Let \( L \), \( M \) and \( N \) be the midpoints on the sides \( BC \), \( AC \) and \( AB\) of a triangle \( ABC \). Points \( D \), \( E \) and \( F \) are taken on the circle circumscribed to the triangle \( LMN \) so that the segments \( LD \), \( ME \) and \( NF \) are diameters of said circumference. Prove that the area of the hexagon \( LENDMF \) is equal to half the area of the triangle \( ABC \)

If $\tan x+\tan y=25$ and $\cot x + \cot y=30$, what is $\tan(x+y)$?

Let $x \ne 1$ be a fixed positive number and $a_1, a_2, a_3,...$ some kind of number sequence. Prove that $x^{a_1},x^{a_2},x^{a_3},...$ is a non-constant geometric sequence if and only if $a_1, a_2, a_3,...$. is a non-constant arithmetic sequence.

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.

Let $ x,y\in\mathbb{R}$ , and $ x,y \in $ $ \left(0,\frac{\pi}{2}\right) $, and $ m \in \left(2,+\infty\right) $ such that $ \tan x * \tan y = m $ . Find the minimum value of the expression $ E(x,y) = \cos x + \cos y $.

Determine all $n \in \mathbb{N}$ for which [list][*] $n$ is not the square of any integer, [*] $\lfloor \sqrt{n}\rfloor ^3$ divides $n^2$. [/list]

Two sequences $(x_n)_{n\in N}$ and $(y_n)_{n\in N}$ are defined recursively as follows: $x_0 = 2015$ and $x_{n+1} =\left \lfloor x_n \cdot \frac{y_{n+1}}{y_{n-1}} \right \rfloor$ for all $n \ge 0$, $y_0 = 307$ and $y_{n+1} = y_n + 1$ for all $n \ge 0$. Compute $\lim_{n\to \infty} \frac{x_n}{(y_n)^2}$.

Let $M$ be the point of intersection of the diagonals of the inscribed quadrilateral $ABCD$, and let the angle $\angle AMB$ be an acute angle. On the side $BC$, as a base, an isosceles triangle $BCK$ is constructed in the direction external to the quadrilateral such that $\angle KBC+\angle AMB= 90^o$. Prove that line $KM$ is perpendicular to $AD$.

Determine all non-negative integers $n$ for which there exist two relatively prime non-negative integers $x$ and $y$ and a positive integer $k\geqslant 2$ such that $3^n=x^k+y^k$.

A tetrahedron is called a [i]MEMO-tetrahedron[/i] if all six sidelengths are different positive integers where one of them is $ 2$ and one of them is $ 3$. Let $ l(T)$ be the sum of the sidelengths of the tetrahedron $ T$. (a) Find all positive integers $ n$ so that there exists a MEMO-Tetrahedron $ T$ with $ l(T)\equal{}n$. (b) How many pairwise non-congruent MEMO-tetrahedrons $ T$ satisfying $ l(T)\equal{}2007$ exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).

Let $a_1, a_2, \ldots, a_{2024}$ be positive integers such that $a_{i+1}+1$ is a multiple of $a_i$ for all $i = 1, 2, \ldots , 2024$, with indices taken modulo $2024$. Find the maximum possible value of $a_1 + a_2 + \ldots + a_{2024}$. [i](Proposed by Ivan Chan Guan Yu)[/i]

A soccer coach named $C$ does a header drill with two players $A$ and $B$, but they all forgot to put sunscreen on their foreheads. They solve this issue by dunking the ball into a vat of sunscreen before starting the drill. Coach $C$ heads the ball to $A$, who heads the ball back to $C$, who then heads the ball to $B$, who heads the ball back to $C$; this pattern $CACBCACB\ldots\,$ repeats ad infinitum. Each time a person heads the ball, $1/10$ of the sunscreen left on the ball ends up on the person's forehead. In the limit, what fraction of the sunscreen originally on the ball will end up on the coach's forehead?

What is the area of a circle with a circumference of $8$?

Let $P(x)= x^n+a_{1}x^{n-1}+...+a_{n-1}x+(-1)^{n}$ , $a_{i} \in C$ , $n\geq 2$ with all roots having same modulo. Prove that $P(-1) \in R$

Given two convex polygons, $A_1A_2...A_n$ and $B_1B_2...B_n$ such that $A_1A_2 = B_1B_2$, $A_2A_3 = B_2B_3$,$ ...$, $A_nA_1 = B_nB_1$ and $n - 3$ angles of one polygon are equal to the respective angles of the other. Find whether these polygons are equal.

Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.

In $\vartriangle ABC$, the external bisectors of $\angle A$ and $\angle B$ meet at a point $D$. Prove that the circumcentre of $\vartriangle ABD$ and the points $C, D$ lie on the same straight line.

In a $2024 \times 2024$ grid of squares, each square is colored either black or white. An ant starts at some black square in the grid and starts walking parallel to the sides of the grid. During this walk, it can choose (not required) to turn $90^\circ$ clockwise or counterclockwise if it is currently on a black square, otherwise it must continue walking in the same direction. A coloring of the grid is called [i]simple[/i] if it is [b]not[/b] possible for the ant to arrive back at its starting location after some time. How many simple colorings of the grid are maximal, in the sense that adding any black square results in a coloring that is not simple?

$33$ counters are shown in the left grid below. Choose a counter to start at and remove it from the grid. At each subsequent step, choose a direction (up, down, left, or right), move along the grid line from your current position to the nearest counter in that direction, and remove that counter. You cannot choose a direction that reverses your previous one (e.g., left then right is not allowed). Your goal is to pick up all $33$ counters in a single sequence of steps. When you find the right sequence, write the numbers $1$ to $33$ on the counters so that $N$ is written on the $N$th counter you removed. A smaller example of a solved grid is shown to the right below. (Note that the final move from $8$ to $9$ is possible because counters $3, 4,$ and $5$ have been removed in earlier steps.) 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.) [asy] unitsize(0.5cm); string[][] x = { {"-","-","0","0","0","0","0","0","0","0","0"}, {"0","0","-","0","0","0","0","0","0","0","0"}, {"0","-","0","0","-","0","0","0","0","0","0"}, {"-","0","0","-","0","-","0","0","0","0","0"}, {"-","-","-","-","0","-","0","0","0","0","0"}, {"0","0","0","-","0","-","-","-","-","0","0"}, {"0","0","0","0","-","0","0","0","0","-","0"}, {"0","0","0","0","0","0","0","-","-","0","0"}, {"0","0","0","0","0","0","-","0","0","0","-"}, {"0","0","0","0","0","0","-","-","-","-","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}, {"0","0","0","0","0","0","0","0","0","0","-"}}; void drawcircle(int x, int y) { filldraw(circle((x,y),0.45),white); } for(int i = 0; i < 11; ++i) { draw((i,-0.7)--(i,13.7),dashed); } for(int i = 0; i < 14; ++i) { draw((-0.7,i)--(10.7,i),dashed); } drawcircle(10,0); drawcircle(10,1); drawcircle(10,2); drawcircle(10,3); drawcircle(10,4); drawcircle(10,5); drawcircle(9,4); drawcircle(8,4); drawcircle(7,4); drawcircle(6,4); drawcircle(6,5); drawcircle(7,6); drawcircle(8,6); drawcircle(9,7); drawcircle(8,8); drawcircle(7,8); drawcircle(6,8); drawcircle(5,8); drawcircle(4,7); drawcircle(5,9); drawcircle(5,10); drawcircle(4,11); drawcircle(3,10); drawcircle(3,9); drawcircle(3,8); drawcircle(2,9); drawcircle(1,9); drawcircle(0,9); drawcircle(0,10); drawcircle(1,11); drawcircle(2,12); drawcircle(1,13); drawcircle(0,13); for(int k = 0; k<14; ++k){ for(int l = 0; l<11; ++l){ if(x[k][l]!="0"){ label((x[k][l]),(l,-k+13),fontsize(10pt)); } } } [/asy]

Suppose that a point in the plane is called [i]good[/i] if it has rational coordinates. Prove that all good points can be divided into three sets satisfying: 1) If the centre of the circle is good, then there are three points in the circle from each of the three sets. 2) There are no three collinear points that are from each of the three sets.

Let $S$ be the number of bijective functions $f:\{0,1,\dots,288\}\rightarrow\{0,1,\dots,288\}$ such that $f((m+n)\pmod{17})$ is divisible by $17$ if and only if $f(m)+f(n)$ is divisible by $17$. Compute the largest positive integer $n$ such that $2^n$ divides $S$.

Fifteen freshmen are sitting in a circle around a table, but the course assistant (who remains standing) has made only six copies of today’s handout. No freshman should get more than one handout, and any freshman who does not get one should be able to read a neighbor’s. If the freshmen are distinguishable but the handouts are not, how many ways are there to distribute the six handouts subject to the above conditions?

Find all odd primes $ p$, if any, such that $ p$ divides $ \sum_{n\equal{}1}^{103}n^{p\minus{}1}$