Olympiad problems

2000 IMO Shortlist, 1

Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that \[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1. \]

2012 Online Math Open Problems, 20

Tags: geometry , rectangle , similar triangles

Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle? [i]Author: Ray Li[/i]

2016 CHMMC (Fall), 9

Tags: number theory

Find the sum of all $3$-digit numbers whose digits, when read from left to right, form a strictly increasing sequence. (Numbers with a leading zero, e.g. ”$087$” or ”$002$”, are not counted as having $3$ digits.)

2000 Romania National Olympiad, 3

Tags: abstract algebra , group theory , superior algebra , morphism , homomorphism

We say that the abelian group $ G $ has property [i](P)[/i] if, for any commutative group $ H, $ any $ H’\le H $ and any momorphism $ \mu’:H\longrightarrow G, $ there exists a morphism $ \mu :H\longrightarrow G $ such that $ \mu\bigg|_{H’} =\mu’ . $ Show that: [b]a)[/b] the group $ \left( \mathbb{Q}^*,\cdot \right) $ hasn’t property [i](P).[/i] [b]b)[/b] the group $ \left( \mathbb{Q}, +\right) $ has property [i](P).[/i]

1995 Israel Mathematical Olympiad, 4

Tags: diophantine , diophantine equation , number theory

Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

2011 Miklós Schweitzer, 7

Tags: real analysis , sequence

prove that for any sequence of nonnegative numbers $(a_n)$, $$\liminf_{n\to\infty} (n^2(4a_n(1-a_{n-1})-1))\leq\frac{1}{4}$$

2020 AIME Problems, 5

Tags: counting , distinguishability

Six cards numbered 1 through 6 are to be lined up in a row. Find the number of arrangements of these six cards where one of the cards can be removed leaving the remaining five cards in either ascending or descending order.

2012 Grigore Moisil Intercounty, 4

Tags: complex number , circumcircle

[b]a)[/b] Let $ A $ denote the complex numbers of modulus $ 1/3, $ and $ B $ denote the complex numbers of modulus at least $ 1/2. $ Show that $ A+B=AB\neq\mathbb{C} . $ [b]b)[/b] Prove that there is no family $ Y $ of complex numbers that satisfies $ X+Y=XY\neq\mathbb{C} , $ where $ X $ denotes the complex numbers of modulus $ 1. $

2020 Canada National Olympiad, 4

Tags: number theory

$S= \{1,4,8,9,16,...\} $is the set of perfect integer power. ( $S=\{ n^k| n, k \in Z, k \ge 2 \}$. )We arrange the elements in $S$ into an increasing sequence $\{a_i\}$ . Show that there are infinite many $n$, such that $9999|a_{n+1}-a_n$

2007 All-Russian Olympiad, 4

Tags: circumcircle , asymptote , cyclic quadrilateral , angle bisector , geometry

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

PEN J Problems, 21

Tags: divisor function

Show that for any positive integer $n$, \[\frac{\sigma(n!)}{n!}\ge \sum_{k=1}^{n}\frac{1}{k}.\]

1950 Miklós Schweitzer, 3

Tags: combinatorics

Let $ E$ be a system of $ n^2 \plus{} 1$ closed intervals of the real line. Show that $ E$ has either a subsystem consisting of $ n \plus{} 1$ elements which are monotonically ordered with respect to inclusion or a subsystem consisting of $ n \plus{} 1$ elements none of which contains another element of the subsystem.

1974 IMO Shortlist, 4

Tags: inequality , polynomial , distance , minimization , optimization

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

2004 ITAMO, 6

Tags: geometry

Let $P$ be a point inside a triangle $ABC$. Lines $AP,BP,CP$ meet the opposite sides of the triangle at points $A',B',C'$ respectively. Denote $x =\frac{AP}{PA'}, y = \frac{BP}{PB'}$ and $z = \frac{CP}{PC'}$. Prove that $xyz = x+y+z+2$.

2006 Princeton University Math Competition, 1

Tags:

Given that $x^2+5x+6=20$, find the value of $3x^2+15x+17$.

2024 CMIMC Algebra and Number Theory, 7

Tags: algebra

Let $x_0$, $x_1$, $x_2$, and $x_3$ be complex numbers forming a square centered at $0$ in the complex plane with side length $2$. For each $0\leq k\leq 3$, there are four more complex numbers $z_{4k}, z_{4k+1}$, $z_{4k+2}$, and $z_{4k+3}$ forming a square centered at $x_k$ with side length $\sqrt 2$. Given that $\prod_{i=0}^{15} z_i$ is a positive integer, how many possible values could it take? [i]Proposed by Hari Desikan[/i]

2011 National Olympiad First Round, 11

Tags: algebra , polynomial

The sum of distinct real roots of the polynomial $x^5+x^4-4x^3-7x^2-7x-2$ is $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ -2 \qquad\textbf{(E)}\ 7$

2021/2022 Tournament of Towns, P3

Tags: combinatorics , board

In a checkered square of size $2021\times 2021$ all cells are initially white. Ivan selects two cells and paints them black. At each step, all the cells that have at least one black neighbor by side are painted black simultaneously. Ivan selects the starting two cells so that the entire square is painted black as fast as possible. How many steps will this take? [i]Ivan Yashchenko[/i]

2022 Swedish Mathematical Competition, 2

Tags: algebra , functional equation

Find all functions $f : R \to R$ such that $$f(x + zf(y)) = f(x) + zf(y), $$ for all $x, y, z \in R$.

XMO (China) 2-15 - geometry, 11.1

Tags: geometry

Let $\triangle ABC$ be connected to the circle $\Gamma$. The angular bisector of $\angle BAC$ intersects $BC$ to $D$. Straight line $BP$ intersects $AC$ to $E$, and straight line $CP$ intersects $AB$ to $F$. Let the tangent of the circle $\Gamma$ at point $A$ intersect the line $EF$ at the point $Q$. Proof: $PQ\parallel BC$.

2013 Ukraine Team Selection Test, 5

Tags: inequalities , algebra

For positive $x, y$, and $z$ that satisfy the condition $xyz = 1$, prove the inequality $$\sqrt[3]{\frac{x+y}{2z}}+\sqrt[3]{\frac{y+z}{2x}}+\sqrt[3]{\frac{z+x}{2y}}\le \frac{5(x+y+z)+9}{8}$$

Kvant 2023, M2777

Tags: combinatorial geometry , homothety , combinatorics , geometry

A convex polygon $\mathcal{P}$ with a center of symmetry $O{}$ is drawn in the plane. Prove that it is possible to place a rhombus in $\mathcal{P}$ whose image following a homothety of factor two centered at $O$ contains $\mathcal{P}$. [i]Proposed by I. Bogdanov, S. Gerdzhikov and N. Nikolov[/i]

2020 CHKMO, 3

Tags: geometry , incircle , circumcircle

Let $\Delta ABC$ be an isosceles triangle with $AB=AC$. The incircle $\Gamma$ of $\Delta ABC$ has centre $I$, and it is tangent to the sides $AB$ and $AC$ at $F$ and $E$ respectively. Let $\Omega$ be the circumcircle of $\Delta AFE$. The two external common tangents of $\Gamma$ and $\Omega$ intersect at a point $P$. If one of these external common tangents is parallel to $AC$, prove that $\angle PBI=90^{\circ}$.

2024 Brazil Undergrad MO, 2

Tags: real analysis , derivative , fixed point , algebra , polynomial

For each pair of integers $ j, k \geq 2 $, define the function $ f_{jk} : \mathbb{R} \to \mathbb{R} $ given by \[ f_{jk}(x) = 1 - (1 - x^j)^k. \] (a) Prove that for any integers $ j, k \geq 2 $, there exists a unique real number $ p_{jk} \in (0, 1) $ such that $ f_{jk}(p_{jk}) = p_{jk} $. Furthermore, defining $ \lambda_{jk} := f'_{jk}(p_{jk}) $, prove that $ \lambda_{jk} > 1 $. (b) Prove that $ p^j_{jk} = 1 - p_{kj} $ for any integers $ j, k \geq 2 $. (c) Prove that $ \lambda_{jk} = \lambda_{kj} $ for any integers $ j, k \geq 2 $.

1969 IMO Shortlist, 30

Tags: composite number , number theory , sophie germain identity

$(GDR 2)^{IMO1}$ Prove that there exist infinitely many natural numbers $a$ with the following property: The number $z = n^4 + a$ is not prime for any natural number $n.$