Found problems: 4776
2003 Kazakhstan National Olympiad, 8
Determine all functions $f: \mathbb R \to \mathbb R$ with the property
\[f(f(x)+y)=2x+f(f(y)-x), \quad \forall x,y \in \mathbb R.\]
1997 Pre-Preparation Course Examination, 4
Let $f : \mathbb N \to \mathbb N$ be an injective function such that there exists a positive integer $k$ for which $f(n) \leq n^k$. Prove that there exist infinitely many primes $q$ such that the equation $f(x) \equiv 0 \pmod q$ has a solution in prime numbers.
2016 IFYM, Sozopol, 8
Prove that there exist infinitely many natural numbers $n$, for which there $\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}$, satisfying the following conditions:
1) $f(x)\neq x$;
2) $f(f(x))=x$;
3) $f(f(f(x+1)+1)+1)=x$ for $\forall x\in \{0,1…n-1\}$.
2010 Today's Calculation Of Integral, 654
A function $f(x)$ defined in $x\geq 0$ satisfies $\lim_{x\to\infty} \frac{f(x)}{x}=1$.
Find $\int_0^{\infty} \{f(x)-f'(x)\}e^{-x}dx$.
[i]1997 Hokkaido University entrance exam/Science[/i]
2011 Macedonia National Olympiad, 4
Find all functions $~$ $f: \mathbb{R} \to \mathbb{R}$ $~$ which satisfy the equation
\[ f(x+yf(x))\, =\, f(f(x)) + xf(y)\, . \]
2011 Pre-Preparation Course Examination, 1
[b]a)[/b] prove that for every compressed set $K$ in the space $\mathbb R^3$, the function $f:\mathbb R^3 \longrightarrow \mathbb R$ that $f(p)=inf\{|p-k|,k\in K\}$ is continuous.
[b]b)[/b] prove that we cannot cover the sphere $S^2\subseteq \mathbb R^3$ with it's three closed sets, such that none of them contain two antipodal points.
1987 Greece National Olympiad, 2
If for function $f$ holds that $$f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0$$ for any $\in\mathbb{R}$, prove that $f$ is periodic and find one period of her.
1997 ITAMO, 2
Let a real function $f$ defined on the real numbers satisfy the following conditions:
(i) $f(10+x) = f(10- x)$
(ii) $f(20+x) = - f(20- x)$
for all $x$. Prove that f is odd and periodic.
2021 IMO Shortlist, A8
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$.
[i]Proposed by Ankan Bhattacharya, USA[/i]
2015 Postal Coaching, Problem 2
Let $ n$ be a positive integer. Find the number of odd coefficients of the polynomial
\[ u_n(x) \equal{} (x^2 \plus{} x \plus{} 1)^n.
\]
2009 National Olympiad First Round, 14
For how many ordered pairs of positive integers $ (m,n)$, $ m \cdot n$ divides $ 2008 \cdot 2009 \cdot 2010$ ?
$\textbf{(A)}\ 2\cdot3^7\cdot 5 \qquad\textbf{(B)}\ 2^5\cdot3\cdot 5 \qquad\textbf{(C)}\ 2^5\cdot3^7\cdot 5 \qquad\textbf{(D)}\ 2^3\cdot3^5\cdot 5^2 \qquad\textbf{(E)}\ \text{None}$
1992 IMO Longlists, 6
Suppose that n numbers $x_1, x_2, . . . , x_n$ are chosen randomly from the set $\{1, 2, 3, 4, 5\}$. Prove that the probability that $x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5$ is at least $\frac 15.$
2018 China Team Selection Test, 4
Let $k, M$ be positive integers such that $k-1$ is not squarefree. Prove that there exist a positive real $\alpha$, such that $\lfloor \alpha\cdot k^n \rfloor$ and $M$ are coprime for any positive integer $n$.
1966 AMC 12/AHSME, 25
If $F(n+1)=\frac{2F(n)+1}{2}$ for $n=1,2,\ldots$, and $F(1)=2$, then $F(101)$ equals:
$\text{(A)} \ 49 \qquad \text{(B)} \ 50 \qquad \text{(C)} \ 51 \qquad \text{(D)} \ 52 \qquad \text{(E)} \ 53$
2003 Putnam, 2
Let $n$ be a positive integer. Starting with the sequence $1,\frac{1}{2}, \frac{1}{3} , \cdots , \frac{1}{n}$, form a new sequence of $n -1$ entries $\frac{3}{4}, \frac{5}{12},\cdots ,\frac{2n -1}{2n(n -1)}$, by taking the averages of two consecutive entries in the first sequence. Repeat the averaging of neighbors on the second sequence to obtain a third sequence of $n -2$ entries and continue until the final sequence consists of a single number $x_n$. Show that $x_n < \frac{2}{n}$.
2012 Middle European Mathematical Olympiad, 1
Let $ \mathbb{R} ^{+} $ denote the set of all positive real numbers. Find all functions $ \mathbb{R} ^{+} \to \mathbb{R} ^{+} $ such that
\[ f(x+f(y)) = yf(xy+1)\]
holds for all $ x, y \in \mathbb{R} ^{+} $.
2011 Iran MO (3rd Round), 4
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove
$\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$.
[i]proposed by Mohammad Ahmadi[/i]
2003 China Team Selection Test, 2
Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.
1994 Flanders Math Olympiad, 1
Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side.
2004 France Team Selection Test, 1
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that
$a_1 + ... + a_n = b_1 + ... + b_n = 1$.
Find the minimal value of
$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
2011 Today's Calculation Of Integral, 766
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and
\[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\]
Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$
2004 Bulgaria Team Selection Test, 1
Find all $k>0$ such that there exists a function $f : [0,1]\times[0,1] \to [0,1]$ satisfying the following conditions:
$f(f(x,y),z)=f(x,f(y,z))$;
$f(x,y) = f(y,x)$;
$f(x,1)=x$;
$f(zx,zy) = z^{k}f(x,y)$, for any $x,y,z \in [0,1]$
2012 Putnam, 1
Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies:
(i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$
(ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$
Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$
PEN H Problems, 86
A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?
2015 Korea - Final Round, 1
Find all functions $f: R \rightarrow R$ such that
$f(x^{2015} + (f(y))^{2015}) = (f(x))^{2015} + y^{2015}$ holds for all reals $x, y$