Find the sum of all solutions of the equation $\frac{1}{x^2-1}+\frac{2}{x^2-2}+\frac{3}{x^2-3}+\frac{4}{x^2-4}=2010x-4$

A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is: $\textbf{(A)}\ \text{equal to}\ \frac{1}{4} \\ \qquad\textbf{(B)}\ \text{equal to or greater than}\ \frac{1}{8}\ \text{and less than}\ \frac{1}{2} \\ \qquad\textbf{(C)}\ \text{greater than}\ \frac{1}{8}\ \text{and less than}\ \frac{1}{2} \\ \qquad\textbf{(D)}\ \text{greater than}\ \frac{1}{8}\ \text{and less than}\ \frac{1}{4} \\ \qquad\textbf{(E)}\ \text{less than}\ \frac{1}{2}$

In $\vartriangle ABC$, let $D$ be on $BC$ such that $\overline{AD} \perp \overline{BC}$. Suppose also that $\tan B = 4 \sin C$, $AB^2 +CD^2 = 17$, and $AC^2 + BC^2 = 21$. Find the measure of $\angle C$ in degrees between $0^o$ and $180^o$ .

Let [i]O[/i] be the center of circle[i] W[/i]. Two equal chords [i]AB[/i] and [i]CD [/i]of[i] W [/i]intersect at [i]L [/i]such that [i]AL>LB [/i]and [i]DL>LC[/i]. Let [i]M [/i]and[i] N [/i]be points on [i]AL[/i] and [i]DL[/i] respectively such that ([i]ALC[/i])=2*([i]MON[/i]). Prove that the chord of [i]W[/i] passing through [i]M [/i]and [i]N[/i] is equal to [i]AB[/i] and [i]CD[/i].

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

Let $O, M, N$ be the circumcenter, the centroid and the Nagel point of a triangle. Prove that angle $MON$ is right if and only if one of the triangle’s angles is equal to $60^o$.

For a fixed natural number $m \geq 2$, prove that [b]a.)[/b] There exists integers $x_1, x_2, \ldots, x_{2m}$ such that \[x_i x_{m + i} = x_{i + 1} x_{m + i - 1} + 1, i = 1, 2, \ldots, m \hspace{2cm}(*)\] [b]b.)[/b] For any set of integers $\lbrace x_1, x_2, \ldots, x_{2m}$ which fulfils (*), an integral sequence $\ldots, y_{-k}, \ldots, y_{-1}, y_0, y_1, \ldots, y_k, \ldots$ can be constructed such that $y_k y_{m + k} = y_{k + 1} y_{m + k - 1} + 1, k = 0, \pm 1, \pm 2, \ldots$ such that $y_i = x_i, i = 1, 2, \ldots, 2m$.

Let $m$ be a positive integer. [b]a.[/b] Show that there exist infinitely many positive integers $k$ such that $1+km^3$ is a perfect cube and $1+kn^3$ is not a perfect cube for all positive integers $n<m$. [b]b.[/b] Let $m=p^r$ where $p \equiv 2 \pmod 3$ is a prime number and $r$ is a positive integer. Find all numbers $k$ satisfying the condition in part a.

Let $AEBC$ be a cyclic quadrilateral. Let $D$ be a point on the ray $AE$ which is outside the circumscribed circumference of $AEBC$. Suppose that $\angle CAB=\angle BAE$. Prove that $AB=BD$ if and only if $DE=AC$.

In Venetiania, the smallest currency is the ducat. The finance minister instructs his officials as follows: "I wish six kinds of banknotes, each worth a whole number of ducats. Those six values must be such that there exists a number N with the following property: Any amount of money of $n$ ducats ($n$ positive and integer) where $n \le N$ may be paid in such a way that no more than two copies of each kind are required either to pay or to return. I also wish those six values to be as large as possible for $N$. Determine those six values and provide proof that all conditions have been met." Solve the problem of those officials

The side length of a cube is increased by $20\%$. The surface area of the cube then increases by $x\%$ and the volume of the cube increases by $y\%$. Find $5(y -x)$.

A biologist watches a chameleon. The chameleon catches flies and rests after each catch. The biologist notices that: [list=1][*]the first fly is caught after a resting period of one minute; [*]the resting period before catching the $2m^\text{th}$ fly is the same as the resting period before catching the $m^\text{th}$ fly and one minute shorter than the resting period before catching the $(2m+1)^\text{th}$ fly; [*]when the chameleon stops resting, he catches a fly instantly.[/list] [list=a][*]How many flies were caught by the chameleon before his first resting period of $9$ minutes in a row? [*]After how many minutes will the chameleon catch his $98^\text{th}$ fly? [*]How many flies were caught by the chameleon after 1999 minutes have passed?[/list]

[b]a)[/b] Calculate $ \lim_{n\to\infty} \int_0^{\alpha } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx , $ for all $ \alpha\in (0,1) . $ [b]b)[/b] Calculate $ \lim_{n\to\infty} \int_0^{1 } \ln \left( 1+x+x^2+\cdots +x^{n-1} \right) dx . $

Given an integer $m > 1$, we call the number $x{}$ dangerous if $x{}$ divides the number $y{}$, which is obtained by writing the digits of $x{}$ in base $m{}$ in reverse order, with $x\neq y$. Prove that if there exists a three-digit (in base $m$) dangerous number for a given $m$, then there exists a two-digit (in base $m$) dangerous number.

There were 20 participants in a chess tournament. Each of them played with each other twice: once as white and once as black. Let us say that participant $X{}$ is no weaker than participant $Y{}$ if $X{}$ has won at least the same number of games playing white as $Y{}$ and also has won at least the same number of games playing black as $Y{}$ . Do there exist for sure two participants $A{}$ and $B{}$ such that $A{}$ is not weaker than $B{}$? [i]Boris Frenkin[/i]

The diagram below shows circles radius $1$ and $2$ externally tangent to each other and internally tangent to a circle radius $3$. There are relatively prime positive integers $m$ and $n$ so that a circle radius $\frac{m}{n}$ is internally tangent to the circle radius $3$ and externally tangent to the other two circles as shown. Find $m+n$. [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); pen dotstyle = black; draw(circle((8,2), 3)); draw(circle((8,1), 2)); draw(circle((8,4), 1)); draw((8,-1)--(8,5)); draw(circle((9.72,3.28), 0.86)); label("$ 2 $",(7.56,1.38),SE*labelscalefactor); label("$ 1 $",(7.6,4.39),SE*labelscalefactor); [/asy]

A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.

Let $P$ and $P '$ be two unequal regular $n-$gons and $A$ and $A'$two points inside $P$ and$ P '$, respectively.Suppose $\{ d_1 , d_2 , \cdots d_n \}$ are the distances from $A $ to the vertices of $P$ and $\{ d'_1 , d'_2 , \cdots d'_n \}$ are defines similarly for $P',A'$. Is it possible for $\{ d'_1 , d'_2 , \cdots d'_n \}$ to be a permutation of $\{ d_1 , d_2 , \cdots d_n \}$ ?

Let $n$ be a positive integer and $a_1,a_2,\ldots a_{2n+1}$ be positive reals. For $k=1,2,\ldots ,2n+1$, denote $b_k = \max_{0\le m\le n}\left(\frac{1}{2m+1} \sum_{i=k-m}^{k+m} a_i \right)$, where indices are taken modulo $2n+1$. Prove that the number of indices $k$ satisfying $b_k\ge 1$ does not exceed $2\sum_{i=1}^{2n+1} a_i$.

Let $P=A_1A_2\cdots A_k$ be a convex polygon in the plane. The vertices $A_1, A_2, \ldots, A_k$ have integral coordinates and lie on a circle. Let $S$ be the area of $P$. An odd positive integer $n$ is given such that the squares of the side lengths of $P$ are integers divisible by $n$. Prove that $2S$ is an integer divisible by $n$.

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?

A 9-digit number $N$ is given, whose digits are non-zero and all different.The sums of all consecutive three-digit segments in the decimal representation of number $N$ are calculated and arranged in increasing order.Is it possible to obtain the following sequences as a result of this operation? $\text{a)}$ $11,15,16,18,19,21,22$ $\text{b)}$ $11,15,16,18,19,21,23$

Let $x_1, x_2, \ldots ,x_n(n\ge 2)$ be real numbers greater than $1$. Suppose that $|x_i-x_{i+1}|<1$ for $i=1, 2,\ldots ,n-1$. Prove that \[\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1\]

Let $ ABCD$ be a trapezoid with $ AB$ and $ CD$ parallel, $ \angle D \equal{} 2 \angle B, AD \equal{} 5,$ and $ CD \equal{} 2.$ Then $ AB$ equals A. 7 B. 8 C. 13/2 D. 27/4 E. $ 5 \plus{} \frac{3 \sqrt{2}}{2}$

Let $f$ be a function defined on $[0, 2022]$, such that $f(0) = f(2022) = 2022$, and $$|f(x) - f(y)| \le 2|x -y|,$$ for all $x, y$ in $[0, 2022]$. Prove that for each $x, y$ in $[0, 2022]$, the distance between $f(x)$ and $f(y)$ does not exceed $2022$.