Found problems: 36
The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$.
(a) How many pages could the notebook originally have been?
(b) What page numbers can be on the torn sheet?
(Walther Janous)
The sequence $(a_n)$ is defined by $a_1=0$,
$$
a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.
$$
Prove that $a_{2016}>{1\over 2}+a_{1000}$.
There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below.
(1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$
(2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$
(3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$
The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$.
(a) How many pages could the notebook originally have been?
(b) What page numbers can be on the torn sheet?
(Walther Janous)
Are there $4$ distinct positive integers such that adding the product of any two of them to $2006$ yields a perfect square?
Find all real $x, y$, satisfying $$(x+1)^2(y+1)^2=27xy$$ and $$(x^2+1)(y^2+1)=10xy.$$
A number plus $4$ is $2003$. What is the number?
Points $M$, $N$, $K$ lie on the sides $BC$, $CA$, $AB$ of a triangle $ABC$, respectively, and are different from its vertices. The triangle $MNK$ is called[i] beautiful[/i] if $\angle BAC=\angle KMN$ and $\angle ABC=\angle KNM$. If in the triangle $ABC$ there are two beautiful triangles with a common vertex, prove that the triangle $ABC$ is right-angled.
[i]Proposed by Nairi M. Sedrakyan, Armenia[/i]
Let $n$ be a positive integer. $n$ letters are written around a circle, each $A$, $B$, or $C$. When the letters are read in clockwise order, the sequence $AB$ appears $100$ times, the sequence $BA$ appears $99$ times, and the sequence $BC$ appears $17$ times. How many times does the sequence $CB$ appear?
Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.
Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that
$$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$