Found problems: 6530
2002 USA Team Selection Test, 1
Let $ ABC$ be a triangle, and $ A$, $ B$, $ C$ its angles. Prove that
\[ \sin\frac{3A}{2}+\sin\frac{3B}{2}+\sin\frac{3C}{2}\leq \cos\frac{A-B}{2}+\cos\frac{B-C}{2}+\cos\frac{C-A}{2}. \]
2001 Romania Team Selection Test, 3
The sides of a triangle have lengths $a,b,c$. Prove that:
\begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
2014 Hanoi Open Mathematics Competitions, 13
Let $a,b, c > 0$ and $abc = 1$. Prove that $\frac{a - 1}{c}+\frac{c - 1}{b}+\frac{b - 1}{a} \ge 0$
1986 Tournament Of Towns, (118) 6
Given the nonincreasing sequence of non-negative numbers in which $a_1 \ge a_2 \ge a_3 \ge ... \ge a_{2n-1}\ge 0$.
Prove that $a^2_1 -a^2_2 + a^2_3 - ... + a^2_{2n- l} \ge (a_1 - a_2 + a_3 - ... + a_{2n- l} )^2$ .
( L . Kurlyandchik , Leningrad )
1992 IMO Longlists, 70
Let two circles $A$ and $B$ with unequal radii $r$ and $R$, respectively, be tangent internally at the point $A_0$. If there exists a sequence of distinct circles $(C_n)$ such that each circle is tangent to both $A$ and $B$, and each circle $C_{n+1}$ touches circle $C_{n}$ at the point $A_n$, prove that
\[\sum_{n=1}^{\infty} |A_{n+1}A_n| < \frac{4 \pi Rr}{R+r}.\]
1991 China Team Selection Test, 1
Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have
\[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]
1987 Romania Team Selection Test, 5
Let $A$ be the set $\{1,2,\ldots,n\}$, $n\geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ with the property: \[ \sum_{i=1}^n \alpha(i) \beta (i) = \dfrac {19}{10} \sum^n_{i=1} \gamma(i)\delta(i) . \]
[i]Marcel Chirita[/i]
2024 Germany Team Selection Test, 3
Let $a_1,a_2,\dots,a_{2023}$ be positive integers such that
[list=disc]
[*] $a_1,a_2,\dots,a_{2023}$ is a permutation of $1,2,\dots,2023$, and
[*] $|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}|$ is a permutation of $1,2,\dots,2022$.
[/list]
Prove that $\max(a_1,a_{2023})\ge 507$.
2019 Switzerland Team Selection Test, 11
Let $n $ be a positive integer. Determine whether there exists a positive real number $\epsilon >0$ (depending on $n$) such that for all positive real numbers $x_1,x_2,\dots ,x_n$, the inequality $$\sqrt[n]{x_1x_2\dots x_n}\leq (1-\epsilon)\frac{x_1+x_2+\dots+x_n}{n}+\epsilon \frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+\dots +\frac{1}{x_n}},$$ holds.
2024 ITAMO, 6
For each integer $n$, determine the smallest real number $M_n$ such that
\[\frac{1}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{n-1}}{a_n} \le M_n\]
for any $n$-tuple $(a_1,a_2,\dots,a_n)$ of integers such that $1<a_1<a_2<\dots<a_n$.
2001 IMO, 2
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]
1995 Nordic, 3
Let $n \ge 2$ and let $x_1, x_2, ..., x_n$ be real numbers satisfying $x_1 +x_2 +...+x_n \ge 0$ and $x_1^2+x_2^2+...+x_n^2=1$. Let $M = max \{x_1, x_2,... , x_n\}$. Show that $M \ge \frac{1}{\sqrt{n(n-1)}}$ (1) .When does equality hold in (1)?
2013 Argentina National Olympiad, 4
Let $x\geq 5, y\geq 6, z\geq 7$ such that $x^2+y^2+z^2\geq 125$. Find the minimum value of $x+y+z$.
2009 China Team Selection Test, 2
Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$
2005 National Olympiad First Round, 11
For the real pairs $(x,y)$ satisfying the equation $x^2 + y^2 + 2x - 6y = 6$, which of the following cannot be equal to $(x-1)^2 + (y-2)^2$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 16
\qquad\textbf{(D)}\ 23
\qquad\textbf{(E)}\ 30
$
1992 Tournament Of Towns, (326) 3
Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater:
$$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$
Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression.
(N. Kurlandchik)
2003 India IMO Training Camp, 9
Let $n$ be a positive integer and $\{A,B,C\}$ a partition of $\{1,2,\ldots,3n\}$ such that $|A|=|B|=|C|=n$. Prove that there exist $x \in A$, $y \in B$, $z \in C$ such that one of $x,y,z$ is the sum of the other two.
2006 Iran Team Selection Test, 4
Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that
\[ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| \]
2024 New Zealand MO, 2
Prove the following inequality $$\dfrac{6}{2024^3} < \left(1-\dfrac{3}{4}\right)\left(1-\dfrac{3}{5}\right)\left(1-\dfrac{3}{6}\right)\left(1-\dfrac{3}{7}\right)\ldots\left(1-\dfrac{3}{2025}\right).$$
2014 Brazil Team Selection Test, 4
Let $n$ be a positive integer, and consider a sequence $a_1 , a_2 , \dotsc , a_n $ of positive integers. Extend it periodically to an infinite sequence $a_1 , a_2 , \dotsc $ by defining $a_{n+i} = a_i $ for all $i \ge 1$. If \[a_1 \le a_2 \le \dots \le a_n \le a_1 +n \] and \[a_{a_i } \le n+i-1 \quad\text{for}\quad i=1,2,\dotsc, n, \] prove that \[a_1 + \dots +a_n \le n^2. \]
2023 ITAMO, 5
Let $a, b, c$ be reals satisfying $a^2+b^2+c^2=6$. Find the maximal values of the expressions
a) $(a-b)^2+(b-c)^2+(c-a)^2$;
b) $(a-b)^2 \cdot (b-c)^2 \cdot (c-a)^2$.
In both cases, describe all triples for which equality holds.
Russian TST 2014, P3
Find the maximum value of real number $k$ such that
\[\frac{a}{1+9bc+k(b-c)^2}+\frac{b}{1+9ca+k(c-a)^2}+\frac{c}{1+9ab+k(a-b)^2}\geq \frac{1}{2}\]
holds for all non-negative real numbers $a,\ b,\ c$ satisfying $a+b+c=1$.
2019 Centers of Excellency of Suceava, 1
Prove that $ \binom{m+n}{\min (m,n)}\le \sqrt{\binom{2m}{m}\cdot \binom{2n}{n}} , $ for nonnegative $ m,n. $
[i]Gheorghe Stoica[/i]
2016 India Regional Mathematical Olympiad, 4
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
2016 Mexico National Olmypiad, 3
Find the minimum real $x$ that satisfies
$$\lfloor x \rfloor <\lfloor x^2 \rfloor <\lfloor x^3 \rfloor < \cdots < \lfloor x^n \rfloor < \lfloor x^{n+1} \rfloor < \cdots$$