Olympiad problems

2013 Federal Competition For Advanced Students, Part 2, 4

For a positive integer $n$, let $a_1, a_2, \ldots a_n$ be nonnegative real numbers such that for all real numbers $x_1>x_2>\ldots>x_n>0$ with $x_1+x_2+\ldots+x_n<1$, the inequality $\sum_{k=1}^na_kx_k^3<1$ holds. Show that \[na_1+(n-1)a_2+\ldots+(n-j+1)a_j+\ldots+a_n\leqslant\frac{n^2(n+1)^2}{4}.\]

2021 China Second Round A1, 3

Tags: algebra , inequalities

Let $\{a_n\}$, $\{b_n\}$ be sequences of positive real numbers satisfying $$a_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} b_{n-j}^2}$$ and $$b_n=\sqrt{\frac{1}{100} \sum\limits_{j=1}^{100} a_{n-j}^2}$$ For all $n\ge 101$. Prove that there exists $m\in \mathbb{N}$ such that $|a_m-b_m|<0.001$ [url=https://zhuanlan.zhihu.com/p/417529866] Link [/url]

2003 AIME Problems, 7

Tags: geometry , perimeter , trigonometry , perpendicular bisector , pythagorean theorem , inequalities

Point $B$ is on $\overline{AC}$ with $AB = 9$ and $BC = 21$. Point $D$ is not on $\overline{AC}$ so that $AD = CD$, and $AD$ and $BD$ are integers. Let $s$ be the sum of all possible perimeters of $\triangle ACD$. Find $s$.

1997 China Team Selection Test, 1

Tags: polynomial , inequalities , vieta , induction , algebra

Find all real-coefficient polynomials $f(x)$ which satisfy the following conditions: [b]i.[/b] $f(x) = a_0 x^{2n} + a_2 x^{2n - 2} + \cdots + a_{2n - 2} x^2 + a_{2n}, a_0 > 0$; [b]ii.[/b] $\sum_{j=0}^n a_{2j} a_{2n - 2j} \leq \left( \begin{array}{c} 2n\\ n\end{array} \right) a_0 a_{2n}$; [b]iii.[/b] All the roots of $f(x)$ are imaginary numbers with no real part.

1971 Czech and Slovak Olympiad III A, 1

Tags: algebra , system , inequalities

Let $a,b,c$ real numbers. Show that there are non-negative $x,y,z,xyz\neq0$ such that \begin{align*} cy-bz &\ge 0, \\ az-cx &\ge 0, \\ bx-ay &\ge 0. \end{align*}

1988 Federal Competition For Advanced Students, P2, 1

Tags: inequalities

If $ a_1,...,a_{1988}$ are positive numbers whose arithmetic mean is $ 1988$, show that: $ \sqrt[1988]{\displaystyle\prod_{i,j\equal{}1}^{1988} \left( 1\plus{}\frac{a_i}{a_j} \right)} \ge 2^{1988}$ and determine when equality holds.

2005 Moldova Team Selection Test, 1

Tags: geometry , circumcircle , inradius , inequalities , trigonometry , trig identities , law of sines

Let $ABC$ and $A_{1}B_{1}C_{1}$ be two triangles. Prove that $\frac{a}{a_{1}}+\frac{b}{b_{1}}+\frac{c}{c_{1}}\leq\frac{3R}{2r_{1}}$, where $a = BC$, $b = CA$, $c = AB$ are the sidelengths of triangle $ABC$, where $a_{1}=B_{1}C_{1}$, $b_{1}=C_{1}A_{1}$, $c_{1}=A_{1}B_{1}$ are the sidelengths of triangle $A_{1}B_{1}C_{1}$, where $R$ is the circumradius of triangle $ABC$ and $r_{1}$ is the inradius of triangle $A_{1}B_{1}C_{1}$.

2012 Korea Junior Math Olympiad, 7

Tags: maximum , positive real , sum , algebra , inequalities

If all $x_k$ ($k = 1, 2, 3, 4, 5)$ are positive reals, and $\{a_1,a_2, a_3, a_4, a_5\} = \{1, 2,3 , 4, 5\}$, find the maximum of $$\frac{(\sqrt{s_1x_1} +\sqrt{s_2x_2}+\sqrt{s_3x_3}+\sqrt{s_4x_4}+\sqrt{s_5x_5})^2}{a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 + a_5x_5}$$ ($s_k = a_1 + a_2 +... + a_k$)

2023 Macedonian Balkan MO TST, Problem 1

Tags: inequalities

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers defined by $a_{1}=1$, $a_{2}=2$ and $$\frac{a_{n+1}^{4}}{a_{n}^3} = 2a_{n+2}-a_{n+1}.$$ Prove that the following inequality holds for every positive integer $N>1$: $$\sum_{k=1}^{N}\frac{a_{k}^{2}}{a_{k+1}}<3.$$ [i]Note: The bound is not sharp.[/i] [i]Authored by Nikola Velov[/i]

2009 Irish Math Olympiad, 5

Tags: inequalities

Hello. Suppose $a$, $b$, $c$ are real numbers such that $a+b+c = 0$ and $a^{2}+b^{2}+c^{2} = 1$. Prove that $a^{2}b^{2}c^{2}\leq \frac{1}{54}$ and determine the cases of equality.

2022 VIASM Summer Challenge, Problem 2

Tags: algebra , inequalities

Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$ a) Assume that $k\in S$. Prove that: $k\ge 2$. b) Prove that: $2\in S$.

2018 Romania National Olympiad, 3

Tags: function , inequalities , calculus , real analysis , integral

Let $f:[a,b] \to \mathbb{R}$ be an integrable function and $(a_n) \subset \mathbb{R}$ such that $a_n \to 0.$ $\textbf{a) }$ If $A= \{m \cdot a_n \mid m,n \in \mathbb{N}^* \},$ prove that every open interval of strictly positive real numbers contains elements from $A.$ $\textbf{b) }$ If, for any $n \in \mathbb{N}^*$ and for any $x,y \in [a,b]$ with $|x-y|=a_n,$ the inequality $\left| \int_x^yf(t)dt \right| \leq |x-y|$ is true, prove that $$\left| \int_x^y f(t)dt \right| \leq |x-y|, \: \forall x,y \in [a,b]$$ [i]Nicolae Bourbacut[/i]

2015 Costa Rica - Final Round, 5

Tags: inequalities , algebra

Let $a,b \in R^+$ with $ab = 1$, prove that $$\frac{1}{a^3 + 3b}+\frac{1}{b^3 + 3a}\le \frac12.$$

1998 Poland - First Round, 2

Tags: inequalities

Show that for all real numbers $ a,b,c,d,$ the following inequality holds: \[ (a\plus{}b\plus{}c\plus{}d)^2 \leq 3 (a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} 6ab\]

1973 All Soviet Union Mathematical Olympiad, 187

Tags: inequalities , algebra

Prove that for every positive $x_1, x_2, x_3, x_4, x_5$ holds inequality: $$(x_1 + x_2 + x_3 + x_4 + x_5)^2 \ge 4(x_1x_2 + x_3x_4 + x_5x_1 + x_2x_3 + x_4x_5)$$

2020 Moldova EGMO TST, 1

Tags: inequalities , number theory

Let[i] $a,b,c$[/i] be positive integers , such that $A=\frac{a^2+1}{bc}+\frac{b^2+1}{ca}+\frac{c^2+1}{ab}$ is, also, an integer. Proof that $\gcd( a, b, c)\leq\lfloor\sqrt[3]{a+ b+ c}\rfloor$.

Russian TST 2017, P2

Tags: inequalities , algebra

Let $a_1, a_2,...,a_n$ be positive real numbers, prove that $$\sum {\frac{a_{i+1}}{a_i}} \ge \sum{\sqrt{\frac{a_{i+1}^2+1}{a_i^2+1}}}$$ $a_{n+1}=a_1$

2019 Saudi Arabia Pre-TST + Training Tests, 5.3

Tags: algebra , min , inequalities

Let $x, y, z, a,b, c$ are pairwise different integers from the set $\{1,2,3, 4,5,6\}$. Find the smallest possible value for expression $xyz + abc - ax - by - cz$.

2014 Putnam, 4

Tags: algebra , polynomial , calculus , inequalities

Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.

1996 French Mathematical Olympiad, Problem 4

Tags: inequality , optimization , inequalities

(a) A function $f$ is defined by $f(x)=x^x$ for all $x>0$. Find the minimum value of $f$. (b) If $x$ and $y$ are two positive real numbers, show that $x^y+y^x>1$.

1996 Kurschak Competition, 1

Tags: inequalities , trapezoid , geometry

Prove that in a trapezoid with perpendicular diagonals, the product of the legs is at least as much as the product of the bases.

2014 Contests, 2

Tags: inequalities , algebra

Define a positive number sequence sequence $\{a_n\}$ by \[a_{1}=1,(n^2+1)a^2_{n-1}=(n-1)^2a^2_{n}.\]Prove that\[\frac{1}{a^2_1}+\frac{1}{a^2_2}+\cdots +\frac{1}{a^2_n}\le 1+\sqrt{1-\frac{1}{a^2_n}} .\]

2008 China Team Selection Test, 5

Tags: function , inequalities , cauchy inequality , algebra

For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where \[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]

2016 India Regional Mathematical Olympiad, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers such that $\dfrac{ab}{1+bc}+\dfrac{bc}{1+ca}+\dfrac{ca}{1+ab}=1$. Prove that $$\dfrac{1}{a^3}+\dfrac{1}{b^3}+\dfrac{1}{c^3} \ge 6\sqrt{2}$$

1998 Brazil National Olympiad, 3

Tags: search , ratio , inequalities , combinatorics

Two mathematicians, lost in Berlin, arrived on the corner of Barbarossa street with Martin Luther street and need to arrive on the corner of Meininger street with Martin Luther street. Unfortunately they don't know which direction to go along Martin Luther Street to reach Meininger Street nor how far it is, so they must go fowards and backwards along Martin Luther street until they arrive on the desired corner. What is the smallest value for a positive integer $k$ so that they can be sure that if there are $N$ blocks between Barbarossa street and Meininger street then they can arrive at their destination by walking no more than $kN$ blocks (no matter what $N$ turns out to be)?