Olympiad problems

2021 Czech-Polish-Slovak Junior Match, 4

Tags: algebra , inequalities

Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$.

2014 Federal Competition For Advanced Students, P2, 5

Tags: inequalities , three variable inequality , 3-variable inequality , algebra

Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?

2004 Korea Junior Math Olympiad, 1

Tags: geometry , inequalities

For positive reals $a_1, a_2, ..., a_5$ such that $a^2_1+a^2_2+...+a^2_5=2$, consider five squares with sides $a_1, a_2, ..., a_5$ respectively. Show that these squares can be placed inside (including boundaries) a square with side length of $2$ so that the square themselves do not overlap each other.

2010 Contests, 3

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

Let $h_a, h_b, h_c$ be the lengths of the altitudes of a triangle $ABC$ from $A, B, C$ respectively. Let $P$ be any point inside the triangle. Show that \[\frac{PA}{h_b+h_c} + \frac{PB}{h_a+h_c} + \frac{PC}{h_a+h_b} \ge 1.\]

2010 Contests, 1

Tags: inequalities

Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.

1963 Polish MO Finals, 4

Tags: algebra , inequalities

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

PEN O Problems, 29

Tags: modular arithmetic , induction , ceiling function , inequalities , number theory

Let $A$ be a set of $N$ residues $\pmod{N^2}$. Prove that there exists a set $B$ of $N$ residues $\pmod{N^2}$ such that the set $A+B=\{a+b \vert a \in A, b \in B \}$ contains at least half of all the residues $\pmod{N^2}$.

2004 Tournament Of Towns, 5

Tags: inequalities , inradius , trigonometry , geometry

Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that [tex]BM\cdot CN>KM \cdot KN[/tex].

2007 Germany Team Selection Test, 1

Tags: inequalities , function , algebra

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]

2005 Germany Team Selection Test, 3

Tags: inradius , trigonometry , inequalities , function , triangle inequality , geometry

Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that [b](a)[/b] $\triangle ABC$ is acute. [b](b)[/b] $a+b+c > r+r_a+r_b+r_c$.

2017 Iran Team Selection Test, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality: $$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$ [i]Proposed by Mohammad Jafari[/i]

1988 Nordic, 2

Tags: inequalities

Let $a, b,$ and $c$ be non-zero real numbers and let $a \ge b \ge c$. Prove the inequality $\frac{a^3 - c^3}{3} \ge abc (\frac{a- b}{c}+ \frac{b- c}{a})$ . When does equality hold?

2023 Euler Olympiad, Round 2, 5

Tags: inequalities , algebra , euler

Find the smallest constant M, so that for any real numbers $a_1, a_2, \dots a_{2023} \in [4, 6]$ and $b_1, b_2, \dots b_{2023} \in [9, 12] $ following inequality holds: $$ \sqrt{a_1^2 + a_2^2 + \dots + a_{2023}^2} \cdot \sqrt{b_1^2 + b_2^2 + \dots + b_{2023}^2} \leq M \cdot \left ( a_1 b_1 + a_2 b_2 + \dots + a_{2023} b_{2023} \right) $$ [i]Proposed by Zaza Meliqidze, Georgia[/i]

2006 District Olympiad, 2

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

Let $n,p \geq 2$ be two integers and $A$ an $n\times n$ matrix with real elements such that $A^{p+1} = A$. a) Prove that $\textrm{rank} \left( A \right) + \textrm{rank} \left( I_n - A^p \right) = n$. b) Prove that if $p$ is prime then \[ \textrm{rank} \left( I_n - A \right) = \textrm{rank} \left( I_n - A^2 \right) = \ldots = \textrm{rank} \left( I_n - A^{p-1} \right) . \]

2009 Costa Rica - Final Round, 5

Tags: polynomial , inequalities , algebra

Suppose the polynomial $ x^{n} \plus{} a_{n \minus{} 1}x^{n \minus{} 1} \plus{} ... \plus{} a_{1} \plus{} a_{0}$ can be factorized as $ (x \plus{} r_{1})(x \plus{} r_{2})...(x \plus{} r_{n})$, with $ r_{1}, r_{2}, ..., r_{n}$ real numbers. Show that $ (n \minus{} 1)a_{n \minus{} 1}^{2}\geq\ 2na_{n \minus{} 2}$

1975 Putnam, B3

Tags: supremum , inequalities

Let $n$ be a positive integer. Let $S=\{a_1,\ldots, a_{k}\}$ be a finite collection of at least $n$ not necessarily distinct positive real numbers. Let $$f(S)=\left(\sum_{i=1}^{k} a_{i}\right)^{n}$$ and $$g(S)=\sum_{1\leq i_{1}<\ldots<i_{n} \leq k} a_{i_{1}}\cdot\ldots\cdot a_{i_{n}}.$$ Determine $\sup_{S} \frac{g(S)}{f(S)}$.

2011 Morocco National Olympiad, 1

Tags: inequalities

Let $a$ and $b$ be two positive real numbers such that $a+b=ab$. Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.

2013 Greece Team Selection Test, 3

Tags: function , inequalities

Find the largest possible value of $M$ for which $\frac{x}{1+\frac{yz}{x}}+\frac{y}{1+\frac{zx}{y}}+\frac{z}{1+\frac{xy}{z}}\geq M$ for all $x,y,z>0$ with $xy+yz+zx=1$

1970 Yugoslav Team Selection Test, Problem 1

Tags: inequalities , number theory

Positive integers $a$ and $b$ have $n$ digits each in their decimal representation. Assume that $m$ is a positive integer such that $\frac n2<m<n$ and assume that each of the leftmost $m$ digits of $a$ is equal to the corresponding digit of $b$. Prove that $$a^{\frac1n}-b^{\frac1n}<\frac1n.$$

2004 Singapore Team Selection Test, 2

Tags: function , inequalities , trigonometry

Let $0 < a, b, c < 1$ with $ab + bc + ca = 1$. Prove that \[\frac{a}{1-a^2} + \frac{b}{1-b^2} + \frac{c}{1-c^2} \geq \frac {3 \sqrt{3}}{2}.\] Determine when equality holds.

2010 China Team Selection Test, 3

Tags: floor function , inequalities , algorithm , combinatorics

Let $n_1,n_2, \cdots, n_{26}$ be pairwise distinct positive integers satisfying (1) for each $n_i$, its digits belong to the set $\{1,2\}$; (2) for each $i,j$, $n_i$ can't be obtained from $n_j$ by adding some digits on the right. Find the smallest possible value of $\sum_{i=1}^{26} S(n_i)$, where $S(m)$ denotes the sum of all digits of a positive integer $m$.

1995 All-Russian Olympiad Regional Round, 11.5

Tags: algebra , inequalities , trigonometry

Angles $\alpha, \beta, \gamma$ satisfy the inequality $\sin \alpha +\sin \beta +\sin \gamma \ge 2$. Prove that $\cos \alpha + \cos \beta +\cos \gamma \le \sqrt5.$

2008 SEEMOUS, Problem 3

Tags: inequalities , functional equation , fe , matrix , linear algebra

Let $\mathcal M_n(\mathbb R)$ denote the set of all real $n\times n$ matrices. Find all surjective functions $f:\mathcal M_n(\mathbb R)\to\{0,1,\ldots,n\}$ which satisfy $$f(XY)\le\min\{f(X),f(Y)\}$$for all $X,Y\in\mathcal M_n(\mathbb R)$.

2018 Pan-African Shortlist, A3

Tags: ratio , algebra , inequalities , combinatorics , geometry , rectangle

Akello divides a square up into finitely many white and red rectangles, each (rectangle) with sides parallel to the sides of the parent square. Within each white rectangle, she writes down the value of its width divided by its height, while within each red rectangle, she writes down the value of its height divided by its width. Finally, she calculates $x$, the sum of these numbers. If the total area of the white rectangles equals the total area of the red rectangles, what is the least possible value of $x$ she can get?

2009 Stars Of Mathematics, 1

Tags: algebra , inequalities , n-variable inequality

Let $x_1, x_2, ... , x_n$ and $y_1, y_2, ..., y_n$ be positive real numbers so that $$x_1 + x_2 + ...+ x_n \ge x_1y_1 + x_2y_2 + ... + x_ny_n.$$ Show that for any non-negative integer $p$ the following inequality holds $$\frac{x_1}{y_1^p} +\frac{ x_2}{y_2^p} + ...+ \frac{x_n}{y_n^p} \ge x_1 + x_2 + ...+ x_n.$$