Olympiad problems

2019 Mathematical Talent Reward Programme, SAQ: P 3

Tags: inequality , inequalities

Suppose $a$, $b$, $c$ are three positive real numbers with $a + b + c = 3$. Prove that $$\frac{a}{b^2 + c}+\frac{b}{c^2 + a}+\frac{c}{a^2 + b}\geq \frac{3}{2}$$

2021 MOAA, 4

Tags: speed , inequalities , algebra

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]

1995 IMO, 5

Tags: geometric inequality , hexagon , inequalities , geometry

Let $ ABCDEF$ be a convex hexagon with $ AB \equal{} BC \equal{} CD$ and $ DE \equal{} EF \equal{} FA$, such that $ \angle BCD \equal{} \angle EFA \equal{} \frac {\pi}{3}$. Suppose $ G$ and $ H$ are points in the interior of the hexagon such that $ \angle AGB \equal{} \angle DHE \equal{} \frac {2\pi}{3}$. Prove that $ AG \plus{} GB \plus{} GH \plus{} DH \plus{} HE \geq CF$.

2012 Sharygin Geometry Olympiad, 6

Tags: 3d geometry , geometry , tetrahedron , inequalities

Consider a tetrahedron $ABCD$. A point $X$ is chosen outside the tetrahedron so that segment $XD$ intersects face $ABC$ in its interior point. Let $A' , B'$ , and $C'$ be the projections of $D$ onto the planes $XBC, XCA$, and $XAB$ respectively. Prove that $A' B' + B' C' + C' A' \le DA + DB + DC$. (V.Yassinsky)

2006 MOP Homework, 3

Tags: divisor , odd , greatest , inequalities , number theory

For positive integer $k$, let $p(k)$ denote the greatest odd divisor of $k$. Prove that for every positive integer $n$, $$\frac{2n}{3} < \frac{p(1)}{1}+ \frac{p(2)}{2}+... +\frac{ p(n)}{n}<\frac{2(n + 1)}{3}$$

2023 Kazakhstan National Olympiad, 4

Tags: algebra , inequalities

Given $x,y>0$ such that $x^2y^2+2x^3y=1$. Find the minimum value of sum $x+y$

2008 Romania National Olympiad, 4

Tags: matrix , linear algebra , algebra , polynomial , inequalities

Let $ A\equal{}(a_{ij})_{1\leq i,j\leq n}$ be a real $ n\times n$ matrix, such that $ a_{ij} \plus{} a_{ji} \equal{} 0$, for all $ i,j$. Prove that for all non-negative real numbers $ x,y$ we have \[ \det(A\plus{}xI_n)\cdot \det(A\plus{}yI_n) \geq \det (A\plus{}\sqrt{xy}I_n)^2.\]

2009 Romanian Masters In Mathematics, 1

Tags: least common multiple , number theory , triangle inequality , inequalities , greatest common divisor , floor function , algebra

For $ a_i \in \mathbb{Z}^ \plus{}$, $ i \equal{} 1, \ldots, k$, and $ n \equal{} \sum^k_{i \equal{} 1} a_i$, let $ d \equal{} \gcd(a_1, \ldots, a_k)$ denote the greatest common divisor of $ a_1, \ldots, a_k$. Prove that $ \frac {d} {n} \cdot \frac {n!}{\prod\limits^k_{i \equal{} 1} (a_i!)}$ is an integer. [i]Dan Schwarz, Romania[/i]

2014 China Team Selection Test, 5

Tags: inequalities , complex number

Let $n$ be a given integer which is greater than $1$ . Find the greatest constant $\lambda(n)$ such that for any non-zero complex $z_1,z_2,\cdots,z_n$ ,have that \[\sum_{k\equal{}1}^n |z_k|^2\geq \lambda(n)\min\limits_{1\le k\le n}\{|z_{k+1}-z_k|^2\},\] where $z_{n+1}=z_1$.

1963 Polish MO Finals, 4

Tags: algebra , inequalities

Prove that for every natural number $ n $ the inequality holds $$ 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \ldots + \frac{1}{\sqrt{n}} > \sqrt{n-1}.$$

1957 AMC 12/AHSME, 34

Tags: parameterization , graphing lines , slope , function , algebra , analytic geometry , inequalities

The points that satisfy the system $ x \plus{} y \equal{} 1,\, x^2 \plus{} y^2 < 25,$ constitute the following set: $ \textbf{(A)}\ \text{only two points} \qquad \\ \textbf{(B)}\ \text{an arc of a circle}\qquad \\ \textbf{(C)}\ \text{a straight line segment not including the end\minus{}points}\qquad \\ \textbf{(D)}\ \text{a straight line segment including the end\minus{}points}\qquad \\ \textbf{(E)}\ \text{a single point}$

1986 China Team Selection Test, 3

Tags: inequalities , quadratic

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

2003 IMO Shortlist, 4

Tags: arithmetic sequence , inequalities

Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

2007 Germany Team Selection Test, 1

Tags: algebra , sequence , summation , integration , inequalities , calculus

The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by \[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]Show that $ a_{n} > 0$ for all $ n\geq 1$. [i]Proposed by Mariusz Skalba, Poland[/i]

2005 Silk Road, 4

Tags: algebra , induction , inequalities

Suppose $\{a(n) \}_{n=1}^{\infty}$ is a sequence that: \[ a(n) =a(a(n-1))+a(n-a(n-1)) \ \ \ \forall \ n \geq 3\] and $a(1)=a(2)=1$. Prove that for each $n \geq 1$ , $a(2n) \leq 2a(n)$.

2008 South africa National Olympiad, 3

Tags: inequalities

Let $a,b,c$ be positive real numbers. Prove that \[(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)\] and determine when equality occurs.

2008 IMAC Arhimede, 3

Tags: trigonometry , trigonometric inequality , inequalities

Let $ 0 \leq x \leq 2\pi$. Prove the inequality $ \sqrt {\frac {\sin^{2}x}{1 + \cos^{2}x}} + \sqrt {\frac {\cos^{2}x}{1 + \sin^{2}x}}\geq 1 $

1971 Spain Mathematical Olympiad, 3

Tags: algebra , inequalities

If $0 < p$, $0 < q$ and $p +q < 1$ prove $$(px + qy)^2 \le px^2 + qy^2$$

2010 Indonesia TST, 4

Tags: greatest common divisor , number theory , inequalities

Prove that for all integers $ m$ and $ n$, the inequality \[ \dfrac{\phi(\gcd(2^m \plus{} 1,2^n \plus{} 1))}{\gcd(\phi(2^m \plus{} 1),\phi(2^n \plus{} 1))} \ge \dfrac{2\gcd(m,n)}{2^{\gcd(m,n)}}\] holds. [i]Nanang Susyanto, Jogjakarta [/i]

2019 JBMO Shortlist, A6

Tags: algebra , inequalities

Let $a, b, c$ be positive real numbers. Prove the inequality $(a^2+ac+c^2) \left( \frac{1}{a+b+c}+\frac{1}{a+c} \right)+b^2 \left( \frac{1}{b+c}+\frac{1}{a+b} \right)>a+b+c$. [i]Proposed by Tajikistan[/i]

2007 Germany Team Selection Test, 1

Tags: function , algebra , inequalities

Prove the inequality: \[\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} \plus{} a_{j}}}\leq \frac {n}{2(a_{1} \plus{} a_{2} \plus{}\cdots \plus{} a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}\] for positive reals $ a_{1},a_{2},\ldots,a_{n}$. [i]Proposed by Dusan Dukic, Serbia[/i]

2008 239 Open Mathematical Olympiad, 8

Tags: inequalities , n-variable inequality

The natural numbers $x_1, x_2, \ldots , x_n$ are such that all their $2^n$ partial sums are distinct. Prove that: $$ {x_1}^2 + {x_2}^2 + \ldots + {x_n}^2 \geq \frac{4^n – 1}{3}. $$

1978 Canada National Olympiad, 3

Tags: inequalities

Determine the largest real number $z$ such that \begin{align*} x + y + z = 5 \\ xy + yz + xz = 3 \end{align*} and $x$, $y$ are also real.

KoMaL A Problems 2024/2025, A. 889

Tags: inequalities , algebra

Let $W,A,B$ be fixed real numbers with $W>0$. Prove that the following statements are equivalent. [list] [*] For all $x, y, z\ge 0$ satisfying $x+y\le z+W, x+z\le y+W, y+z\le x+W$ we have $Axyz+B\ge x^2+y^2+z^2$. [*] $B\ge W^2$ and $AW^3+B\ge 3W^2$. [/list] [i]Proposed by Ákos Somogyi, London[/i]

2002 China Team Selection Test, 1

Tags: polynomial , inequalities , algebra

Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that \begin{align*} P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right), \end{align*} where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]