Olympiad problems

2020 Taiwan TST Round 1, 4

Tags: algebra , inequalities

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

1966 IMO Longlists, 13

Tags: n-variable inequality , inequalities

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality \[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]

2021 China Girls Math Olympiad, 6

Tags: inequalities , subset

Given a finite set $S$, $P(S)$ denotes the set of all the subsets of $S$. For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality:$$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$

2007 Iran Team Selection Test, 2

Tags: analytic geometry , trigonometry , inequalities , circumcircle , geometry

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

2010 Korea - Final Round, 1

Tags: geometry , incenter , perimeter , trigonometry , inradius , inequalities

Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality \[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\] holds.

2022 China Team Selection Test, 5

Tags: inequalities , complex number

Let $C=\{ z \in \mathbb{C} : |z|=1 \}$ be the unit circle on the complex plane. Let $z_1, z_2, \ldots, z_{240} \in C$ (not necessarily different) be $240$ complex numbers, satisfying the following two conditions: (1) For any open arc $\Gamma$ of length $\pi$ on $C$, there are at most $200$ of $j ~(1 \le j \le 240)$ such that $z_j \in \Gamma$. (2) For any open arc $\gamma$ of length $\pi/3$ on $C$, there are at most $120$ of $j ~(1 \le j \le 240)$ such that $z_j \in \gamma$. Find the maximum of $|z_1+z_2+\ldots+z_{240}|$.

2000 Finnish National High School Mathematics Competition, 3

Tags: inequalities , induction , algebra , logarithm

Determine the positive integers $n$ such that the inequality \[n! > \sqrt{n^n}\] holds.

2022 Indonesia TST, A

Tags: inequality , am-gm , simple , algebra , inequalities

Let $a, b, c$ be positive real numbers such that $abc = 1$. Prove that $$(a + b + c)(ab + bc + ca) + 3\ge 4(a + b + c).$$

2011 Today's Calculation Of Integral, 694

Tags: calculus , integration , inequalities , quadratic , calculus computations , logarithm

Prove the following inequality: \[\int_1^e \frac{(\ln x)^{2009}}{x^2}dx>\frac{1}{2010\cdot 2011\cdot2012}\] created by kunny

1988 Bulgaria National Olympiad, Problem 3

Tags: geometry , 3d geometry , tetrahedron , inequalities , geometric inequality

Let $M$ be an arbitrary interior point of a tetrahedron $ABCD$, and let $S_A,S_B,S_C,S_D$ be the areas of the faces $BCD,ACD,ABD,ABC$, respectively. Prove that $$S_A\cdot MA+S_B\cdot MB+S_C\cdot MC+S_D\cdot MD\ge9V,$$where $V$ is the volume of $ABCD$. When does equality hold?

2010 Peru IMO TST, 7

Tags: inequalities

Let $a, b, c$ be positive real numbers such that $a + b + c = 1.$ Prove that $$ \displaystyle{\frac{1}{a + b}+\frac{1}{b + c}+\frac{1}{c + a}+ 3(ab + bc + ca) \geq \frac{11}{2}.}$$

1966 IMO Shortlist, 30

Tags: inequalities , calculus , logarithm

Let $n$ be a positive integer, prove that : [b](a)[/b] $\log_{10}(n + 1) > \frac{3}{10n} +\log_{10}n ;$ [b](b)[/b] $ \log n! > \frac{3n}{10}\left( \frac 12+\frac 13 +\cdots +\frac 1n -1\right).$

1918 Eotvos Mathematical Competition, 3

Tags: algebra , inequalities , trinomial

If $a, b,c,p,q, r $are real numbers such that, for every real number $x,$ $$ax^2 - 2bx + c \ge 0 \ \ and \ \ px^2 + 2qx + r \ge 0;$$ prove that then $$apx^2 + bqx + cr \ge 0$$ for all real $x$.

2013 Serbia National Math Olympiad, 6

Tags: inequalities

Find the largest constant $K\in \mathbb{R}$ with the following property: if $a_1,a_2,a_3,a_4>0$ are numbers satisfying $a_i^2 + a_j^2 + a_k^2 \geq 2 (a_ia_j + a_ja_k + a_ka_i)$, for every $1\leq i<j<k\leq 4$, then \[a_1^2+a_2^2+a_3^2+a_4^2 \geq K (a_1a_2+a_1a_3+a_1a_4+a_2a_3+a_2a_4+a_3a_4).\]

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: algebra , inequalities

If $x, y$ and $z$ are real numbers such that $x^2 + y^2 + z^2 = 2$, prove that $x + y + z \le xyz + 2$.

2015 Purple Comet Problems, 7

Tags: triangle inequality , inequalities , trooga inooga theorem

How many non-congruent isosceles triangles (including equilateral triangles) have positive integer side lengths and perimeter less than 20?

2025 China Team Selection Test, 18

Tags: complex number , inequalities

Find the smallest real number $M$ such that there exist four complex numbers $a,b,c,d$ with $|a|=|b|=|c|=|d|=1$, and for any complex number $z$, if $|z| = 1$, then\[|az^3+bz^2+cz+d|\le M.\]

2022 Philippine MO, 7

Tags: inequalities , algebra

Let $a, b,$ and $c$ be positive real numbers such that $ab + bc + ca = 3$. Show that \[ \dfrac{bc}{1 + a^4} + \dfrac{ca}{1 + b^4} + \dfrac{ab}{1 + c^4} \geq \dfrac{3}{2}. \]

1954 Poland - Second Round, 6

Tags: trigonometry , inequalities

Prove that if $ x_1, x_2, \ldots, x_n $ are angles between $ 0^\circ $ and $ 180^\circ $, and $ n $ is any natural number greater than $ 1 $, then $$ \sin (x_1 + x_2 + \ldots + x_n) < \sin x_1 + \sin x_2 + \ldots + \sin x_n.$$

2019 Hong Kong TST, 3

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]

2011 ISI B.Math Entrance Exam, 4

Tags: function , inequalities , algebra

Let $t_1 < t_2 < t_3 < \cdots < t_{99}$ be real numbers. Consider a function $f: \mathbb{R} \to \mathbb{R}$ given by $f(x)=|x-t_1|+|x-t_2|+...+|x-t_{99}|$ . Show that $f(x)$ will attain minimum value at $x=t_{50}$.

2007 Ukraine Team Selection Test, 1

Tags: inequalities

$\{a,b,c\}\subset\left(\frac{1}{\sqrt6},+\infty\right)$ such that $a^{2}+b^{2}+c^{2}=1.$ Prove that $\frac{1+a^{2}}{\sqrt{2a^{2}+3ab-c^{2}}}+\frac{1+b^{2}}{\sqrt{2b^{2}+3bc-a^{2}}}+\frac{1+c^{2}}{\sqrt{2c^{2}+3ca-b^{2}}}\ge2(a+b+c).$

2007 Irish Math Olympiad, 2

Tags: inequalities

Suppose that $ a,b,$ and $ c$ are positive real numbers. Prove that: $ \frac{a\plus{}b\plus{}c}{3} \le \sqrt{\frac{a^2\plus{}b^2\plus{}c^2}{3}} \le \frac {\frac{ab}{c}\plus{}\frac{bc}{a}\plus{}\frac{ca}{b}}{3}$. For each of the inequalities, find the conditions on $ a,b,$ and $ c$ such that equality holds.

2009 Postal Coaching, 5

Tags: number theory , least common multiple , lcm , inequalities

For positive integers $n, k$ with $1 \le k \le n$, define $$L(n, k) = Lcm \,(n, n - 1, n -2, ..., n - k + 1)$$ Let $f(n)$ be the largest value of $k$ such that $L(n, 1) < L(n, 2) < ... < L(n, k)$. Prove that $f(n) < 3\sqrt{n}$ and $f(n) > k$ if $n > k! + k$.

2014 Thailand TSTST, 3

Tags: inequalities

For all pairwise distinct positive real numbers $a, b, c$ such that $abc = 1$, prove that $$\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1}{(a+b+c+1)^2}+\frac{3}{8}\sqrt[3]{\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}}\geq 1.$$