Found problems: 15925
1962 IMO, 4
Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$
2023 USA IMOTST, 1
Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \]
[i]Carl Schildkraut[/i]
2014 BMT Spring, 14
Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, ... , x_7) = f(x_1) + f(x_2) +... + f(x_7)$. If $x_1, x_2,..., x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, ... , x_7)$ does $g(x_1, x_2, ... , x_7)$ obtain its maximal value.
1997 All-Russian Olympiad, 1
Of the quadratic trinomials $x^2 + px + q$ where $p$; $q$ are integers and $1\leqslant p, q \leqslant 1997$, which are there more of: those having integer roots or those not having real roots?
[i]M. Evdokimov[/i]
2022 VJIMC, 2
For any given pair of positive integers $m>n$ find all $a\in\mathbb R$ for which the polynomial $x^m-ax^n+1$ can be expressed as a quotient of two nonzero polynomials with real nonnegative coefficients.
2015 Hanoi Open Mathematics Competitions, 8
Solve the equation $(x + 1)^3(x - 2)^3 + (x -1)^3(x + 2)^3 = 8(x^2 -2)^3.$
1999 IMO, 2
Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality
\[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C
\left(\sum_{i}x_{i} \right)^4\]
holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.
2012 Today's Calculation Of Integral, 840
Let $x,\ y$ be real numbers. For a function $f(t)=x\sin t+y\cos t$, draw the domain of the points $(x,\ y)$ for which the following inequality holds.
\[\left|\int_{-\pi}^{\pi} f(t)\cos t\ dt\right|\leq \int_{-\pi}^{\pi} \{f(t)\}^2dt.\]
1980 IMO Longlists, 14
Let $\{x_n\}$ be a sequence of natural numbers such that \[(a) 1 = x_1 < x_2 < x_3 < \ldots; \quad (b) x_{2n+1} \leq 2n \quad \forall n.\] Prove that, for every natural number $k$, there exist terms $x_r$ and $x_s$ such that $x_r - x_s = k.$
2015 239 Open Mathematical Olympiad, 4
َA natural number $n$ is given. Let $f(x,y)$ be a polynomial of degree less than $n$ such that for any positive integers $x,y\leq n, x+y \leq n+1$ the equality $f(x,y)=\frac{x}{y}$ holds. Find $f(0,0)$.
2013 Bogdan Stan, 1
Under composition, let be a group of linear polynomials that admit a fixed point . Show that all polynomials of this group have the same fixed point.
[i]Vasile Pop[/i]
2008 Princeton University Math Competition, 10
Consider the sequence $s_0 = (1, 2008)$. Define new sequences $s_i$ inductively by inserting the sum of each pair of adjacent terms in $s_{i-1}$ between them — for instance, $s_1 = (1, 2009, 2008)$. For some $n, s_n$ has exactly one term that appears twice. Find this repeated term.
2020 Greece Junior Math Olympiad, 1
Solve in real numbers $\frac{(x+2)^4}{x^3}-\frac{(x+2)^2}{2x}\ge - \frac{x}{16}$
2015 Grand Duchy of Lithuania, 1
Find all pairs of real numbers $(x, y)$ for which the inequality $y^2 + y + \sqrt{y - x^2 -xy} \le 3xy$ holds.
2002 May Olympiad, 1
A group of men, some of them accompanied by their wives, spent $\$1.000$ on a hotel. Each man spent $\$19$ and each woman $\$13$. Determine how many women and how many men there were.
2004 APMO, 5
Prove that the inequality \[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\] holds for all positive reals $a$, $b$, $c$.
2003 Olympic Revenge, 6
Find all functions $f:R^{*} \rightarrow R$ such that $f(x)\not = x$ and
$$ f(y(f(x)-x))=\frac{f(x)}{y}-\frac{f(y)}{x} $$ for any $x,y \not = 0$.
2008 Vietnam National Olympiad, 1
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$
2017 IFYM, Sozopol, 6
The sequence $a_1,a_2…$ , is defined by the equations $a_1=1$ and $a_n=n.a_{[n/2]}$ for $n>1$. Prove that $a_n>n^2$ for $n>11$.
2015 Hanoi Open Mathematics Competitions, 15
Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$.
Determine the maximum value of $M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$
2009 Romania National Olympiad, 4
Find all functions $ f:[0,1]\longrightarrow [0,1] $ that are bijective, continuous and have the property that, for any continuous function $ g:[0,1]\longrightarrow\mathbb{R} , $ the following equality holds.
$$ \int_0^1 g\left( f(x) \right) dx =\int_0^1 g(x) dx $$
1962 Poland - Second Round, 1
Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equationw
$$x + y + z = a,$$
$$ \frac{1}{x} + \frac{1}{y} + \frac{1}{z} = \frac{1}{a},$$
then at least one of them is equal to $ a $.
2006 Romania Team Selection Test, 1
Let $\{a_n\}_{n\geq 1}$ be a sequence with $a_1=1$, $a_2=4$ and for all $n>1$, \[ a_{n} = \sqrt{ a_{n-1}a_{n+1} + 1 } . \]
a) Prove that all the terms of the sequence are positive integers.
b) Prove that $2a_na_{n+1}+1$ is a perfect square for all positive integers $n$.
[i]Valentin Vornicu[/i]
2009 Peru IMO TST, 6
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$;
(ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$.
Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list]
[i]Proposed by Hans Zantema, Netherlands[/i]
2001 Slovenia National Olympiad, Problem 2
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?