Found problems: 85335
On the parliament of Sikinia, for any two deputies, there is third deputy, which knows exactly one of the two. Every deputy belongs to one of the two ruling parties. Every day, he president tells a certain group of deputies to change the party that they belong, and all the deputies which which know at least one of the deputies of the group has to change their party too. Prove that, the president can reach any configuration of deputies between two parties.(The president himself isn't a member of the parliament of Sikinia).
[I]Proposed by S. Berlov[/i]
The number $N$ is the product of $200$ distinct positive integers. Prove that it has at least 19901 distinct divisors (including 0 and itself).
[I]Proposed by A. Golovanov[/i]
[b]H[/b]orizontal parallel segments $AB=10$ and $CD=15$ are the bases of trapezoid $ABCD$. Circle $\gamma$ of radius $6$ has center within the trapezoid and is tangent to sides $AB$, $BC$, and $DA$. If side $CD$ cuts out an arc of $\gamma$ measuring $120^{\circ}$, find the area of $ABCD$.
Let $n\geq 3$ be integer. Given two pairs of $n$ cards numbered from 1 to $n$. Mix the $2n$ cards up and take the card 3 times every one card. Denote $X_1,\ X_2,\ X_3$ the numbers of the cards taken out in this order taken the cards. Find the probabilty such that $X_1<X_2<X_3$. Note that once a card taken out, it is not taken a back.
The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.
Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing \right\}$. Find the number of elements of $K(2004,2004)$.
Let $ABCDEF$ be a convex hexagon whose vertices lie on a circle. Suppose that $AB\cdot CD\cdot EF = BC\cdot DE\cdot FA$. Show that the diagonals $AD, BE$ and $CF$ are concurrent.
Let $ABC$ be an acute triangle and let $M$, $N$, and $ P$ be on $CB$, $AC$, and $AB$, respectively, such that $AB = AN + PB$, $BC = BP + MC$, $CA = CM + AN$. Let $\ell$ be a line in a different half plane than $C$ with respect to to the line $AB$ such that if $X, Y$ are the projections of $A, B$ on $\ell$ respectively, $AX = AP$ and $BY = BP$. Prove that $NXYM$ is a cyclic quadrilateral.
Let $f$ be a function defined from $((x,y) : x,y$ real, $xy\ne 0)$ to the set of all positive real numbers such that
$ (i) f(xy,z)= f(x,z)\cdot f(y,z)$ for all $x,y \ne 0$
$ (ii) f(x,yz)= f(x,y)\cdot f(x,z)$ for all $x,y \ne 0$
$ (iii) f(x,1-x) = 1 $ for all $x \ne 0,1$
Prove that
$ (a) f(x,x) = f(x,-x) = 1$ for all $x \ne 0$
$(b) f(x,y)\cdot f(y,x) = 1 $ for all $x,y \ne 0$
The condition (ii) was left out in the paper leading to an incomplete problem during contest.
For every positive integer $n$, define $S_n$ to be the sum
\[
S_n = \sum_{k = 1}^{2010} \left( \cos \frac{k! \, \pi}{2010} \right)^n .
\]
As $n$ approaches infinity, what value does $S_n$ approach?
Find all the polynomials $P(x)$ of a degree $\leq n$ with real non-negative coefficients such that $P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2$ , $ \forall x>0$.
Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert \equal{}\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.
For any given non-negative integer $m$, let $f(m)$ be the number of $1$'s in the base $2$ representation of $m$. Let $n$ be a positive integer. Prove that the integer $$\sum^{2^n - 1}_{m = 0} \Big( (-1)^{f(m)} \cdot 2^m \Big)$$ contains at least $n!$ positive divisors.
Twenty players participated in a chess tournament. Each player faced every other player exactly once and each match ended with either player winning or a draw. In this tournament, it was noticed that for every match that ended in a draw, each of the other $18$ players won at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players as $P_1, P_2, ...., P_{20}$ so that player $P_k$ beat player $P_{k+1}$, to each $k \in \{ 1, 2, 3,... , 19\}$.
Let $\triangle ABC$ be an isosceles triangle such that $AB=AC$. Let $\omega$ be a circle centered at $A$ with a radius strictly less than $AB$. Draw a tangent from $B$ to $\omega$ at $P$, and draw a tangent from $C$ to $\omega$ at $Q$. Suppose that the line $PQ$ intersects the line $BC$ at point $M$. Prove that $M$ is the midpoint of $BC$.
[b]8.[/b] Let the function $f(x)$ be periodic with the period $1$, non-negative, concave in the interval $(0,1)$ and continuous at the point $0$. Prove that $f(nx)\leq nf(x)$ for every real $x$ and positive integer $n$. [b](R. 6)[/b]
In triangle $ABC$, points $D$ and $E$ lie on the interior of segments $AB$ and $AC$, respectively,such that $AD = 1$, $DB = 2$, $BC = 4$, $CE = 2$ and $EA = 3$. Let $DE$ intersect $BC$ at $F$. Determine the length of $CF$.
Which one of the following gives an odd integer?
$\textbf{(A)}\ 6^2 \qquad \textbf{(B)}\ 23-17 \qquad \textbf{(C)}\ 9\times24 \qquad \textbf{(D)}\ 96\div8 \qquad \textbf{(E)}\ 9\times41$
In the plane, $n$ lines are drawn in general position (that is, there are neither two of them parallel nor three of them passing through the same point). Prove that it is possible to put a positive integer in each region (finite or infinite) determined by these lines so that for each line the sum of the numbers in the regions of a sdemiplane is equal to the sum of the numbers in the regions of the other semiplane.
Note: A region is a set of points such that the straight line connecting any two of them it does not intersect any of the lines. For example, a line divides the plane into $2$ infinite regions and three lines into general position divide the plane into $7$ regions, some finite(s) and others infinite.
Let $x<0.1$ be a positive real number. Let the [i]foury series[/i] be $4+4x+4x^2+4x^3+\dots$, and let the [i]fourier series[/i] be $4+44x+444x^2+4444x^3+\dots$. Suppose that the sum of the fourier series is four times the sum of the foury series. Compute $x$.
Given a $19 \times 19$ matrix where each component is either $1$ or $-1$. Let $b_i$ be the product of all components in the $i$-th row, and $k_i$ be the product of all components in the $i$-th column, for all $1 \le i \le 19$. Prove that for any such matrix, $b_1 + k_1 + b_2 + k_2 + \cdots + b_{19} + k_{19} \neq 0$.
Let $ABCDEF$ be a convex hexagon, such that the triangles $ABC$ and $DEF$ are equilateral and the diagonals $AD, BE$ and $CF$ are concurrent. Prove that $AC\parallel DF$ or $BE=AD+CF.$
Let $A$ be a finite ring. Show that there exists two natural numbers $m,p$ where $m> p\ge 1$, such that $a^m=a^p$ for all $a\in A$.
$n$ coins lies in the circle. If two neighbour coins lies both head up or both tail up, then we can flip both. How many variants of coins are available that can not be obtained from each other by applying such operations?
What is the smallest result that can be obtained from the following process?
*Choose three different numbers from the set $\{3,5,7,11,13,17\}$.
*Add two of these numbers.
*Multiply their sum by the third number.
$\text{(A)}\ 15 \qquad \text{(B)}\ 30 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 50 \qquad \text{(E)}\ 56$