Found problems: 15925
2012 District Olympiad, 2
[b]a)[/b] Solve in $ \mathbb{R} $ the equation $ 2^x=x+1. $
[b]b)[/b] If a function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ has the property that
$$ (f\circ f)(x)=2^x-1,\quad\forall x\in\mathbb{R} , $$
then $ f(0)+f(1)=1. $
2010 Regional Competition For Advanced Students, 2
Solve the following in equation in $\mathbb{R}^3$:
\[4x^4-x^2(4y^4+4z^4-1)-2xyz+y^8+2y^4z^4+y^2z^2+z^8=0.\]
2006 IMC, 6
Find all sequences $a_{0}, a_{1},\ldots, a_{n}$ of real numbers such that $a_{n}\neq 0$, for which the following statement is true:
If $f: \mathbb{R}\to\mathbb{R}$ is an $n$ times differentiable function
and $x_{0}<x_{1}<\ldots <x_{n}$ are real numbers such that
$f(x_{0})=f(x_{1})=\ldots =f(x_{n})=0$ then there is $h\in (x_{0}, x_{n})$ for which \[a_{0}f(h)+a_{1}f'(h)+\ldots+a_{n}f^{(n)}(h)=0.\]
2017 Miklós Schweitzer, 6
Let $I$ and $J$ be intervals. Let $\varphi,\psi:I\to\mathbb{R}$ be strictly increasing continuous functions and let $\Phi,\Psi:J\to\mathbb{R}$ be continuous functions. Suppose that $\varphi(x)+\psi(x)=x$ and $\Phi(u)+\Psi(u)=u$ holds for all $x\in I$ and $u\in J$. Show that if $f:I\to J$ is a continuous solution of the functional inequality
$$f\big(\varphi(x)+\psi(y)\big)\le \Phi\big(f(x)\big)+\Psi\big(f(y)\big)\qquad (x,y\in I),$$then $\Phi\circ f\circ \varphi^{-1}$ and $\Psi\circ f\circ \psi^{-1}$ are convex functions.
2020 Abels Math Contest (Norwegian MO) Final, 3
Show that the equation $x^2 \cdot (x - 1)^2 \cdot (x - 2)^2 \cdot ... \cdot (x - 1008)^2 \cdot (x- 1009)^2 = c$ has $2020$ real solutions, provided $0 < c <\frac{(1009 \cdot1007 \cdot ... \cdot 3\cdot 1)^4}{2^{2020}}$ .
2018 South East Mathematical Olympiad, 5
Let $\{a_n\}$ be a nonnegative real sequence. Define
$$X_k = \sum_{i=1}^{2^k}a_i, Y_k = \sum_{i=1}^{2^k}\left\lfloor \frac{2^k}{i}\right\rfloor a_i, k=0,1,2,...$$
Prove that $X_n\le Y_n - \sum_{i=0}^{n-1} Y_i \le \sum_{i=0}^n X_i$ for all positive integer $n$. Here $\lfloor\alpha\rfloor$ denotes the largest integer that does not exceed $\alpha$.
2014 Thailand Mathematical Olympiad, 5
Determine the maximal value of $k$ such that the inequality
$$\left(k +\frac{a}{b}\right) \left(k + \frac{b}{c}\right)\left(k + \frac{c}{a}\right)
\le \left( \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\right) \left( \frac{b}{a}+ \frac{c}{b}+ \frac{a}{c}\right)$$
holds for all positive reals $a, b, c$.
1998 All-Russian Olympiad Regional Round, 9.5
The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?
2018 Estonia Team Selection Test, 10
A sequence of positive real numbers $a_1, a_2, a_3, ... $ satisfies $a_n = a_{n-1} + a_{n-2}$ for all $n \ge 3$. A sequence $b_1, b_2, b_3, ...$ is defined by equations
$b_1 = a_1$ ,
$b_n = a_n + (b_1 + b_3 + ...+ b_{n-1})$ for even $n > 1$ ,
$b_n = a_n + (b_2 + b_4 + ... +b_{n-1})$ for odd $n > 1$.
Prove that if $n\ge 3$, then $\frac13 < \frac{b_n}{n \cdot a_n} < 1$
1981 IMO Shortlist, 16
A sequence of real numbers $u_1, u_2, u_3, \dots$ is determined by $u_1$ and the following recurrence relation for $n \geq 1$:
\[4u_{n+1} = \sqrt[3]{ 64u_n + 15.}\]
Describe, with proof, the behavior of $u_n$ as $n \to \infty.$
2024 Mathematical Talent Reward Programme, 1
Hari the milkman delivers milk to his customers everyday by travelling on his cycle. Each litre of milk costs him Rs. $20$, and he sells it at Rs. $24$. One day while riding his cycle with $20$L, Hari trips and loses $5$L of it, and he decides to mix some water with the rest of the milk. His customers can detect if the milk is more than $10$% impure ($1$L water in $10$L misture). Given that he doesn't wish to make his customers angry, what is his maximum profit for the day?
$(A)$ Rs $12$ profit
$(B)$ Rs $24$ profit
$(C)$ No profit
$(D)$ Rs $12$ loss
2003 Gheorghe Vranceanu, 1
Solve in $ \mathbb{R}^2 $ the equation $ \lfloor x/y-y/x \rfloor =x^2/y+y/x^2. $
1999 Akdeniz University MO, 3
Let $a$,$b$,$c$ and $d$ positive reals. Prove that
$$\frac{1}{a+b+c+d} \leq \frac{1}{64}(\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d})$$
2000 IMC, 6
Let $A$ be a real $n\times n$ Matrix and define $e^{A}=\sum_{k=0}^{\infty} \frac{A^{k}}{k!}$
Prove or disprove that for any real polynomial $P(x)$ and any real matrices $A,B$,
$P(e^{AB})$ is nilpotent if and only if $P(e^{BA})$ is nilpotent.
2008 Thailand Mathematical Olympiad, 4
Prove that $$\sqrt{a^2 + b^2 -\sqrt2 ab} +\sqrt{b^2 + c^2 -\sqrt2 bc} \ge \sqrt{a^2 + c^2}$$ for all real numbers $a, b, c > 0$
2009 Stanford Mathematics Tournament, 5
Compute $\int_{0}^{\infty} t^5e^{-t}dt$
2023 BMT, 9
The boxes in the expression below are filled with the numbers $3$, $4$, $5$, $6$, $7$, and $8$, so that each number is used exactly once. What is the least possible value of the expression?
$$\square \times \square +\square \times \square -\square \times \square$$
2005 Federal Math Competition of S&M, Problem 1
Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.
2020 USA EGMO Team Selection Test, 6
Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.
2008 German National Olympiad, 3
Find all functions $ f$ defined on non-negative real numbers having the following properties:
(i) For all non-negative $ x$ it is $ f(x) \geq 0$.
(ii) It is $ f\left(1\right)\equal{}\frac 12$.
(iii) For all non-negative numbers $ x,y$ it is $ f\left( y \cdot f(x) \right) \cdot f(x) \equal{} f(x\plus{}y)$.
2005 China Team Selection Test, 2
Determine whether $\sqrt{1001^2+1}+\sqrt{1002^2+1}+ \cdots + \sqrt{2000^2+1}$ be a rational number or not?
1968 IMO, 5
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
2018 Brazil EGMO TST, 2
(a) Let $x$ be a real number with $x \ge 1$. Prove that $x^3 - 5x^2 + 8x - 4 \ge 0$.
(b) Let $a, b \ge 1$ real numbers. Find the minimum value of the expression $ab(a + b - 10) + 8(a + b)$. Determine
also the real number pairs $(a, b)$ that make this expression equal to this minimum value.
2025 Kosovo National Mathematical Olympiad`, P2
Find the smallest natural number $k$ such that the system of equations
$$x+y+z=x^2+y^2+z^2=\dots=x^k+y^k+z^k $$
has only one solution for positive real numbers $x$, $y$ and $z$.
2010 QEDMO 7th, 2
Let $c: Q-\{0\} \to Q-\{0\}$ a function with the following properties (for all $x,y, a, b \in Q-\{0\}$ and $x \ne 1$):
a) $c (x, 1- x) = 1$
b) $c (ab,y) = c (a,y)c(b, y)$
c) $c (y,ab) = c (y, a)c(y,b)$
Show that then $c (a,b) c(b,a) = 1 = c(a,-a)$ also holds.