Olympiad problems

2009 Today's Calculation Of Integral, 403

Evaluate $ \int_0^1 \frac{2e^{2x}\plus{}xe^x\plus{}3e^x\plus{}1}{(e^x\plus{}1)^2(e^x\plus{}x\plus{}1)^2}\ dx$.

2012 Online Math Open Problems, 48

Tags: Online Math Open , modular arithmetic , quadratics , Gauss , geometric series

Suppose that \[\sum_{i=1}^{982} 7^{i^2}\] can be expressed in the form $983q + r$, where $q$ and $r$ are integers and $0 \leq r \leq 492$. Find $r$. [i]Author: Alex Zhu[/i]

2020 Purple Comet Problems, 11

Tags: geometry

Two circles have radius $9$, and one circle has radius $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown. The sine of the base angle of the triangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/c34ff6bcaf6f07e6ba81a7d256e15a61f0e1fa.png[/img]

2006 Princeton University Math Competition, 9

Tags: combinatorics

A stick of length $10$ is marked with $9$ evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing either right or left. Suddenly, all the ants start walking simultaneously at a rate of $ 1$ unit per second. If two ants collide head-on, they immediately reverse direction (assume that turning takes no time). Ants fall off the stick as soon as they walk past the endpoints (so the two on the end don’t fall off immediately unless they are facing outwards). On average, how long (in seconds) will it take until all of the ants fall off of the stick?

2002 National High School Mathematics League, 1

Tags: function

The increasing interval of $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is $\text{(A)}(-\infty,-1)\qquad\text{(B)}(-\infty,1)\qquad\text{(C)}(1,+\infty)\qquad\text{(D)}(3,+\infty)$

2010 AIME Problems, 9

Tags: 3D geometry , quadratics , inequalities , AMC , AIME , function

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.

2024 Turkey Olympic Revenge, 1

Tags: combinatorics

Let $m,n$ be positive integers. An $n\times n$ board has rows and columns numbered $1,2,\dots,n$ from left to right and top to bottom, respectively. This board is colored with colors $r_1,r_2,\dots,r_m$ such that the cell at the intersection of $i$th row and $j$th column is colored with $r_{i+j-1}$ where indices are taken modulo $m$. After the board is colored, Ahmet wants to put $n$ stones to the board so that each row and column has exactly one stone, also he wants to put the same amount of stones to each color. Find all pairs $(m,n)$ for which he can accomplish his goal. Proposed by [i]Sena Başaran[/i]

2022 CHMMC Winter (2022-23), 1

Tags: combinatorics , CHMMC

Yor and Fiona are playing a match of tennis against each other. The first player to win $6$ games wins the match (while the other player loses the match). Yor has currently won $2$ games, while Fiona has currently won $0$ games. Each game is won by one of the two players: Yor has a probability of $\frac23$ to win each game, while Fiona has a probability of $\frac13$ to win each game. Then, $\frac{m}{n}$ is the probability Fiona wins the tennis match, for relatively prime integers $m,n$. Compute $m$.

2001 National Olympiad First Round, 18

Tags: inequalities , triangle inequality , geometry , perpendicular bisector

A convex polygon has at least one side with length $1$. If all diagonals of the polygon have integer lengths, at most how many sides does the polygon have? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 7 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ \text{None of the preceding} $

1989 Romania Team Selection Test, 4

Tags: function , Sets , divisor , combinatorics

A family of finite sets $\left\{ A_{1},A_{2},.......,A_{m}\right\} $is called [i]equipartitionable [/i] if there is a function $\varphi:\cup_{i=1}^{m}$$\rightarrow\left\{ -1,1\right\} $ such that $\sum_{x\in A_{i}}\varphi\left(x\right)=0$ for every $i=1,.....,m.$ Let $f\left(n\right)$ denote the smallest possible number of $n$-element sets which form a non-equipartitionable family. Prove that a) $f(4k +2) = 3$ for each nonnegative integer $k$, b) $f\left(2n\right)\leq1+m d\left(n\right)$, where $m d\left(n\right)$ denotes the least positive non-divisor of $n.$

2022 Bulgarian Spring Math Competition, Problem 11.3

Tags: combinatorics , table , inequalities

In every cell of a table with $n$ rows and $m$ columns is written one of the letters $a$, $b$, $c$. Every two rows of the table have the same letter in at most $k\geq 0$ positions and every two columns coincide at most $k$ positions. Find $m$, $n$, $k$ if \[\frac{2mn+6k}{3(m+n)}\geq k+1\]

1972 AMC 12/AHSME, 27

Tags: geometry , trigonometry , AMC

If the area of $\triangle ABC$ is $64$ square units and the geometric mean (mean proportional) between sides $AB$ and $AC$ is $12$ inches, then $\sin A$ is equal to $\textbf{(A) }\dfrac{\sqrt{3}}{2}\qquad\textbf{(B) }\frac{3}{5}\qquad\textbf{(C) }\frac{4}{5}\qquad\textbf{(D) }\frac{8}{9}\qquad \textbf{(E) }\frac{15}{17}$

1970 IMO Longlists, 37

Tags: algebra proposed , algebra

Solve the set of simultaneous equations \begin{align*} v^2+ w^2+ x^2+ y^2 &= 6 - 2u, \\ u^2+ w^2+ x^2+ y^2 &= 6 - 2v, \\ u^2+ v^2+ x^2+ y^2 &= 6- 2w, \\ u^2+ v^2+ w^2+ y^2 &= 6 - 2x, \\ u^2+ v^2+ w^2+ x^2 &= 6- 2y. \end{align*}

2003 Tournament Of Towns, 2

Tags: geometry unsolved , geometry

In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?

2016 PUMaC Number Theory B, 2

Tags: number theory

For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$. If $n$ is a two-digit positive integer such that $\frac{n}{s(n)}$ is a multiple of $3$, compute the sum of all possible values of $n$.

2016 ASDAN Math Tournament, 15

Tags: 2016 , Guts Round

Let $a$ be the least positive integer with $20$ positive divisors and $b$ be the least positive integer with $16$ positive divisors. What is $a+b$? (Note that for any integer $n$, both $1$ and $n$ are considered divisors of $n$.)

2013 Junior Balkan MO, 1

Tags: JBMO , JBMO Shortlist , number theory

Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers.

2012 JBMO ShortLists, 3

Tags:

Let $AB$ and $CD$ be chords in a circle of center $O$ with $A , B , C , D$ distinct , and with the lines $AB$ and $CD$ meeting at a right angle at point $E$. Let also $M$ and $N$ be the midpoints of $AC$ and $BD$ respectively . If $MN \bot OE$ , prove that $AD \parallel BC$.

2017 May Olympiad, 1

Tags: number theory , Digits

We shall call a positive integer [i]ascending [/i] if its digits read from left to right they are in strictly increasing order. For example, $458$ is ascending and $2339$ is not. Find the largest ascending number that is a multiple of $56$.

2021 MOAA, 12

Tags: MOAA 2021 , Gunga

Andy wishes to open an electronic lock with a keypad containing all digits from $0$ to $9$. He knows that the password registered in the system is $2469$. Unfortunately, he is also aware that exactly two different buttons (but he does not know which ones) $\underline{a}$ and $\underline{b}$ on the keypad are broken $-$ when $\underline{a}$ is pressed the digit $b$ is registered in the system, and when $\underline{b}$ is pressed the digit $a$ is registered in the system. Find the least number of attempts Andy needs to surely be able to open the lock. [i]Proposed by Andrew Wen[/i]

1983 IMO Longlists, 48

Tags: inequalities , geometry , parallelogram , inequalities proposed

Prove that in any parallelepiped the sum of the lengths of the edges is less than or equal to twice the sum of the lengths of the four diagonals.

1973 IMO, 1

Tags: geometry , triangle inequality , minimization , Triangle , optimization , IMO , IMO 1973

A soldier needs to check if there are any mines in the interior or on the sides of an equilateral triangle $ABC.$ His detector can detect a mine at a maximum distance equal to half the height of the triangle. The soldier leaves from one of the vertices of the triangle. Which is the minimum distance that he needs to traverse so that at the end of it he is sure that he completed successfully his mission?

2005 Italy TST, 2

Tags: geometry , geometric transformation , reflection , power of a point , geometry unsolved

The circle $\Gamma$ and the line $\ell$ have no common points. Let $AB$ be the diameter of $\Gamma$ perpendicular to $\ell$, with $B$ closer to $\ell$ than $A$. An arbitrary point $C\not= A$, $B$ is chosen on $\Gamma$. The line $AC$ intersects $\ell$ at $D$. The line $DE$ is tangent to $\Gamma$ at $E$, with $B$ and $E$ on the same side of $AC$. Let $BE$ intersect $\ell$ at $F$, and let $AF$ intersect $\Gamma$ at $G\not= A$. Let $H$ be the reflection of $G$ in $AB$. Show that $F,C$, and $H$ are collinear.

1964 AMC 12/AHSME, 1

Tags: logarithms , AMC

What is the value of $[\log_{10}(5\log_{10}100)]^2$? ${{ \textbf{(A)}\ \log_{10}50 \qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 2}\qquad\textbf{(E)}\ 1 } $

2013 Stanford Mathematics Tournament, 1

Tags:

Nick is a runner, and his goal is to complete four laps around a circuit at an average speed of $10$ mph. If he completes the first three laps at a constant speed of only $9$ mph, what speed does he need to maintain in miles per hour on the fourth lap to achieve his goal?