In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is [asy] size(180); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,1), B=(0,-1), C=(2,-1), D=(2,1), E=(1,-1), F=(1,1), G=(.8,.6); pair X=(4,sqrt(5)), Y=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5)); draw(A--B--C--D--A^^B--F^^E--D^^A--G^^rightanglemark(A,G,F)); draw(X--Y--Z--W--X^^T--V--X^^Y--U); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, S); label("F", F, N); label("G", G, E); label("X", X, NW); label("Y", Y, SW); label("Z", Z, SE); label("W", W, NE); [/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 1+2\sqrt{3}\qquad\textbf{(C)}\ 2\sqrt{5}\qquad\textbf{(D)}\ \frac{8+4\sqrt{3}}{3}\qquad\textbf{(E)}\ 5 $

Let $n \geq 2$ be a positive integer. There are $2n$ cities $M_1, M_2, \ldots, M_{2n}$ in the country of Mathlandia. Currently there roads only between $M_1$ and $M_2, M_3, \ldots, M_n$ and the king wants to build more roads so that it is possible to reach any city from every other city. The cost to build a road between $M_i$ and $M_j$ is $k_{i, j}>0$. Let $$K=\sum_{j=n+1}^{2n} k_{1,j}+\sum_{2 \leq i<j \leq 2n} k_{i, j}.$$Prove that the king can fulfill his plan at cost no more than $\frac{2K}{3n-1}$.

One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of ​​this triangle to the area of ​​the quadrilateral formed by three such lines and one of the sides of the triangle.

A “number triangle” $(t_{n, k}) (0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1 (n \ge 0),$ \[t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n)\] Prove that all $t_{n,m}$ are integers.

Let $ a_{1},a_{2},...,a_{n} $ be real positive numbers and $ k>m, k,m $ natural numbers. Prove that $(n-1)(a_{1}^m +a_{2}^m+...+a_{n}^m)\leq\frac{a_{2}^k+a_{3}^k+...+a_{n}^k}{a_{1}^{k-m}}+\frac{a_{1}^k+a_{3}^k+...+a_{n}^k}{a_2^{k-m}}+...+\frac{a_{1}^k+a_{2}^k+...+a_{n-1}^k}{a_{n}^{k-m}} $

Find a polynomial $ p\left(x\right)$ with real coefficients such that $ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$ for all real $ x$ and $ p\left(1\right)\equal{}210$.

The incircle of the quadrilateral $ABCD$ touches $AB,BC, CD$ and $DA$ at $E, F,G$ and $H$ respectively. Prove that the line joining the incentres of triangles $HAE$ and $FCG$ is perpendicular to the line joining the incentres of triangles $EBF$ and $GDH$. (4)

What is the radius of the largest sphere that fits inside the tetrahedron whose vertices are the points $(0, 0, 0)$, $(1, 0, 0)$, $(0, 1, 0)$, $(0, 0, 1)$?

The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

Let $p \geq 5$ be a prime number. Prove that there exists a positive integer $a < p-1$ such that neither of $a^{p-1}-1$ and $(a+1)^{p-1}-1$ is divisible by $p^{2}$ .

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$, $$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$ and, for each positive integer $k$ such that $b_{k-1}\geq 2$, $$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$ It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$. Prove that the numbers of terms in the two sequences are always equal.

Let $\triangle AB\varGamma$ be an acute-angled triangle with $AB < A\varGamma$, and let $O$ be the center of the circumcircle of the triangle. On the sides $AB$ and $A \varGamma$ we select points $T$ and $P$ respectively such that $OT=OP$. Let $M,K$ and $\varLambda$ be the midpoints of $PT,PB$ and $\varGamma T$ respectively. Prove that $\angle TMK = \angle M\varLambda K$.

John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kopeks wins. Which player has a winning strategy? [i](5 points)[/i]

In an acute triangle $ABC$, the circle with diameter $AC$ intersects $AB$ and $AC$ at $K$ and $L$ different from $A$ and $C$ respectively. The circumcircle of $ABC$ intersects the line $CK$ at the point $F$ different from $C$ and the line $AL$ at the point $D$ different from $A$. A point $E$ is choosen on the smaller arc of $AC$ of the circumcircle of $ABC$ . Let $N$ be the intersection of the lines $BE$ and $AC$ . If $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$ prove that $\angle KNB= \angle BNL$ .

$\triangle ABC$ is isosceles $AB = AC$. $P$ is a point inside $\triangle ABC$ such that $\angle BCP = 30$ and $\angle APB = 150$ and $\angle CAP = 39$. Find $\angle BAP$.

Of the following sets, the one that includes all values of $ x$ which will satisfy $ 2x \minus{} 3 > 7 \minus{} x$ is: $ \textbf{(A)}\ x > 4 \qquad \textbf{(B)}\ x < \frac {10}{3} \qquad \textbf{(C)}\ x \equal{} \frac {10}{3} \qquad \textbf{(D)}\ x > \frac {10}{3} \qquad \textbf{(E)}\ x < 0$

Find, with proof, all triples of integers $ (a,b,c)$ such that $ a, b$ and $ c$ are the lengths of the sides of a right angled triangle whose area is $ a \plus{} b \plus{} c$

Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.

The square $ABCD$ has to be decomposed into $n$ triangles (which are not overlapping) and which have all angles acute. Find the smallest integer $n$ for which there exist a solution of that problem and for such $n$ construct at least one decomposition. Answer whether it is possible to ask moreover that (at least) one of these triangles has the perimeter less than an arbitrarily given positive number.

Two nonadjacent vertices of a rectangle are $(4,3)$ and $(-4,-3)$, and the coordinates of the other two vertices are integers. The number of such rectangles is $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

What is the smallest integer larger than $(\sqrt{3}+\sqrt{2})^6$? $ \textbf{(A)}\ 972 \qquad\textbf{(B)}\ 971 \qquad\textbf{(C)}\ 970 \qquad\textbf{(D)}\ 969 \qquad\textbf{(E)}\ 968 $

Let $n$ be a integer and let $n=d_1>d_2>\cdots>d_k=1$ its positive divisors. a) Prove that $$d_1-d_2+d_3-\cdots+(-1)^{k-1}d_k=n-1$$ iff $n$ is prime or $n=4$. b) Determine the three positive integers such that $$d_1-d_2+d_3-...+(-1)^{k-1}d_k=n-4.$$

Find one real value of $x$ satisfying $\frac{x^7}{7}=1+\sqrt[7]{10}x\left(x^2-\sqrt[7]{10}\right)^2$.

Circles $ \omega_1$ and $ \omega_2$ meet at $ P$ and $ Q$. Segments $ AC$ and $ BD$ are chords of $ \omega_1$ and $ \omega_2$ respectively, such that segment $ AB$ and ray $ CD$ meet at $ P$. Ray $ BD$ and segment $ AC$ meet at $ X$. Point $ Y$ lies on $ \omega_1$ such that $ PY \parallel BD$. Point $ Z$ lies on $ \omega_2$ such that $ PZ \parallel AC$. Prove that points $ Q,X,Y,Z$ are collinear.