Olympiad problems

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 $$a - \frac{1}{b}=b - \frac{1}{c}=c - \frac{1}{a}.$$

2016 CMIMC, 7

Tags: combinatorics

There are eight people, each with their own horse. The horses are arbitrarily arranged in a line from left to right, while the people are lined up in random order to the left of all the horses. One at a time, each person moves rightwards in an attempt to reach their horse. If they encounter a mounted horse on their way to their horse, the mounted horse shouts angrily at the person, who then scurries home immediately. Otherwise, they get to their horse safely and mount it. The expected number of people who have scurried home after all eight people have attempted to reach their horse can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

2019 HMNT, 9

Tags: geometry

For dinner, Priya is eating grilled pineapple spears. Each spear is in the shape of the quadrilateral $PINE$, with $PI = 6$ cm, $IN = 15$ cm, $NE = 6$ cm, $EP = 25$ cm, and $\angle NEP + \angle EPI = 60^o$: What is the area of each spear, in cm$^2$?

2014 Moldova Team Selection Test, 1

Tags: inequalities , algebra

Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.

2020 Online Math Open Problems, 15

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Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {{2020}\choose{k}} \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For example, if it were true that $n=12=2\times 2\times 3$, then the answer would be $2+2+3=7$.) [i]Proposed by Ankit Bisain[/i]

2013 Austria Beginners' Competition, 2

Tags: combinatorics

The following figure is given: [img]https://cdn.artofproblemsolving.com/attachments/9/b/97a30e248fcd6f098a900c89721a2e1b3b3f0e.png[/img] Determine the number of paths from the starting square $A$ to the target square $Z$, where a path consists of steps from a square to its top or right neighbor square . (W. Janous, WRG Ursulinen, Innsbruck)

1962 Putnam, B6

Tags: trigonometry , fourier

Let $$f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,$$ where $a_k$ and $b_k$ are constants. Show that if $|f(x)| \leq 1$ for $x \in [0, 2 \pi]$ and there exist $0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi$ with $|f(x_i )|=1,$ then $f(x)= \cos(nx +a)$ for some constant $a.$

2019 BMT Spring, 9

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Let $ a_n $ be the product of the complex roots of $ x^{2n} = 1 $ that are in the first quadrant of the complex plane. That is, roots of the form $ a + bi $ where $ a, b > 0 $. Let $ r = a_1 \cdots a_2 \cdot \ldots \cdot a_{10} $. Find the smallest integer $ k $ such that $ r $ is a root of $ x^k = 1 $.