Observe that \[\frac{1}{1}= \frac{1}{2}+\frac{1}{2};\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\quad \frac{1}{4}= \frac{1}{5}+\frac{1}{20}. \] State a general law suggested by these examples, and prove it. Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that \[\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}. \] [hide="Remark."] It seems that this is a two-part problem. [/hide]

Determine the pairs of sets $X,Y\subset\mathbb{R}$ for which the following is true: if $f(x, y)$ is a function on $X\times Y{}$ such that for every $x\in X$ it is equal to a polynomial in $y$ on $Y$ and for every $y\in Y$ it is equal to a polynomial in $x$ on $X$ then $f$ is a bivariate polynomial on $X\times Y.$

Let $y=f(x)$ be a function of the graph of broken line connected by points $(-1,\ 0),\ (0,\ 1),\ (1,\ 4)$ in the $x$ -$y$ plane. Find the minimum value of $\int_{-1}^1 \{f(x)-(a|x|+b)\}^2dx.$ [i]2010 Tohoku University entrance exam/Economics, 2nd exam[/i]

Let A be a subset of natural numbers and let k , r be positive integers. Suppose that for any r different elements selected from A , their greatest common divisor has at most k different prime factors. Prove that A can be partitioned into B and C , where any element of B has at most k + 1 different prime divisors and $$\sum_{n\in C} \frac{1}{n} <\infty$$

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.

Let $Q(x)$ be a polynomial with integer coefficients. Prove that there exists a polynomial $P(x)$ with integer coefficients such that for every integer $n\ge\deg{Q}$, \[\sum_{i=0}^{n}\frac{!i P(i)}{i!(n-i)!} = Q(n),\]where $!i$ denotes the number of derangements (permutations with no fixed points) of $1,2,\ldots,i$. [i]Calvin Deng.[/i]

In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line? $ \textbf{(A) }2021 \qquad \textbf{(B) }2022 \qquad \textbf{(C) }2023 \qquad \textbf{(D) }2024 \qquad \textbf{(E) }2025 \qquad $

A circle center $O$ meets a circle center $O'$ at $A$ and $B.$ The line $TT'$ touches the first circle at $T$ and the second at $T'$. The perpendiculars from $T$ and $T'$ meet the line $OO'$ at $S$ and $S'$. The ray $AS$ meets the first circle again at $R$, and the ray $AS'$ meets the second circle again at $R'$. Show that $R, B$ and $R'$ are collinear.

Given $n$ points $A_1, A_2, \ldots, A_n,$ no three collinear, show that the $n$- gon $A_1 A_2 \ldots A_n,$ is inscribed in a circle if and only if $A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots$ $+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n$ $= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n$, where $XY$ denotes the length of the segment $XY.$

Let $S$ be a subset of $\{1,2,3,\ldots,1989\}$ such that no two members of $S$ differ by $4$ or $7$. What is the largest number of elements $S$ can have?

Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.

Consider the sequence $2, 3, 5, 6, 7, 8, 10, ...$ of all positive integers that are not perfect squares. Determine the $2011^{th}$ term of the sequence.

Let $A=\{1,2,...,100\}$ and $f(k), k\in N$ be the size of the largest subset of $A$ such that no two elements differ by $k$. How many solutions are there to $f(k)=50$?

Let $p$ be an odd prime, and let $(p^p)!=mp^k$ for some positive integers $m$ and $k$. Find in terms of $p$ the number of ordered pairs $(m,k)$ satisfying $m+k\equiv0\pmod p$.

Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?

$ABC$ is a triangle with $AB = 33$, $AC = 21$ and $BC = m$, an integer. There are points $D$, $E$ on the sides $AB$, $AC$ respectively such that $AD = DE = EC = n$, an integer. Find $m$.

In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$ SM \mid \mid AC \Leftrightarrow OM \perp BS$$

For each positive integer $n$, let $f_1(n)$ be twice the number of positive integer divisors of $n$, and for $j \ge 2$, let $f_j(n) = f_1(f_{j-1}(n))$. For how many values of $n \le 50$ is $f_{50}(n) = 12?$ $\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }9\qquad\textbf{(D) }10\qquad\textbf{(E) }11$

The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism. A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box? $\textbf{(A) }\frac{24}{5}\qquad\textbf{(B) }\frac{42}{5}\qquad\textbf{(C) }\frac{81}{5}\qquad\textbf{(D) }30\qquad\textbf{(E) }48$

Find all positive integers $n$ such that $$n(2^n-1)$$ is a perfect square

Let be two real reducible quadratic polynomials $ P,Q $ in one variable. Prove that if $ P-Q $ is irreducible, then $ P+Q $ is reducible.

Let positive real numbers $ a,b$ satisfy $ b \minus{} a > 2.$ Prove that for any two distinct integers $ m,n$ belonging to $ [a,b),$ there always exists non-empty set $ S$ consisting of certain integers belonging to $ [ab,(a \plus{} 1)(b \plus{} 1))$ such that $ \frac {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.

A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$. [i]S. Berlov[/i]

Luna and Sam have access to a windowsill with three plants. On the morning of January $1$, $2011$, the plants were sitting in the order of cactus, dieffenbachia, and orchid, from left to right. Every afternoon, when Luna waters the plants, she swaps the two plants sitting on the left and in the center. Every evening, when Sam waters the plants, he swaps the two plants sitting on the right and in the center. What was the order of the plants on the morning of January $1$, $2012$, $365$ days later, from left to right? $\text{(A) cactus, orchid, dieffenbachia}\qquad\text{(B) dieffenbachia, cactus, orchid}$ $\text{(C) dieffenbachia, orchid, cactus}\qquad\text{(D) orchid, dieffenbachia, cactus}$ $\text{(E) orchid, cactus, dieffenbachia}$

Let $\{a_n\}_{n\geq 0}$ be a non-decreasing, unbounded sequence of non-negative integers with $a_0=0$. Let the number of members of the sequence not exceeding $n$ be $b_n$. Prove that \[ (a_0 + a_1 + \cdots + a_m)( b_0 + b_1 + \cdots + b_n ) \geq (m + 1)(n + 1). \]