Olympiad problems

1975 IMO Shortlist, 12

Consider on the first quadrant of the trigonometric circle the arcs $AM_1 = x_1,AM_2 = x_2,AM_3 = x_3, \ldots , AM_v = x_v$ , such that $x_1 < x_2 < x_3 < \cdots < x_v$. Prove that \[\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})\]

2001 Korea Junior Math Olympiad, 6

Tags: algebra , inequality

For real variables $0 \leq x, y, z, w \leq 1$, find the maximum value of $$x(1-y)+2y(1-z)+3z(1-w)+4w(1-x)$$

2006 Petru Moroșan-Trident, 2

Tags: induction , inequalities , inequality

Let be an increasing, infinite sequence of natural numbers $ \left( a_n \right)_{n\ge 1} . $ [b]a)[/b] Prove that if $ a_n=n, $ for any natural numbers $ n, $ then $$ -2+2\sqrt{1+n} <\frac{1}{\sqrt{a_1}} +\frac{1}{\sqrt{a_2}} +\cdots +\frac{1}{\sqrt{a_n}} <2\sqrt n , $$ for any natural numbers $ n. $ [b]b)[/b] Disprove the converse of [b]a).[/b] [i]Vasile Radu[/i]

2012 Israel National Olympiad, 3

Tags: algebra , inequality , symmetric inequality , inequalities

Let $a,b,c$ be real numbers such that $a^3(b+c)+b^3(a+c)+c^3(a+b)=0$. Prove that $ab+bc+ca\leq0$.

1983 IMO Longlists, 60

Tags: algebra , summation , series summation , approximation , inequality

Find the greatest integer less than or equal to $\sum_{k=1}^{2^{1983}} k^{\frac{1}{1983} -1}.$

1992 IMO Longlists, 49

Tags: algebra , recurrence relation , inequality , sequence

Given real numbers $x_i \ (i = 1, 2, \cdots, 4k + 2)$ such that \[\sum_{i=1}^{4k +2} (-1)^{i+1} x_ix_{i+1} = 4m \qquad ( \ x_1=x_{4k+3} \ )\] prove that it is possible to choose numbers $x_{k_{1}}, \cdots, x_{k_{6}}$ such that \[\sum_{i=1}^{6} (-1)^{i} k_i k_{i+1} > m \qquad ( \ x_{k_{1}} = x_{k_{7}} \ )\]

1976 Vietnam National Olympiad, 6

Tags: inequality , algebra , inequalities

Show that $\frac{1}{x_1^n} + \frac{1}{x_2^n} +...+ \frac{1}{x_k^n} \ge k^{n+1}$ for positive real numbers $x_i $ with sum $1$.

2020 JBMO Shortlist, 3

Tags: shortlist , algebra , inequality , minimum

Find all triples of positive real numbers $(a, b, c)$ so that the expression $M = \frac{(a + b)(b + c)(a + b + c)}{abc}$ gets its least value.

1990 IMO Longlists, 88

Tags: cauchy schwarz , holder , inequality , 4-variable inequality , algebra

Let $ w, x, y, z$ are non-negative reals such that $ wx \plus{} xy \plus{} yz \plus{} zw \equal{} 1$. Show that $ \frac {w^3}{x \plus{} y \plus{} z} \plus{} \frac {x^3}{w \plus{} y \plus{} z} \plus{} \frac {y^3}{w \plus{} x \plus{} z} \plus{} \frac {z^3}{w \plus{} x \plus{} y}\geq \frac {1}{3}$.

2022 JBMO Shortlist, N2

Tags: lcm , shortlist , inequality , number theory

Let $a < b < c < d < e$ be positive integers. Prove that $$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$ where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?

2017 Pan-African Shortlist, I4

Tags: inequality , maximum value , minimum value , permutation , combinatorics

Find the maximum and minimum of the expression \[ \max(a_1, a_2) + \max(a_2, a_3), + \dots + \max(a_{n-1}, a_n) + \max(a_n, a_1), \] where $(a_1, a_2, \dots, a_n)$ runs over the set of permutations of $(1, 2, \dots, n)$.

2015 China Girls Math Olympiad, 7

Tags: inequality , algebra , inequalities

Let $x_1,x_2,\cdots,x_n \in(0,1)$ , $n\geq2$. Prove that$$\frac{\sqrt{1-x_1}}{x_1}+\frac{\sqrt{1-x_2}}{x_2}+\cdots+\frac{\sqrt{1-x_n}}{x_n}<\frac{\sqrt{n-1}}{x_1 x_2 \cdots x_n}.$$

2001 Moldova National Olympiad, Problem 2

Tags: inequality , optimization , inequalities

If $n\in\mathbb N$ and $a_1,a_2,\ldots,a_n$ are arbitrary numbers in the interval $[0,1]$, find the maximum possible value of the smallest among the numbers $a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1$.

2005 Korea Junior Math Olympiad, 7

Tags: algebra , inequalities , inequality , n-variable inequality

If positive reals $ x_1,x_2,\cdots,x_n $ satisfy $\sum_{i=1}^{n}x_i=1.$ Prove that$$\sum_{i=1}^{n}\frac{1}{1+\sum_{j=1}^{i}x_j}<\sqrt{\frac{2}{3}\sum_{i=1}^{n}\frac{1}{x_i}} $$

1971 IMO Shortlist, 11

Tags: linear algebra , matrix , inequality , combinatorial inequality

The matrix \[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\] satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that \[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]

2018 South East Mathematical Olympiad, 1

Tags: algebra , inequality , inequalities

Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.

2021 Azerbaijan IZhO TST, 1

Tags: algebra , inequality

Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.

2020 Iran MO (2nd Round), P2

Tags: inequality , floor function , inequalities , algebra

let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the $$\max( [x]y + [y]z + [z]x ) $$ ( $[a]$ is the biggest integer not exceeding $a$)

2002 IMO Shortlist, 6

Tags: algebra , inequality , reciprocal sum , additive number theory

Let $A$ be a non-empty set of positive integers. Suppose that there are positive integers $b_1,\ldots b_n$ and $c_1,\ldots,c_n$ such that - for each $i$ the set $b_iA+c_i=\left\{b_ia+c_i\colon a\in A\right\}$ is a subset of $A$, and - the sets $b_iA+c_i$ and $b_jA+c_j$ are disjoint whenever $i\ne j$ Prove that \[{1\over b_1}+\,\ldots\,+{1\over b_n}\leq1.\]

1984 IMO Shortlist, 5

Tags: inequality , three variable inequality , calculus , maximization

Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

2004 Bosnia and Herzegovina Team Selection Test, 3

Tags: inequality , positive real , algebra , inequalities

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$

2022 IFYM, Sozopol, 5

Tags: algebra , inequality , sum

Prove that $\sum_{n=1}^{2022^{2022}} \frac{1}{\sqrt{n^3+2n^2+n}}<\frac{19}{10}$.

2022 Bulgarian Spring Math Competition, Problem 8.3

Tags: algebra , inequality , inequalities

Given the inequalities: $a)$ $\left(\frac{2a}{b+c}\right)^2+\left(\frac{2b}{a+c}\right)^2+\left(\frac{2c}{a+b}\right)^2\geq \frac{a}{c}+\frac{b}{a}+\frac{c}{b}$ $b)$ $\left(\frac{a+b}{c}\right)^2+\left(\frac{b+c}{a}\right)^2+\left(\frac{c+a}{b}\right)^2\geq \frac{a}{b}+\frac{b}{c}+\frac{c}{a}+9$ For each of them either prove that it holds for all positive real numbers $a$, $b$, $c$ or present a counterexample $(a,b,c)$ which doesn't satisfy the inequality.

2023 Romania National Olympiad, 1

Tags: fractional part , inequality , algebra

For natural number $n$ we define \[ a_n = \{ \sqrt{n} \} - \{ \sqrt{n + 1} \} + \{ \sqrt{n + 2} \} - \{ \sqrt{n + 3} \}. \] a) Show that $a_1 > 0,2$. b) Show that $a_n < 0$ for infinity many values of $n$ and $a_n > 0$ for infinity values of natural numbers of $n$ as well. ( We denote by $\{ x \} $ the fractional part of $x.$)

2019 Balkan MO Shortlist, A4

Tags: inequality , algebra , inequalities , n-variable inequality

Let $a_{ij}, i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ be positive real numbers. Prove that \[ \sum_{i = 1}^m \left( \sum_{j = 1}^n \frac{1}{a_{ij}} \right)^{-1} \le \left( \sum_{j = 1}^n \left( \sum_{i = 1}^m a_{ij} \right)^{-1} \right)^{-1} \]