Found problems: 15925
1968 IMO Shortlist, 11
Find all solutions $(x_1, x_2, . . . , x_n)$ of the equation
\[1 +\frac{1}{x_1} + \frac{x_1+1}{x{}_1x{}_2}+\frac{(x_1+1)(x_2+1)}{x{}_1{}_2x{}_3} +\cdots + \frac{(x_1+1)(x_2+1) \cdots (x_{n-1}+1)}{x{}_1x{}_2\cdots x_n} =0\]
2010 Postal Coaching, 5
Let $p$ be a prime and $Q(x)$ be a polynomial with integer coefficients such that $Q(0) = 0, \ Q(1) = 1$ and the remainder of $Q(n)$ is either $0$ or $1$ when divided by $p$, for every $n \in \mathbb{N}$. Prove that $Q(x)$ is of degree at least $p - 1$.
2018 Auckland Mathematical Olympiad, 1
For two non-zero real numbers $a, b$ , the equation, $a(x-a)^2 + b(x-b)^2 = 0$ has a unique solution.
Prove that $a=\pm b$.
2018 Purple Comet Problems, 5
One afternoon at the park there were twice as many dogs as there were people, and there were twice as many people as there were snakes. The sum of the number of eyes plus the number of legs on all of these dogs, people, and snakes was $510$. Find the number of dogs that were at the park.
1995 Vietnam Team Selection Test, 3
Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$.
$ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$.
$ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.
2004 Estonia Team Selection Test, 1
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.
2013 Romanian Master of Mathematics, 2
Does there exist a pair $(g,h)$ of functions $g,h:\mathbb{R}\rightarrow\mathbb{R}$ such that the only function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(g(x))=g(f(x))$ and $f(h(x))=h(f(x))$ for all $x\in\mathbb{R}$ is identity function $f(x)\equiv x$?
1973 Poland - Second Round, 4
Let $ x_n = (p + \sqrt{q})^n - [(p + \sqrt{q})^n] $ for $ n = 1, 2, 3, \ldots $. Prove that if $ p $, $ q $ are natural numbers satisfying the condition $ p - 1 < \sqrt{q} < p $, then $ \lim_{n\to \infty} x_n = 1 $.
Attention. The symbol $ [a] $ denotes the largest integer not greater than $ a $.
2020 China Second Round Olympiad, 2
Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$ and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$
2006 District Olympiad, 1
Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]
1983 Austrian-Polish Competition, 5
Let $a_1 < a_2 < a_3 < a_4$ be given positive numbers. Find all real values of parameter $c$ for which the system
$$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\
a_1x_1 + a_2 x_2 + a_3x_3 + a_4 x_4 = c \\
a_1^2x_1 + a_2^2 x_2 + a_3^2x_3 + a_4^2 x_4 = c^2\end{cases}$$
has a solution in nonnegative $(x_1,x_2,x_3,x_4)$ real numbers.
2001 Romania National Olympiad, 2
We consider a matrix $A\in M_n(\textbf{C})$ with rank $r$, where $n\ge 2$ and $1\le r\le n-1$.
a) Show that there exist $B\in M_{n,r}(\textbf{C}), C\in M_{r,n}(\textbf{C})$, with $%Error. "rank" is a bad command.
B=%Error. "rank" is a bad command.
C = r$, such that $A=BC$.
b) Show that the matrix $A$ verifies a polynomial equation of degree $r+1$, with complex coefficients.
2019 LIMIT Category B, Problem 10
$\frac1{1+\sqrt3}+\frac1{\sqrt3+\sqrt5}+\frac1{\sqrt5+\sqrt7}+\ldots+\frac1{\sqrt{2017}+\sqrt{2019}}=?$
$\textbf{(A)}~\frac{\sqrt{2019}-1}2$
$\textbf{(B)}~\frac{\sqrt{2019}+1}2$
$\textbf{(C)}~\frac{\sqrt{2019}-1}4$
$\textbf{(D)}~\text{None of the above}$
2011 VJIMC, Problem 3
Let $p$ and $q$ be complex polynomials with $\deg p>\deg q$ and let $f(z)=\frac{p(z)}{q(z)}$. Suppose that all roots of $p$ lie inside the unit circle $|z|=1$ and that all roots of $q$ lie outside the unit circle. Prove that
$$\max_{|z|=1}|f'(z)|>\frac{\deg p-\deg q}2\max_{|z|=1}|f(z)|.$$
2007 Hanoi Open Mathematics Competitions, 3
Find the number of di erent positive integer triples (x; y; z) satsfying the equations
x+y-z=1 and $x^2+y^2-z^2=1$.
1986 China Team Selection Test, 2
Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent:
[b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$
[b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.
2004 China Team Selection Test, 3
Find all positive integer $ n$ satisfying the following condition: There exist positive integers $ m$, $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$, such that $ \displaystyle n \equal{} \sum_{i\equal{}1}^{m\minus{}1} a_i(m\minus{}a_i)$, where $ a_1$, $ a_2$, $ \cdots$, $ a_{m\minus{}1}$ may not distinct and $ 1 \leq a_i \leq m\minus{}1$.
2016 Peru MO (ONEM), 3
Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ such that
\[f(x + y) + f(x + z) - f(x)f(y + z) \ge 1\]
for all $x,y,z \in \mathbb{R}$
2013 Junior Balkan Team Selection Tests - Moldova, 5
The real numbers $a, b, c$ are positive, and the real numbers $p, q, r \in [0,1/2]$ satisfy equality $p + q + r = 1$. Prove the inequality $$pab + qbc + rca \le \frac18 (a + b + c)^2.$$
Taiwan TST 2015 Round 1, 1
Let $a,b,c,d$ be any real numbers such that $a+b+c+d=0$, prove that
\[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]
2017 BMT Spring, 5
Find the value of $y$ such that the following equation has exactly three solutions.
$$||x -1|-4|= y.$$
2024 China Team Selection Test, 5
Find all functions $f:\mathbb N_+\to \mathbb N_+,$ such that for all positive integer $a,b,$
$$\sum_{k=0}^{2b}f(a+k)=(2b+1)f(f(a)+b).$$
[i]Created by Liang Xiao, Yunhao Fu[/i]
2021 Malaysia IMONST 1, 13
Jasmin has a mobile phone that runs on a battery. When the battery is dead, it takes $2$ hours to recharge it fully, if she is not using the phone. If she uses the phone while recharging, $75\%$ of the charge obtained is immediately consumed and the remaining is stored in the battery. One day, her battery died. Jasmin took $2$ hours $30$ minutes to recharge the battery fully. For how many minutes did she use the phone while recharging?
1949-56 Chisinau City MO, 53
Solve the equation: $\sqrt[3]{a+\sqrt{x}}+\sqrt[3]{a-\sqrt{x}}=\sqrt[3]{b}$
2005 Greece National Olympiad, 2
The sequence $(a_n)$ is defined by $a_1=1$ and $a_n=a_{n-1}+\frac{1}{n^3}$ for $n>1.$
(a) Prove that $a_n<\frac{5}{4}$ for all $n.$
(b) Given $\epsilon>0$, find the smallest natural number $n_0$ such that ${\mid a_{n+1}-a_n}\mid<\epsilon$ for all $n>n_0.$