Find the largest possible number of intersection points of the diagonals of a convex $n$-gon.

Evaluate the following $$\prod^{50}_{j=1} \left( 2 cos \left( \frac{4\pi j}{101} \right) + 1\right).$$

The ancient One-Dimensional Empire was located along a straight line. Initially, there were no cities. A total of $n$ different point-like cities were founded one by one; from the second onwards, each newly founded city and the nearest existing city (the older one, if there were two) were declared sister cities. The surviving map of the empire shows the cities and the distances between them, but not the order in which they were founded. Historians have tried to deduce from the map that each city had at most 41 sister cities. [list=a] [*] For $n=10^6$, give a map from which this deduction can be made. [*] Prove that for $n=10^{13}$, this conclusion cannot be drawn from any map. [/list]

Let $n$ be an odd natural number. We consider an $n\times n$ grid which is made up of $n^2$ unit squares and $2n(n+1)$ edges. We colour each of these edges either $\color{red} \textit{red}$ or $\color{blue}\textit{blue}$. If there are at most $n^2$ $\color{red} \textit{red}$ edges, then show that there exists a unit square at least three of whose edges are $\color{blue}\textit{blue}$.

Given a rectangular strip of measure $12 \times 1$. Paste this strip in two layers over the cube with edge $1$ (the strip can be bent, but cannot be cut). (V. Shevyakov)

Does there exist an arithmetic progression with $2017$ terms such that each term is not a perfect power, but the product of all $2017$ terms is?

A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.

A mass is attached to an ideal spring. At time $t = \text{0}$ the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is $T$. At what time is the power delivered [i]to[/i] the mass by the spring first a maximum? (A) $t = \text{0}$ (B) $t = T/\text{8}$ (C) $t = T/\text{4}$ (D) $t = \text{3}T/\text{8}$ (E) $t = T/\text{2}$

Let $x$ be a real number with $0<x<\pi $.Prove that, for all natural number $n$ ,\[sinx+\frac{sin3x}{3}+\frac{sin5x}{5}+\cdots+\frac{sin(2n-1)x}{2n-1}>0.\]

Let $OP$ and $OQ$ be the perpendiculars from the circumcenter $O$ of a triangle $ABC$ to the internal and external bisectors of the angle $B$. Prove that the line$ PQ$ divides the segment connecting midpoints of $CB$ and $AB$ into two equal parts. (Artemiy Sokolov)

Flasks A, B, and C each have a circular base with a radius of 2 cm. An equal volume of water is poured into each flask, and none overflow. Rank the force of water F on the base of the flask from greatest to least. A) $F_A > F_B > F_C$ B) $F_A > F_C > F_B$ C) $F_B > F_C > F_A$ D) $F_C > F_A > F_B$ E) $F_A = F_B = F_C$

In triangle $ABC$, angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that $$ TB\cdot TC=TI_b^2.$$

Let $a_1<a_2<a_3<\dots$ be positive integers such that $a_{k+1}$ divides $2(a_1+a_2+\dots+a_k)$ for every $k\geqslant 1$. Suppose that for infinitely many primes $p$, there exists $k$ such that $p$ divides $a_k$. Prove that for every positive integer $n$, there exists $k$ such that $n$ divides $a_k$.

The incircle of triangle $ABC$ is tangent to sides $BC$, $AC$, and $AB$ at points $D$, $E$ and $F$, respectively. Let $A'$ denote the point of the incircle for which circle $(A'BC)$ is tangent to the incircle. Define points $B'$ and $C'$ similarly. Prove that lines $A'D$, $BE'$, and $CF'$ are concurrent. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Given that $\log_{10}(4) = 0.6021$ to the nearest ten-thousandth, find $\log_{10}(5)$ to the nearest thousandth. [i]Lightning 3.3[/i]

Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$

Prove that for any natural numbers$a,b$ there exist infinity many prime numbers $p$ so that $Ord_p(a)=Ord_p(b)$(Proving that there exist infinity prime numbers $p$ so that $Ord_p(a) \ge Ord_p(b)$ will get a partial mark)

Mr. Lopez has a choice of two routes to get to work. Route A is $6$ miles long, and his average speed along this route is $30$ miles per hour. Route B is $5$ miles long, and his average speed along this route is $40$ miles per hour, except for a $\frac{1}{2}$-mile stretch in a school zone where his average speed is $20$ miles per hour. By how many minutes is Route B quicker than Route A? $\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\ 5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}$

For which integers $0 \le k \le 9$ do there exist positive integers $m$ and $n$ so that the number $3^m + 3^n + k$ is a perfect square?

Denote $X,Y$ two convex polygons, such that $X$ is contained inside $Y$. Denote $S (X)$, $P (X)$, $S (Y)$, $P (Y)$ the area and perimeter of the first and second polygons, respectively. Prove that $$ \frac{S(X)}{P(X)}<2 \frac{S(Y)}{P(Y)}.$$

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

Two ants crawl along the sides of the $49$ squares of a $7 * 7$ board. Each ant passes through all $64$ vertices exactly once and returns to its starting point. What is the smallest possible number of sides covered by both ants?

Five points are given in space, no four of which are coplanar. Each of the segments connecting two of them is painted in white, green or red, so that all the colors are used and no three segments of the same color form a triangle. Prove that among these five points there is one at which segments of all the three colors meet.

Given triangle $ABC$ with $AB<AC$ and its circumcircle $\Omega$. Let $I$ be the incenter of $ABC$, and the feet from $I$ to $BC$ is $D$. The circle with center $A$ and radius $AI$ intersects $\Omega$ at $E$ and $F$. $P$ is a point on $EF$ such that $DP$ is parallel to $AI$. Prove that $AP$ and $MI$ intersects on $\Omega$ where $M$ is the midpoint of arc $BAC$. [hide = Remark] In the test, the incenter called $O$ and the circumcircle called $Luna$ $Cabrera$ You have to prove $AP \cap MO \in Luna$ $Cabrera$ [/hide] [i]Proposed by BlessingOfHeaven[/i]