Found problems: 15925
2004 Estonia Team Selection Test, 4
Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$
For which positive integers $m$ is $f(m)$ rational?
2012 Thailand Mathematical Olympiad, 5
Determine all functions $f : R \to R$ satisfying $f(f(x) + xf(y))= 3f(x) + 4xy$ for all real numbers $x,y$.
2020 Romanian Master of Mathematics, 4
Let $\mathbb N$ be the set of all positive integers. A subset $A$ of $\mathbb N$ is [i]sum-free[/i] if, whenever $x$ and $y$ are (not necessarily distinct) members of $A$, their sum $x+y$ does not belong to $A$. Determine all surjective functions $f:\mathbb N\to\mathbb N$ such that, for each sum-free subset $A$ of $\mathbb N$, the image $\{f(a):a\in A\}$ is also sum-free.
[i]Note: a function $f:\mathbb N\to\mathbb N$ is surjective if, for every positive integer $n$, there exists a positive integer $m$ such that $f(m)=n$.[/i]
1990 Romania Team Selection Test, 3
Find all polynomials $P(x)$ such that $2P(2x^2 -1) = P(x)^2 -1$ for all $x$.
2020 Balkan MO Shortlist, A2
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that
$\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$.
[i]Albania[/i]
2010 IFYM, Sozopol, 7
Does there exist a function $f: \mathbb{R}\rightarrow \mathbb{R}$ such that:
$f(f(x))=-x$, for all $x\in \mathbb{R}$?
2018 Pan African, 5
Let $a$, $b$, $c$ and $d$ be non-zero pairwise different real numbers such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} = 4 \text{ and } ac = bd.
$$
Show that
$$
\frac{a}{c} + \frac{b}{d} + \frac{c}{a} + \frac{d}{b} \leq -12
$$
and that $-12$ is the maximum.
2023 Denmark MO - Mohr Contest, 2
The numbers $1, 2, 3, . . . , 16$ must be placed in the $16$ squares in such a way that the sum of the numbers in each of the four rows and columns is the same. What is the smallest possible sum of the four numbers in the corner squares?
[img]https://cdn.artofproblemsolving.com/attachments/c/2/fad1837625fd71e8ea333f9f9477f0bd120e05.png[/img]
2007 Today's Calculation Of Integral, 200
Evaluate the following definite integral.
\[\int_{0}^{\pi}\frac{\cos nx}{2-\cos x}dx\ (n=0,\ 1,\ 2,\ \cdots)\]
2000 Vietnam National Olympiad, 3
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$.
2023 Indonesia TST, 2
Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that
\[2^{j-i}x_ix_j>2^{s-3}.\]
2021 Kosovo National Mathematical Olympiad, 2
Find all functions $f:\mathbb R\to\mathbb R$ so that the following relation holds for all $x, y\in\mathbb R$.
$$f(f(x)f(y)-1) = xy - 1$$
2009 Postal Coaching, 3
Let $n \ge 3$ be a positive integer. Find all nonconstant real polynomials $f_1(x), f_2(x), ..., f_n(x)$ such that $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$, $1 \le k \le n$ for all real x. [All suffixes are taken modulo $n$.]
2019 Junior Balkan Team Selection Tests - Romania, 2
Let $a, b, c, d \ge 0$ such that $a^2 + b^2 + c^2 + d^2 = 4$. Prove that $$\frac{a + b + c + d}{2} \ge 1 + \sqrt{abcd}$$ When does the equality hold?
Leonard Giugiuc and Valmir B. Krasniqi
2025 Bangladesh Mathematical Olympiad, P8
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
2003 China Team Selection Test, 1
$m$ and $n$ are positive integers. Set $A=\{ 1, 2, \cdots, n \}$. Let set $B_{n}^{m}=\{ (a_1, a_2 \cdots, a_m) \mid a_i \in A, i= 1, 2, \cdots, m \}$ satisfying:
(1) $|a_i - a_{i+1}| \neq n-1$, $i=1,2, \cdots, m-1$; and
(2) at least three of $a_1, a_2, \cdots, a_m$ ($m \geq 3$) are pairwise distince.
Find $|B_n^m|$ and $|B_6^3|$.
2013 BMT Spring, 17
Let $N \ge 1$ be a positive integer and $k$ be an integer such that $1 \le k \le N$. Define the recurrence $x_n =
\frac{x_{n-1} + x_{n-2} +... + x_{n-N}}{N}$ for $n > N$ and $x_k = 1$, $x_1 = x_2 = ... = x_{k-1} =x_{k+1} =.. = x_N = 0$. As $n$ approaches infinity, $x_n$ approaches some value. What is this value?
2020 IMC, 4
A polynomial $p$ with real coefficients satisfies $p(x+1)-p(x)=x^{100}$ for all $x \in \mathbb{R}.$ Prove that $p(1-t) \ge p(t)$ for $0 \le t \le 1/2.$
2017 Singapore MO Open, 2
Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$ be real numbers with $p >- 1$. Prove that
$$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$
2017 India IMO Training Camp, 1
Let $P_c(x)=x^4+ax^3+bx^2+cx+1$ and $Q_c(x)=x^4+cx^3+bx^2+ax+1$ with $a,b$ real numbers, $c \in \{1,2, \dots, 2017\}$ an integer and $a \ne c$. Define $A_c=\{\alpha | P_c(\alpha)=0\}$ and $B_c=\{\beta | P(\beta)=0\}$.
(a) Find the number of unordered pairs of polynomials $P_c(x), Q_c(x)$ with exactly two common roots.
(b) For any $1 \le c \le 2017$, find the sum of the elements of $A_c \Delta B_c$.
2007 Princeton University Math Competition, 7
Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$,
\[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\]
find $\lim_{n \to \infty} \frac{x_n}{y_n}$.
2023 ELMO Shortlist, A2
Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]
[i]Proposed by Luke Robitaille[/i]
2014 District Olympiad, 2
Let real numbers $a,b,c$ such that $\left| a-b \right|\ge \left| c \right|,\left| b-c \right|\ge \left| a \right|,\left| c-a \right|\ge \left| b \right|.$
Prove that $a=b+c$ or $b=c+a$ or $c=a+b.$
1968 German National Olympiad, 3
Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation
$$a \cdot f(x^n) + f(-x^n) = bx$$
suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.
2014 Federal Competition For Advanced Students, P2, 2
Let $S$ be the set of all real numbers greater than or equal to $1$.
Determine all functions$ f: S \to S$, so that for all real numbers $x ,y \in S$ with $x^2 -y^2 \in S$ the condition $f (x^2 -y^2) = f (xy)$ is fulfilled.