Olympiad problems

2020 AIME Problems, 7

Two congruent right circular cones each with base radius $3$ and height $8$ have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies inside both cones. The maximum possible value for $r^2$ is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2010 Contests, 3

Tags: circumcircle , trigonometry , perpendicular bisector , angle bisector , geometry

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

2003 IMO Shortlist, 4

Tags: linear algebra , matrix , combinatorics

Let $x_1,\ldots, x_n$ and $y_1,\ldots, y_n$ be real numbers. Let $A = (a_{ij})_{1\leq i,j\leq n}$ be the matrix with entries \[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\] Suppose that $B$ is an $n\times n$ matrix with entries $0$, $1$ such that the sum of the elements in each row and each column of $B$ is equal to the corresponding sum for the matrix $A$. Prove that $A=B$.

1986 Poland - Second Round, 5

Tags: algebra , polynomial

Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .

1955 AMC 12/AHSME, 41

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A train traveling from Aytown to Beetown meets with an accident after $ 1$ hr. It is stopped for $ \frac{1}{2}$ hr., after which it proceeds at four-fifths of its usual rate, arriving at Beetown $ 2$ hr. late. If the train had covered $ 80$ miles more before the accident, it would have been just $ 1$ hr. late. The usual rate of the train is: $ \textbf{(A)}\ \text{20 mph} \qquad \textbf{(B)}\ \text{30 mph} \qquad \textbf{(C)}\ \text{40 mph} \qquad \textbf{(D)}\ \text{50 mph} \qquad \textbf{(E)}\ \text{60 mph}$

2004 Putnam, A2

Tags: function , trigonometry , area of a triangle , heron's formula , geometry

For $i=1,2,$ let $T_i$ be a triangle with side length $a_i,b_i,c_i,$ and area $A_i.$ Suppose that $a_1\le a_2, b_1\le b_2, c_1\le c_2,$ and that $T_2$ is an acute triangle. Does it follow that $A_1\le A_2$?

2015 Denmark MO - Mohr Contest, 4

Tags: algebra , system of equations , system

Determine all numbers $x, y$ and $z$ satisfying the system of equations $$\begin{cases} x^2 + yz = 1 \\ y^2 - xz = 0 \\ z^2 + xy = 1\end{cases}$$

2015 Paraguay Mathematical Olympiad, 3

Tags: geometry

A cube is divided into $8$ smaller cubes of the same size, as shown in the figure. Then, each of these small cubes is divided again into $8$ smaller cubes of the same size. This process is done $4$ more times to each resulting cube. What is the ratio between the sum of the total areas of all the small cubes resulting from the last division and the total area of the initial cube?

2014 IMO Shortlist, A5

Tags: polynomial , algebra , functional inequality

Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has \[|y^2-P(x)|\le 2|x|\quad\text{if and only if}\quad |x^2-P(y)|\le 2|y|.\] Determine all possible values of $P(0)$. [i]Proposed by Belgium[/i]

2005 MOP Homework, 4

Tags: inequalities

Let $x_1$, $x_2$, ..., $x_5$ be nonnegative real numbers such that $x_1+x_2+x_3+x_4+x_5=5$. Determine the maximum value of $x_1x_2+x_2x_3+x_3x_4+x_4x_5$.

1988 USAMO, 3

Tags: pigeonhole principle , ceiling function , function , boolean lattice method

A function $f(S)$ assigns to each nine-element subset of $S$ of the set $\{1,2,\ldots, 20\}$ a whole number from $1$ to $20$. Prove that regardless of how the function $f$ is chosen, there will be a ten-element subset $T\subset\{1,2,\ldots, 20\}$ such that $f(T - \{k\})\neq k$ for all $k\in T$.

2008 Baltic Way, 4

Tags: polynomial , algebra

The polynomial $P$ has integer coefficients and $P(x)=5$ for five different integers $x$. Show that there is no integer $x$ such that $-6\le P(x)\le 4$ or $6\le P(x)\le 16$.

2020 Estonia Team Selection Test, 3

Tags: number theory , remainder

The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .

2024 Lusophon Mathematical Olympiad, 5

Tags: combinatorics

In a $9\times9$ board, the squares are labeled from 11 to 99, with the first digit indicating the row and the second digit indicating the column. One would like to paint the squares in black or white in a way that each black square is adjacent to at most one other black square and each white square is adjacent to at most one other white square. Two squares are adjacent if they share a common side. How many ways are there to paint the board such that the squares $44$ and $49$ are both black?

2023 CCA Math Bonanza, T4

Tags:

Triangle $ABC$ has side lengths $AB=7, BC=8, CA=9.$ Let $E$ be the foot from $B$ to $AC$ and $F$ be the foot from $C$ to $AB.$ Denote $M$ the midpoint of $BC.$ The circumcircles of $\triangle BMF$ and $\triangle CME$ meet at another point $G.$ Compute the length of $GC.$ [i]Team #4[/i]

2011 HMNT, 3

Tags: combinatorics

Alberto, Bernardo, and Carlos are collectively listening to three different songs. Each is simultaneously listening to exactly two songs, and each song is being listened to by exactly two people. In how many ways can this occur?

MOAA Team Rounds, 2021.1

Tags: team

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2004 ITAMO, 5

Tags: number theory

Decide if the following statement is true or false: For every sequence $\{x_n\}_{n\in \mathbb{N}}$ of non-negative real numbers, there exist sequences $\{a_n\}_{n\in\mathbb{N}}$ and $\{b_n\}_{n\in\mathbb{N}}$ of non-negative real numbers such that: (a) $x_n = a_n + b_n$ for all $n$; (b) $a_1 + \cdots + a_n \le n$ for infinitely many values of $n$; (c) $b_1 + \cdots + b_n \le n$ for infinitely many values of $n$.

Gheorghe Țițeica 2025, P2

Tags: function

Let $n\geq 2$ and consider the functions $f,g:\{1,2,\dots ,n\}\rightarrow\{1,2,\dots ,n\}$ such that $$g(k)=|\{i\mid f(i)\leq f(k)\}|$$ for all $1\leq k\leq n$. [list=a] [*] Show that $f$ is bijective if and only if $g$ is bijective. [*] If $g$ is a given function, find how many functions $f$ (in terms of $g$) satisfy the hypothesis. [/list] [i]Silviu Cristea[/i]

2020 USOMO, 1

Tags: minimize , geometry , olympiad geometry , area of a triangle

Let $ABC$ be a fixed acute triangle inscribed in a circle $\omega$ with center $O$. A variable point $X$ is chosen on minor arc $AB$ of $\omega$, and segments $CX$ and $AB$ meet at $D$. Denote by $O_1$ and $O_2$ the circumcenters of triangles $ADX$ and $BDX$, respectively. Determine all points $X$ for which the area of triangle $OO_1O_2$ is minimized. [i]Proposed by Zuming Feng[/i]

2016 Indonesia TST, 6

Tags: geometry

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

LMT Speed Rounds, 2010.4

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Determine the largest positive integer that is a divisor of all three of $A=2^{2010}\times3^{2010}, B=3^{2010}\times5^{2010},$ and $C=5^{2010}\times2^{2010}.$

2020 Greece Team Selection Test, 1

Tags: functional , functional equation , algebra

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

2021 HMNT, 6

Tags: number theory

Let $n$ be the answer to this problem. $a$ and $b$ are positive integers satisfying $$3a + 5b \equiv 19 \,\,\, (mod \,\,\, n + 1)$$ $$4a + 2b \equiv 25 \,\,\, (mod \,\,\, n + 1)$$ Find $ 2a + 6b$.

2001 AMC 12/AHSME, 3

Tags:

The state income tax where Kristin lives is levied at the rate of $ p \%$ of the first $ \$28000$ of annual income plus $ (p \plus{} 2) \%$ of any amount above $ \$28000$. Kristin noticed that the state income tax she paid amounted to $ (p \plus{} 0.25) \%$ of her annual income. What was her annual income? $ \textbf{(A)} \ \$28000 \qquad \textbf{(B)} \ \$32000 \qquad \textbf{(C)} \ \$35000 \qquad \textbf{(D)} \ \$42000 \qquad \textbf{(E)} \ \$56000$