Found problems: 15925
1986 Poland - Second Round, 1
Determine all functions $ f : \mathbb{R} \to \mathbb{R} $ continuous at zero and such that for every real number $ x $ the equality holds $$ 2f(2x) = f(x) + x.$$
1981 Bulgaria National Olympiad, Problem 5
Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.
1995 IMO Shortlist, 4
Find all $ x,y$ and $ z$ in positive integer: $ z \plus{} y^{2} \plus{} x^{3} \equal{} xyz$ and $ x \equal{} \gcd(y,z)$.
1998 Czech And Slovak Olympiad IIIA, 2
Given any set of $14$ (different) natural numbers, prove that for some $k$ ($1 \le k \le 7$) there exist two disjoint $k$-element subsets $\{a_1,...,a_k\}$ and $\{b_1,...,b_k\}$ such that $A =\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}$ and $B =\frac{1}{b_1}+\frac{1}{b_2}+...+\frac{1}{b_k}$ differ by less than $0.001$, i.e. $|A-B| < 0.001$
2021 Princeton University Math Competition, A7
Consider the following expression
$$S = \log_2 \left( \sum^{2019}_{k=1}\sum^{2020}_{j=2}\log_{2^{1/k}} (j) \log_{j^2} \left(\sin \frac{\pi k}{2020}\right) \right).$$
Find the smallest integer $n$ which is bigger than $S$ (i.e. find $\lceil S \rceil$).
2011 NIMO Problems, 14
In circle $\theta_1$ with radius $1$, circles $\phi_1, \phi_2, \dots, \phi_8$, with equal radii, are drawn such that for $1 \le i \le 8$, $\phi_i$ is tangent to $\omega_1$, $\phi_{i-1}$, and $\phi_{i+1}$, where $\phi_0 = \phi_8$ and $\phi_1 = \phi_9$. There exists a circle $\omega_2$ such that $\omega_1 \neq \omega_2$ and $\omega_2$ is tangent to $\phi_i$ for $1 \le i \le 8$. The radius of $\omega_2$ can be expressed in the form $a - b\sqrt{c} -d\sqrt{e - \sqrt{f}} + g \sqrt{h - j \sqrt{k}}$ such that $a, b, \dots, k$ are positive integers and the numbers $e, f, k, \gcd(h, j)$ are squarefree. What is $a+b+c+d+e+f+g+h+j+k$.
[i]Proposed by Eugene Chen
[/i]
2006 Moldova National Olympiad, 10.4
Find all real values of the real parameter $a$ such that the equation \[ 2x^{2}-6ax+4a^{2}-2a-2+\log_{2}(2x^{2}+2x-6ax+4a^{2})= \] \[ =\log_{2}(x^{2}+2x-3ax+2a^{2}+a+1). \]
has a unique solution.
2008 Ukraine Team Selection Test, 8
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition
\[ f(m \plus{} n) \geq f(m) \plus{} f(f(n)) \minus{} 1
\]
for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$
[i]Author: Nikolai Nikolov, Bulgaria[/i]
2011 Bosnia And Herzegovina - Regional Olympiad, 1
Determine value of real parameter $\lambda$ such that equation $$\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda $$ has root in interval $\left(0,\frac{\pi}{2}\right)$
1993 Nordic, 1
Let $F$ be an increasing real function defined for all $x, 0 \le x \le 1$, satisfying the conditions
(i) $F (\frac{x}{3}) = \frac{F(x)}{2}$.
(ii) $F(1- x) = 1 - F(x)$.
Determine $F(\frac{173}{1993})$ and $F(\frac{1}{13})$ .
IV Soros Olympiad 1997 - 98 (Russia), 10.10
The football tournament, held in one round, involved $16$ teams, each two of which scored a different number of points. ($3$ points were given for a victory, $1$ point for a draw, $0$ points for a defeat.) It turned out that the Chisel team lost to all the teams that ultimately scored fewer points. What is the best result that the Chisel team could achieve (insert location)?
1975 Bundeswettbewerb Mathematik, 1
Let $a, b, c, d$ be distinct positive real numbers. Prove that if one of the numbers $c, d$ lies between $a$ and $b$, or one of $a, b$ lies between $c$ and $d$, then
$$\sqrt{(a+b)(c+d)} >\sqrt{ab} +\sqrt{cd}$$
and that otherwise, one can choose $a, b, c, d$ so that this inequality is false.
2021 Junior Balkan Team Selection Tests - Moldova, 6
Solve the system of equations
$$\begin{cases} (x+y)(x^2-y^2)=32 \\ (x-y)(x^2+y^2)=20 \end{cases}$$
2005 China Team Selection Test, 1
Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.
2002 Baltic Way, 1
Solve the system of simultaneous equations
\[\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}\]
in real numbers.
2020 Kosovo National Mathematical Olympiad, 3
Let $a$ and $b$ be real numbers such that $a+b=\log_2( \log_2 3)$. What is the minimum value of $2^a + 3^b$ ?
2002 Argentina National Olympiad, 1
On the computer screen there are initially two $1$'s written. The [i] insert [/i] program causes the sum of those numbers to be inserted between each pair of numbers by pressing the $Enter$ key.
In the first step a number is inserted and we obtain $1-2-1$; In the second step two numbers are inserted and we have $1-3-2-3-1$; In the third, four numbers are inserted and you have $1-4-3-5-2-5-3-4-1$; etc Find the sum of all the numbers that appear on the screen at the end of step number $25$.
1974 Swedish Mathematical Competition, 1
Let $a_n = 2^{n-1}$ for $n > 0$. Let
\[
b_n = \sum\limits_{r+s \leq n} a_ra_s
\]
Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.
2014 China National Olympiad, 1
Let $n=p_1^{a_1}p_2^{a_2}\cdots p_t^{a_t}$ be the prime factorisation of $n$. Define $\omega(n)=t$ and $\Omega(n)=a_1+a_2+\ldots+a_t$. Prove or disprove:
For any fixed positive integer $k$ and positive reals $\alpha,\beta$, there exists a positive integer $n>1$ such that
i) $\frac{\omega(n+k)}{\omega(n)}>\alpha$
ii) $\frac{\Omega(n+k)}{\Omega(n)}<\beta$.
2004 USA Team Selection Test, 6
Define the function $f: \mathbb N \cup \{0\} \to \mathbb{Q}$ as follows: $f(0) = 0$ and \[ f(3n+k) = -\frac{3f(n)}{2} + k , \] for $k = 0, 1, 2$. Show that $f$ is one-to-one and determine the range of $f$.
2011 VJIMC, Problem 2
Let $k$ be a positive integer. Compute
$$\sum_{n_1=1}^\infty\sum_{n_2=1}^\infty\cdots\sum_{n_k=1}^\infty\frac1{n_1n_2\cdots n_k(n_1+n_2+\ldots+n_k+1)}.$$
1969 IMO Longlists, 8
Find all functions $f$ defined for all $x$ that satisfy the condition $xf(y) + yf(x) = (x + y)f(x)f(y),$ for all $x$ and $y.$ Prove that exactly two of them are continuous.
2018 Junior Balkan Team Selection Tests - Romania, 1
Determine all triples of real numbers $(a,b,c)$ that satisfy simultaneously the equations:
$$\begin{cases} a(b^2 + c) = c(c + ab) \\ b(c^2 + a) = a(a + bc) \\ c(a^2 + b) = b(b + ca) \end{cases}$$
2012 Online Math Open Problems, 46
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]
2000 Tournament Of Towns, 4
(a) Does there exist an infinite sequence of real numbers such that the sum of every ten successive numbers is positive, while for every $n$ the sum of the first $10n + 1$ successive numbers is negative?
(b) Does there exist an infinite sequence of integers with the same properties?
(AK Tolpygo)