Which is greater: $300!$ or $100^{300}$?

If $x$ and $y$ are real numbers such that $x^{2013}+y^{2013}>x^{2012}+y^{2012}$, prove that $x^{2014}+y^{2014}>x^{2013}+y^{2013}$

(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors. (b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.

Prove that: \[ \frac{1989}{2}\minus{}\frac{1988}{3}\plus{}\frac{1987}{4}\minus{}\cdots\minus{}\frac{2}{1989}\plus{}\frac{1}{1990}\equal{}\frac{1}{996}\plus{}\frac{3}{997}\plus{}\frac{5}{998}\plus{}\cdots\plus{}\frac{1989}{1990}\]

Two non-zero real numbers, $a$ and $b$, satisfy $ab=a-b$. Which of the following is a possible value of $\frac ab+\frac ba-ab$? $\text{(A)}\ -2\qquad\text{(B)}\ -\frac12\qquad\text{(C)}\ \frac13\qquad\text{(D)}\ \frac12\qquad\text{(E)}\ 2$

In the acute triangle $ABC$, the sum of the distances from the vertices $B$ and $C$ to of the orthocenter $H$ is equal to $4r,$ where $r$ is the radius of the circle inscribed in this triangle. Find the perimeter of triangle $ABC$ if it is known that $BC=a$. (Gryhoriy Filippovskyi)

Given that $a,b,c$ are integers with $abc = 60$, and that complex number $\omega \neq 1$ satisfies $\omega^3=1$, find the minimum possible value of $|a + b\omega + c\omega^2|$.  

Find all triplets of prime numbers $(m, n, p)$, that satisfy the system of equations: $$\left\{\begin{matrix} 2m-n+13p=2072,\\3m+11n+13p=2961.\end{matrix}\right.$$

There are $30$ children standing in a circle. For each girl, it turns out that among the five people following her clockwise, there are more boys than girls. Find the greatest number of girls that can stand in a circle.

Let $b$ be a positive integer. Find all $2002$−tuples $(a_1, a_2,\ldots , a_{2002})$, of natural numbers such that \[\sum_{j=1}^{2002} a_j^{a_j}=2002b^b.\]

Determine all functions $f: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ such that \[ f(x,y) + f(y,z) + f(z,x) = \max \{ x,y,z \} - \min \{ x,y,z \} \] for every $x,y,z \in \mathbb{R}$ and there exists some real $a$ such that $f(x,a) = f(a,x) $ for every $x \in \mathbb{R}$.

Let $p, q$ be real numbers. Knowing that there are positive real numbers $a, b, c$, different two by two, such that $$p=\frac{a^2}{(b-c)^2}+\frac{b^2}{(a-c)^2}+\frac{c^2}{(a-b)^2},$$ $$q=\frac{1}{(b-c)^2}+\frac{1}{(a-c)^2}+\frac{1}{(b-a)^2}$$ calculate the value of $$\frac{a}{(b-c)^2}+\frac{b}{(a-c)^2}+\frac{c}{(b-a)^2}$$ in terms of $p, q$.

Consider the $5$ by $5$ equilateral triangular grid as shown: [img]https://cdn.artofproblemsolving.com/attachments/1/2/cac43ae24fd4464682a7992e62c99af4acaf8f.png[/img] How many equilateral triangles are there with sides along the gridlines?

Let $P$ be a point in the plane and let $\gamma$ be a circle which does not contain $P$. Two distinct variable lines $\ell$ and $\ell'$ through $P$ meet the circle $\gamma$ at points $X$ and $Y$, and $X'$ and $Y'$, respectively. Let $M$ and $N$ be the antipodes of $P$ in the circles $PXX'$ and $PYY'$, respectively. Prove that the line $MN$ passes through a fixed point. [i]Mihai Chis[/i]

Let be a triangle $ ABC $ and the points $ E,F,M,N $ positioned in this way: $ E,F $ on the segment $ BC $ (excluding its endpoints), $ M $ on the segment $ AC $ (excluding its endpoints) and $ N $ on the segment $ AC $ (excluding its endpoints). Knowing that $ BAE $ is similar to $ FAC $ and that $ BE=BM,FC=CN,AM=AN, $ show that $ ABC $ is isosceles.

Euler's formula states that for a convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces, $V - E + F = 2$. A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V$ vertices, $T$ triangular faces and $P$ pentagonal faces meet. What is the value of $100P + 10T + V$?

How many minimal left ideals does the full matrix ring $M_n(K)$ of $n\times n$ matrices over a field $K$ have?

Thirteen birds arrive and sit down in a plane. It's known that from each 5-tuple of birds, at least four birds sit on a circle. Determine the greatest $M \in \{1, 2, ..., 13\}$ such that from these 13 birds, at least $M$ birds sit on a circle, but not necessarily $M + 1$ birds sit on a circle. (prove that your $M$ is optimal)

Let $ABCD$ be a convex quadrilateral. The rays $AB$ and $DC$ intersect in point $E$. Rays $AD$ and $BC$ intersect in point $F$. The angle bisector of $\angle DCF$ intersects $EF$ in point $K$. Let $I_1$ and $I_2$ be the centers of the inscribed circles in $\Delta ECB$ and $\Delta FCD$. $M$ is the projection of $I_2$ on line $CF$ and $N$ is the projection of $I_1$ on line $BC$. Let $P$ be the reflection of $N$ in $I_1$. If $P,M,K$ are colinear, prove that $ABCD$ is tangential.

Let $AM$ and $AN$ be the tangents to a circle $\Gamma$ drawn from a point $A$ ($M$ and $N$ lie on the circle). A line passing through $A$ cuts $\Gamma$ at $B$ and $C$, with B between $A$ and $C$ such that $AB: BC = 2: 3$. If $P$ is the intersection point of $AB$ and $MN$, calculate the ratio $AP: CP$ .

If $a$ and $b$ are positive integers such that $ab \mid a^2+b^2$ prove that $a=b$

Prove that there is no positive integers $x,y,z$ satisfying \[ x^2 y^4 - x^4 y^2 + 4x^2 y^2 z^2 +x^2 z^4 -y^2 z^4 =0 \]

In a triangle $ABC$ $A_0$,$B_0$ and $C_0$ are the midpoints of the sides $BC$,$CA$ and $AB$.$A_1$,$B_1$,$C_1$ are the midpoints of the broken lines $BAC,CAB,ABC$.Show that $A_0A_1,B_0B_1,C_0C_1$ are concurrent.

If $a_1, a_2, \ldots$ is a sequence of real numbers such that for all $n$, $$\sum_{k = 1}^n a_k \left( \frac{k}{n} \right)^2 = 1,$$ find the smallest $n$ such that $a_n < \frac{1}{2018}$.

A uniform rectangular wood block of mass $M$, with length $b$ and height $a$, rests on an incline as shown. The incline and the wood block have a coefficient of static friction, $\mu_s$. The incline is moved upwards from an angle of zero through an angle $\theta$. At some critical angle the block will either tip over or slip down the plane. Determine the relationship between $a$, $b$, and $\mu_s$ such that the block will tip over (and not slip) at the critical angle. The box is rectangular, and $a \neq b$. [asy] draw((-10,0)--(0,0)--20/sqrt(3)*dir(150)); label("$\theta$",(0,0),dir(165)*6); real x = 3; fill((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle,grey); draw((0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)--(3,3)*dir(60)+(-x*sqrt(3),x)--(0,3)*dir(60)+(-x*sqrt(3),x)--cycle); label("$a$",(0,0)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x)); label("$b$",(3,3)*dir(60)+(-x*sqrt(3),x)--(3,0)*dir(60)+(-x*sqrt(3),x),dir(60)); [/asy] (A) $\mu_s > a/b$ (B) $\mu_s > 1-a/b$ (C) $\mu_s >b/a$ (D) $\mu_s < a/b$ (E) $\mu_s < b/a-1$