Found problems: 15925
2022 Thailand TST, 1
For each integer $n\ge 1,$ compute the smallest possible value of \[\sum_{k=1}^{n}\left\lfloor\frac{a_k}{k}\right\rfloor\] over all permutations $(a_1,\dots,a_n)$ of $\{1,\dots,n\}.$
[i]Proposed by Shahjalal Shohag, Bangladesh[/i]
2007 German National Olympiad, 1
Determine all real numbers $x$ such that for all positive integers $n$ the inequality $(1+x)^n \leq 1+(2^n -1)x$ is true.
2016 Kyrgyzstan National Olympiad, 1
If $a+b+c=0$ ,then find the value of
$(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})$
1979 Brazil National Olympiad, 1
Show that if $a < b$ are in the interval $\left[0, \frac{\pi}{2}\right]$ then $a - \sin a < b - \sin b$. Is this true for $a < b$ in the interval $\left[\pi,\frac{3\pi}{2}\right]$?
2011 Baltic Way, 5
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that
\[f(f(x))=x^2-x+1\]
for all real numbers $x$. Determine $f(0)$.
1989 IMO Longlists, 37
There are n cars waiting at distinct points of a circular race track. At the starting signal each car starts. Each car may choose arbitrarily which of the two possible directions to go. Each car has the same constant speed. Whenever two cars meet they both change direction (but not speed). Show that at some time each car is back at its starting point.
2014 Baltic Way, 1
Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]
2025 Malaysian APMO Camp Selection Test, 1
A sequence is defined as $a_1=2025$ and for all $n\ge 2$, $$a_n=\frac{a_{n-1}+1}{n}$$ Determine the smallest $k$ such that $\displaystyle a_k<\frac{1}{2025}$.
[i]Proposed by Ivan Chan Kai Chin[/i]
2005 MOP Homework, 4
Let $p$ be an odd prime. Prove that \[\sum^{p-1}_{k=1} k^{2p-1} \equiv \frac{p(p+1)}{2}\pmod{p^2}.\]
2019 BMT Spring, 7
Let $ r_1 $, $ r_2 $, $ r_3 $ be the (possibly complex) roots of the polynomial $ x^3 + ax^2 + bx + \dfrac{4}{3} $. How many pairs of integers $ a $, $ b $ exist such that $ r_1^3 + r_2^3 + r_3^3 = 0 $?
2011 JBMO Shortlist, 1
Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.
1999 Ukraine Team Selection Test, 4
If $n \in N$ and $0 < x <\frac{\pi}{2n}$, prove the inequality $\frac{\sin 2x}{\sin x}+\frac{\sin 3x}{\sin 2x} +...+\frac{\sin (n+1)x}{\sin nx} < 2\frac{\cos x}{\sin^2 x}$.
.
1987 Tournament Of Towns, (144) 1
Suppose $p(x)$ is a polynomial with integer coefficients. It is known that $p(a) - p(b) = 1$ (where $a$ and $b$ are integers). Prove that $a$ and $b$ differ by $1$ .
(Folklore)
2007 Italy TST, 3
Find all $f: R \longrightarrow R$ such that
\[f(xy+f(x))=xf(y)+f(x)\]
for every pair of real numbers $x,y$.
1972 Polish MO Finals, 1
Polynomials $u_i(x) = a_ix+b_i$ ($a_i,b_i \in R$, $ i = 1,2,3$) satisfy
$$u_1(x)^n +u_2(x)^n = u_3(x)^n$$
for some integer $n \ge 2.$
Prove that there exist real numbers $A$,$B$,$c_1$,$c_2$,$c_3$ such that $u_i(x) = c_i(Ax+B)$ for $i = 1,2,3$.
2006 IMO Shortlist, 4
Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$.
2007 Pre-Preparation Course Examination, 2
Let $\{A_{1},\dots,A_{k}\}$ be matrices which make a group under matrix multiplication. Suppose $M=A_{1}+\dots+A_{k}$. Prove that each eigenvalue of $M$ is equal to $0$ or $k$.
2010 IberoAmerican Olympiad For University Students, 7
(a) Prove that, for any positive integers $m\le \ell$ given, there is a positive integer $n$ and positive integers $x_1,\cdots,x_n,y_1,\cdots,y_n$ such that the equality \[ \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k\] holds for every $k=1,2,\cdots,m-1,m+1,\cdots,\ell$, but does not hold for $k=m$.
(b) Prove that there is a solution of the problem, where all numbers $x_1,\cdots,x_n,y_1,\cdots,y_n$ are distinct.
[i]Proposed by Ilya Bogdanov and Géza Kós.[/i]
2010 N.N. Mihăileanu Individual, 2
If at least one of the integers $ a,b $ is not divisible by $ 3, $ then the polynom $ X^2-abX+a^2+b^2 $ is irreducible over the integers.
[i]Ion Cucurezeanu[/i]
2019 Hanoi Open Mathematics Competitions, 14
Let $a, b, c$ be nonnegative real numbers satisfying $a + b + c =3$.
a) If $c > \frac32$, prove that $3(ab + bc + ca) - 2abc < 7$.
b) Find the greatest possible value of $M =3(ab + bc + ca) - 2abc $.
2019 Brazil Team Selection Test, 2
Given any set $S$ of positive integers, show that at least one of the following two assertions holds:
(1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$;
(2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.
2022 District Olympiad, P2
$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$
$b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$
2016 Latvia Baltic Way TST, 4
Find all functions $f : R \to R$ defined for real numbers, take real values and for all real $x$ and $y$ the equality holds:
$$f(2^x+2y) =2^yf(f(x))f(y).$$
2017 Turkey MO (2nd round), 5
Let $x_0,\dots,x_{2017}$ are positive integers and $x_{2017}\geq\dots\geq x_0=1$ such that $A=\{x_1,\dots,x_{2017}\}$ consists of exactly $25$ different numbers. Prove that $\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623$, and find the number of sequences that holds the case of equality.
2018 Dutch IMO TST, 2
Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.