Olympiad problems

2021 MOAA, 2

Add one pair of brackets to the expression \[1+2\times 3+4\times 5+6\] so that the resulting expression has a valid mathematical value, e.g., $1+2\times (3 + 4\times 5)+6=53$. What is the largest possible value that one can make? [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.13

Tags: MOAA 2021 , Gunga

Determine the greatest power of $2$ that is a factor of $3^{15}+3^{11}+3^{6}+1$. [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.8

Tags: MOAA 2021 , Gunga

Compute the number of triangles of different sizes which contain the gray triangle in the figure below. [asy] size(5cm); real n = 4; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } filldraw((1.5,0.866)--(2,2*0.866)--(2.5,0.866)--cycle, gray); [/asy] [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.20

Tags: MOAA 2021 , Gunga

In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$. [i]Proposed by Andrew Wen[/i]

MOAA Gunga Bowls, 2021.23

Tags: MOAA 2021 , Gunga

Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

2021 MOAA, 24

Tags: MOAA 2021 , Gunga

Freddy the Frog is situated at 1 on an infinitely long number line. On day $n$, where $n\ge 1$, Freddy can choose to hop 1 step to the right, stay where he is, or hop $k$ steps to the left, where $k$ is an integer at most $n+1$. After day 5, how many sequences of moves are there such that Freddy has landed on at least one negative number? [i]Proposed by Andy Xu[/i]

2021 MOAA, 19

Tags: MOAA 2021 , Gunga

Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\] can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 23

Tags: MOAA 2021 , Gunga

Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

MOAA Gunga Bowls, 2021.19

Tags: MOAA 2021 , Gunga

Let $S$ be the set of triples $(a,b,c)$ of non-negative integers with $a+b+c$ even. The value of the sum \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\] can be expressed as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.2

Tags: MOAA 2021 , Gunga

Add one pair of brackets to the expression \[1+2\times 3+4\times 5+6\] so that the resulting expression has a valid mathematical value, e.g., $1+2\times (3 + 4\times 5)+6=53$. What is the largest possible value that one can make? [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.7

Tags: MOAA 2021 , Gunga

Andover has a special weather forecast this week. On Monday, there is a $\frac{1}{2}$ chance of rain. On Tuesday, there is a $\frac{1}{3}$ chance of rain. This pattern continues all the way to Sunday, when there is a $\frac{1}{8}$ chance of rain. The probability that it doesn't rain in Andover all week can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

MOAA Gunga Bowls, 2021.15

Tags: MOAA 2021 , Gunga

Let $a,b,c,d$ be the four roots of the polynomial \[x^4+3x^3-x^2+x-2.\] Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}$ and $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}$, the value of \[\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 10

Tags: MOAA 2021 , Gunga

We say that an ordered pair $(a,b)$ of positive integers with $a>b$ is square-ish if both $a+b$ and $a-b$ are perfect squares. For example, $(17,8)$ is square-ish because $17+8=25$ and $17-8=9$ are both perfect squares. How many square-ish pairs $(a,b)$ with $a+b<100$ are there? [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 21

Tags: MOAA 2021 , Gunga

King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

MOAA Gunga Bowls, 2021.14

Tags: MOAA 2021 , Gunga

Sinclair starts with the number $1$. Every minute, he either squares his number or adds $1$ to his number, both with equal probability. What is the expected number of minutes until his number is divisible by $3$? [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 3

Tags: MOAA 2021 , Gunga

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

2021 MOAA, 5

Tags: MOAA 2021 , Gunga

Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2021 MOAA, 22

Tags: MOAA 2021 , Gunga

Let $p$ and $q$ be positive integers such that $p$ is a prime, $p$ divides $q-1$, and $p+q$ divides $p^2+2020q^2$. Find the sum of the possible values of $p$. [i]Proposed by Andy Xu[/i]

2021 MOAA, 18

Tags: MOAA 2021 , Gunga

Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$. [i]Proposed by Andy Xu[/i]

MOAA Gunga Bowls, 2021.3

Tags: MOAA 2021 , Gunga

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

2021 MOAA, 4

Tags: MOAA 2021 , Gunga

How many of the following capital English letters look the same when rotated $180^\circ$ about their center? [center]A B C D E F G H I J K L M N O P Q R S T U V W X Y Z[/center] [i]Proposed by William Yue[/i]

MOAA Gunga Bowls, 2021.12

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]

2021 MOAA, 13

Tags: MOAA 2021 , Gunga

Determine the greatest power of $2$ that is a factor of $3^{15}+3^{11}+3^{6}+1$. [i]Proposed by Nathan Xiong[/i]