Olympiad problems

2010 Sharygin Geometry Olympiad, 3

Tags: geometry , convex polygon , Inequality , geometric inequality , diagonals , diagonal

All sides of a convex polygon were decreased in such a way that they formed a new convex polygon. Is it possible that all diagonals were increased?

2001 All-Russian Olympiad, 2

Tags: algebra , polynomial , function , Inequality

The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.

2021 Indonesia TST, A

Tags: Inequality , inequalities , algebra

A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

2021 Taiwan TST Round 1, A

Tags: algebra , Inequality , IMO Shortlist , IMO Shortlist 2020 , 4-variable inequality , inequalities

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

2018 ITAMO, 3

Tags: Inequality

Let $x_1,x_2, ... , x_n$ be positive integers,Asumme that in their decimal representations no $x_i$ "prolongs" $x_j$.For instance , $123$ prolongs $12$ , $459$ prolongs $4$ , but $124$ does not prolog $123$. Prove that : $\frac {1}{x_1}+\frac {1}{x_2}+...+\frac {1}{x_n} < 3$.

2017 Azerbaijan BMO TST, 1

Tags: algebra , Inequality , BMO Shortlist

Let $a, b,c$ be positive real numbers. Prove that $ \sqrt{a^3b+a^3c}+\sqrt{b^3c+b^3a}+\sqrt{c^3a+c^3b}\ge \frac43 (ab+bc+ca)$

2021 Winter Stars of Mathematics, 1

Tags: algebra , Inequality , romania

Let $a_1,a_2,a_3,a_4$ be positive real numbers satisfying \[\sum_{i<j}a_ia_j=1.\]Prove that \[\sum_{\text{sym}}\frac{a_1a_2}{1+a_3a_4}\geq\frac{6}{7}.\][i]* * *[/i]

2021 Austrian MO National Competition, 1

Tags: algebra , AZE CMO TST , AZE EGMO TST , Inequality , TST

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

2015 Saudi Arabia JBMO TST, 1

Tags: 3-variable inequality , inequalities , three variable inequality , algebraic inequality , Inequality , inequalities proposed

Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

2021 Serbia JBMO TSTs, 1

Tags: algebra , Inequality , TST

Prove that for positive real numbers $a, b, c$ the following inequality holds: \begin{align*} \frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2} \end{align*} When does equality occur?

2015 Indonesia MO Shortlist, A3

Tags: Inequality , inequalities

Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$

KoMaL A Problems 2022/2023, A. 852

Tags: Inequality , algebra , inequalities , komal

Let $(a_i,b_i)$ be pairwise distinct pairs of positive integers for $1\le i\le n$. Prove that \[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)>\frac29 n^3,\] and show that the statement is sharp, i.e. for an arbitrary $c>\frac29$ it is possible that \[(a_1+a_2+\ldots+a_n)(b_1+b_2+\ldots+b_n)<cn^3.\] [i]Submitted by Péter Pál Pach, Budapest, based on an OKTV problem[/i]

2022 Thailand Mathematical Olympiad, 2

Tags: Inequality , algebra , inequalities , number theory

Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that $$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} \\ -1 &\text{if }m,n\text{ have different parity}\end{cases}$$ for every positive integers $m,n$. Determine the minimum possible value of $$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$ across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$.

2006 Singapore Team Selection Test, 2

Tags: Inequality , rearrangement inequality

Let n be an integer greater than 1 and let $x_1, x_2, . . . , x_n$ be real numbers such that $|x_1| + |x_2| + ... + |x_n| = 1$ and $x_1 + x_2 + ... + x_n = 0$ Prove that $\left| \frac{x_1}{1}+\frac{x_2}{2}+\cdots+\frac{x_n}{n} \right| \leq \frac{1}{2} \left(1-\frac{1}{n}\right)$

2018 Mexico National Olympiad, 4

Tags: algebra , Inequality , maximum value , positive integers

Let $n\geq 2$ be an integer. For each $k$-tuple of positive integers $a_1, a_2, \ldots, a_k$ such that $a_1+a_2+\cdots +a_k=n$, consider the sums $S_i=1+2+\ldots +a_i$ for $1\leq i\leq k$. Determine, in terms of $n$, the maximum possible value of the product $S_1S_2\cdots S_k$. [i]Proposed by Misael Pelayo[/i]

1977 IMO Shortlist, 7

Tags: function , trigonometry , Trigonometric inequality , Inequality , IMO , IMO 1977

Let $a,b,A,B$ be given reals. We consider the function defined by \[ f(x) = 1 - a \cdot \cos(x) - b \cdot \sin(x) - A \cdot \cos(2x) - B \cdot \sin(2x). \] Prove that if for any real number $x$ we have $f(x) \geq 0$ then $a^2 + b^2 \leq 2$ and $A^2 + B^2 \leq 1.$

2021 Science ON grade VII, 3

Tags: algebra , Inequality , symmetric sums , Newton Sums

Are there any real numbers $a,b,c$ such that $a+b+c=6$, $ab+bc+ca=9$ and $a^4+b^4+c^4=260$? What about if we let $a^4+b^4+c^4=210$? [i] (Andrei Bâra)[/i]

1969 IMO Longlists, 14

Tags: quadratics , algebra , Inequality , roots , polynomial , IMO Longlist , IMO Shortlist

$(CZS 3)$ Let $a$ and $b$ be two positive real numbers. If $x$ is a real solution of the equation $x^2 + px + q = 0$ with real coefficients $p$ and $q$ such that $|p| \le a, |q| \le b,$ prove that $|x| \le \frac{1}{2}(a +\sqrt{a^2 + 4b})$ Conversely, if $x$ satisfies the above inequality, prove that there exist real numbers $p$ and $q$ with $|p|\le a, |q|\le b$ such that $x$ is one of the roots of the equation $x^2+px+ q = 0.$