Olympiad problems

2023 China Western Mathematical Olympiad, 7

For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$

2022 Taiwan TST Round 2, A

Tags: inequality , algebra , inequalities

Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]

2023 China Northern MO, 2

Tags: inequality , inequalities , algebra

Let $ a,b,c \in (0,1) $ and $ab+bc+ca=4abc .$ Prove that $$\sqrt{a+b+c}\geq \sqrt{1-a}+\sqrt{1-b}+\sqrt{1-c}$$

1995 IMO Shortlist, 8

Tags: prime number , inequality , number theory , minimization

Let $ p$ be an odd prime. Determine positive integers $ x$ and $ y$ for which $ x \leq y$ and $ \sqrt{2p} \minus{} \sqrt{x} \minus{} \sqrt{y}$ is non-negative and as small as possible.

1973 IMO Shortlist, 10

Tags: sequence , inequality , geometric sequence , algebra , construction

Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that: [i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$ [i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$ [i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$

2009 Iran Team Selection Test, 3

Tags: three variable inequality , inequality , function , algebra , calculus

Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that : $ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$

2019 Moldova Team Selection Test, 6

Tags: inequality , inequalities

Let $a,b,c \ge 0$ such that $a+b+c=1$ and $s \ge 5$. Prove that $s(a^2+b^2+c^2) \le 3(s-3)(a^3+b^3+c^3)+1$

2018 Serbia Team Selection Test, 2

Tags: inequality , inequalities , algebra

Let $n$ be a fixed positive integer and let $x_1,\ldots,x_n$ be positive real numbers. Prove that $$x_1\left(1-x_1^2\right)+x_2\left(1-(x_1+x_2)^2\right)+\cdots+x_n\left(1-(x_1+...+x_n)^2\right)<\frac{2}{3}.$$

2018 Azerbaijan IZhO TST, 2

Tags: inequality , algebra

Problem 4. Let a,b be positive real numbers and let x,y be positive real numbers less than 1, such that: a/(1-x)+b/(1-y)=1 Prove that: ∛ay+∛bx≤1.

2018 JBMO Shortlist, A6

Tags: inequality , inequalities , algebra

For $a,b,c$ positive real numbers such that $ab+bc+ca=3$, prove: $ \frac{a}{\sqrt{a^3+5}}+\frac{b}{\sqrt{b^3+5}}+\frac{c}{\sqrt{c^3+5}} \leq \frac{\sqrt{6}}{2}$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2022 District Olympiad, P2

Tags: inequality , cyclic inequality , inequalities , algebra

$a)$ Prove that $2x^3-3x^2+1\geq 0,~(\forall)x\geq0.$ $b)$ Let $x,y,z\geq 0$ such that $\frac{2}{1+x^3}+\frac{2}{1+y^3}+\frac{2}{1+z^3}=3.$ Prove that $\frac{1-x}{1-x+x^2}+\frac{1-y}{1-y+y^2}+\frac{1-z}{1-z+z^2}\geq 0.$

2020 Stars of Mathematics, 1

Tags: inequality , algebra

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]

1969 IMO Shortlist, 14

Tags: quadratic , inequality , root , polynomial , algebra

$(CZS 3)$ 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 $x^2+px+ q = 0.$

1983 IMO Shortlist, 2

Tags: inequality , divisor , sum of divisors , number theory

Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$ Prove that there exists an infinity of [i]superabundant[/i] numbers.

2015 JBMO TST - Turkey, 4

Tags: inequality

Prove that $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \ge \dfrac{a}{b}+\dfrac{b}{c}+\dfrac{c}{a}+2(a+b+c)$$ for the all $a,b,c$ positive real numbers satisfying $a^2+b^2+c^2+2abc \le 1$.

2021 Macedonian Mathematical Olympiad, Problem 1

Tags: sequence , inequality , am-gm , recursive , factorial , inequalities

Let $(a_n)^{+\infty}_{n=1}$ be a sequence defined recursively as follows: $a_1=1$ and $$a_{n+1}=1 + \sum\limits_{k=1}^{n}ka_k$$ For every $n > 1$, prove that $\sqrt[n]{a_n} < \frac {n+1}{2}$.

1975 IMO Shortlist, 12

Tags: trigonometric inequality , trigonometric identities , inequality , optimization , trigonometry

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 \[\sum_{i=0}^{v-1} \sin 2x_i - \sum_{i=0}^{v-1} \sin (x_i- x_{i+1}) < \frac{\pi}{2} + \sum_{i=0}^{v-1} \sin (x_i + x_{i+1})\]

2012 Kosovo National Mathematical Olympiad, 2

Tags: inequality , geometry

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

1979 IMO Longlists, 54

Tags: recurrence relation , inequality , sequence , number theory

Consider the sequences $(a_n), (b_n)$ defined by \[a_1=3, \quad b_1=100 , \quad a_{n+1}=3^{a_n} , \quad b_{n+1}=100^{b_n} \] Find the smallest integer $m$ for which $b_m > a_{100}.$

2022 Azerbaijan JBMO TST, A2

Tags: inequality , algebra

For positive real numbers $a,b,c$, $\frac{1}{a}+\frac{1}{b} + \frac{1}{c} \ge \frac{3}{abc}$ is true. Prove that: $$ \frac{a^2+b^2}{a^2+b^2+1}+\frac{b^2+c^2}{b^2+c^2+1}+\frac{c^2+a^2}{c^2+a^2+1} \ge 2$$

1969 IMO Shortlist, 56

Tags: decimal representation , inequality , calculus

Let $a$ and $b$ be two natural numbers that have an equal number $n$ of digits in their decimal expansions. The first $m$ digits (from left to right) of the numbers $a$ and $b$ are equal. Prove that if $m >\frac{n}{2},$ then $a^{\frac{1}{n}} -b^{\frac{1}{n}} <\frac{1}{n}$

2023 JBMO Shortlist, A4

Tags: inequality , algebra

Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that $$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$

2024 Azerbaijan JBMO TST, 4

Tags: inequality , algebra

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

1984 IMO Shortlist, 5

Tags: inequality , three variable inequality , maximization , calculus

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

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P2

Tags: inequality , n variables , strict , summation , algebra

Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that $\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$. [i]Proposed by Mirko Petrusevski[/i]