Found problems: 15925
2023 Indonesia TST, A
Let $a,b,c$ positive real numbers and $a+b+c = 1$. Prove that
\[a^2 + b^2 + c^2 + \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}} \ge 2(ab + bc + ac)\]
2024 JBMO TST - Turkey, 3
Find all $x,y,z \in R^+$ such that the sets
$(23x+24y+25z,23y+24z+25x,23z+24x+25y)$ and $(x^5+y^5,y^5+z^5,z^5+x^5)$
are same
2007 iTest Tournament of Champions, 5
A polynomial $p(x)$ of degree $1000$ is such that $p(n) = (n+1)2^n$ for all nonnegative integers $n$ such that $n\leq 1000$. Given that \[p(1001) = a\cdot 2^b - c,\] where $a$ is an odd integer, and $0 < c < 2007$, find $c-(a+b)$.
2011 Dutch IMO TST, 2
Find all functions $f : R\to R$ satisfying $xf(x + xy) = xf(x) + f(x^2)f(y)$ for all $x, y \in R$.
Kvant 2025, M2828
Maxim has guessed a polynomial $f(x)$ of degree $n$. Sasha wants to guess it (knowing $n$). During a turn, Sasha can name a certain segment $[a;b]$ and Maxim will give in response the maximum value of $f(x)$ on the segment $[a;b]$. Will Sasha be able to guess $f(x)$ in a finite number of steps?
[i]M. Didin[/i]
2018 Azerbaijan Junior NMO, 5
For a positive integer $n$, define $f(n)=n+P(n)$ and $g(n)=n\cdot S(n)$, where $P(n)$ and $S(n)$ denote the product and sum of the digits of $n$, respectively. Find all solutions to $f(n)=g(n)$
1976 Miklós Schweitzer, 4
Let $ \mathbb{Z}$ be the ring of rational integers. Construct an integral domain $ I$ satisfying the following conditions:
a)$ \mathbb{Z} \varsubsetneqq I$;
b) no element of $ I \minus{} \mathbb{Z}$ (only in $ I$) is algebraic over $ \mathbb{Z}$ (that is, not a root of a polynomial with coefficients in $ \mathbb{Z}$);
c) $ I$ only has trivial endomorphisms.
[i]E. Fried[/i]
2004 Bulgaria Team Selection Test, 1
Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.
2016 Indonesia TST, 3
Let $n$ be a positive integer greater than $1$. Evaluate the following summation:
\[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]
Oliforum Contest V 2017, 6
Fix reals $x, y,z > 0$ such that $x + y + z = \sqrt[5]{x} + \sqrt[5]{y} +\sqrt[5]{z}$ . Prove that $x^x y^y z^z \ge 1$.
(Paolo Leonetti)
2014 Hanoi Open Mathematics Competitions, 12
Find a polynomial $Q(x)$ such that $(2x^2 - 6x + 5)Q(x)$ is a polynomial with all positive coefficients.
2006 Moldova MO 11-12, 1
Let $n\in\mathbb{N}^*$. Prove that \[ \lim_{x\to 0}\frac{ \displaystyle (1+x^2)^{n+1}-\prod_{k=1}^n\cos kx}{ \displaystyle x\sum_{k=1}^n\sin kx}=\frac{2n^2+n+12}{6n}. \]
1933 Eotvos Mathematical Competition, 1
Let $a, b,c$ and $d$ be rea] numbers such that $a^2 + b^2 = c^2 + d^2 = 1$ and $ac + bd = 0$. Determine the value of $ab + cd$.
2011 India IMO Training Camp, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2023 Junior Balkan Team Selection Tests - Romania, P4
Let be $a$ be positive real number. Prove that there are no real numbers $b$ and $c$, with $b < c$, so that for any distinct numbers $x, y \in (b, c)$ we have $|\frac{x+y} {x-y}| \leq a$.
2024 Chile Classification NMO Seniors, 2
Find all real numbers $x$ such that:
\[
2^x + 3^x + 6^x - 4^x - 9^x = 1,
\]
and prove that there are no others.
2024 Junior Balkan Team Selection Tests - Romania, P1
For positive real numbers $x,y,z$ with $xy+yz+zx=1$, prove that
$$\frac{2}{xyz}+9xyz \geq 7(x+y+z)$$
1949-56 Chisinau City MO, 14
Prove that if the numbers $a, b, c$ are related by the relation $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}= \frac{1}{a+b+c}$ then the sum of some two of them is equal to zero.
2020 Austrian Junior Regional Competition, 1
Let $a$ be a real number and $b$ a real number with $b\neq-1$ and $b\neq0. $ Find all pairs $ (a, b)$ such that $$\frac{(1 + a)^2 }{1 + b}\leq 1 + \frac{a^2}{b}.$$ For which pairs (a, b) does equality apply?
(Walther Janous)
1999 Baltic Way, 5
The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.
2017 Latvia Baltic Way TST, 1
Prove that for all real $x > 0$ holds the inequality $$\sqrt{\frac{1}{3x+1}}+\sqrt{\frac{x}{x+3}}\ge 1.$$
For what values of $x$ does the equality hold?
2021 BMT, 6
Let $f$ be a real function such that for all $x\ne 0$, $x\ne 1$,
$$f (x) + f \left(- \frac{1}{x - 1} \right) =\frac{9}{4x^2} + f\left(1 - \frac{1}{x} \right) .$$
Compute $f \left( \frac{1}{2}\right).$
.
2015 Indonesia MO Shortlist, A6
Let functions $f, g: \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following:
\[ f(g(x)y + f(x)) = (y+2015)f(x) \]
for every $x,y \in \mathbb{R}^+$.
(a) Prove that $g(x) = \frac{f(x)}{2015}$ for every $x \in \mathbb{R}^+. $
(b) State an example of function that satisfy the equation above and $f(x), g(x) \ge 1$ for every $x \in \mathbb{R}^+$.
the 13th XMO, P2
Given $n\in\mathbb N_+,n\ge 3,a_1,a_2,\cdots ,a_n\in\mathbb R_+.$ Let $b_1,b_2,\cdots ,b_n\in\mathbb R_+$ satisfy that for $\forall k\in\{1,2,\cdots ,n\},$
$$\sum_{\substack{i,j\in\{1,2,\cdots ,n\}\backslash \{k\}\\i\neq j}}a_ib_j=0.$$
Prove that $b_1=b_2=\cdots =b_n=0.$
2006 Iran Team Selection Test, 2
Let $n$ be a fixed natural number.
[b]a)[/b] Find all solutions to the following equation :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-1 \]
[b]b)[/b] Find the number of solutions to the following equation ($m$ is a fixed natural) :
\[ \sum_{k=1}^n [\frac x{2^k}]=x-m \]