Found problems: 15925
1997 Iran MO (3rd Round), 3
Let $S = \{x_0, x_1,\dots , x_n\}$ be a finite set of numbers in the interval $[0, 1]$ with $x_0 = 0$ and $x_1 = 1$. We consider pairwise distances between numbers in $S$. If every distance that appears, except the distance $1$, occurs at least twice, prove that all the $x_i$ are rational.
2021 Olympic Revenge, 1
Let $a$, $b$, $c$, $k$ be positive reals such that $ab+bc+ca \leq 1$ and $0 < k \leq \frac{9}{2}$. Prove that:
\[\sqrt[3]{ \frac{k}{a} + (9-3k)b} + \sqrt[3]{\frac{k}{b} + (9-3k)c} + \sqrt[3]{\frac{k}{c} + (9-3k)a } \leq \frac{1}{abc}.\]
[i]Proposed by Zhang Yanzong and Song Qing[/i]
2023 Kyiv City MO Round 1, Problem 2
For any given real $a, b, c$ solve the following system of equations:
$$\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right.$$
[i]Proposed by Oleksiy Masalitin, Bogdan Rublov[/i]
1986 China Team Selection Test, 3
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
Kvant 2024, M2808
Some participants of the tournament are friends with each other, and everyone has at least one friend. Each participant of the tournament was given a T-shirt with the number of his friends at the tournament written on it. Prove that at least one participant has the arithmetic mean of the numbers written on his friends' T-shirts, not less than the arithmetic mean of the numbers on all T-shirts.
[i] From Czech-Slovak Olympiad 1991 [/i]
1994 All-Russian Olympiad, 5
Prove the equality
$$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}=\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $$
(R. Zhenodarov)
1995 Grosman Memorial Mathematical Olympiad, 6
(a) Prove that there is a unique function $f : Q \to Q$ satisfying:
(i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$
(ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$
(iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$
(b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$
1990 IMO Shortlist, 16
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
1998 Italy TST, 4
Find all polynomials $P(x) = x^n +a_1x^{n-1} +...+a_n$ whose zeros (with their multiplicities) are exactly $a_1,a_2,...,a_n$.
2018 China Team Selection Test, 6
Suppose $a_i, b_i, c_i, i=1,2,\cdots ,n$, are $3n$ real numbers in the interval $\left [ 0,1 \right ].$ Define $$S=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k<1 \right \}, \; \; T=\left \{ \left ( i,j,k \right ) |\, a_i+b_j+c_k>2 \right \}.$$ Now we know that $\left | S \right |\ge 2018,\, \left | T \right |\ge 2018.$ Try to find the minimal possible value of $n$.
1973 Swedish Mathematical Competition, 6
$f(x)$ is a real valued function defined for $x \geq 0$ such that $f(0) = 0$, $f(x+1)=f(x)+\sqrt{x}$ for all $x$, and
\[
f(x) < \frac{1}{2}f\left(x - \frac{1}{2}\right)+\frac{1}{2}f\left(x + \frac{1}{2}\right) \quad \text{for all} \quad x \geq \frac{1}{2}
\]
Show that $f\left(\frac{1}{2}\right)$ is uniquely determined.
1996 Rioplatense Mathematical Olympiad, Level 3, 3
The real numbers $x, y, z$, distinct in pairs satisfy $$\begin{cases} x^2=2 + y \\ y^2=2 + z \\ z^2=2 + x.\end{cases}$$
Find the possible values of $x^2 + y^2 + z^2$.
2018 Moscow Mathematical Olympiad, 1
The graphs of a square trinomial and its derivative divide the coordinate plane into four parts. How many roots does this
square trinomial has?
2015 Dutch IMO TST, 3
Let $n$ be a positive integer.
Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$.
Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\
b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$
Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.
1983 Tournament Of Towns, (032) O1
A pedestrian walked for $3.5$ hours. In every period of one hour’s duration he walked $5$ kilometres. Is it true that his average speed was $5$ kilometres per hour?
(NN Konstantinov, Moscow)
2025 Poland - First Round, 1
Let $f(x)=ax^2+bx+c$ be a quadratic function, the graph of which doesn't intersect the x-axis. Prove that
$$a(2a+3b+6c)>0.$$
2004 Romania Team Selection Test, 4
Let $D$ be a closed disc in the complex plane. Prove that for all positive integers $n$, and for all complex numbers $z_1,z_2,\ldots,z_n\in D$ there exists a $z\in D$ such that $z^n = z_1\cdot z_2\cdots z_n$.
2012 AMC 8, 7
Isabella must take four 100-point tests in her math class. Her goal is to achieve an average grade of 95 on the tests. Her first two test scores were 97 and 91. After seeing her score on the third test, she realized she can still reach her goal. What is the lowest possible score she could have made on the third test?
$\textbf{(A)}\hspace{.05in}90 \qquad \textbf{(B)}\hspace{.05in}92 \qquad \textbf{(C)}\hspace{.05in}95 \qquad \textbf{(D)}\hspace{.05in}96 \qquad \textbf{(E)}\hspace{.05in}97 $
1996 French Mathematical Olympiad, Problem 2
Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise.
(a) Prove that one can find an element of $u_n$ smaller than $a$.
(b) Prove that the sequence is eventually periodic.
1998 Junior Balkan Team Selection Tests - Romania, 1
Solve in $ \mathbb{Z}^2 $ the following equation:
$$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$
[i]Adrian Zanoschi[/i]
2024 Ukraine National Mathematical Olympiad, Problem 4
Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following:
$$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$
[i]Proposed by Mykhailo Shtandenko[/i]
1974 Kurschak Competition, 3
Let $$p_k(x) = 1 -x + \frac{x^2}{2! } - \frac{x^3}{3!}+ ... + \frac{(-x)^{2k}}{(2k)!}$$ Show that it is non-negative for all real $x$ and all positive integers $k$.
2019 Dutch IMO TST, 2
Write $S_n$ for the set $\{1, 2,..., n\}$. Determine all positive integers $n$ for which there exist functions $f : S_n \to S_n$ and $g : S_n \to S_n$ such that for every $x$ exactly one of the equalities $f(g(x)) = x$ and $g(f(x)) = x$ holds.
2025 District Olympiad, P2
Solve in $\mathbb{R}$ the equation $$\frac{1}{x}+\frac{1}{\lfloor x\rfloor} + \frac{1}{\{x\}} = 0.$$
[i]Mathematical Gazette[/i]
2007 District Olympiad, 3
Find all functions $ f:\mathbb{N}\longrightarrow\mathbb{N} $ that satisfy the following relation:
$$ f(x)^2+y\vdots x^2+f(y) ,\quad\forall x,y\in\mathbb{N} . $$