Olympiad problems

2012 CentroAmerican, 3

Tags: inequalities

Let $a,b,c$ be real numbers that satisfy $\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{a+c} =1$ and $ab+bc+ac >0$. Show that \[ a+b+c - \frac{abc}{ab+bc+ac} \ge 4 \]

2024 Indonesia MO, 6

Tags: geometry , polygon , inequality , convex , inequalities

Suppose $A_1 A_2 \ldots A_n$ is an $n$-sided polygon with $n \geq 3$ and $\angle A_j \leq 180^{\circ}$ for each $j$ (in other words, the polygon is convex or has fewer than $n$ distinct sides). For each $i \leq n$, suppose $\alpha_i$ is the smallest possible value of $\angle{A_i A_j A_{i+1}}$ where $j$ is neither $i$ nor $i+1$. (Here, we define $A_{n+1} = A_1$.) Prove that \[ \alpha_1 + \alpha_2 + \cdots + \alpha_n \leq 180^{\circ} \] and determine all equality cases.

2015 Moldova Team Selection Test, 1

Tags: trigonometry , trigonometric inequality , algebra , inequalities , function

Let $c\in \Big(0,\dfrac{\pi}{2}\Big) , a = \Big(\dfrac{1}{sin(c)}\Big)^{\dfrac{1}{cos^2 (c)}}, b = \Big(\dfrac{1}{cos(c)}\Big)^{\dfrac{1}{sin^2 (c)}}$. \\Prove that at least one of $a,b$ is bigger than $\sqrt[11]{2015}$.

2019 Belarus Team Selection Test, 1.2

Tags: inequalities , geometry

Points $M$ and $N$ are the midpoints of the sides $BC$ and $AD$, respectively, of a convex quadrilateral $ABCD$. Is it possible that $$ AB+CD>\max(AM+DM,BN+CN)? $$ [i](Folklore)[/i]

1949 Kurschak Competition, 1

Tags: inequalities , algebra , trigonometry

Prove that $\sin x + \frac12 \sin 2x + \frac13 \sin 3x > 0$ for $0 < x < 180^o$.

2024 South Africa National Olympiad, 5

Tags: geometric inequality , geometry , inequalities

Consider three circles $\Gamma_1$, $\Gamma_2$, and $\Gamma_3$, with centres $O_1$, $O_2$ and $O_3$, respectively, such that each pair of circles is externally tangent. Suppose we have another circle $\Gamma$ with centre $O$ on the line segment $O_1O_3$ such that $\Gamma_1$, $\Gamma_2$ and $\Gamma_3$ are each internally tangent to $\Gamma$. Show that $\angle O_1O_2O_3$ measures less than $90^\circ$.

2018 China Western Mathematical Olympiad, 6

Tags: inequalities

Let $n \geq 2$ be an integer. Positive reals satisfy $a_1\geq a_2\geq \cdots\geq a_n.$ Prove that $$\left(\sum_{i=1}^n\frac{a_i}{a_{i+1}}\right)-n \leq \frac{1}{2a_1a_n}\sum_{i=1}^n(a_i-a_{i+1})^2,$$ where $a_{n+1}=a_1.$

2011 Morocco National Olympiad, 4

Tags: geometry , inequalities

Let $ABCD$ be a convex quadrilateral with angles $\angle ABC$ and $\angle BCD$ not less than $120^{\circ}$. Prove that \[AC + BD> AB+BC+CD\]

2012 APMO, 1

Tags: geometry , polynomial , inequalities , algebra

Let $ P $ be a point in the interior of a triangle $ ABC $, and let $ D, E, F $ be the point of intersection of the line $ AP $ and the side $ BC $ of the triangle, of the line $ BP $ and the side $ CA $, and of the line $ CP $ and the side $ AB $, respectively. Prove that the area of the triangle $ ABC $ must be $ 6 $ if the area of each of the triangles $ PFA, PDB $ and $ PEC $ is $ 1 $.

2013 Dutch IMO TST, 5

Tags: algebra , inequalities

Let $a, b$, and $c$ be positive real numbers satisfying $abc = 1$. Show that $a + b + c \ge \sqrt{\frac13 (a + 2)(b + 2)(c + 2)}$

2021 China National Olympiad, 3

Tags: number theory , inequalities , algebra , bashing

Let $n$ be positive integer such that there are exactly 36 different prime numbers that divides $n.$ For $k=1,2,3,4,5,$ $c_n$ be the number of integers that are mutually prime numbers to $n$ in the interval $[\frac{(k-1)n}{5},\frac{kn}{5}] .$ $c_1,c_2,c_3,c_4,c_5$ is not exactly the same.Prove that$$\sum_{1\le i<j\le 5}(c_i-c_j)^2\geq 2^{36}.$$

2011 Lusophon Mathematical Olympiad, 2

Tags: modular arithmetic , inequalities , number theory

A non-negative integer $n$ is said to be [i]squaredigital[/i] if it equals the square of the sum of its digits. Find all non-negative integers which are squaredigital.

2006 China Team Selection Test, 1

Tags: induction , ratio , inequalities , limit , algebra

Two positive valued sequences $\{ a_{n}\}$ and $\{ b_{n}\}$ satisfy: (a): $a_{0}=1 \geq a_{1}$, $a_{n}(b_{n+1}+b_{n-1})=a_{n-1}b_{n-1}+a_{n+1}b_{n+1}$, $n \geq 1$. (b): $\sum_{i=1}^{n}b_{i}\leq n^{\frac{3}{2}}$, $n \geq 1$. Find the general term of $\{ a_{n}\}$.

2020 Taiwan TST Round 1, 1

Tags: inequality , inequalities

Let $a$, $b$, $c$, $d$ be real numbers satisfying \begin{align*} (a + c)(b + d) = \sqrt{2}(ac - 2bd - 1). \end{align*} Show that \begin{align*} (ab - 1)^2 + (bc - 1)^2 + (cd - 1)^2 + (da - 1)^2 + (ac - 1)^2 + (2bd + 1)^2 \ge 4. \end{align*}

2008 Harvard-MIT Mathematics Tournament, 10

Tags: inequalities , geometry , geometric transformation , reflection , triangle inequality

Let $ ABC$ be an equilateral triangle with side length 2, and let $ \Gamma$ be a circle with radius $ \frac {1}{2}$ centered at the center of the equilateral triangle. Determine the length of the shortest path that starts somewhere on $ \Gamma$, visits all three sides of $ ABC$, and ends somewhere on $ \Gamma$ (not necessarily at the starting point). Express your answer in the form of $ \sqrt p \minus{} q$, where $ p$ and $ q$ are rational numbers written as reduced fractions.

2018 IMO Shortlist, A7

Tags: inequalities

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

PEN R Problems, 7

Tags: limit , calculus , integration , 3d geometry , inequalities , triangle inequality , geometry

Show that the number $r(n)$ of representations of $n$ as a sum of two squares has $\pi$ as arithmetic mean, that is \[\lim_{n \to \infty}\frac{1}{n}\sum^{n}_{m=1}r(m) = \pi.\]

2021 MOAA, 4

Tags: speed , algebra , inequalities

Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$. [i]Proposed by Andrew Wen[/i]

2009 USAMO, 4

Tags: inequalities , induction , quadratic , floor function

For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that \[ (a_1 \plus{} a_2 \plus{} \cdots \plus{} a_n)\left(\frac {1}{a_1} \plus{} \frac {1}{a_2} \plus{} \cdots \plus{} \frac {1}{a_n}\right) \leq \left(n \plus{} \frac {1}{2}\right)^2. \] Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.

2019 Taiwan TST Round 3, 1

Tags: inequalities

Find the maximal value of \[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\] where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$. [i]Proposed by Evan Chen, Taiwan[/i]

2024 Dutch IMO TST, 3

Tags: algebra , inequalities

Let $a,b,c$ be real numbers such that $0 \le a \le b \le c$ and $a+b+c=1$. Show that \[ab\sqrt{b-a}+bc\sqrt{c-b}+ac\sqrt{c-a}<\frac{1}{4}.\]

2016 Irish Math Olympiad, 5

Tags: positive integer , inequalities , algebra

Let $a_1, a_2, ..., a_m$ be positive integers, none of which is equal to $10$, such that $a_1 + a_2 + ...+ a_m = 10m$. Prove that $(a_1a_2a_3 \cdot ...\cdot a_m)^{1/m} \le 3\sqrt{11}$.

2011 IMO Shortlist, 7

Tags: inequalities , algebra

Let $a,b$ and $c$ be positive real numbers satisfying $\min(a+b,b+c,c+a) > \sqrt{2}$ and $a^2+b^2+c^2=3.$ Prove that \[\frac{a}{(b+c-a)^2} + \frac{b}{(c+a-b)^2} + \frac{c}{(a+b-c)^2} \geq \frac{3}{(abc)^2}.\] [i]Proposed by Titu Andreescu, Saudi Arabia[/i]

2005 BAMO, 3

Tags: trigonometry , inequalities , pigeonhole principle , ceiling function

Let $ n\ge12$ be an integer, and let $ P_1,P_2,...P_n, Q$ be distinct points in a plane. Prove that for some $ i$, at least $ \frac{n}{6}\minus{}1$ of the distances $ P_1P_i,P_2P_i,...P_{i\minus{}1}P_i,P_{i\plus{}1}P_i,...P_nP_i$ are less than $ P_iQ$.

2006 Petru Moroșan-Trident, 3

Tags: inequalities , am-gm

Let a ,b and c be positive real numbers such that $a^2+b^2+c^2=3$. Prove that for whatever positive real numbers x y and z, the inequality below holds. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge \sqrt{xy}+\sqrt{yz}+\sqrt{zx}$ At first I noticed $\sqrt{xy}+\sqrt{yz}+\sqrt{zx}\le \sqrt{x+y+z}\sqrt{x+y+z}=x+y+z$, so perhaps the next move is to prove $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge x+y+z$, but I don't see how to do that, the best thing that I can do with the LHS of this inequality is to prove it by AM-GM in the way that $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\ge 3\left(\frac{xyz}{abc}\right)^{\frac{1}{3}}\ge 3(xyz)^{\frac{1}{3}}$, but this isn't going to be helpful...