Olympiad problems

2021-2022 OMMC, 8

Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$th time, for any nonnegative integer $n$, he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\tfrac ab$ for coprime positive integers $a$ and $b$. Find $a + b$. [i]Proposed by Isaac Chen[/i]

2020-2021 OMMC, 3

Tags: ommc

Define $f(x)$ as $\frac{x^2-x-2}{x^2+x-6}$. $f(f(f(f(1))))$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p,q$. Find $10p+q$.

2020-2021 OMMC, 8

Tags: ommc

The function $g\left(x\right)$ is defined as $\sqrt{\dfrac{x}{2}}$ for all positive $x$. $ $\\ $$g\left(g\left(g\left(g\left(g\left(\frac{1}{2}\right)+1\right)+1\right)+1\right)+1\right)$$ $ $\\ can be expressed as $\cos(b)$ using degrees, where $0^\circ < b < 90^\circ$ and $b = p/q$ for some relatively prime positive integers $p, q$. Find $p+q$.

2021-2022 OMMC, 20

Tags: ommc

Let \[\mathcal{S} = \sum_{i=1}^{\infty}\left(\prod_{j=1}^i \dfrac{3j - 2}{12j}\right).\] Then $(\mathcal{S} + 1)^3 = \tfrac mn$ with $m$ and $n$ coprime positive integers. Find $10m + n$. [i]Proposed by Justin Lee and Evan Chang[/i]

OMMC POTM, 2023 9

Tags: inequalities , algebra , ommc

Show that for any $8$ distinct positive real numbers, one can choose a quadraple of them $(a,b,c,d)$ , all distinct such that $$(ac+bd)^2 \ge \frac{2+\sqrt3}{4}\left(a^2+b^2 \right)\left(c^2+d^2 \right)$$ [i]Proposed by Evan Chang (squareman), USA[/i]

2020-2021 OMMC, 11

Tags: ommc

In equilateral triangle $XYZ$ with side length $10$, define points $A, B$ on $XY,$ points $C, D$ on $YZ,$ and points $E, F$ on $ZX$ such that $ABDE$ and $ACEF$ are rectangles, $XA<XB,$ $YC<YD,$ and $ZE<ZF$. The area of hexagon $ABCDEF$ can be written as $\sqrt{x}$ for some positive integer $x$. Find $x$.

OMMC POTM, 2023 6

Tags: combinatorics , ommc

Choose a permutation of$ \{1,2, ..., 20\}$ at random. Let $m$ be the amount of numbers in the permutation larger than all numbers before it. Find the expected value of $2^m$. [i]Proposed by Evan Chang (squareman), USA[/i]

OMMC POTM, 2022 12

Tags: ommc , geometry , circumcircle

Let $\triangle ABC$ be such that the midpoint of $BC$ is $D$. Let $E$ be the point on the opposite side of $AC$ as $B$ on the circumcircle of $\triangle ABC$ such that $\angle DEA = \angle DEC$ and let $\omega$ be the circumcircle of $\triangle CED$. If $\omega$ intersects $AE$ at $X$ and the tangent to $\omega$ at $D$ intersects $AB$ at $Y$, show that $XY$ is parallel to $BC$. [i]Proposed by Taco12[/i]

2020-2021 OMMC, 12

Tags: ommc , Polynomials

Let $P(x) = x^3 + 8x^2 - x + 3$ and let the roots of $P$ be $a, b,$ and $c.$ The roots of a monic polynomial $Q(x)$ are $ab - c^2, ac - b^2, bc - a^2.$ Find $Q(-1).$

2020-2021 OMMC, 6

Tags: ommc

In square $ABCD$ with $AB = 10$, point $P, Q$ are chosen on side $CD$ and $AD$ respectively such that $BQ \perp AP,$ and $R$ lies on $CD$ such that $RQ \parallel PA.$ $BC$ and $AP$ intersect at $X,$ and $XQ$ intersects the circumcircle of $PQD$ at $Y$. Given that $\angle PYR = 105^{\circ},$ $AQ$ can be expressed in simplest radical form as $b\sqrt{c}-a$ where $a, b, c$ are positive integers. Find $a+b+c.$

2021-2022 OMMC, 2

Tags: ommc

In a room, each person is an painter and/or a musician. $2$ percent of the painters are musicians, and $5$ percent of the musicians are painters. Only one person is both an painter and a musician. How many people are in the room? [i]Proposed by Evan Chang[/i]

2020-2021 OMMC, 7

Tags: ommc

Derek fills a square $10$ by $10$ grid with $50$ $1$s and $50$ $2$s. He takes the product of the numbers in each of the $10$ rows. He takes the product of the numbers in each of the $10$ columns. He then sums these $20$ products up to get an integer $N.$ Find the minimum possible value of $N.$

2021-2022 OMMC, 1

Tags: ommc , mean , median , number theory

Find the sum of all positive integers $n$ where the mean and median of $\{20, 42, 69, n\}$ are both integers. [i]Proposed by bissue[/i]

OMMC POTM, 2022 10

Tags: geometry , rhombus , ommc , squareman

Define a convex quadrilateral $\mathcal{P}$ on the plane. In a turn, it is allowed to take some vertex of $\mathcal{P}$, move it perpendicular to the current diagonal of $\mathcal{P}$ not containing it, so long as it never crosses that diagonal. Initially $\mathcal{P}$ is a parallelogram and after several turns, it is similar but not congruent to its original shape. Show that $\mathcal P$ is a rhombus. [i]Proposed by Evan Chang (squareman), USA[/i]

OMMC POTM, 2022 9

Tags: number theory , least common multiple , inequalities , ommc

For positive integers $a_1 < a_2 < \dots < a_n$ prove that $$\frac{1}{\operatorname{lcm}(a_1, a_2)}+\frac{1}{\operatorname{lcm}(a_2, a_3)}+\dots+\frac{1}{\operatorname{lcm}(a_{n-1}, a_n)} \leq 1-\frac{1}{2^{n-1}}.$$ [i]Proposed by Evan Chang (squareman), USA[/i]

OMMC POTM, 2022 4

Tags: number theory , ommc , prime factorization

Define a function $P(n)$ from the set of positive integers to itself, where $P(1)=1$ and if an integer $n > 1$ has prime factorization $$n = p_1^{a_1}p_2^{a_2} \dots p_k^{a_k}$$ then $$P(n) = a_1^{p_1}a_2^{p_2} \dots a_k^{p_k}.$$ Prove that $P(P(n)) \le n$ for all positive integers $n.$ [i]Proposed by Evan Chang (squareman), USA[/i]

2021-2022 OMMC, 21

Tags: ommc

For some real number $a$, define two parabolas on the coordinate plane with equations $x = y^2 + a$ and $y = x^2 + a$. Suppose there are $3$ lines, each tangent to both parabolas, that form an equilateral triangle with positive area $s$. If $s^2 = \tfrac pq$ for coprime positive integers $p$, $q$, find $p + q$. [i]Proposed by Justin Lee[/i]

OMMC POTM, 2022 2

Tags: ommc , algebra , functional equation

Find all functions $f:\mathbb R \to \mathbb R$ (from the set of real numbers to itself) where$$f(x-y)+xf(x-1)+f(y)=x^2$$for all reals $x,y.$ Proposed by [b]cj13609517288[/b]

2020-2021 OMMC, 13

Tags: ommc

Find the number of nonnegative integers $n < 29$ such that there exists positive integers $x,y$ where $$x^2+5xy-y^2$$ has remainder $n$ when divided by $29$.

OMMC POTM, 2022 5

Tags: combinatorics , ommc

A unit square is given. Evan places a series of squares inside this unit square according to the following rules: $\bullet$ The $n$th square he places has side length $\frac{1}{n+1}.$ $\bullet$ At any point, no two placed squares can overlap. Can he place squares indefinitely? [i]Proposed by Evan Chang (squareman), USA[/i]

OMMC POTM, 2022 7

Tags: ommc , Diophantine equation , number theory

Find all ordered triples of positive integers $(a,b,c)$ where $$\left(a+\frac{1}{a}\right)\left(b+\frac{1}{b}\right)=c+\frac{1}{c}.$$ [i]Proposed by vsamc[/i]

2021-2022 OMMC, 15

Tags: ommc

Let $1 = x_{1} < x_{2} < \dots < x_{k} = n$ denote the sequence of all divisors $x_{1}, x_{2} \dots x_{k}$ of $n$ in increasing order. Find the smallest possible value of $n$ such that $$n = x_{1}^{2} + x_{2}^{2} +x_{3}^{2} + x_{4}^{2}.$$ [i]Proposed by Justin Lee[/i]

2021-2022 OMMC, 17

Tags: ommc

Find the number of positive integer divisors of \[\sum_{k=0}^{50}(-3)^k\dbinom{100}{2k}.\] [i]Proposed by Serena Xu[/i]

2021-2022 OMMC, 4

Tags: ommc

How many sequences of real numbers $a_1,a_2,\ldots a_9$ satisfy \[|a_1-1|=|a_2-a_1|=\cdots=|a_9-a_8|=|1-a_9|=1?\] [i]Proposed by Evan Chang [/i]

OMMC POTM, 2022 1

Tags: number theory , divisible , ommc

The digits $2,3,4,5,6,7,8,9$ are written down in some order. When read in that order, the digits form an $8$-digit, base $10$ positive integer. if this integer is divisible by $44$, how many ways could the digits have been initially ordered? [i]Proposed by Evan Chang (squareman), USA[/i]