Found problems: 15925
2024 Korea Junior Math Olympiad (First Round), 19.
For all integers $ {a}_{0},{a}_{1}, \cdot\cdot\cdot {a}_{100}$, find the maximum of ${a}_{5}-2{a}_{40}+3{a}_{60}-4{a}_{95} $
$\bigstar$ 1) ${a}_{0}={a}_{100}=0$
2) for all $i=0,1,\cdot \cdot \cdot 99, $ $|{a}_{i+1}-{a}_{i}|\le1$
3) $ {a}_{10}={a}_{90} $
2016 Azerbaijan BMO TST, 4
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$
1992 Poland - Second Round, 2
Given a natural number $ n \geq 2 $. Let $ a_1, a_2, \ldots , a_n $, $ b_1, b_2, \ldots , b_n $ be real numbers. Prove that the following conditions are equivalent:
- For any real numbers $ x_1 \leq x_2 \leq \ldots \leq x_n $ holds the inequality
$$\sum_{i=1}^n a_i x_i \leq \sum_{i=1}^n b_i x_i.$$
- For every natural number $ k\in \{1,2,\ldots, n-1\} $ holds the inequality
$$
\sum_{i=1}^k a_i \geq \sum_{i=1}^k b_i, \ \ \text{ and } \\ \ \sum_{i=1}^n a_i = \sum_{i=1 }^n b_i.$$
2011 ELMO Shortlist, 1
Let $n$ be a positive integer. There are $n$ soldiers stationed on the $n$th root of unity in the complex plane. Each round, you pick a point, and all the soldiers shoot in a straight line towards that point; if their shot hits another soldier, the hit soldier dies and no longer shoots during the next round. What is the minimum number of rounds, in terms of $n$, required to eliminate all the soldiers?
[i]David Yang.[/i]
2011 Bosnia And Herzegovina - Regional Olympiad, 1
Find the real number coefficient $c$ of polynomial $x^2+x+c$, if his roots $x_1$ and $x_2$ satisfy following: $$\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1$$
2001 Regional Competition For Advanced Students, 2
Find all real solutions to the equation
$$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$
2015 BMT Spring, 17
There exist real numbers $x$ and $y$ such that $x(a^3 + b^3 + c^3) + 3yabc \ge (x + y)(a^2b + b^2c + c^2a)$ holds for all positive real numbers $a, b$, and $c$. Determine the smallest possible value of $x/y$.
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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]
2008 IMO Shortlist, 3
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a [i]Spanish Couple[/i] on $ S$, if they satisfy the following conditions:
(i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$;
(ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$.
Decide whether there exists a Spanish Couple [list][*] on the set $ S \equal{} \mathbb{N}$ of positive integers; [*] on the set $ S \equal{} \{a \minus{} \frac {1}{b}: a, b\in\mathbb{N}\}$[/list]
[i]Proposed by Hans Zantema, Netherlands[/i]
1899 Eotvos Mathematical Competition, 3
Prove that, for any natural number $n$, the expression $$A = 2903^n-803^n-464^n+261^n$$ is divisible by $1897$.
2007 Bulgarian Autumn Math Competition, Problem 9.2
Let $a$, $b$, $c$ be real numbers, such that $a+b+c=0$ and $a^4+b^4+c^4=50$. Determine the value of $ab+bc+ca$.
2010 Indonesia MO, 7
Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$.
[i]Raja Oktovin, Pekanbaru[/i]
2024 Belarusian National Olympiad, 11.2
$29$ quadratic polynomials $f_1(x), \ldots, f_{29}(x)$ and $15$ real numbers $x_1<x_2<\ldots<x_{15}$ are given. Prove that for some two given polynomials $f_i(x)$ and $f_j(x)$ the following inequality holds: $$\sum_{k=1}^{14} (f_i(x_{k+1})-f_i(x_k))(f_j(x_{k+1})-f_j(x_k))>0$$
[i]A. Voidelevich[/i]
2024 New Zealand MO, 5
A shop sells golf balls, golf clubs and golf hats. Golf balls can be purchased at a rate of $25$ cents for two balls. Golf hats cost $\$1$ each. Golf clubs cost $\$10$ each. At this shop, Ross purchased $100$ items for a total cost of exactly $\$100$ (Ross purchased at least one of each type of item). How many golf hats did Ross purchase?
2007 All-Russian Olympiad, 1
Given reals numbers $a$, $b$, $c$. Prove that at least one of three equations $x^{2}+(a-b)x+(b-c)=0$, $x^{2}+(b-c)x+(c-a)=0$, $x^{2}+(c-a)x+(a-b)=0$ has a real root.
[i]O. Podlipsky[/i]
1989 French Mathematical Olympiad, Problem 4
For natural numbers $x_1,\ldots,x_k$, let $[x_k,\ldots,x_1]$ be defined recurrently as follows: $[x_2,x_1]=x_2^{x_1}$ and, for $k\ge3$, $[x_k,x_{k-1},\ldots,x_1]=x_k^{[x_{k-1},\ldots,x_1]}$.
(a) Let $3\le a_1\le a_2\le\ldots\le a_n$be integers. For a permutation $\sigma$ of the set $\{1,2,\ldots,n\}$, we set $P(\sigma)=[a_{\sigma(n)},a_{\sigma(n-1)},\ldots,a_{\sigma(1)}]$. Find the permutations $\sigma$ for which $P(\sigma)$ is minimal or maximal.
(b) Find all integers $a,b,c,d$, greater than or equal to $2$, for which $[178,9]\le[a,b,c,d]\le[198,9]$.
2011 Serbia National Math Olympiad, 1
Let $n \ge 2$ be integer. Let $a_0$, $a_1$, ... $a_n$ be sequence of positive reals such that:
$(a_{k-1}+a_k)(a_k+a_{k+1})=a_{k-1}-a_{k+1}$, for $k=1, 2, ..., n-1$.
Prove $a_n< \frac{1}{n-1}$.
1996 Romania National Olympiad, 3
Prove that $ \forall x\in \mathbb{R} $ , $ \cos ^7x+\cos ^7(x+\frac {2\pi}{3})+\cos ^7(x+\frac {4\pi}{3})=\frac {63}{64}\cos 3x $
2006 Italy TST, 3
Find all functions $f : \mathbb{Z} \rightarrow \mathbb{Z}$ such that for all integers $m,n$,
\[f(m - n + f(n)) = f(m) + f(n).\]
2017 Indonesia MO, 5
A polynomial $P$ has integral coefficients, and it has at least 9 different integral roots. Let $n$ be an integer such that $|P(n)| < 2017$. Prove that $P(n) = 0$.
2021 Brazil National Olympiad, 4
A set \(A\) of real numbers is framed when it is bounded and, for all \(a, b \in A\), not necessarily distinct, \((a-b)^{2} \in A\). What is the smallest real number that belongs to some framed set?
2023 Indonesia TST, 3
Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.
2020 Iranian Our MO, 6
Find all functions $f:\mathbb{R}^+ \to \mathbb{R}^+$ and plynomials $P(x),Q(x),R(x)$ with positive real coefficients such that $Q(-1)=-1$ and for all positive reals $x,y$:$$f(\frac{x}{y}+R(y))=\frac{f(x)}{Q(y)}+P(y).$$
[i]Proposed by Alireza Danaie, Ali Mirazaie Anari[/i] [b]Rated 2[/b]
1996 May Olympiad, 3
Natalia and Marcela count $1$ by $1$ starting together at $1$, but Marcela's speed is triple that of Natalia (when Natalia says her second number, Marcela says the fourth number). When the difference of the numbers that they say in unison is any of the multiples of $ 29$, between $500$ and $600$, Natalia continues counting normally and Marcela begins to count downwards in such a way that, at one point, the two say in unison the same number. What is said number?
2009 Moldova Team Selection Test, 2
$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.