One of Euler's conjectures was disproved in then 1960s by three American mathematicians when they showed there was a positive integer $ n$ such that \[133^5 \plus{} 110^5 \plus{} 84^5 \plus{} 27^5 \equal{} n^5.\] Find the value of $ n$.

Recall that a palindrome is a word which is the same when we read it forward or backward. (a) We have an infinite number of cards with words $\{abc, bca, cab\}$. A word is made from them in the following way. The initial word is an arbitrary card. At each step we obtain a new word either gluing a card (from the right or from the left) to the existing word or making a cut between any two of its letters and gluing a card between both parts. Is it possible to obtain a palindrome this way? [i](4 points)[/i] (b) We have an infinite number of red cards with words $\{abc, bca, cab\}$ and of blue cards with words $\{cba, acb, bac\}$. A palindrome was formed from them in the same way as in part (a). Is it necessarily true that the number of red and blue cards used was equal? [i](6 points)[/i] [i]Alexandr Gribalko, Ivan Mitrofanov [/i]

A pen costs $11$ € and a notebook costs $13$ €. Find the number of ways in which a person can spend exactly $1000$ € to buy pens and notebooks.

Let $n \in \mathbb{N} , n \ge 2$ and $ a_0,a_1,a_2,\cdots,a_n \in \mathbb{C} ; a_n \not = 0 $. Then: [b][size=100][i]P.[/i][/size][/b] $|a_nz^n + a_{n-1}z^z{n-1} + \cdots + a_1z + a_0 | \le |a_n+a_0|$ for any $z \in \mathbb{C}, |z|=1$ [b][size=100][i]Q[/i][/size][/b]. $a_1=a_2=\cdots=a_{n-1}=0$ and $a_0/a_n \in [0,\infty)$ Prove that $ P \Longleftrightarrow Q$

Let $k$ be a fixed positive integer. Prove that the sequence $\binom{2}{1},\binom{4}{2},\binom{8}{4},\ldots, \binom{2^{n+1}}{2^n},\ldots$ is eventually constant modulo $2^k$. [i]Proposed by V. Rastorguyev[/i]

A rectangle is said to be inscribed in a parallelogram if its vertices lie one on each side of the parallelogram. On the larger side $AB$ of the parallelogram $ABCD$, find all those points $K$ that are the vertices of the rectangles inscribed in $ABCD$.

Let $C',B',A'$ be points respectively on sides $AB,AC,BC$ of $\triangle ABC$ satisfying $ \tfrac{AB'}{B'C}= \tfrac{BC'}{C'A}=\tfrac{CA'}{A'B}=k$. Prove that the ratio of the area of the triangle formed by the lines $AA',BB',CC'$ over the area of $\triangle ABC$ is $\tfrac{(k-1)^2}{(k^2+k+1)}$.

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

Find the number of all sequences $ (x_1,...,x_n)$ of letters $ a,b,c$ that satisfy $ x_1\equal{}x_n\equal{}a$ and $ x_i \not\equal{} x_{i\plus{}1}$ for $ 1 \le i \le n\minus{}1.$

A coast artillery gun can fire at every angle of elevation between $0^{\circ}$ and $90^{\circ}$ in a fixed vertical plane. If air resistance is neglected and the muzzle velocity is constant ($=v_0 $), determine the set $H$ of points in the plane and above the horizontal which can be hit.

Let $\triangle ABC$ be an equilateral triangle and $P$ a point on $\overline{BC}$. If $PB=50$ and $PC=30$, compute $PA$.

Prove that every infinite sequence $S$ of distinct positive integers contains either an infinite subsequence such that for every pair of terms, neither term ever divides the other, or an infinite subsequence such that in every pair of terms, one always divides the other.

A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$: $A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$. However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.

Decide whether any rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1.

Let $ABC$ and $A'B'C'$ be two triangles so that the midpoints of $\overline{AA'}, \overline{BB'}, \overline{CC'}$ form a triangle as well. Suppose that for any point $X$ on the circumcircle of $ABC$, there exists exactly one point $X'$ on the circumcircle of $A'B'C'$ so that the midpoints of $\overline{AA'}, \overline{BB'}, \overline{CC'}$ and $\overline{XX'}$ are concyclic. Show that $ABC$ is similar to $A'B'C'$. [i]Proposed by usjl[/i]

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.

Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.

Two cells in a $20 \times 20$ board are adjacent if they have a common edge (a cell is not considered adjacent to itself). What is the maximum number of cells that can be marked in a $20 \times 20$ board such that every cell is adjacent to at most one marked cell?

A board that has some of its squares painted black is called [i]acceptable [/i] if there are no four black squares that form a $2 \times 2$ subboard. Find the largest real number $\lambda$ such that for every positive integer $n$ the following proposition holds: mercy: if an $n \times n$ board is acceptable and has fewer than $\lambda n^2$ dark squares, then an additional square black can be painted so that the board is still acceptable.

Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB, Q, R$ on $BC$ and $S$ on $AC$. If the area of the square $PQRS$ is $m +n\sqrt{k}$ where $m, n$ are integers and $k$ is a prime number then determine the value of $\sqrt{\frac{m+n}{k^2}}$.

Let $ABCD$ be a trapezoid with $AB\parallel CD$. Its diagonals intersect at a point $P$. The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$, respectively. Exterior angle bisectors of angles $DBA$, $DCA$ intersect at $X$. Let $S$ be the foot of $X$ onto $BC$. Prove that if quadrilaterals $ABPQ$, $CDQP$ are circumcribed, then $PR=PS$. [i]Proposed by Dominik Burek, Poland[/i]

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

The sequence $p_1, p_2, p_3, ...$ is defined as follows. $p_1$ and $p_2$ are primes. $p_n$ is the greatest prime divisor of $p_{n-1} + p_{n-2} + 2000$. Show that the sequence is bounded.

Thaddeus is given a $2013 \times 2013$ array of integers each between $1$ and $2013$, inclusive. He is allowed two operations: 1. Choose a row, and subtract $1$ from each entry. 2. Chooses a column, and add $1$ to each entry. He would like to get an array where all integers are divisible by $2013$. On how many arrays is this possible?

A function $f$ satisfies the following condition for each $n\in N$: $f (1)+ f (2)+...+ f (n) = n^2 f (n)$. Find $f (1997)$ if $f (1) = 999$.