Let $x_1$ and $x_2$ be the roots of the equation $x^2-px+1=0$ where $p>2$ is a prime number. Prove that $x_1^p+x_2^p$ is an integer divisible by $p^2$. [i]From the folklore[/i]

A positive integer is called [i]decimal palindrome[/i] if its decimal representation $z_n...z_0$ with $z_n\ne 0$ is mirror symmetric, i.e. if $z_k = z_{n-k}$ applies to all $k= 0, ..., n$. Show that each integer that is not divisible by $10$ has a positive multiple, which is a decimal palindrome.

$f(n,k)$ is defined by (1) $f(n,0) = f(n,n) = 1$ and (2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$. How many times do we need to use (2) to find $f(3991,1993)$?

In $3$-dimensional space there is given a point $O$ and a finite set $A$ of segments with the sum of lengths equal to $1988$. Prove that there exists a plane disjoint from $A$ such that the distance from it to $O$ does not exceed $574$.

A $7 \times 7$ square is made up of $16$ $1 \times 3$ tiles and $1$ $1 \times 1$ tile. Prove that the $1 \times 1$ tile lies either at the centre of the square or adjoins one of its boundaries .

Define the Fibonacci numbers by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n\geq 3$. Let $k$ be a positive integer. Suppose that for every positive integer $m$ there exists a positive integer $n$ such that $m \mid F_n-k$. Must $k$ be a Fibonacci number? [i]Proposed by Fedir Yudin.[/i]

Show that $1994$ divides $10^{900}-2^{1000}$.

Let $D$ be a dodecahedron which can be inscribed in a sphere with radius $R$. Let $I$ be an icosahedron which can also be inscribed in a sphere of radius $R$. Which has the greater volume, and why? Note: A regular [i]polyhedron [/i] is a geometric solid, all of whose faces are congruent regular polygons, in which the same number of polygons meet at each vertex. A regular dodecahedron is a polyhedron with $12$ faces which are regular pentagons and a regular icosahedron is a polyhedron with $20$ faces which are equilateral triangles. A polyhedron is inscribed in a sphere if all of its vertices lie on the surface of the sphere. The illustration below shows a dodecahdron and an icosahedron, not necessarily to scale. [img]https://cdn.artofproblemsolving.com/attachments/7/5/9873b42aacf04bb5daa0fe70d4da3bf0b7be38.png[/img]

Andrea and Lauren are $ 20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $ 1$ kilometer per minute. After $ 5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea? $ \textbf{(A)}\ 20 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 55 \qquad \textbf{(D)}\ 65 \qquad \textbf{(E)}\ 80$

In the parallelogram $ABCD$, all sides are equal, and $\angle A = 60^o$. Let $F$ be a point on line $AD, H$ a point on line $DC$, and $G$ a point on diagonal $AC$ such that $DFGH$ is a parallelogram. Show that then $\vartriangle BHF$ is equilateral.

A company with $2004$ workers celebrated its anniversary by inviting everyone to a lunch served at a round table. When the $2004$ workers sat around this table, they formed a circle of people and soon discovered that they all had salaries. different and also that the difference between the salaries of any two neighbors, at the round table, was $2000$ or $3000$ pesos. Calculate the maximum difference that can exist between the wages of these workers.

The circumcircle of triangle $ABC$ has centre $O$. $P$ is the midpoint of $\widehat{BAC}$ and $QP$ is the diameter. Let $I$ be the incentre of $\triangle ABC$ and let $D$ be the intersection of $PI$ and $BC$. The circumcircle of $\triangle AID$ and the extension of $PA$ meet at $F$. The point $E$ lies on the line segment $PD$ such that $DE=DQ$. Let $R,r$ be the radius of the inscribed circle and circumcircle of $\triangle ABC$, respectively. Show that if $\angle AEF=\angle APE$, then $\sin^2\angle BAC=\dfrac{2r}R$

Let $r > 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue. [list][*]If $r > \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that $AB = \pi$ and $A,B$ have the same color [*]Does the conclusion in (a) hold if $r > \frac{\pi}{2}$?[/list] [i]Proposed by S Muralidharan[/i]

Two mathematicians take a morning coffee break each day. They arrive at the cafeteria independently, at random times between 9 a.m. and 10 a.m., and stay for exactly $m$ mintues. The probability that either one arrives while the other is in the cafeteria is $40 \%,$ and $m=a-b\sqrt{c},$ where $a, b,$ and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

Define a function $f:\mathbb{N}\rightarrow\mathbb{N}$, \[f(1)=p+1,\] \[f(n+1)=f(1)\cdot f(2)\cdots f(n)+p,\] where $p$ is a prime number. Find all $p$ such that there exists a natural number $k$ such that $f(k)$ is a perfect square.

$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$

Three friends have a total of $6$ identical pencils, and each one has at least one pencil. In how many ways can this happen? $\textbf{(A)}\ 1\qquad \textbf{(B)}\ 3\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ 12$

Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$. [i]Remark:[/i] For a nonnegative integer $k$ and an integer $n$, $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$. Proposed by Adrian Beker.

Depei is imprisoned by an evil wizard and is coerced to play the following game. Every turn, Depei flips a fair coin. Then, the following events occur in this order: $\bullet$ The wizard computes the difference between the total number of heads and the total number of tails Depei has flipped. If that number is greater than or equal to $4$ or less than or equal to $-3$, then Depei is vaporized by the wizard. $\bullet$ The wizard determines if Depei has flipped at least $10$ heads or at least $10$ tails. If so, then the wizard releases Depei from the prison. The probability that Depei is released by the evil wizard equals $\frac{m}{2^k}$ , where $m, k$ are positive integers. Compute $m+k$.

Let $a,b$ positive real numbers. Prove that $$\frac{(a+b)^3}{a^2b}\ge \frac{27}{4}.$$ When does equality occur?

Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.

Prove the following two statements: (a) If a triangle is isosceles, then two of its bisectors are of equal length. (b) If two angle bisectors in a triangle are of equal length, then it is isosceles.

Let $P$ be a point in the interior of triangle $ABC$. Let $X$, $Y$, and $Z$ be points on sides $BC$, $AC$, and $AB$ respectively, such that $<PXC=<PYA=<PZB$. Let $U$, $V$, and $W$ be points on sides $BC$, $AC$, and $AB$, respectively, or on their extensions if necessary, with $X$ in between $B$ and $U$, $Y$ in between $C$ and $V$, and $Z$ in between $A$ and $W$, such that $PU=2PX$, $PV=2PY$, and $PW=2PZ$. If the area of triangle $XYZ$ is $1$, find the area of triangle $UVW$.

On the plane 1000 lines are drawn, among which there are no parallel lines. From any seven of these lines, some three pass through one point. But no more than 500 lines pass through each point. Prove that there are three points such that each line contains at least of of them.

Let a positive integer number $n$ has $k$ different prime divisors. Prove that there exists a positive integer number $x \in \left(1, \frac{n}{k}+1 \right)$ such that $x^2-x$ divides by $n$.