Found problems: 6530
2019 Kosovo National Mathematical Olympiad, 2
Show that for any positive real numbers $a,b,c$ the following inequality is true:
$$4(a^3+b^3+c^3+3)\geq 3(a+1)(b+1)(c+1)$$
When does equality hold?
2010 IMO Shortlist, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2025 Turkey EGMO TST, 2
Does there exist a sequence of positive real numbers $\{a_i\}_{i=1}^{\infty}$ satisfying:
\[
\sum_{i=1}^{n} a_i \geq n^2 \quad \text{and} \quad \sum_{i=1}^{n} a_i^2 \leq n^3 + 2025n
\]
for all positive integers $n$.
2005 Taiwan National Olympiad, 2
In triangle $ABC$, $D$ is the midpoint of side $AB$. $E$ and $F$ are points arbitrarily chosen on segments $AC$ and $BC$, respectively. Show that $[DEF] < [ADE] + [BDF]$.
2007 Grigore Moisil Intercounty, 4
Solve in the set of real numbers the fractional part inequality $ \{ x \}\le\{ nx \} , $ where $ n $ is a fixed natural number.
2019 Hong Kong TST, 6
If $57a + 88b + 125c \geq 1148$, where $a,b,c > 0$, what is the minimum value of
\[ a^3 + b^3 + c^3 + 5a^2 + 5b^2 + 5c^2? \]
2016 Junior Balkan Team Selection Test, 4
Let $a,b,c\in \mathbb{R}^+$, prove that: $$\frac{2a}{\sqrt{3a+b}}+\frac{2b}{\sqrt{3b+c}}+\frac{2c}{\sqrt{3c+a}}\leq \sqrt{3(a+b+c)}$$
2002 All-Russian Olympiad Regional Round, 10.4
(10.4) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $0 \le a_{k+1}- a_k \le 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2$$
(11.3) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $a_{k+1} \ge a_k + 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2$$
2005 All-Russian Olympiad Regional Round, 8.8
8.8, 9.8, 11.8
a) 99 boxes contain apples and oranges. Prove that we can choose 50 boxes in such a way that they contain at least half of all apples and half of all oranges.
b) 100 boxes contain apples and oranges. Prove that we can choose 34 boxes in such a way that they contain at least a third of all apples and a third of all oranges.
c) 100 boxes contain apples, oranges and bananas. Prove that we can choose 51 boxes in such a way that they contain at least half of all apples, and half of all oranges and half of all bananas.
([i]I. Bogdanov, G. Chelnokov, E. Kulikov[/i])
2018 Moldova Team Selection Test, 10
The positive real numbers $a,b, c,d$ satisfy the equality $ \frac {1}{a+1} + \frac {1}{b+1} + \frac {1}{c+1} + \frac{ 1}{d+1} = 3 $ . Prove the inequality $\sqrt [3]{abc} + \sqrt [3]{bcd} + \sqrt [3]{cda} + \sqrt [3]{dab} \le \frac {4}{3} $.
2009 International Zhautykov Olympiad, 3
For a convex hexagon $ ABCDEF$ with an area $ S$, prove that:
\[ AC\cdot(BD\plus{}BF\minus{}DF)\plus{}CE\cdot(BD\plus{}DF\minus{}BF)\plus{}AE\cdot(BF\plus{}DF\minus{}BD)\geq 2\sqrt{3}S
\]
2010 China National Olympiad, 1
Let $m,n\ge 1$ and $a_1 < a_2 < \ldots < a_n$ be integers. Prove that there exists a subset $T$ of $\mathbb{N}$ such that
\[|T| \leq 1+ \frac{a_n-a_1}{2n+1}\]
and for every $i \in \{1,2,\ldots , m\}$, there exists $t \in T$ and $s \in [-n,n]$, such that $a_i=t+s$.
2019 Taiwan TST Round 2, 1
Prove that for any positive reals $ a,b,c,d $ with $ a+b+c+d = 4 $, we have $$ \sum\limits_{cyc}{\frac{3a^3}{a^2+ab+b^2}}+\sum\limits_{cyc}{\frac{2ab}{a+b}} \ge 8 $$
2011 Belarus Team Selection Test, 3
Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$
[i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]
2018 Math Prize for Girls Olympiad, 4
For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that
\[
\sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} )
\le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, .
\]
1963 Putnam, B5
Let $(a_n )$ be a sequence of real numbers satisfying the inequalities
$$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$
and such that the series
$$\sum_{n=0}^{\infty} a_n $$
converges. Prove that
$$\lim_{n\to \infty} n a_n = 0.$$
2010 Federal Competition For Advanced Students, Part 1, 2
For a positive integer $n$, we define the function $f_n(x)=\sum_{k=1}^n |x-k|$ for all real numbers $x$. For any two-digit number $n$ (in decimal representation), determine the set of solutions $\mathbb{L}_n$ of the inequality $f_n(x)<41$.
[i](41st Austrian Mathematical Olympiad, National Competition, part 1, Problem 2)[/i]
2005 Federal Math Competition of S&M, Problem 3
If $x,y,z$ are nonnegative numbers with $x+y+z=3$, prove that
$$\sqrt x+\sqrt y+\sqrt z\ge xy+yz+xz.$$
2019 China Team Selection Test, 3
Let $n$ be a given even number, $a_1,a_2,\cdots,a_n$ be non-negative real numbers such that $a_1+a_2+\cdots+a_n=1.$ Find the maximum possible value of $\sum_{1\le i<j\le n}\min\{(i-j)^2,(n+i-j)^2\}a_ia_j .$
KoMaL A Problems 2022/2023, A. 854
Prove that
\[\sum_{k=0}^n\frac{2^{2^k}\cdot 2^{k+1}}{2^{2^k}+3^{2^k}}<4\]
holds for all positive integers $n$.
[i]Submitted by Béla Kovács, Szatmárnémeti[/i]
2011 JBMO Shortlist, 6
Let $\displaystyle {x_i> 1, \forall i \in \left \{1, 2, 3, \ldots, 2011 \right \}}$. Show that:$$\displaystyle{\frac{x^2_1}{x_2-1}+\frac{x^2_2}{x_3-1}+\frac{x^2_3}{x_4-1}+\ldots+\frac{x^2_{2010}}{x_{2011}-1}+\frac{x^2_{2011}}{x_1-1}\geq 8044}$$
When the equality holds?
2012 Belarus Team Selection Test, 3
Given a polynomial $P(x)$ with positive real coefficients.
Prove that $P(1)P(xy) \ge P(x)P(y)$ for all $x\ge1, y \ge 1$.
(K. Gorodnin)
1966 IMO Longlists, 30
Let $n$ be a positive integer, prove that :
[b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$
[b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$
2014 Abels Math Contest (Norwegian MO) Final, 1a
Assume that $x, y \ge 0$. Show that $x^2 + y^2 + 1 \le \sqrt{(x^3 + y + 1)(y^3 + x + 1)}$.
1980 USAMO, 5
Prove that for numbers $a,b,c$ in the interval $[0,1]$, \[\frac{a}{b+c+1}+\frac{b}{c+a+1}+\frac{c}{a+b+1}+(1-a)(1-b)(1-c) \le 1.\]