Olympiad problems

2002 BAMO, 4

For $n \ge 1$, let $a_n$ be the largest odd divisor of $n$, and let $b_n = a_1+a_2+...+a_n$. Prove that $b_n \ge \frac{ n^2 + 2}{3}$, and determine for which $n$ equality holds. For example, $a_1 = 1, a_2 = 1, a_3 = 3, a_4 = 1, a_5 = 5, a_6 = 3$, thus $b_6 = 1 + 1 + 3 + 1 + 5 + 3 = 14 \ge \frac{ 6^2 + 2}{3}= 12\frac23$ .

2002 China Team Selection Test, 1

Tags: inequalities , trigonometry , geometry

In acute triangle $ ABC$, show that: $ \sin^3{A}\cos^2{(B \minus{} C)} \plus{} \sin^3{B}\cos^2{(C \minus{} A)} \plus{} \sin^3{C}\cos^2{(A \minus{} B)} \leq 3\sin{A} \sin{B} \sin{C}$ and find out when the equality holds.

2005 Vietnam National Olympiad, 2

Tags: parallelogram , geometric transformation , geometry , inequalities , trapezoid , reflection

Let $(O)$ be a fixed circle with the radius $R$. Let $A$ and $B$ be fixed points in $(O)$ such that $A,B,O$ are not collinear. Consider a variable point $C$ lying on $(O)$ ($C\neq A,B$). Construct two circles $(O_1),(O_2)$ passing through $A,B$ and tangent to $BC,AC$ at $C$, respectively. The circle $(O_1)$ intersects the circle $(O_2)$ in $D$ ($D\neq C$). Prove that: a) \[ CD\leq R \] b) The line $CD$ passes through a point independent of $C$ (i.e. there exists a fixed point on the line $CD$ when $C$ lies on $(O)$).

1994 Tournament Of Towns, (419) 7

Tags: inequalities , geometry

Consider an arbitrary “figure” $F$ (non convex polygon). A chord of $F$ is defined to be a segment which lies entirely within $ F$ and whose ends are on its boundary. (a) Does there always exist a chord of $F$ that divides its area in half? (b) Prove that for any $F$ there exists a chord such that the area of each of the two parts of $F$ is not less than $ 1/3$ of the area of $F$. (c) Can the number $1/3$ in (b) be changed to a greater one? (V Proizvolov)

2025 Romania National Olympiad, 2

Tags: function , inequalities , calculus , integration

Let $f \colon [0,1] \to \mathbb{R} $ be a differentiable function such that its derivative is an integrable function on $[0,1]$, and $f(1)=0$. Prove that \[ \int_0^1 (xf'(x))^2 dx \geq 12 \cdot \left( \int_0^1 xf(x) dx\right)^2 \]

1983 Bulgaria National Olympiad, Problem 2

Tags: inequalities

Let $b_1\ge b_2\ge\ldots\ge b_n$ be nonnegative numbers, and $(a_1,a_2,\ldots,a_n)$ be an arbitrary permutation of these numbers. Prove that for every $t\ge0$, $$(a_1a_2+t)(a_3a_4+t)\cdots(a_{2n-1}a_{2n}+t)\le(b_1b_2+t)(b_3b_4+t)\cdots(b_{2n-1}b_{2n}+t).$$

PEN J Problems, 22

Tags: inequalities , divisor function

Let $n$ be an odd positive integer. Prove that $\sigma(n)^3 <n^4$.

2006 Switzerland - Final Round, 9

Tags: algebra , inequalities

Let $a, b, c, d$ be real numbers. Prove that is $$(a^2 + b^2 + 1)(c^2 + d^2 + 1) \ge 2(a + c)(b + d).$$

2010 AIME Problems, 14

Tags: function , inequalities , algebra , logarithm , floor function , search

For each positive integer n, let $ f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor$. Find the largest value of n for which $ f(n) \le 300$. [b]Note:[/b] $ \lfloor x \rfloor$ is the greatest integer less than or equal to $ x$.

2002 Austrian-Polish Competition, 4

Tags: algebra , inequalities

For each positive integer $n$ find the largest subset $M(n)$ of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

2007 Bosnia Herzegovina Team Selection Test, 4

Tags: polynomial , inequalities , vieta , algebra

Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?

2013 Sharygin Geometry Olympiad, 11

Tags: inequalities , geometry

a) Let $ABCD$ be a convex quadrilateral and $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABC, BCD, CDA, DAB$. Can the inequality $r_4 > 2r_3$ hold? b) The diagonals of a convex quadrilateral $ABCD$ meet in point $E$. Let $r_1 \le r_2 \le r_3 \le r_4$ be the radii of the incircles of triangles $ABE, BCE, CDE, DAE$. Can the inequality $r_2 > 2r_1$ hold?

2024 Korea Junior Math Olympiad, 2

Tags: inequalities , combinatorics

$99$ different points $P_1, P_2, ..., P_{99}$ are marked on circle $O$. For each $P_i$, define $n_i$ as the number of marked points you encounter starting from $P_i$ to its antipode, moving clockwise. Prove the following inequality. $$n_1+n_2+\cdots+n_{99} \leq \frac{99\cdot 98}{2}+49=4900$$

1978 Swedish Mathematical Competition, 1

Tags: algebra , inequalities

Let $a,b,c,d$ be real numbers such that $a>b>c>d\geq 0$ and $a + d = b + c$. Show that \[ x^a + x^d \geq x^b + x^c \] for $x>0$.

2017 India PRMO, 7

Tags: algebra , radical , inequalities

Find the number of positive integers $n$, such that $\sqrt{n} + \sqrt{n + 1} < 11$.

2000 IMC, 4

Tags: inequalities , real analysis

Let $(x_i)$ be a decreasing sequence of positive reals, then show that: (a) for every positive integer $n$ we have $\sqrt{\sum^n_{i=1}{x_i^2}} \leq \sum^n_{i=1}\frac{x_i}{\sqrt{i}}$. (b) there is a constant C for which we have $\sum^{\infty}_{k=1}\frac{1}{\sqrt{k}}\sqrt{\sum^{\infty}_{i=k}x_i^2} \le C\sum^{\infty}_{i=1}x_i$.

VMEO III 2006, 10.3

Tags: inequalities

Prove that for all non negative real numbers $a,b,c$ we have \[a^2+b^2+c^2\leq\sqrt{(b^2-bc+c^2)(c^2-ca+a^2)}+\sqrt{(c^2-ca+a^2)(a^2-ab+b^2)}+\sqrt{(a^2-ab+b^2)(b^2-bx+c^2)} \]

1985 Brazil National Olympiad, 3

Tags: cyclic quadrilateral , geometry , perimeter , inequalities

A convex quadrilateral is inscribed in a circle of radius $1$. Show that the its perimeter less the sum of its two diagonals lies between $0$ and $2$.

2017 Kyiv Mathematical Festival, 4

Tags: inequalities

Real numbers $x,y$ are such that $x^2\ge y$ and $y^2\ge x.$ Prove that $\frac{x}{y^2+1}+\frac{y}{x^2+1}\le1.$

2009 India Regional Mathematical Olympiad, 5

Tags: geometry , function , induction , circumcircle , conic , inequalities , ellipse

A convex polygon is such that the distance between any two vertices does not exceed $ 1$. $ (i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $ 1$. $ (ii)$ If $ X$ and $ Y$ are two distinct points inside the polygon, prove that there exists a point $ Z$ on the boundary of the polygon such that $ XZ \plus{} YZ\le1$.

Russian TST 2016, P1

Tags: algebra , inequalities

The positive numbers $a, b, c$ are such that $a^2<16bc, b^2<16ca$ and $c^2<16ab$. Prove that \[a^2+b^2+c^2<2(ab+bc+ca).\]

2000 Romania Team Selection Test, 2

Tags: inequalities

Let $n\ge 1$ be a positive integer and $x_1,x_2\ldots ,x_n$ be real numbers such that $|x_{k+1}-x_k|\le 1$ for $k=1,2,\ldots ,n-1$. Prove that \[\sum_{k=1}^n|x_k|-\left|\sum_{k=1}^nx_k\right|\le\frac{n^2-1}{4}\] [i]Gh. Eckstein[/i]

2004 Romania National Olympiad, 2

Tags: floor function , inequalities , linear algebra

Let $n \in \mathbb N$, $n \geq 2$. (a) Give an example of two matrices $A,B \in \mathcal M_n \left( \mathbb C \right)$ such that \[ \textrm{rank} \left( AB \right) - \textrm{rank} \left( BA \right) = \left\lfloor \frac{n}{2} \right\rfloor . \] (b) Prove that for all matrices $X,Y \in \mathcal M_n \left( \mathbb C \right)$ we have \[ \textrm{rank} \left( XY \right) - \textrm{rank} \left( YX \right) \leq \left\lfloor \frac{n}{2} \right\rfloor . \] [i]Ion Savu[/i]

1996 Portugal MO, 2

Tags: algebra , inequalities

Prove that given three positive numbers, we can choose two of them, say $x$ and $y,$ with $x >y$ such that $$\frac{x-y}{1 +xy }<1.$$ Prove also that if the number $1$ that appears in the second member of the previous inequality is replaced by a lower number, even if very close to $1$, the previous proposition is false.

2018 Saudi Arabia GMO TST, 1

Tags: algebra , sequence , inequalities , recurrence relation , reciprocal

Let $\{x_n\}$ be a sequence defined by $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ for $n \ge 1$. Prove that $$1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}$$ for all $n$