Olympiad problems

1995 Chile National Olympiad, 7

In a semicircle of radius $4$ three circles are inscribed, as indicated in the figure. Larger circles have radii $ R_1 $ and $ R_2 $, and the larger circle has radius $ r $. a) Prove that $ \dfrac {1} {\sqrt{r}} = \dfrac {1} {\sqrt{R_1}} + \dfrac {1} {\sqrt{R_2}} $ b) Prove that $ R_1 + R_2 \le 8 (\sqrt{2} -1) $ c) Prove that $ r \le \sqrt{2} -1 $ [img]https://cdn.artofproblemsolving.com/attachments/0/9/aaaa65d1f4da4883973751e1363df804b9944c.jpg[/img]

2013 Princeton University Math Competition, 3

Tags: modular arithmetic , number theory

Find the smallest positive integer $x$ such that [list] [*] $x$ is $1$ more than a multiple of $3$, [*] $x$ is $3$ more than a multiple of $5$, [*] $x$ is $5$ more than a multiple of $7$, [*] $x$ is $9$ more than a multiple of $11$, and [*] $x$ is $2$ more than a multiple of $13$.[/list]

LMT Team Rounds 2010-20, B7

Tags: number theory

Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$

2022 Germany Team Selection Test, 1

Tags: algebra

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

1999 AMC 12/AHSME, 6

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What is the sum of the digits of the decimal form of the product $ 2^{1999}\cdot 5^{2001}$? $ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 10$

2007 QEDMO 4th, 12

Tags: geometry , reflection , incenter , circumcircle , radical axis

Let $ABC$ be a triangle, and let $D$, $E$, $F$ be the points of contact of its incircle $\omega$ with its sides $BC$, $CA$, $AB$, respectively. Let $K$ be the point of intersection of the line $AD$ with the incircle $\omega$ different from $D$, and let $M$ be the point of intersection of the line $EF$ with the line perpendicular to $AD$ passing through $K$. Prove that $AM$ is parallel to $BC$.

2015 India Regional MathematicaI Olympiad, 5

Tags: geometry , incenter , circumcircle , ratio

Let ABC be a right triangle with $\angle B = 90^{\circ}$.Let E and F be respectively the midpoints of AB and AC.Suppose the incentre I of ABC lies on the circumcircle of triangle AEF,find the ratio BC/AB.

2010 LMT, 13

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Let $ABC$ be a non-degenerate triangle inscribed in a circle, such that $AB$ is the diameter of the circle. Let the angle bisectors of the angles at $A$ and $B$ meet at $P.$ Determine the maximum possible value of $\angle APB,$ in degrees.

2016 Purple Comet Problems, 13

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One afternoon Elizabeth noticed that twice as many cars on the expressway carried only a driver as compared to the number of cars that carried a driver and one passenger. She also noted that twice as many cars carried a driver and one passenger as those that carried a driver and two passengers. Only 10% of the cars carried a driver and three passengers, and no car carried more than four people. Any car containing at least three people was allowed to use the fast lane. Elizabeth calculated that $\frac{m}{n}$ of the people in cars on the expressway were allowed to ride in the fast lane, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2012 AMC 10, 9

Tags: probability

A pair of six-sided fair dice are labeled so that one die has only even numbers (two each of $2$, $4$, and $6$), and the other die has only odd numbers (two each of $1$, $3$, and $5$). The pair of dice is rolled. What is the probability that the sum of the numbers on top of the two dice is $7$? $ \textbf{(A)}\ \dfrac{1}{6} \qquad\textbf{(B)}\ \dfrac{1}{5} \qquad\textbf{(C)}\ \dfrac{1}{4} \qquad\textbf{(D)}\ \dfrac{1}{3} \qquad\textbf{(E)}\ \dfrac{1}{2} $

1991 AMC 8, 21

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For every $3^\circ $ rise in temperature, the volume of a certain gas expands by $4$ cubic centimeters. If the volume of the gas is $24$ cubic centimeters when the temperature is $32^\circ $, what was the volume of the gas in cubic centimeters when the temperature was $20^\circ $? $\text{(A)}\ 8 \qquad \text{(B)}\ 12 \qquad \text{(C)}\ 15 \qquad \text{(D)}\ 16 \qquad \text{(E)}\ 40$

1963 German National Olympiad, 5

Tags: geometry , locus , midpoint , square

Given is a square with side length $a$. A distance $PQ$ of length $p$, where $p < a$, moves so that its end points are always on the sides of the square. What is the geometric locus of the midpoints of the segments $PQ$?

Gheorghe Țițeica 2025, P1

Tags: combinatorics , combinatorial geometry

Let there be $2n+1$ distinct points on a circle. Consider the set of distances between any two out of the $2n+1$ points. What is the smallest size of this set? [i]Radu Bumbăcea[/i]

1999 Brazil Team Selection Test, Problem 2

Tags: geometric inequality , inequalities , geometry , ratio

In a triangle $ABC$, the bisector of the angle at $A$ of a triangle $ABC$ intersects the segment $BC$ and the circumcircle of $ABC$ at points $A_1$ and $A_2$, respectively. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that $$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$

1992 Tournament Of Towns, (329) 6

Tags: combinatorics , combinatorial geometry

A circle is divided into $n$ sectors. Pawns stand on some of the sectors; the total number of pawns equals $n + 1$. This configuration is changed as follows. Any two of the pawns standing on the same sector move simultaneously to the neighbouring sectors in different directions. Prove that after several such transformations a configuration in which no less than half of the sectors are occupied by pawns, will inevitably appear. (D. Fomin, St Petersburg)

2013 Hitotsubashi University Entrance Examination, 5

Tags: probability , function , algebra , polynomial , geometric series , probability and stats

Throw a die $n$ times, let $a_k$ be a number shown on the die in the $k$-th place. Define $s_n$ by $s_n=\sum_{k=1}^n 10^{n-k}a_k$. (1) Find the probability such that $s_n$ is divisible by 4. (2) Find the probability such that $s_n$ is divisible by 6. (3) Find the probability such that $s_n$ is divisible by 7. Last Edited Thanks, jmerry & JBL

2007 IMAC Arhimede, 4

Tags: divide , sum , sums and products , combinatorics , pigeonhole principle , number theory

Prove that for any given number $a_k, 1 \le k \le 5$, there are $\lambda_k \in \{-1, 0, 1\}, 1 \le k \le 5$, which are not all equal zero, such that $11 | \lambda_1a_1^2+\lambda_2a_2^2+\lambda_3a_3^2+\lambda_4a_4^2+\lambda_5a_5^2$

2022/2023 Tournament of Towns, P5

Tags: combinatorics , game

Alice has 8 coins. She knows for sure only that 7 of these coins are genuine and weigh the same, while the remaining one is counterfeit and is either heavier or lighter than any of the other 7. Bob has a balance scale. The scale shows which plate is heavier but does not show by how much. For each measurement, Alice pays Bob beforehand a fee of one coin. If a genuine coin has been paid, Bob tells Alice the correct weighing outcome, but if a counterfeit coin has been paid, he gives a random answer. Alice wants to identify 5 genuine coins and not to give any of these coins to Bob. Can Alice achieve this result for sure?

1969 IMO Longlists, 54

Tags: algebra , polynomial , modular arithmetic , number theory , divisibility

$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

2004 District Olympiad, 3

Tags: algebra , inequalities

One considers the set $$A = \left\{ n \in N^* \big | 1 < \sqrt{1 + \sqrt{n}} < 2 \right\}$$ a) Find the set $A$. b) Find the set of numbers $n \in A$ such that $$\sqrt{n} \cdot \left| 1-\sqrt{1 + \sqrt{n}}\right| <1 ?$$

2019 BMT Spring, 8

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Let $ \triangle ABC $ be a triangle with $ AB = 13 $, $ BC = 14 $, and $ CA = 15 $. Let $ G $ denote the centroid of $ \triangle ABC $, and let $ G_A $ denote the image of $ G $ under a reflection across $ \overline{BC} $, with $ G_B $ the image of $ G $ under a reflection across $ \overline{AC} $, and $ G_C $ the image of $ G $ under a reflection across $ \overline{AB} $. Let $ O_G $ be the circumcenter of $ \triangle G_AG_BG_C $ and let $ X $ be the intersection of $ \overline{AO_G} $ with $ \overline{BC} $. Let $ Y $ denote the intersection of $ \overline{AG} $ with $ \overline{BC} $. Compute $ XY $.

2020 Bulgaria Team Selection Test, 6

Tags: geometry

In triangle $\triangle ABC$, $BC>AC$, $I_B$ is the $B$-excenter, the line through $C$ parallel to $AB$ meets $BI_B$ at $F$. $M$ is the midpoint of $AI_B$ and the $A$-excircle touches side $AB$ at $D$. Point $E$ satisfies $\angle BAC=\angle BDE, DE=BC$, and lies on the same side as $C$ of $AB$. Let $EC$ intersect $AB,FM$ at $P,Q$ respectively. Prove that $P,A,M,Q$ are concyclic.

2007 Stanford Mathematics Tournament, 5

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How many five-letter "words" can you spell using the letters $S$, $I$, and $T$, if a "word" is defines as any sequence of letters that does not contain three consecutive consonants?

2003 China Team Selection Test, 3

Tags: function , combinatorics

Let $A= \{a_1,a_2, \cdots, a_n \}$ and $B=\{b_1,b_2 \cdots, b_n \}$ be two positive integer sets and $|A \cap B|=1$. $C= \{ \text{all the 2-element subsets of A} \} \cup \{ \text{all the 2-element subsets of B} \}$. Function $f: A \cup B \to \{ 0, 1, 2, \cdots 2 C_n^2 \}$ is injective. For any $\{x,y\} \in C$, denote $|f(x)-f(y)|$ as the $\textsl{mark}$ of $\{x,y\}$. If $n \geq 6$, prove that at least two elements in $C$ have the same $\textsl{mark}$.

2016 Regional Olympiad of Mexico Northeast, 5

Tags: algebra , system of equations

Find all triples of reals $(a, b, c)$ such that $$a - \frac{1}{b}=b - \frac{1}{c}=c - \frac{1}{a}.$$