 which it's both ends are integer points. A polystick is called [b]wise[/b] if using it and it's rotations or flips we can cover all grid segments without overlapping, otherwise it's called [b]unwise[/b].
(c) Prove that there are at least $2^{n-6}$ different unwise $n$-sticks.
(d) Prove that any polystick which is in form of a path only going up and right is wise.
(e) Extra points: Prove that for large enough $n$ we have $3^n \leq S_n \leq 12^n$
Time allowed for this exam was 2 hours.
Let $ A$ be the numbers of 5-digit positive numbers satisfying following condition:
The first digit is odd. Remaining $ 0$, or $ 2$ or $ 4$ digit/digits are even.
Let $ B$ be the numbers of 5-digit positive numbers satisfying following condition:
The first digit is even. Remaining $ 0$, or $ 2$ or $ 4$ digit/digits are even.
$ A \minus{} B \equal{} ?$
$\textbf{(A)}\ 5000 \qquad\textbf{(B)}\ 4640 \qquad\textbf{(C)}\ 3200 \qquad\textbf{(D)}\ 0 \qquad\textbf{(E)}\ \text{None}$
Let $ABC$ be an acute triangle orthocenter angle $H$. Let $\omega_1$ be the circle tangent to $BC$ at $B$ and passing through $H$ and $\omega_2$ the circle tangent to $BC$ at $C$ and passing through through $H$. A line $\ell$ passing through $H$ intersects the circles $\omega_1$ and $\omega_2$ at points $D$ and $E$, respectively (with $D$ and $E$ other than $H$). Lines $BD$ and $CE$ intersect at $F$, the lines $\ell$ and $AF$ intersect at $X$ and the circles $\omega_1$ and $\omega_2$ intersect at the points $P$ and $H$. Prove that the points $A, H, P$ and $X$ are still on the same circle.
In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.
The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges.
Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.
$P$ is a point inside the triangle $ABC$. The line through $P$ parallel to $AB$ meets $AC$ $A_0$ and $BC$ at $B_0$. Similarly, the line through $P$ parallel to $CA$ meets $AB$ at $A_1$ and $BC$ at $C_1$, and the line through P parallel to BC meets $AB$ at $B_2$ and $AC$ at $C_2$. Find the point $P$ such that $A_0B_0 = A_1B_1 = A_2C_2$.
Determine if there exists a positive integer $n$ such that $n$ has exactly $2000$ prime divisors and $2^{n}+1$ is divisible by $n$.