The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 \equal{} 1, F_2 \equal{} 1$ and $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n \equal{} mn$.

Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that $$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$. let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$. prove that $\widehat{BAD}=\widehat{CAX}$

In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the previous term to the previous one: $$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8,…$$ As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?

Let $ABC$ a triangle and $O$ his circumcentre.The lines $OA$ and $BC$ intersect each other at $M$ ; the points $N$ and $P$ are defined in an analogous way.The tangent line in $A$ at the circumcircle of triangle $ABC$ intersect $NP$ in the point $X$ ; the points $Y$ and $Z$ are defined in an analogous way.Prove that the points $X$ , $Y$ and $Z$ are collinear.

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

For any integers $m,n$, we have the set $A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the integer set. Does there exist three distinct elements $a,b,c$ which belong to $A(m,n)$ and satisfy the equality $a=bc$?

Let $ n$ be a positive integer, and $ k\leq n\minus{}1$, $ k\in \mathbb{N}$. Denote $ a_k\equal{}k!(1\plus{}\frac12\plus{}\frac13\plus{}\cdots\plus{}\frac1k)$. Prove that the number $ k! \cdot\left[\binom{n\minus{}1}{k}\minus{}(\minus{}1)^k\right]\plus{}(\minus{}1)^k\cdot a_k \cdot n$ is divisible by $ n^2$.

Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$)

Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

Give two congruent regular triangular pyramids, stick their bottom surfaces together. Then ,it becomes a hexahedron with all dihedral angles equal. The length of the shortest edge of the hexahedron is $2$. Then, the furthest distance between two vertexes is________.

Giiven $\Delta ABC$, $\angle CAB=75^{\circ}$ and $\angle ACB=45^{\circ}$. $BC$ is extended to $T$ so that $BC=CT$. Let $M$ be the midpoint of the segment $AT$. Find $\angle BMC$.

An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]

For each integer $ n$ of the form $ n\equal{}p_1 p_2 p_3 p_4$, where $ p_1,p_2,p_3,p_4$ are distinct primes, let $ 1\equal{}d_1<d_2<...<d_{15}<d_{16}\equal{}n$ be the divisors of $ n$. Prove that if $ n<1995$, then $ d_9\minus{}d_8 \not\equal{} 22$.

Given a rectangle paper of size $15$ cm $\times$  $20$ cm, fold it along a diagonal. Determine the area of the common part of two halfs of the paper?

Show that there exist (at least) a rearrangement $a_0, a_1, a_2,..., a_{63}$ of the numbers $0,1, 2,..., 63$, such that $a_i - a_j \ne a_j - a_k$, for any $i < j < k \in \{0,1, 2,..., 63\}$.

If $(x^2-x-1)^n=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+...+a_1x+a_0$, where $a_i,i\in\{0,1,2,..,2n\}$, find $a_1+a_3+...+a_{2n-1}$ and $a_0+a_2+a_4+...+a_{2n}$.

Find all pairs $(x,y)$ of real numbers such that $ |x|+ |y|=1340$ and $x^{3}+y^{3}+2010xy= 670^{3}$ .

Suppose $ABCD$ is a square piece of cardboard with side length $a$. On a plane are two parallel lines $\ell_1$ and $\ell_2$, which are also $a$ units apart. The square $ABCD$ is placed on the plane so that sides $AB$ and $AD$ intersect $\ell_1$ at $E$ and $F$ respectively. Also, sides $CB$ and $CD$ intersect $\ell_2$ at $G$ and $H$ respectively. Let the perimeters of $\triangle AEF$ and $\triangle CGH$ be $m_1$ and $m_2$ respectively. Prove that no matter how the square was placed, $m_1+m_2$ remains constant.

Let $n$ be a natural number. Find all solutions $x$ of the system of equations $$\left\{\begin{matrix} sinx+cosx=\frac{\sqrt{n}}{2}\\tg\frac{x}{2}=\frac{\sqrt{n}-2}{3}\end{matrix}\right.$$ On interval $\left[0,\frac{\pi}{4}\right).$

On each of the 2006 cards a natural number is written. Cards are placed arbitrarily in a row. 2 players take in turns a card from any end of the row until all the cards are taken. After that each player calculates sum of the numbers written of his cards. If the sum of the first player is not less then the sum of the second one then the first player wins. Show that there's a winning strategy for the first player.

On the legs of $\angle AOB$, the segments $OA$ and $OB$ lie, $OA > OB$. Points $M$ and $N$ on lines $OA$ and $OB$, respectively, are such that $AM = BN = x$. Find $x$ for which the length of $MN$ is minimal.

The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled? [asy] draw((0,0)--(8,0)--(8,6)--(0,0)); draw(Circle((4.5,1),1)); draw((4.5,2.5)..(5.55,2.05)..(6,1), EndArrow); dot((0,0)); dot((8,0)); dot((8,6)); dot((4.5,1)); label("A", (0,0), SW); label("B", (8,0), SE); label("C", (8,6), NE); label("8", (4,0), S); label("6", (8,3), E); label("10", (4,3), NW); label("P", (4.5,1), NW); [/asy] $ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17 $

A polynomial $P(x)$ has degree at most $2k$, where $k = 0, 1,2,\cdots$. Given that for an integer $i$, the inequality $-k \le i \le k$ implies $|P(i)| \le 1$, prove that for all real numbers $x$, with $-k \le x \le k$, the following inequality holds: \[|P(x)| < (2k + 1)\dbinom{2k}{k}\]

Does there exist a convex polyhedron having equal number of edges and diagonals? [i](A diagonal of a polyhedron is a segment through two vertices not lying on the same face) [/i]