Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n = 5, m = 7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$? [asy] int i; for(i=0; i<8; i=i+1) { dot((i,0)^^(i,5)); } for(i=1; i<5; i=i+1) { dot((0,i)^^(7,i)); } draw(origin--(7,0)--(7,5)--(0,5)--cycle, linewidth(0.8)); pair P=(3.5, 2.5); draw((0,4)--P--(0,3)^^(2,0)--P--(3,0)); label("$B$", (2.3,0), NE); label("$A$", (0,3.7), SE);[/asy] $\text{(A)} \ 1 \qquad \text{(B)} \ m/n \qquad \text{(C)} \ n/m \qquad \text{(D)} \ 2m/n \qquad \text{(E)} \ 2n/m$

Let $M_A, M_B, M_C$ be the midpoints of the sides $BC, CA, AB$ respectively of a non-isosceles triangle $ABC$. Points $H_A, H_B, H_C$ lie on the corresponding sides, different from $M_A, M_B, M_C$ such that $M_AH_B=M_AH_C, $ $M_BH_A=M_BH_C,$ and $M_CH_A=M_CH_B$. Prove that $H_A, H_B, H_C$ are the feet of the corresponding altitudes.

On side $AB$ of triangle $ABC$, point $M$ is selected. A straight line passing through $M$ intersects the segment $AC$ at point $N$ and the ray $CB$ at point $K$. The circumscribed circle of the triangle $AMN$ intersects $\omega$, the circumscribed circle of the triangle $ABC$, at points $A$ and $S$. Straight lines $SM$ and $SK$ intersect with $\omega$ for the second time at points $P$ and $Q$, respectively. Prove that $AC = PQ$.

Let $\vartriangle ABC$ be a triangle such that the area$ [ABC] = 10$ and $\tan (\angle ABC) = 5$. If the smallest possible value of $(\overline{AC})^2$ can be expressed as $-a + b\sqrt{c}$ for positive integers $a, b, c$, what is $a + b + c$?

In triangle $ABC$, $AB=13$, $BC=14$ and $CA=15$. Segment $BC$ is split into $n+1$ congruent segments by $n$ points. Among these points are the feet of the altitude, median, and angle bisector from $A$. Find the smallest possible value of $n$. [i]Proposed by Evan Chen[/i]

Let $x_0, x_1, x_2 \dots$ be a sequence of positive real numbers such that for all $n \geq 0$, $$x_{n+1} = \dfrac{(n^2+1)x_n^2}{x_n^3+n^2}$$ For which values of $x_0$ is this sequence bounded?

Amy and Bob play a game. They alternate turns, with Amy going first. At the start of the game, there are $20$ cookies on a red plate and $14$ on a blue plate. A legal move consists of eating two cookies taken from one plate, or moving one cookie from the red plate to the blue plate (but never from the blue plate to the red plate). The last player to make a legal move wins; in other words, if it is your turn and you cannot make a legal move, you lose, and the other player has won. Which player can guarantee that they win no matter what strategy their opponent chooses? Prove that your answer is correct.

Find all $3$-tuples $(a, b, c)$ of positive integers, with $a \geq b \geq c$, such that $a^2 + 3b$, $b^2 + 3c$, and $c^2 + 3a$ are all squares.

In the sequence of the natural (i.e. positive integers) numbers every member from the third equals the absolute value of the difference of the two previous. What is the maximal possible length of such a sequence, if every member is less or equal to $1967$?

Prove that for all $x \in \left( 0, \frac{\pi}{3} \right)$ inequality $sin2x+cosx>1$ holds.

Given $(2m+1)$ different integers, each absolute value is not greater than $(2m-1)$. Prove that it is possible to choose three numbers among them, with their sum equal to zero.

Consider a natural number $ n\equiv 9\pmod {25}. $ Prove that there exist three nonnegative integers $ a,b,c $ having the property that: $$ n=\frac{a(a+1)}{2} +\frac{b(b+1)}{2} +\frac{c(c+1)}{2} $$

Let $ a_1,a_2,...,a_n$ be nonnegative real numbers. Prove that \[ ( \sum_{i=1}^na_i)( \sum_{i=1}^na_i^{n-1}) \leq n \prod_{i=1}^na_i+ (n-1) ( \sum_{i=1}^na_i^n).\] [i]J. Suranyi[/i]

$f:R->R$ such that : $f(1)=1$ and for any $x\in R$ i) $f(x+5)\geq f(x)+5$ ii)$f(x+1)\leq f(x)+1$ If $g(x)=f(x)+1-x$ find g(2016)

A ball moves endlessly on a circular billiard table. When it hits the edge it is reflected. Show that if it passes through a point on the table three times, then it passes through it infinitely many times.

Quadrangle $ABCD$ is inscribed in a circle with diameter $AC$. Points $K$ and $M$ are projections of vertices $A$ and $C$, respectively, onto line $BD$. A line parallel to $BC$ is drawn through point $K$ and intersecting $AC$ at point $P$. Prove that angle $KPM$ is a right angle.

In triangle $BEN$ shown below with its altitudes intersecting at $X$, $NA = 7$, $EA = 3$, $AX = 4$, and $NS = 8$. Find the area of $BEN$. [img]https://cdn.artofproblemsolving.com/attachments/5/7/7e6dcbe6aa220821cb5020824b8aa6d4fc597d.png[/img]

Determine all functions $f : \mathbb{R} \to \mathbb{R}$ satisfying $$f(x^2 + f(x)f(y)) = xf(x + y)$$ for all real numbers $x$ and $y$.

Do there exist postive integers $a_1<a_2<\cdots<a_{100}$ such that for $2\le k\le100$ the greatest common divisor of $a_{k-1}$ and $a_k$ is greater than the greatest common divisor of $a_k$ and $a_{k+1}$?

A collection of \( n \) balls is distributed in several boxes, with no box containing 100 or more balls. In one move, we can remove several (at least one, possibly all) balls from one box. Find the smallest positive integer \( n \) with the following property: regardless of the distribution, we can make moves such that each non-empty box contains the same number of balls and the total number of remaining balls is at least 100.

A steel cube has edges of length $3$ cm, and a cone has a diameter of $8$ cm and a height of $24$ cm. The cube is placed in the cone so that one of its interior diagonals coincides with the axis of the cone. What is the distance, in cm, between the vertex of the cone and the closest vertex of the cube? [NEEDS DIAGRAM] $ \mathrm{(A) \ } \frac{12\sqrt6-3\sqrt3}{4} \qquad \mathrm{(B) \ } \frac{9\sqrt6-3\sqrt3}{2} \qquad \mathrm {(C) \ } 5\sqrt3 \qquad \mathrm{(D) \ } 6\sqrt6 - \sqrt3 \qquad \mathrm{(E) \ } 6\sqrt6$

Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\ell_1||\ell_2$, $F_1$ lies on $\mathcal{P}_2$, and $F_2$ lies on $\mathcal{P}_1$. The two parabolas intersect at distinct points $A$ and $B$. Given that $F_1F_2=1$, the value of $AB^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by Yannick Yao

There are two piles with $72$ and $30$ candies. Two students alternate taking candies from one of the piles. Each time the number of candies taken from a pile must be a multiple of the number of candies in the other pile. Which student can always assure taking the last candy from one of the piles?

Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy $$\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}$$

It is known that the number $\pi$ is transcendental, that is, it is not a root of any polynomial with integer coefficients. Using this fact, prove that the same is true for the number $\pi + \sqrt2$.