Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others. $5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice?

In the lower-left $3\times3$ square of an $8\times8$ chessboard there are nine pawns. Every pawn can jump horizontally or vertically over a neighboring pawn to the cell across it if that cell is free. Is it possible to arrange the nine pawns in the upperleft $3\times3$ square of the chessboard using finitely many such moves?

We define the following sequences: • Sequence $A$ has $a_n = n$. • Sequence $B$ has $b_n = a_n$ when $a_n \not\equiv 0$ (mod 3) and $b_n = 0$ otherwise. • Sequence $C$ has $c_n =\sum_{i=1}^{n} b_i$ .• Sequence $D$ has $d_n = c_n$ when $c_n \not\equiv 0$ (mod 3) and $d_n = 0$ otherwise. • Sequence $E$ has $e_n =\sum_{i=1}^{n}d_i$ Prove that the terms of sequence E are exactly the perfect cubes.

Let $ABCD$ be a quadrilateral inscribed in a circle $\Gamma$ such that $AB=BC=CD$. Let $M$ and $N$ be the midpoints of $AD$ and $AB$ respectively. The line $CM$ meets $\Gamma$ again at $E$. Prove that the tangent at $E$ to $\Gamma$, the line $AD$ and the line $CN$ are concurrent.

Let $P$ be a point on the median $AM$ of a triangle $ABC$. Suppose that the tangents to the circumcircles of $ABP$ and $ACP$ at $B$ and $C$ respectively meet at $Q$. Show that $\angle PAB = \angle CAQ$.

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive real numbers, and $s$ be a positive integer, such that \[a_n = \max \{ a_k + a_{n-k} \mid 1 \leq k \leq n-1 \} \ \textrm{ for all } \ n > s.\] Prove there exist positive integers $\ell \leq s$ and $N$, such that \[a_n = a_{\ell} + a_{n - \ell} \ \textrm{ for all } \ n \geq N.\] [i]Proposed by Morteza Saghafiyan, Iran[/i]

Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?

Points $ A,B,C$ and $ D$ lie on a line, in that order, with $ AB\equal{}CD$ and $ BC\equal{}12$. Point $ E$ is not on the line, and $ BE\equal{}CE\equal{}10$. The perimeter of $ \triangle AED$ is twice the perimeter of $ \triangle BEC$. Find $ AB$. $ \text{(A)}\ 15/2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 17/2 \qquad \text{(D)}\ 9 \qquad \text{(E)}\ 19/2$

A square $ABCD$ is given. A point $P$ is chosen inside the triangle $ABC$ such that $\angle CAP = 15^\circ = \angle BCP$. A point $Q$ is chosen such that $APCQ$ is an isosceles trapezoid: $PC \parallel AQ$, and $AP=CQ, AP\nparallel CQ$. Denote by $N$ the midpoint of $PQ$. Find the angles of the triangle $CAN$.

Fill in each empty cell of the grid with a digit from 1 to 8 so that every row and every column contains each of these digits exactly once. Some diagonally adjacent cells have been joined together. For these pairs of joined cells, the same number must be written in both. [asy] filldraw((0,0)--(0,8)--(8,8)--(8,0)--cycle,white); path removex(pair p) { return ((p.x-0.5, p.y)--(p.x+0.5,p.y)); } path removey(pair p) { return ((p.x, p.y-0.5)--(p.x,p.y+0.5)); } unitsize(1cm); draw((0,0)--(8,0)--(8,8)--(0,8)--cycle, linewidth(2)); for(int i = 0; i < 8; ++i){ draw((0,i)--(8,i)); } for(int j = 0 ; j<8; ++j){ draw((j,0)--(j,8)); } pair [] pointsa = {(1,2),(3,1),(5,7),(7,6)}; pair [] pointsb= {(1,5),(4,4),(2,7),(6,1),(7,3)}; for(int q = 0; q<4; ++q){ draw(removex(pointsa[q]), white+linewidth(2)); draw(removey(pointsa[q]),white+linewidth(2)); draw(arc(pointsa[q]+(0.5,-0.5),0.5,90,180)); draw(arc(pointsa[q]-(0.5,-0.5),0.5,270,0,CCW)); draw(pointsa[q]+(-0.5,0)--pointsa[q]+(-1,0)); draw(pointsa[q]+(0.5,0)--pointsa[q]+(1,0)); draw(pointsa[q]+(0,0.5)--pointsa[q]+(0,1)); draw(pointsa[q]+(0,-0.5)--pointsa[q]+(0,-1)); } for(int q = 0; q<5; ++q){ draw(removex(pointsb[q]), white+linewidth(2)); draw(removey(pointsb[q]),white+linewidth(2)); draw(arc(pointsb[q]+(0.5,0.5),0.5,180,270,CCW)); draw(arc(pointsb[q]-(0.5,0.5),0.5,0,90,CCW)); draw(pointsb[q]+(-0.5,0)--pointsb[q]+(-1,0)); draw(pointsb[q]+(0.5,0)--pointsb[q]+(1,0)); draw(pointsb[q]+(0,0.5)--pointsb[q]+(0,1)); draw(pointsb[q]+(0,-0.5)--pointsb[q]+(0,-1)); } int [][] x = { {1,0,0,0,0,0,0,0}, {2,3,0,0,0,0,0,0}, {0,4,5,0,0,0,0,0}, {0,0,6,0,1,0,0,0}, {0,0,0,7,0,1,0,0}, {0,0,0,0,0,3,4,0}, {0,0,0,0,0,0,2,8}, {0,0,0,0,0,0,0,5} }; for(int k = 0; k<8; ++k){ for(int l = 0; l<8; ++l){ if(x[k][l]!=0){ label(string(x[k][l]), (l+0.5,-k+7.5), fontsize(24pt)); } } } [/asy] There is a unique solution, but you do not need to prove that your answer is the only one possible. You merely need to find an answer that satisfies the constraints above. (Note: In any other USAMTS problem, you need to provide a full proof. Only in this problem is an answer without justification acceptable.)

A polynomial $P (x)$ with real coefficients and of degree $n \ge 3$ has $n$ real roots $x_1 <x_2 < \cdots < x_n$ such that \[x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1} \] Prove that the maximum value of $|P (x)|$ on the interval $[x_1 , x_n ]$ is attained in the interval $[x_{n-1} , x_n ]$.

Each BMT, every student chooses one of three focus rounds to take. Bob plans to attend BMT for the next $4$ years and wants to gure out what focus round to take each year. Given that he wants to take each focus round at least once, how many ways can he choose which round to take each year?

Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

Consider function $ f: R\to R$ which satisfies the conditions for any mutually distinct real numbers $ a,b,c,d$ satisfying $ \frac {a \minus{} b}{b \minus{} c} \plus{} \frac {a \minus{} d}{d \minus{} c} \equal{} 0$, $ f(a),f(b),f(c),f(d)$ are mutully different and $ \frac {f(a) \minus{} f(b)}{f(b) \minus{} f(c)} \plus{} \frac {f(a) \minus{} f(d)}{f(d) \minus{} f(c)} \equal{} 0.$ Prove that function $ f$ is linear

Inside the $7\times8$ rectangle below, one point is chosen a distance $\sqrt2$ from the left side and a distance $\sqrt7$ from the bottom side. The line segments from that point to the four vertices of the rectangle are drawn. Find the area of the shaded region. [asy] import graph; size(4cm); pair A = (0,0); pair B = (9,0); pair C = (9,7); pair D = (0,7); pair P = (1.5,3); draw(A--B--C--D--cycle,linewidth(1.5)); filldraw(A--B--P--cycle,rgb(.76,.76,.76),linewidth(1.5)); filldraw(C--D--P--cycle,rgb(.76,.76,.76),linewidth(1.5)); [/asy]

Jesse and Tjeerd are playing a game. Jesse has access to $n\ge 2$ stones. There are two boxes: in the black box there is room for half of the stones (rounded down) and in the white box there is room for half of the stones (rounded up). Jesse and Tjeerd take turns, with Jesse starting. Jesse grabs in his turn, always one new stone, writes a positive real number on the stone and places put him in one of the boxes that isn't full yet. Tjeerd sees all these numbers on the stones in the boxes and on his turn may move any stone from one box to the other box if it is not yet full, but he may also choose to do nothing. The game stops when both boxes are full. If then the total value of the stones in the black box is greater than the total value of the stones in the white box, Jesse wins; otherwise win Tjeerd. For every $n \ge 2$, determine who can definitely win (and give a corresponding winning strategy).

In the language of wolves has two letters $F$ and $P$, any finite sequence which forms a word. А word $Y$ is called 'subpart' of word $X$ if Y is obtained from X by deleting some letters (for example, the word $FFPF$ has 8 'subpart's: F, P, FF, FP, PF, FFP, FPF, FFF). Determine $n$ such that the $n$ is the greatest number of 'subpart's can have n-letter word language of wolves. F. Petrov, V. Volkov

Prove that : For each integer $n \ge 3$, there exists the positive integers $a_1<a_2< \cdots <a_n$ , such that for $ i=1,2,\cdots,n-2 $ , With $a_{i},a_{i+1},a_{i+2}$ may be formed as a triangle side length , and the area of the triangle is a positive integer.

Let $x_1, x_2, \ldots, x_n$ be real positive numbers such that $x_1\cdot x_2\cdots x_n=1$. Prove the inequality $\frac1{x_1(x_1+1)}+\frac1{x_2(x_2+1)}+\cdots+\frac1{x_n(x_n+1)}\geq\frac n2$

Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \ln \left(1\plus{}\frac{k^a}{n^{a\plus{}1}}\right).$

If $x$ is the number of solutions to the equation $a^2 + b^2 + c^2 = d^2$ of the form $(a,b,c,d)$ such that $\{a,b,c\}$ are three consecutive square numbers and $d$ is also a square number, find $x$.

The altitudes $AN$ and $BM$ are drawn in triangle $ABC$. Prove that the perpendicular bisector to the segment $NM$ divides the segment $AB$ in half.

Let $ABC$ be an acute triangle with orthocenter $H$. The circumcircle of the triangle $BHC$ intersects $AC$ a second time in point $P$ and $AB$ a second time in point $Q$. Prove that $H$ is the circumcenter of the triangle $APQ$. [i](Karl Czakler)[/i]

Nine lines, parallel to the base of a triangle, divide the other sides into 10 equal segments and the area into 10 distinct parts. Find the area of the original triangle, if the area of the largest of these parts is 76.