Olympiad problems

2006 Iran Team Selection Test, 5

Tags: circumcircle , trigonometry , inequalities , function , incenter , geometry

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

2005 All-Russian Olympiad Regional Round, 11.6

Tags: geometry , geometric transformation , reflection , vector , trigonometry , inequalities , area of a triangle

11.6 Construct for each vertex of the quadrilateral of area $S$ a symmetric point wrt to the diagonal, which doesn't contain this vertex. Let $S'$ be an area of the obtained quadrilateral. Prove that $\frac{S'}{S}<3$. ([i]L. Emel'yanov[/i])

2020 Jozsef Wildt International Math Competition, W48

Tags: inequalities , geometry , geometric inequalities

Let $ABC$ be a triangle such that $$S^2=2R^2+8Rr+3r^2$$ Then prove that $\frac Rr=2$ or $\frac Rr\ge\sqrt2+1$. [i]Proposed by Marian Cucoanoeş and Marius Drăgan[/i]

2006 Princeton University Math Competition, 9

Tags: algebra , inequalities

Suppose $a,b,c$ are real numbers so that $a+b+c=15$ and $ab+ac+bc=27$. Find the range of values that may be obtained by the expression $abc$.

2003 India Regional Mathematical Olympiad, 3

Tags: inequalities

Let $a,b,c$ be three positive real numbers such that $a + b +c =1$ . prove that among the three numbers $a-ab, b - bc, c-ca$ there is one which is at most $\frac{1}{4}$ and there is one which is at least $\frac{2}{9}$.

1969 Miklós Schweitzer, 4

Tags: inequalities , logarithm , real analysis

Show that the following inequality hold for all $ k \geq 1$, real numbers $ a_1,a_2,...,a_k$, and positive numbers $ x_1,x_2,...,x_k.$ \[ \ln \frac {\sum\limits_{i \equal{} 1}^kx_i}{\sum\limits_{i \equal{} 1}^kx_i^{1 \minus{} a_i}} \leq \frac {\sum\limits_{i \equal{} 1}^ka_ix_i \ln x_i}{\sum\limits_{i \equal{} 1}^kx_i} . \] [i]L. Losonczi[/i]

2020 Jozsef Wildt International Math Competition, W50

Tags: calculus , integration , inequalities

Let $f:[0,1]\to\mathbb R$ be a differentiable function, while $f'$ is continuous on $[0,1]$ and $|f'(x)|\le1$, $(\forall)x\in[0,1]$. If $$2\left|\int^1_0f(x)dx\right|\le1$$ Show that: $$(n+2)\left|\int^1_0x^nf(x)dx\right|\le1,~(\forall)x\ge1$$ [i]Proposed by Florin Stănescu and Şerban Cioculescu[/i]

2007 IberoAmerican Olympiad For University Students, 3

Tags: calculus , integration , inequalities , function , real analysis , integral inequality , logarithm

Let $f:\mathbb{R}\to\mathbb{R}^+$ be a continuous and periodic function. Prove that for all $\alpha\in\mathbb{R}$ the following inequality holds: $\int_0^T\frac{f(x)}{f(x+\alpha)}dx\ge T$, where $T$ is the period of $f(x)$.

2024 Junior Balkan Team Selection Tests - Moldova, 1

Tags: inequalities

Let $a,b,c,x,y,z$ be positive real numbers, such that $a+b+c=xyz=1$ Prove that: $$ \frac{x^2}{3a+2}+\frac{y^2}{3b+2}+\frac{z^2}{3c+2} \ge 1 $$ When does equality hold?

1985 Tournament Of Towns, (085) 1

Tags: trigonometry , geometry , geometric inequality , inequalities

$a, b$ and $c$ are sides of a triangle, and $\gamma$ is its angle opposite $c$. Prove that $c \ge (a + b) \sin \frac{\gamma}{2}$ (V. Prasolov )

2000 Baltic Way, 12

Tags: inequalities , algebra

Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that \[\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. \]

Kvant 2024, M2797

Tags: inequality , algebra , inequalities

For real numbers $0 \leq a_1 \leq a_2 \leq ... \leq a_n$ and $0 \leq b_1 \leq b_2 \leq ... \leq b_n$ prove that \[ \left( \frac{a_1}{1 \cdot 2}+\frac{a_2}{2 \cdot 3}+...+\frac{a_n}{n(n+1)} \right) \times \left( \frac{b_1}{1 \cdot 2}+\frac{b_2}{2 \cdot 3}+...+\frac{b_n}{n(n+1)} \right) \leq \frac{a_1b_1}{1 \cdot 2}+\frac{a_2b_2}{2 \cdot 3}+...+\frac{a_nb_n}{n(n+1)}.\] [i]Proposed by A. Antropov[/i]

2011 Uzbekistan National Olympiad, 2

Tags: floor function , limit , inequalities , algebra

Prove that $ \forall n\in\mathbb{N}$,$ \exists a,b,c\in$$\bigcup_{k\in\mathbb{N}}(k^{2},k^{2}+k+3\sqrt 3) $ such that $n=\frac{ab}{c}$.

1974 Dutch Mathematical Olympiad, 3

Tags: algebra , number theory , inequalities

Proove that in every five positive numbers there is a pair, say $a,b$, for which $$\left| \frac{1}{a+25}- \frac{1}{b+25}\right| <\frac{1}{100}.$$

2013 Macedonia National Olympiad, 4

Tags: inequalities

Let $ a,b,c $ be positive real numbers such that $ a^4+b^4+c^4=3 $. Prove that \[ \frac{9}{a^2+b^4+c^6}+\frac{9}{a^4+b^6+c^2}+\frac{9}{a^6+b^2+c^4}\leq\ a^6+b^6+c^6+6 \]

2023 IMO, 4

Tags: inequalities

Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that \[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\] is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$

2007 Hanoi Open Mathematics Competitions, 10

Tags: minimum , inequalities , algebra

What is the smallest possible value of $x^2+2y^2-x-2y-xy$?

2023 Brazil Team Selection Test, 3

Tags: inequalities , inequality , arithmetic mean , geometric mean , max

Show that for all positive real numbers $a, b, c$, we have that $$\frac{a+b+c}{3}-\sqrt[3]{abc} \leq \max\{(\sqrt{a}-\sqrt{b})^2, (\sqrt{b}-\sqrt{c})^2, (\sqrt{c}-\sqrt{a})^2\}$$

2017 Kyrgyzstan Regional Olympiad, 3

Tags: inequalities

If $ {|x|}<{1}$ and ${|y|}<1$ then prove that $|\frac{x-y}{1-xy}|<1$

1999 IMO Shortlist, 1

Tags: inequalities , algebra , maximization

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

2018 IFYM, Sozopol, 8

Tags: algebra , inequality , inequalities

Prove that for every positive integer $n \geq 2$ the following inequality holds: $e^{n-1}n!<n^{n+\frac{1}{2}}$

2011-2012 SDML (High School), 7

Tags: inequalities

Let $x$ and $y$ be nonnegative real numbers such that $x+y=1$. Find the maximum value of $x^4y+xy^4$.

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: inequalities , sum , algebra , absolute value

Let $a_1, a_2, ..., a_n$ real numbers such that $a_1 + a_2 + ... + a_n = 0$ and $| a_1 | + | a_2 | + ... + | a_n | = 1$. Show that: $| a _ 1 + 2 a _ 2 + ... + n a _ n | \le \frac {n-1} {2}$.

2018 Vietnam National Olympiad, 4

Tags: analytic geometry , inequalities

On the Cartesian plane the curve $(C)$ has equation $x^2=y^3$. A line $d$ varies on the plane such that $d$ always cut $(C)$ at three distinct points with $x$-coordinates $x_1,\, x_2,\, x_3$. a. Prove that the following quantity is a constant: $$\sqrt[3]{\frac{x_1x_2}{x_3^2}}+\sqrt[3]{\frac{x_2x_3}{x_1^2}}+\sqrt[3]{\frac{x_3x_1}{x_2^2}}.$$ b. Prove the following inequality: $$\sqrt[3]{\frac{x_1^2}{x_2x_3}}+\sqrt[3]{\frac{x_2^2}{x_3x_1}}+\sqrt[3]{\frac{x_3^2}{x_3x_1}}<-\frac{15}{4}.$$

2011 ELMO Shortlist, 5

Tags: geometry , inequalities

Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. [i]Victor Wang.[/i]