Found problems: 85
2020-2021 OMMC, 1
There are $20$ people in a particular social network. Each person follows exactly $2$ others in this network, and also has $2$ people following them as well. What is the maximum possible number of people that can be placed into a subset of the network such that no one in this subset follows someone else in the subset?
2020-2021 OMMC, 9
There is a $4 \times 4$ array of integers $A$, all initially equal to $0$. An operation may be performed on the array for any row or column such that every number in that row or column has $1$ added to it, and then is replaced with its remainder modulo $3$. Given a random $4 \times 4$ array of integers between $0$ and $2$ not identical to $A$, the probability that it can be reached through a series of operations on $A$ is $\frac{p}{q},$ where $p,q$ are relatively prime positive integers. Find $p$.
2020-2021 OMMC, 4
Robert tiles a $420 \times 420$ square grid completely with $1 \times 2$ blocks, then notices that the two diagonals of the grid pass through a total of $n$ blocks. Find the sum of all possible values of $n$.
2021-2022 OMMC, 23
A $39$-tuple of real numbers $(x_1,x_2,\ldots x_{39})$ satisfies
\[2\sum_{i=1}^{39} \sin(x_i) = \sum_{i=1}^{39} \cos(x_i) = -34.\]
The ratio between the maximum of $\cos(x_1)$ and the maximum of $\sin(x_1)$ over all tuples $(x_1,x_2,\ldots x_{39})$ satisfying the condition is $\tfrac ab$ for coprime positive integers $a$, $b$ (these maxima aren't necessarily achieved using the same tuple of real numbers). Find $a + b$.
[i]Proposed by Evan Chang[/i]
2020-2021 OMMC, 2
There are a family of $5$ siblings. They have a pile of at least $2$ candies and are trying to split them up
amongst themselves. If the $2$ oldest siblings share the candy equally, they will have $1$ piece of candy left over.
If the $3$ oldest siblings share the candy equally, they will also have $1$ piece of candy left over. If all $5$ siblings
share the candy equally, they will also have $1$ piece left over. What is the minimum amount of candy required
for this to be true?
2020-2021 OMMC, 9
The difference between the maximum and minimum values of $$2\cos 2x +7\sin x$$
over the real numbers equals $\frac{p}{q}$ for relatively prime positive integers $p, q.$ Find $p+q.$
2021-2022 OMMC, 13
$ABCD$ is a rhombus where $\angle BAD = 60^\circ$. Point $E$ lies on minor arc $\widehat{AD}$ of the circumcircle of $ABD$, and $F$ is the intersection of $AC$ and the circle circumcircle of $EDC$. If $AF = 4$ and the circumcircle of $EDC$ has radius $14$, find the squared area of $ABCD$.
[i]Proposed by Vivian Loh [/i]
2021-2022 OMMC, 1
The integers from $1$ through $9$ inclusive, are placed in the squares of a $3 \times 3$ grid. Each square contains a different integer. The product of the integers in the first and second rows are $60$ and $96$ respectively. Find the sum of the integers in the third row.
[i]Proposed by bissue [/i]
2020-2021 OMMC, 3
Two real numbers $x, y$ are chosen randomly and independently on the interval $(1, r)$ where $r$ is some real number between $1024$ and $2048$. Let $P$ be the probability that $\lfloor \log_2x \rfloor > \lfloor \log_2y \rfloor .$ The value of $P$ is maximized when $r = \frac{p}{q}$ where $p,q$ are relatively prime positive integers. Find $p+q.$
OMMC POTM, 2021 11
Find the sum of all positive integers $x$ such that $$|x^2-x-6|$$ has exactly $4$ positive integer divisors.
[i]Proposed by Evan Chang (squareman), USA[/i]
2020-2021 OMMC, 4
In 3-dimensional space, two spheres centered at points $O_1$ and $O_2$ with radii $13$ and $20$ respectively intersect in a circle. Points $A, B, C$ lie on that circle, and lines $O_1A$ and $O_1B$ intersect sphere $O_2$ at points $D$ and $E$ respectively. Given that $O_1O_2 = AC = BC = 21,$ $DE$ can be expressed as $\frac{a\sqrt{b}}{c}$ where $a,b,c$ are positive integers. Find $a+b+c$.
2020-2021 OMMC, 1
Find the remainder when $$20^{20}+21^{21}-21^{20}-20^{21}$$ is divided by $100$.
2021-2022 OMMC, 10
A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find \[\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3.\]
[i]Proposed by Evan Chang[/i]
2020-2021 OMMC, 10
An [i]indivisible tiling [/i]is a tiling of an $m \times n$ rectangular grid using only rectangles with a width and/or length of 1, such that nowhere in the tiling is a smaller complete tiling of a rectangle with more than 1 tile. Find the smallest integer $a$ such that an indivisible tiling of an $a \times a$ square may contain exactly $2021$ $1 \times 1$ tiles.
2020-2021 OMMC, 7
Find the number of ordered triples of integers $(a,b,c)$ such that $$a^2 + b^2 + c^2 - ab - bc - ca - 1 \le 4042b - 2021a - 2021c - 2021^2$$ and $|a|, |b|, |c| \le 2021.$
2021-2022 OMMC, 12
Katelyn is building an integer (in base $10$). She begins with $9$. Each step, she appends a randomly chosen digit from $0$ to $9$ inclusive to the right end of her current integer. She stops immediately when the current integer is $0$ or $1$ (mod $11$). The probability that the final integer ends up being $0$ (mod $11$) is $\tfrac ab$ for coprime positive integers $a$, $b$. Find $a + b$.
[i]Proposed by Evan Chang[/i]
2020-2021 OMMC, 15
A point $X$ exactly $\sqrt{2}-\frac{\sqrt{6}}{3}$ away from the origin is chosen randomly. A point $Y$ less than $4$ away from the origin is chosen randomly. The probability that a point $Z$ less than $2$ away from the origin exists such that $\triangle XYZ$ is an equilateral triangle can be expressed as $\frac{a\pi + b}{c \pi}$ for some positive integers $a, b, c$ with $a$ and $c$ relatively prime. Find $a+b+c$.
2020-2021 OMMC, 2
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5,$ $b_1 = 7,$
$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$
and
$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$
$ $ \\
How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?
OMMC POTM, 2023 2
Find all functions $f$ from the set of reals to itself so that for all reals $x,y,$
$$f(x)f(f(x)+y) = f(x^2) + f(xy).$$
[i]Proposed by Culver Kwan[/i]
OMMC POTM, 2024 3
Define acute triangle $ABC$ with $AB = AC$ and circumcenter $O$. Define point $D$ inside $ABC$ on the circumcircle of $BOC$. Prove that the distance from $A$ to line $DO$ is half $BD+DC$..
2021-2022 OMMC, 6
Calvin makes a number. He starts with $1$, and on each move, he multiplies his current number by $3$, then adds $5$. After $10$ moves, find the sum of the digits (in base $10$) when Calvin's resulting number is expressed in base $9$.
[i]Proposed by Calvin Wang [/i]
2021-2022 OMMC, 25
Let $K > 0$ be an integer. An integer $k \in [0,K]$ is randomly chosen. A sequence of integers is defined starting on $k$ and ending on $0$, where each nonzero term $t$ is followed by $t$ minus the largest Lucas number not exceeding $t$.
The probability that $4$, $5$, or $6$ is in this sequence approaches $\tfrac{a - b \sqrt c}{d}$ for arbitrarily large $K$, where $a$, $b$, $c$, $d$, are positive integers, $\gcd(a,b,d) = 1$, and $c$ is squarefree. Find $a + b + c + d$.
[i](Lucas numbers are defined as the members of the infinite integer sequence $2$, $1$, $3$, $4$, $7$, $\ldots$ where each term is the sum of the two before it.)[/i]
[i]Proposed by Evan Chang[/i]
2020-2021 OMMC, 7
An infinitely large grid is filled such that each grid square contains exactly one of the digits $\{ 1,2,3,4\},$ each digit appears at least once, and the digit in each grid square equals the digit located $5$ squares above it as well as the digit located $5$ squares to the right. A group of $4$ horizontally adjacent digits or $4$ vertically adjacent digits is chosen randomly, and depending on its orientation is read left to right or top to bottom to form an $4$-digit integer. The expected value of this integer is also a $4$-digit integer $N$. Given this, find the last three digits of the sum of all possible values of $N$.
2020-2021 OMMC, 10
Positive integers $a,b,c$ exist such that $a+b+c+1$, $a^2+b^2+c^2 +1$, $a^3+b^3+c^3+1,$ and $a^4+b^4+c^4+7459$ are all multiples of $p$ for some prime $p$. Find the sum of all possible values of $p$ less than $1000$.
OMMC POTM, 2022 8
The positive integers are partitioned into two infinite sets so that the sum of any $2023$ distinct integers in one set is also in that set. Prove that one set contains all the odd positive integers, and one set contains all the even positive integers.
[i]Proposed by Evan Chang (squareman), USA[/i]