Olympiad problems

2023 MOAA, 11

Tags: MOAA 2023

Let $s(n)$ denote the sum of the digits of $n$ and let $p(n)$ be the product of the digits of $n$. Find the smallest integer $k$ such that $s(k)+p(k)=49$ and $s(k+1)+p(k+1)=68$. [i]Proposed by Anthony Yang[/i]

MOAA Gunga Bowls, 2023.24

Tags: MOAA 2023

Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$. [i]Proposed by Harry Kim[/i]

MOAA Team Rounds, 2023.11

Tags: MOAA 2023

Let the quadratic $P(x)=x^2+5x+1$. Two distinct real numbers $a,b$ satisfy \[P(a+b)=ab\] \[P(ab)=a+b\] Find the sum of all possible values of $a^2$. [i]Proposed by Harry Kim[/i]

MOAA Accuracy Rounds, 2023.2

Tags: MOAA 2023

Let $ABCD$ be a square. Let $M$ be the midpoint of $BC$ and $N$ be the point on $AB$ such that $2AN=BN$. If the area of $\triangle DMN$ is 15, find the area of square $ABCD$. [i]Proposed by Harry Kim[/i]

MOAA Gunga Bowls, 2023.8

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]

2023 MOAA, 24

Tags: MOAA 2023

Circle $\omega$ is inscribed in acute triangle $ABC$. Let $I$ denote the center of $\omega$, and let $D,E,F$ be the points of tangency of $\omega$ with $BC, CA, AB$ respectively. Let $M$ be the midpoint of $BC$, and $P$ be the intersection of the line through $I$ perpendicular to $AM$ and line $EF$. Suppose that $AP=9$, $EC=2EA$, and $BD=3$. Find the sum of all possible perimeters of $\triangle ABC$. [i]Proposed by Harry Kim[/i]

MOAA Team Rounds, 2023.13

Tags: MOAA 2023

If real numbers $x$, $y$, and $z$ satisfy $x^2-yz = 1$ and $y^2-xz = 4$ such that $|x+y+z|$ is minimized, then $z^2-xy$ can be expressed in the form $\sqrt{a}-b$ where $a$ and $b$ are positive integers. Find $a+b$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 9

Tags: MOAA 2023

Let $ABCDEF$ be an equiangular hexagon. Let $P$ be the point that is a distance of 6 from $BC$, $DE$, and $FA$. If the distances from $P$ to $AB$, $CD$, and $EF$ are $8$, $11$, and $5$ respectively, find $(DE-AB)^2$. [i]Proposed by Andy Xu[/i]

MOAA Team Rounds, 2023.12

Tags: MOAA 2023

Let $N$ be the number of $105$-digit positive integers that contain the digit 1 an odd number of times. Find the remainder when $N$ is divided by $1000$. [i]Proposed by Harry Kim[/i]

MOAA Team Rounds, 2023.15

Tags: MOAA 2023

Triangle $ABC$ has circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $AD$ intersect $\omega$ at $E \neq A$. Let $M$ be the midpoint of $AD$. If $\angle{BMC} = 90^\circ$, $AB = 9$ and $AE = 10$, the area of $\triangle{ABC}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 18

Tags: MOAA 2023

Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. [i]Proposed by Anthony Yang and Andy Xu[/i]

2023 MOAA, 7

Tags: MOAA 2023

In a cube, let $M$ be the midpoint of one of the segments. Choose two vertices of the cube, $A$ and $B$. What is the number of distinct possible triangles $\triangle AMB$ up to congruency? [i]Proposed by Harry Kim[/i]

MOAA Gunga Bowls, 2023.17

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, 2023.3

Tags: MOAA 2023

After the final exam, Mr. Liang asked each of his 17 students to guess the average final exam score. David, a very smart student, received a 100 and guessed the average would be 97. Each of the other 16 students guessed $30+\frac{n}{2}$ where $n$ was that student’s score. If the average of the final exam scores was the same as the average of the guesses, what was the average score on the final exam? [i]Proposed by Eric Wang[/i]

2023 MOAA, 15

Tags: MOAA 2023

Triangle $ABC$ has $AB = 5$, $BC = 7$, $CA = 8$. Let $M$ be the midpoint of $BC$ and let points $P$ and $Q$ lie on $AB$ and $AC$ respectively such that $MP \perp AB$ and $MQ \perp AC$. If $H$ is the orthocenter of $\triangle{APQ}$ then the area of $\triangle{HPM}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a$ and $c$ are relatively prime positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Harry Kim[/i]

MOAA Individual Speed General Rounds, 2023.2

Tags: MOAA 2023

In the coordinate plane, the line passing through points $(2023,0)$ and $(-2021,2024)$ also passes through $(1,c)$ for a constant $c$. Find $c$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 10

Tags: MOAA 2023

If $x,y,z$ satisfy the system of equations \[xy+yz+zx=23\] \[\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1\] \[\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202\] Find the value of $x^2+y^2+z^2$. [i]Proposed by Harry Kim[/i]

MOAA Team Rounds, 2023.10

Tags: MOAA 2023

Let $S$ be the set of lattice points $(a,b)$ in the coordinate plane such that $1\le a\le 30$ and $1\le b\le 30$. What is the maximum number of lattice points in $S$ such that no four points form a square of side length 2? [i]Proposed by Harry Kim[/i]

2023 MOAA, 5

Tags: MOAA 2023

Angeline starts with a 6-digit number and she moves the last digit to the front. For example, if she originally had $100823$ she ends up with $310082$. Given that her new number is $4$ times her original number, find the smallest possible value of her original number. [i]Proposed by Angeline Zhao[/i]

2023 MOAA, 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]

MOAA Individual Speed General Rounds, 2023.8

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]

MOAA Individual Speed General Rounds, 2023.4

Tags: MOAA 2023

A number is called \textit{super odd} if it is an odd number divisible by the square of an odd prime. For example, $2023$ is a \textit{super odd} number because it is odd and divisible by $17^2$. Find the sum of all \textit{super odd} numbers from $1$ to $100$ inclusive. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2023.22

Tags: MOAA 2023

Harry the knight is positioned at the origin of the Cartesian plane. In a "knight hop", Harry can move from the point $(i,j)$ to a point with integer coordinates that is a distance of $\sqrt{5}$ away from $(i,j)$. What is the number of ways that Harry can return to the origin after 6 knight hops? [i]Proposed by Harry Kim[/i]

2023 MOAA, 15

Tags: MOAA 2023

Triangle $ABC$ has circumcircle $\omega$. Let $D$ be the foot of the altitude from $A$ to $BC$ and let $AD$ intersect $\omega$ at $E \neq A$. Let $M$ be the midpoint of $AD$. If $\angle{BMC} = 90^\circ$, $AB = 9$ and $AE = 10$, the area of $\triangle{ABC}$ can be expressed in the form $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers and $b$ is square-free. Find $a+b+c$. [i]Proposed by Andy Xu[/i]

2023 MOAA, 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]