Olympiad problems

MOAA Accuracy Rounds, 2023.1

Tags: MOAA 2023

Compute $$\left(20+\frac{1}{23}\right)\cdot\left(23+\frac{1}{20}\right)-\left(20-\frac{1}{23}\right)\cdot\left(23-\frac{1}{20}\right)$$ [i]Proposed by Andy Xu[/i]

2023 MOAA, 13

Tags: MOAA 2023

Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 1

Tags: MOAA 2023

Find the last two digits of $2023+202^3+20^{23}$. [i]Proposed by Anthony Yang[/i]

Two consecutive positive integers $n$ and $n+1$ have the property that they both have $6$ divisors but a different number of distinct prime factors. Find the sum of the possible values of $n$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 3

Tags: MOAA 2023

Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 7

Tags: MOAA 2023

Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$. If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $m+n$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 14

Tags: MOAA 2023

For a positive integer $n$, let function $f(n)$ denote the number of positive integers $a\leq n$ such that $\gcd(a,n) = \gcd(a+1,n) = 1$. Find the sum of all $n$ such that $f(n)=15$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 17

Tags: MOAA 2023

Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$. [i]Proposed by Anthony Yang[/i]

MOAA Team Rounds, 2023.14

Tags: MOAA 2023

For a positive integer $n$, let function $f(n)$ denote the number of positive integers $a\leq n$ such that $\gcd(a,n) = \gcd(a+1,n) = 1$. Find the sum of all $n$ such that $f(n)=15$. [i]Proposed by Harry Kim[/i]

2023 MOAA, 19

Tags: MOAA 2023

Compute the remainder when $\binom{205}{101}$ is divded by $101 \times 103$. [i]Proposed by Brandon Xu[/i]

2023 MOAA, 8

Tags: MOAA 2023

Let $ABCD$ be a parallelogram with area 160. Let diagonals $AC$ and $BD$ intersect at $E$. Point $P$ is on $\overline{AE}$ such that $EC = 4EP$. If line $DP$ intersects $AB$ at $F$, find the area of $BFPC$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2023.4

Tags: MOAA 2023

An equilateral triangle with side length 2023 has area $A$ and a regular hexagon with side length 289 has area $B$. If $\frac{A}{B}$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime, find $m+n$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 8

Tags: MOAA 2023

In the coordinate plane, Yifan the Yak starts at $(0,0)$ and makes $11$ moves. In a move, Yifan can either do nothing or move from an arbitrary point $(i,j)$ to $(i+1,j)$, $(i,j+1)$ or $(i+1,j+1)$. How many points $(x,y)$ with integer coordinates exist such that the number of ways Yifan can end on $(x,y)$ is odd? [i]Proposed by Yifan Kang[/i]

2023 MOAA, 20

Tags: MOAA 2023

Big Bad Brandon is assigning groups of his Big Bad Burglars to attack 7 different towers. Each Burglar can only belong to one attack group and Brandon takes over a tower if the number of Burglars attacking the tower strictly exceeds the number of knights guarding it. He knows there the total number of knights guarding the towers is 99 but does not know the exact number of knights guarding each tower. What is the minimum number of Burglars that Brandon needs to guarantee he can take over at least 4 of the 7 towers? [i]Proposed by Eric Wang[/i]

2023 MOAA, 9

Tags: MOAA 2023

Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $BC \perp CD$. There exists a point $P$ on $BC$ such that $\triangle{PAD}$ is equilateral. If $PB = 20$ and $PC = 23$, the area of $ABCD$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $b$ is square-free and $a$ and $c$ are relatively prime. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 9

Tags: MOAA 2023

Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. [color=#00f]Note that this problem is null because a diagram is impossible.[/color] [i]Proposed by Andy Xu[/i]

2023 MOAA, 1

Tags: MOAA 2023

Compute $$\left(20+\frac{1}{23}\right)\cdot\left(23+\frac{1}{20}\right)-\left(20-\frac{1}{23}\right)\cdot\left(23-\frac{1}{20}\right)$$ [i]Proposed by Andy Xu[/i]

MOAA Accuracy Rounds, 2023.10

Tags: MOAA 2023

Let $S$ be a set of integers such that if $a$ and $b$ are in $S$ then $3a-2b$ is also in $S$. How many ways are there to construct $S$ such that $S$ contains exactly $4$ elements in the interval $[0,40]$? [i]Proposed by Harry Kim[/i]

2023 MOAA, 6

Tags: MOAA 2023

Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$. [i]Proposed by Anthony Yang[/i]

MOAA Gunga Bowls, 2023.20

Tags: MOAA 2023

Big Bad Brandon is assigning groups of his Big Bad Burglars to attack 7 different towers. Each Burglar can only belong to one attack group and Brandon takes over a tower if the number of Burglars attacking the tower strictly exceeds the number of knights guarding it. He knows there the total number of knights guarding the towers is 99 but does not know the exact number of knights guarding each tower. What is the minimum number of Burglars that Brandon needs to guarantee he can take over at least 4 of the 7 towers? [i]Proposed by Eric Wang[/i]

MOAA Gunga Bowls, 2023.23

Tags: MOAA 2023

For every positive integer $n$ let $$f(n) = \frac{n^4+n^3+n^2-n+1}{n^6-1}$$ Given $$\sum_{n = 2}^{20} f(n) = \frac{a}{b}$$ for relatively prime positive integers $a$ and $b$, find the sum of the prime factors of $b$. [i]Proposed by Harry Kim[/i]

MOAA Gunga Bowls, 2023.7

Tags: MOAA 2023

Written in mm/dd format, a date is called [i]cute[/i] if the month is divisible by the day. For example, the date [i]cute[/i] is a [i]cute[/i] date because $8$ is divisible by $2$. Find the number of [i]cute[/i] dates in a year. [i]Proposed by Andy Xu[/i]

MOAA Team Rounds, 2023.6

Tags: MOAA 2023

Call a set of integers [i]unpredictable[/i] if no four elements in the set form an arithmetic sequence. How many unordered [i]unpredictable[/i] sets of five distinct positive integers $\{a, b, c, d, e\}$ exist such that all elements are strictly less than $12$? [i]Proposed by Anthony Yang[/i]

2023 MOAA, 10

Tags: MOAA 2023

Let $S$ be a set of integers such that if $a$ and $b$ are in $S$ then $3a-2b$ is also in $S$. How many ways are there to construct $S$ such that $S$ contains exactly $4$ elements in the interval $[0,40]$? [i]Proposed by Harry Kim[/i]

MOAA Gunga Bowls, 2023.21

Tags: MOAA 2023

In obtuse triangle $ABC$ where $\angle B > 90^\circ$ let $H$ and $O$ be its orthocenter and circumcenter respectively. Let $D$ be the foot of the altitude from $A$ to $HC$ and $E$ be the foot of the altitude from $B$ to $AC$ such that $O,E,D$ lie on a line. If $OC=8$ and $OE=4$, find the area of triangle $HAB$. [i]Proposed by Harry Kim[/i]