Found problems: 6530
2010 Estonia Team Selection Test, 3
Let the angles of a triangle be $\alpha, \beta$, and $\gamma$, the perimeter $2p$ and the radius of the circumcircle $R$. Prove the inequality $\cot^2 \alpha + \cot^2 \beta + \cot^2 \gamma \ge 3 \left(\frac{9R^2}{p^2}-1\right)$. When is the equality achieved?
2010 Contests, 2
Let $x$ be a real number such that $0<x<\frac{\pi}{2}$. Prove that
\[\cos^2(x)\cot (x)+\sin^2(x)\tan (x)\ge 1\]
2011 Korea National Olympiad, 4
Let $ x_1, x_2, \cdots, x_{25} $ real numbers such that $ 0 \le x_i \le i (i=1, 2, \cdots, 25) $. Find the maximum value of
\[x_{1}^{3}+x_{2}^{3}+\cdots +x_{25}^{3} - ( x_1x_2x_3 + x_2x_3x_4 + \cdots x_{25}x_1x_2 ) \]
1987 All Soviet Union Mathematical Olympiad, 462
Prove that for every natural $n$ the following inequality is held: $$(2n + 1)^n \ge (2n)^n + (2n - 1)^n$$
2013 Korea Junior Math Olympiad, 1
Compare the magnitude of the following three numbers.
$$
\sqrt[3]{\frac{25}{3}} ,\, \sqrt[3]{\frac{1148}{135}} ,\, \frac{\sqrt[3]{25}}{3} + \sqrt[3]{\frac{6}{5}}
$$
2023-IMOC, A6
We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\]
Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\] hold for all reals $x,y,z$ satisfying $x+y+z=0$.
[i]Proposed by Untro368.[/i]
2021 Israel TST, 1
Which is greater: \[\frac{1^{-3}-2^{-3}}{1^{-2}-2^{-2}}-\frac{2^{-3}-3^{-3}}{2^{-2}-3^{-2}}+\frac{3^{-3}-4^{-3}}{3^{-2}-4^{-2}}-\cdots +\frac{2019^{-3}-2020^{-3}}{2019^{-2}-2020^{-2}}\]
or \[1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\frac{1}{9}-\cdots +\frac{1}{5781}?\]
2000 District Olympiad (Hunedoara), 1
[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $
[b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality
$$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$
2013 USA TSTST, 2
A finite sequence of integers $a_1, a_2, \dots, a_n$ is called [i]regular[/i] if there exists a real number $x$ satisfying \[ \left\lfloor kx \right\rfloor = a_k \quad \text{for } 1 \le k \le n. \] Given a regular sequence $a_1, a_2, \dots, a_n$, for $1 \le k \le n$ we say that the term $a_k$ is [i]forced[/i] if the following condition is satisfied: the sequence \[ a_1, a_2, \dots, a_{k-1}, b \] is regular if and only if $b = a_k$. Find the maximum possible number of forced terms in a regular sequence with $1000$ terms.
2002 Iran Team Selection Test, 9
$\pi(n)$ is the number of primes that are not bigger than $n$. For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$. Does exist infinitely many integers $n$ that $\pi(n)|n$?
1989 IMO Longlists, 61
Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds:
\[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]
2011 Romania National Olympiad, 2
[color=darkred]Let $u:[a,b]\to\mathbb{R}$ be a continuous function that has finite left-side derivative $u_l^{\prime}(x)$ in any point $x\in (a,b]$ . Prove that the function $u$ is monotonously increasing if and only if $u_l^{\prime}(x)\ge 0$ , for any $x\in (a,b]$ .[/color]
2008 AMC 12/AHSME, 19
A function $ f$ is defined by $ f(z) \equal{} (4 \plus{} i) z^2 \plus{} \alpha z \plus{} \gamma$ for all complex numbers $ z$, where $ \alpha$ and $ \gamma$ are complex numbers and $ i^2 \equal{} \minus{} 1$. Suppose that $ f(1)$ and $ f(i)$ are both real. What is the smallest possible value of $ | \alpha | \plus{} |\gamma |$?
$ \textbf{(A)} \; 1 \qquad \textbf{(B)} \; \sqrt {2} \qquad \textbf{(C)} \; 2 \qquad \textbf{(D)} \; 2 \sqrt {2} \qquad \textbf{(E)} \; 4 \qquad$
2023 South East Mathematical Olympiad, 1
Let $a, b>0$. Prove that:$$ (a^3+b^3+a^3b^3)(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{a^3b^3} ) +27 \ge 6(a+b+\frac{1}{a} +\frac{1}{b} +\frac{a}{b} +\frac{b}{a}) $$
2022 Indonesia TST, A
Given a monic quadratic polynomial $Q(x)$, define \[ Q_n (x) = \underbrace{Q(Q(\cdots(Q(x))\cdots))}_{\text{compose $n$ times}} \]
for every natural number $n$. Let $a_n$ be the minimum value of the polynomial $Q_n(x)$ for every natural number $n$. It is known that $a_n > 0$ for every natural number $n$ and there exists some natural number $k$ such that $a_k \neq a_{k+1}$.
(a) Prove that $a_n < a_{n+1}$ for every natural number $n$.
(b) Is it possible to satisfy $a_n < 2021$ for every natural number $n$?
[i]Proposed by Fajar Yuliawan[/i]
2012 Finnish National High School Mathematics Competition, 4
Let $k,n\in\mathbb{N},0<k<n.$ Prove that \[\sum_{j=1}^k\binom{n}{j}=\binom{n}{1}+ \binom{n}{2}+\ldots + \binom{n}{k}\leq n^k.\]
1995 China National Olympiad, 1
Let $a_1,a_2,\cdots ,a_n; b_1,b_2,\cdots ,b_n (n\ge 3)$ be real numbers satisfying the following conditions:
(1) $a_1+a_2+\cdots +a_n= b_1+b_2+\cdots +b_n $;
(2) $0<a_1=a_2, a_i+a_{i+1}=a_{i+2}$ ($i=1,2,\cdots ,n-2$);
(3) $0<b_1\le b_2, b_i+b_{i+1}\le b_{i+2}$ ($i=1,2,\cdots ,n-2$).
Prove that $a_{n-1}+a_n\le b_{n-1}+b_n$.
2019 Junior Balkan Team Selection Tests - Moldova, 4
Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$
2004 South East Mathematical Olympiad, 1
Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.
MathLinks Contest 1st, 3
Prove that in any acute triangle with sides $a, b, c$ circumscribed in a circle of radius $R$ the following inequality holds:
$$\frac{\sqrt2}{4} <\frac{Rp}{2aR + bc} <\frac{1}{2}$$ where $p$ represents the semi-perimeter of the triangle.
2011 Bosnia Herzegovina Team Selection Test, 2
Let $a, b, c$ be positive reals such that $a+b+c=1$. Prove that the inequality
\[a \sqrt[3]{1+b-c} + b\sqrt[3]{1+c-a} + c\sqrt[3]{1+a-b} \leq 1\]
holds.
2015 IMC, 2
For a positive integer $n$, let $f(n)$ be the number obtained by
writing $n$ in binary and replacing every 0 with 1 and vice
versa. For example, $n=23$ is 10111 in binary, so $f(n)$ is 1000 in
binary, therefore $f(23) =8$. Prove that
\[\sum_{k=1}^n f(k) \leq \frac{n^2}{4}.\]
When does equality hold?
(Proposed by Stephan Wagner, Stellenbosch University)
2014 Benelux, 3
For all integers $n\ge 2$ with the following property:
[list]
[*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]
1999 Israel Grosman Mathematical Olympiad, 2
Find the smallest positive integer $n$ for which $0 <\sqrt[4]{n}- [\sqrt[4]{n}]< 10^{-5}$
.
1997 Irish Math Olympiad, 5
Let $ S$ be the set of odd integers greater than $ 1$. For each $ x \in S$, denote by $ \delta (x)$ the unique integer satisfying the inequality $ 2^{\delta (x)}<x<2^{\delta (x) \plus{}1}$. For $ a,b \in S$, define:
$ a \ast b\equal{}2^{\delta (a)\minus{}1} (b\minus{}3)\plus{}a.$
Prove that if $ a,b,c \in S$, then:
$ (a)$ $ a \ast b \in S$ and
$ (b)$ $ (a \ast b)\ast c\equal{}a \ast (b \ast c)$.