The vertices $M$, $N$, $K$ of rectangle $KLMN$ lie on the sides $AB$, $BC$, $CA$ respectively of a regular triangle $ABC$ in such a way that $AM = 2$, $KC = 1$. The vertex $L$ lies outside the triangle. Find the value of $\angle KMN$.

Determine the largest integer $N$, for which there exists a $6\times N$ table $T$ that has the following properties: $*$ Every column contains the numbers $1,2,\ldots,6$ in some ordering. $*$ For any two columns $i\ne j$, there exists a row $r$ such that $T(r,i)= T(r,j)$. $*$ For any two columns $i\ne j$, there exists a row $s$ such that $T(s,i)\ne T(s,j)$. (Proposed by Gerhard Woeginger, Austria)

If the remainder is $2013$ when a polynomial with coefficients from the set $\{0,1,2,3,4,5\}$ is divided by $x-6$, what is the least possible value of the coefficient of $x$ in this polynomial? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1 $

Three positive integers are written on the board. In every minute, instead of the numbers $a, b, c$, Elbek writes $a+\gcd(b,c), b+\gcd(a,c), c+\gcd(a,b)$ . Prove that there will be two numbers on the board after some minutes, such that one is divisible by the other. Note. $\gcd(x,y)$ - Greatest common divisor of numbers $x$ and $y$ [i]Proposed by Sergey Berlov, Russia[/i]

Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.

Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$

Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

Given points $P_1, P_2,\cdots,P_7$ on a straight line, in the order stated (not necessarily evenly spaced). Let $P$ be an arbitrarily selected point on the line and let $s$ be the sum of the undirected lengths \[PP_1, PP_2, \cdots , PP_7.\] Then $s$ is smallest if and only if the point $P$ is: $\textbf{(A)}\ \text{midway between }P_1\text{ and }P_7\qquad \textbf{(B)}\ \text{midway between }P_2\text{ and }P_6\qquad \textbf{(C)}\ \text{midway between }P_3\text{ and }P_5 \qquad$ $ \textbf{(D)}\ \text{at }P_4 \qquad \textbf{(E)}\ \text{at }P_1$

The function $g$, with domain and real numbers, fulfills the following: $\bullet$ $g (x) \le x$, for all real $x$ $\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$ Find $g (1990)$.

A terrain ( $ABCD$ ) has a rectangular trapezoidal shape. The angle in $A$ measures $90^o$. $AB$ measures $30$ m, $AD$ measures $20$ m and $DC$ measures 45 m. This land must be divided into two areas of the same area, drawing a parallel to the $AD$ side . At what distance from $D$ do we have to draw the parallel? [img]https://1.bp.blogspot.com/-DnyNY3x4XKE/XNYvRUrLVTI/AAAAAAAAKLE/gohd7_S9OeIi-CVUVw-iM63uXE5u-WmGwCK4BGAYYCw/s400/image002.gif[/img]

Let $ABC$ be an acute triangle with circumcenter $O$ and orthocenter $H$. The circle with center $X_A$ passes through the points $A$ and $H$ and is tangent to the circumcircle of the triangle $ABC$. Similarly, define the points $X_B$ and $X_C$. Let $O_A$, $O_B$ and $O_C$ be the reflections of $O$ with respect to sides $BC$, $CA$ and $AB$, respectively. Prove that the lines $O_AX_A$, $O_BX_B$ and $O_CX_C$ are concurrent.

Two candles of the same height are lighted at the same time. The first is consumed in $ 4$ hours and the second in $ 3$ hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second? $ \textbf{(A)}\ \frac {3}{4} \qquad\textbf{(B)}\ 1\frac {1}{2} \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 2\frac {2}{5} \qquad\textbf{(E)}\ 2\frac {1}{2}$

Show that for any $n\geq 2$, $2^{2^n}+1$ ends with 7

Let $ D,E,F $ be the feet of the interior bisectors from $ A,B, $ respectively $ C, $ and let $ A',B',C' $ be the symmetric points of $ A,B, $ respectively, $ C, $ to $ D,E, $ respectively $ F, $ such that $ A,B,C $ lie on $ B'C',A'C', $ respectively, $ A'B'. $ Show that the $ ABC $ is equilateral. [i]Marius Beceanu[/i]

In $\vartriangle ABC$ on the sides $BC, CA, AB$ mark feet of altitudes $H_1, H_2, H_3$ and the midpoint of sides $M_1, M_3, M_3$. Let $H$ be orthocenter $\vartriangle ABC$. Suppose that $X_2, X_3$ are points symmetric to $H_1$ wrt $BH_2$ and $CH_3$. Lines $M_3X_2$ and $M_2X_3$ intersect at point $X$. Similarly, $Y_3,Y_1$ are points symmetric to $H_2$ wrt $C_3H$ and $AH_1$.Lines $M_1Y_3$ and $M_3Y_1$ intersect at point $Y.$ Finally, $Z_1,Z_2$ are points symmetric to $H_3$ wrt $AH_1$ and $BH_2$. Lines $M_1Z_2$ and $M_2Z_1$ intersect at the point $Z$ Prove that $H$ is the incenter $\vartriangle XYZ$ .

Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.

Given $5n$ real numbers $r_i, s_i, t_i, u_i, v_i \geq 1 (1 \leq i \leq n)$, let $R = \frac {1}{n} \sum_{i=1}^{n} r_i$, $S = \frac {1}{n} \sum_{i=1}^{n} s_i$, $T = \frac {1}{n} \sum_{i=1}^{n} t_i$, $U = \frac {1}{n} \sum_{i=1}^{n} u_i$, $V = \frac {1}{n} \sum_{i=1}^{n} v_i$. Prove that $\prod_{i=1}^{n}\frac {r_i s_i t_i u_i v_i + 1}{r_i s_i t_i u_i v_i - 1} \geq \left(\frac {RSTUV +1}{RSTUV - 1}\right)^n$.

Let $\alpha \in (1, +\infty)$ be a real number, and let $P(x) \in \mathbb{R}[x]$ be a monic polynomial with degree $24$, such that (i) $P(0) = 1$. (ii) $P(x)$ has exactly $24$ positive real roots that are all less than or equal to $\alpha$. Show that $|P(1)| \le \left( \frac{19}{5}\right)^5 (\alpha-1)^{24}$.

Prove or disprove: If $x$ and $y$ are real numbers with $y\geq0$ and $y(y+1) \leq (x+1)^2$, then $y(y-1)\leq x^2$.

Let $f(x) = ax^3 + bx^2 + cx + d$ be a polynomial with real coefficients. Given that $f(x)$ has three real positive roots and that $f(0) < 0$, prove that $2b^3+ 9a^2 d - 7abc \le 0$.

Let real angles $\theta_1, \theta_2, \theta_3, \theta_4$ satisfy \begin{align*} \sin\theta_1+\sin\theta_2+\sin\theta_3+\sin\theta_4 &= 0, \\ \cos\theta_1+\cos\theta_2+\cos\theta_3+\cos\theta_4 &= 0. \end{align*} If the maximum possible value of the sum \[\sum_{i<j}\sqrt{1-\sin\theta_i\sin\theta_j-\cos\theta_i\cos\theta_j}\] for $i, j \in \{1, 2, 3, 4\}$ can be expressed as $a+b\sqrt{c}$, where $c$ is square-free and $a,b,c$ are positive integers, find $a+b+c$ [i]Proposed by Alex Li[/i]

Let $a_1,a_2 \cdots a_{2n}$ be an arithmetic progression of positive real numbers with common difference $d$. Let $(i)$ $\sum_{i=1}^{n}a_{2i-1}^2 =x$ $(ii)$ $\sum _{i=1}^{n}a_{2i}^2=y$ $(iii)$ $a_n+a_{n+1}=z$ Express $d$ in terms of $x,y,z,n$

Ten square cardboards of $3$ centimeters on a side are cut by a line, as indicated in the figure. After the cuts, there are $20$ pieces: $10$ triangles and $10$ trapezoids. Assemble a square that uses all $20$ pieces without overlaps or gaps. [img]https://cdn.artofproblemsolving.com/attachments/7/9/ec2242cca617305b02eef7a5409e6a6b482d66.gif[/img]

Given triangle $ABC$, let $D$ be an inner point of segment $BC$. Let $P$ and $Q$ be distinct inner points of the segment $AD$. Let $K=BP\cap AC, L=CP\cap AB, E=BQ\cap AC, F=CQ\cap AB$. Given that $KL\parallel EF$, find all possible values of the ratio $BD:DC$.

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.)