Found problems: 6530
2002 Moldova National Olympiad, 2
Let $ a,b,c\geq 0$ such that $ a\plus{}b\plus{}c\equal{}1$. Prove that:
$ a^2\plus{}b^2\plus{}c^2\geq 4(ab\plus{}bc\plus{}ca)\minus{}1$
1989 Polish MO Finals, 3
Show that for positive reals $a, b, c, d$ we have
\[ \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2 \]
2005 Uzbekistan National Olympiad, 1
Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$.
Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$
1983 Vietnam National Olympiad, 2
$(a)$ Prove that $\sqrt{2}(\sin t + \cos t) \ge 2\sqrt[4]{\sin 2t}$ for $0 \le t \le\frac{\pi}{2}.$
$(b)$ Find all $y, 0 < y < \pi$, such that $1 +\frac{2 \cot 2y}{\cot y} \ge \frac{\tan 2y}{\tan y}$.
.
2007 Romania Team Selection Test, 4
The points $M, N, P$ are chosen on the sides $BC, CA, AB$ of a triangle $\Delta ABC$, such that the triangle $\Delta MNP$ is acute-angled. We denote with $x$ the length of the shortest altitude of the triangle $\Delta ABC$, and with $X$ the length of the longest altitudes of the triangle $\Delta MNP$. Prove that $x \leq 2X$.
2023 Korea - Final Round, 6
For positive integer $n\geq 3$ and real numbers $a_1,...,a_n,b_1,...,b_n$, prove the following.
$$\sum_{i=1}^n a_i(b_i-b_{i+3})\leq\frac{3n}{8}\sum_{i=1}^n((a_i-a_{i+1})^2+(b_i-b_{i+1})^2)$$
($a_{n+1}=a_1$, and for $i=1,2,3$ $b_{n+i}=b_i$.)
2022 China Girls Math Olympiad, 8
Let $x_1, x_2, \ldots, x_{11}$ be nonnegative reals such that their sum is $1$. For $i = 1,2, \ldots, 11$, let
\[ y_i = \begin{cases} x_{i} + x_{i + 1}, & i \, \, \textup{odd} ,\\ x_{i} + x_{i + 1} + x_{i + 2}, & i \, \, \textup{even} ,\end{cases} \]
where $x_{12} = x_{1}$. And let $F (x_1, x_2, \ldots, x_{11}) = y_1 y_2 \ldots y_{11}$.
Prove that $x_6 < x_8$ when $F$ achieves its maximum.
2011 Hanoi Open Mathematics Competitions, 4
Prove that $1 + x + x^2 + x^3 + ...+ x^{2011} \ge 0$ for every $x \ge - 1$ .
2012 Turkmenistan National Math Olympiad, 1
Find the max and min value of $a\cos^2 x+b\sin x\cos x+c\sin^2 x$.
Mathematical Minds 2024, P4
Let $a$, $b$, $c$ be positive real numbers such that $a+b+c=3$. Prove that $$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+a^3}{2}}\leqslant a^2+b^2+c^2.$$
[i]Proposed by Andrei Vila[/i]
2010 Today's Calculation Of Integral, 644
For a constant $p$ such that $\int_1^p e^xdx=1$, prove that
\[\left(\int_1^p e^x\cos x\ dx\right)^2+\left(\int_1^p e^x\sin x\ dx\right)^2>\frac 12.\]
Own
2025 China National Olympiad, 6
Let $a_1, a_2, \ldots, a_n$ be real numbers such that $\sum_{i=1}^n a_i = n$, $\sum_{i = 1}^n a_i^2 = 2n$, $\sum_{i=1}^n a_i^3 = 3n$.
(i) Find the largest constant $C$, such that for all $n \geqslant 4$, \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C. \]
(ii) Prove that there exists a positive constant $C_2$, such that \[ \max \left\{ a_1, a_2, \ldots, a_n \right\} - \min \left\{ a_1, a_2, \ldots, a_n \right\} \geqslant C + C_2 n^{-\frac 32}, \]where $C$ is the constant determined in (i).
2017 Balkan MO Shortlist, A4
Let $M = \{(a,b,c)\in R^3 :0 <a,b,c<\frac12$ with $a+b+c=1 \}$ and $f: M\to R$ given as $$f(a,b,c)=4\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)-\frac{1}{abc}$$
Find the best (real) bounds $\alpha$ and $\beta$ such that $f(M) = \{f(a,b,c): (a,b,c)\in M\}\subseteq [\alpha,\beta]$ and determine whether any of them is achievable.
2018 Tuymaada Olympiad, 7
Prove the inequality $$(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)$$ for $x, y, z \geq 1$.
[i]Proposed by K. Kokhas[/i]
2018 Czech-Polish-Slovak Junior Match, 3
Calculate all real numbers $r $ with the following properties:
If real numbers $a, b, c$ satisfy the inequality$ | ax^2 + bx + c | \le 1$ for each $x \in [ - 1, 1]$, then they also satisfy the inequality $| cx^2 + bx + a | \le r$ for each $ x \in [- 1, 1]$.
2014 Contests, 1
Numbers $1$ through $2014$ are written on a board. A valid operation is to erase two numbers $a$ and $b$ on the board and replace them with the greatest common divisor and the least common multiple of $a$ and $b$.
Prove that, no matter how many operations are made, the sum of all the numbers that remain on the board is always larger than $2014$ $\times$ $\sqrt[2014]{2014!}$
2024 Korea Junior Math Olympiad, 7
Let $A_k$ be the number of pairs $(a_1, a_2, ..., a_{2k})$ for $k\leq 50$, where $a_1, a_2, ..., a_{2k}$ are all different positive integers that satisfy the following.
[b]$\cdot$[/b] $a_1, a_2, ..., a_{2k} \leq 100$
[b]$\cdot$[/b] For an odd number less or equal than $2k-1$, we have $a_i > a_{i+1}$
[b]$\cdot$[/b] For an even number less or equal than $2k-2$, we have $a_i < a_{i+1}$
Prove that $A_1 \leq A_2 \leq \cdots \leq A_{49}$.
2017 Saudi Arabia JBMO TST, 1
Let $a,b,c>0$ and $a^2+b^2+c^2=3$ . Prove that $$ \frac{a(a-b^2)}{a+b^2}+\frac{b(b-c^2)}{b+c^2}+\frac{c(c-a^2)}{c+a^2}\ge 0.$$
2012 China Team Selection Test, 3
Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial
\[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\]
with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality
\[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]
MathLinks Contest 2nd, 3.1
Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$
1986 AMC 8, 14
If $ 200 \le a \le 400$ and $ 600 \le b \le 1200$, then the largest value of the quotient $ \frac{b}{a}$ is
\[ \textbf{(A)}\ \frac{3}{2} \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 300 \qquad
\textbf{(E)}\ 600 \qquad
\]
2023 India IMO Training Camp, 1
Let $\mathbb{Z}_{\ge 0}$ be the set of non-negative integers and $\mathbb{R}^+$ be the set of positive real numbers. Let $f: \mathbb{Z}_{\ge 0}^2 \rightarrow \mathbb{R}^+$ be a function such that $f(0, k) = 2^k$ and $f(k, 0) = 1$ for all integers $k \ge 0$, and $$f(m, n) = \frac{2f(m-1, n) \cdot f(m, n-1)}{f(m-1, n)+f(m, n-1)}$$ for all integers $m, n \ge 1$. Prove that $f(99, 99)<1.99$.
[i]Proposed by Navilarekallu Tejaswi[/i]
2004 IMC, 2
Let $f,g:[a,b]\to [0,\infty)$ be two continuous and non-decreasing functions such that each $x\in [a,b]$ we have
\[ \int^x_a \sqrt { f(t) }\ dt \leq \int^x_a \sqrt { g(t) }\ dt \ \ \textrm{and}\ \int^b_a \sqrt {f(t)}\ dt = \int^b_a \sqrt { g(t)}\ dt. \]
Prove that
\[ \int^b_a \sqrt { 1+ f(t) }\ dt \geq \int^b_a \sqrt { 1 + g(t) }\ dt. \]
1993 Vietnam Team Selection Test, 3
Let's consider the real numbers $x_1, x_2, x_3, x_4$ satisfying the condition
\[ \dfrac{1}{2}\le x_1^2+x_2^2+x_3^2+x_4^2\le 1 \]
Find the maximal and the minimal values of expression:
\[ A = (x_1 - 2 \cdot x_2 + x_3)^2 + (x_2 - 2 \cdot x_3 + x_4)^2 + (x_2 - 2 \cdot x_1)^2 + (x_3 - 2 \cdot x_4)^2 \]
1987 Traian Lălescu, 2.1
Let $ \lambda \in (0,2) $ and $ a,b,c,d\in\mathbb{R} $ so that $ a\le b\le c. $ Prove the inequality:
$$ (a+b+c+d)^2\ge 4\lambda (ac+bd). $$