$x_1, x_2,\cdots,x_{n+1}$ are posive real numbers satisfying the equation $\frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_{n+1}} =1$ Prove that $x_1x_2 \cdots x_{n+1} \geq n^{n+1}$.

Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: ${{ \textbf{(A)}\ \text{infinite} \qquad\textbf{(B)}\ \text{finite, but greater than 2} \qquad\textbf{(C)}\ \text{Two} \qquad\textbf{(D)}\ \text{One} }\qquad\textbf{(E)}\ \text{Zero} } $

Let \( n \geq 3 \) be an integer. For every two adjacent vertices \( A \) and \( B \) of a convex \( n \)-gon, we find a vertex \( C \) such that the angle \( \angle ACB \) is the largest, and write down the measure in degrees. Find the smallest possible value of the sum of the written \( n \) numbers.

Let $ABC$ be a triangle and its bisector at $A$ cuts its circumcircle at $D.$ Let $I$ be the incenter of triangle $ABC,$ $M$ be the midpoint of $BC,$ $P$ is the symmetric to $I$ with respect to $M$ (Assuming $P$ is in the circumcircle). Extend $DP$ until it cuts the circumcircle again at $N.$ Prove that among segments $AN, BN, CN$, there is a segment that is the sum of the other two.

Find all positive integer $n$, such that there exists $n$ points $P_1,\ldots,P_n$ on the unit circle , satisfying the condition that for any point $M$ on the unit circle, $\sum_{i=1}^n MP_i^k$ is a fixed value for \\a) $k=2018$ \\b) $k=2019$.

Let $a,b,c\ge0$ and $a+b+c=3.$ Prove that $(3a-bc)(3b-ac)(3c-ab)\le8.$ (O. Rudenko)

Four points $B,E,A,F$ lie on line $AB$ in order, four points $C,G,D,H$ lie on line $CD$ in order, satisfying: $$\frac{AE}{EB}=\frac{AF}{FB}=\frac{DG}{GC}=\frac{DH}{HC}=\frac{AD}{BC}.$$ Prove that $FH\perp EG$.

Find all positive real numbers \((a,b,c) \leq 1\) which satisfy \[ \huge \min \Bigg\{ \sqrt{\frac{ab+1}{abc}}\, \sqrt{\frac{bc+1}{abc}}, \sqrt{\frac{ac+1}{abc}} \Bigg \} = \sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\]

If $ a,b,c $ are all in the interval $ (0,1) $ or all in the interval $ \left( 1,\infty \right), $ then $$ 1\le\sum_{\text{cyc}} \frac{\log_a^7 b\cdot \log_b^3c}{\log_c a +2\log_a b} . $$ [i]Gheorghe Duță[/i]

A solid in Euclidean $3$-space extends from $z = \frac{-h}{2}$ to $z = \frac{+h}{2}$ and the area of the section $z = k$ is a polynomial in $k$ of degree at most $3$. Show that the volume of the solid is $\frac{h(B + 4M + T)}{6},$ where $B$ is the area of the bottom $(z = \frac{-h}{2})$, $M$ is the area of the middle section $(z = 0),$ and $T$ is the area of the top $(z = \frac{h}{2})$. Derive the formulae for the volumes of a cone and a sphere.

[b](a)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ ends with the string $2004$, followed by a number of digits from the set $\left\{0;\;4\right\}$ ? [b](b)[/b] Does there exist a positive integer $n$ such that the decimal representation of $n!$ starts with the string $2004$ ?

Let $n>2$ be a positive integer. Consider all numbers $S$ of the form \begin{align*} S= a_1 a_2 + a_2 a_3 + \ldots + a_{k-1} a_k \end{align*} with $k>1$ and $a_i$ begin positive integers such that $a_1+a_2+ \ldots + a_k=n$. Determine all the numbers that can be represented in the given form.

Alex scans the list of integers between $1$ and $2020$ inclusive using the following algorithm. First, he reads off perfect squares between $1$ and $2020$ in ascending order and removes these numbers from the list. Next, he reads off numbers now at perfect square indices in ascending order, which are $2$, $6$, $12$, $...$, and removes these numbers from the list. He repeats this algorithm until he reads off $2020$, which is the nth number he has read o so far. Compute $n$.

Two concentric circles have radii $1$ and $4$. Six congruent circles form a ring where each of the six circles is tangent to the two circles adjacent to it as shown. The three lightly shaded circles are internally tangent to the circle with radius $4$ while the three darkly shaded circles are externally tangent to the circle with radius $1$. The radius of the six congruent circles can be written $\textstyle\frac{k+\sqrt m}n$, where $k,m,$ and $n$ are integers with $k$ and $n$ relatively prime. Find $k+m+n$. [asy] size(150); defaultpen(linewidth(0.8)); real r = (sqrt(133)-9)/2; draw(circle(origin,1)^^circle(origin,4)); for(int i=0;i<=2;i=i+1) { filldraw(circle(dir(90 + i*120)*(4-r),r),gray); } for(int j=0;j<=2;j=j+1) { filldraw(circle(dir(30+j*120)*(1+r),r),darkgray); } [/asy]

Points $C\neq D$ lie on the same side of line $AB$ so that $\triangle ABC$ and $\triangle BAD$ are congruent with $AB = 9$, $BC=AD=10$, and $CA=DB=17$. The intersection of these two triangular regions has area $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $ABC$ be a triangle with $\angle A=60^o$ and $T$ be a point such that $\angle ATB=\angle BTC=\angle ATC$. A circle passing through $B,C$ and $T$ meets $AB$ and $AC$ for the second time at points $K$ and $L$.Prove that the distances from $K$ and $L$ to $AT$ are equal.

Given any arc $AB$ on a circle and points $C$ and $D$ on segment $AB$, such that $$CD = DB = 2AC.$$ Find the ratio $\frac{CM}{MD}$, where $M$ is a point on arc $AB$, such that $\angle CMD$ is maximized. [img]https://i.imgur.com/NfjRpgP.png[/img] [i] Proposed by Andria Gvaramia, Georgia [/i]

Ten evenly spaced vertical lines in the plane are labeled $\ell_1,\ell_2, \ldots,\ell_{10}$ from left to right. A set $\{a,b,c,d\}$ of four distinct integers $a,b,c,d \in \{1,2,\ldots,10\}$ is [i]squarish[/i] if some square has one vertex on each of the lines $\ell_a,\ell_b,\ell_c,$ and $\ell_d.$ Find the number of squarish sets.

Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female? $\textbf{(A) }\dfrac13\qquad\textbf{(B) }\dfrac4{11}\qquad\textbf{(C) }\dfrac25\qquad\textbf{(D) }\dfrac49\qquad\textbf{(E) }\dfrac12$

Find all pairs $(x,y)$ of integers such that $y^3-1=x^4+x^2$.

If Scott rolls four fair six-sided dice, what is the probability that he rolls more 2's than 1's? $\text{(A) }\frac{8}{27}\qquad\text{(B) }\frac{25}{81}\qquad\text{(C) }\frac{103}{324}\qquad\text{(D) }\frac{421}{1296}\qquad\text{(E) }\frac{65}{162}$

All faces of the parallelepiped $ABCDA_1B_1C_1D_1$ are equal rhombuses. Plane angles at vertex $A$ are equal. Points $K$ and $M$ are taken on the edges $A_1B_1$ and $A_1D_1$. It is known that $A_1K = a$, $A_1M = b$, and$ a + b$ is an edge of the parallelepiped. Prove that the plane $AKM$ touches the sphere inscribed in the parallelepiped. Let us denote by $Q$ the touchpoint of this sphere with the plane $AKM $. In what ratio does the straight line $AQ$ divide the segment $KM$?

The edges of a graph with $2n$ vertices ($n \ge 4$) are colored in blue and red such that there is no blue triangle and there is no red complete subgraph with $n$ vertices. Find the least possible number of blue edges.

Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.

In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that $$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$