Found problems: 106
2021 MMATHS, 11
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$.
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2024 MMATHS, 1
Let $ab^2=126, bc^2=14, cd^2=128, da^2=12.$ Find $\tfrac{bd}{ac}.$
2024 MMATHS, 2
Define the [i]factorial function[/i] of $n,$ denoted $\partial (n),$ as the sum of the factorials of the digits of $n.$ For example, $\partial(2024)=2!+0!+2!+4!=29.$ There are four positive integers such that $\partial(\partial(n))=n$ and $\partial(n) \neq n.$ Given that $n=871$ is one of them, compute the sum of the other three.
2021 Girls in Math at Yale, 9
Ali defines a [i]pronunciation[/i] of any sequence of English letters to be a partition of those letters into substrings such that each substring contains at least one vowel. For example, $\text{A } \vert \text{ THEN } \vert \text{ A}$, $\text{ATH } \vert \text{ E } \vert \text{ NA}$, $\text{ATHENA}$, and $\text{AT } \vert \text{ HEN } \vert \text{ A}$ are all pronunciations of the sequence $\text{ATHENA}$. How many distinct pronunciations does $\text{YALEMATHCOMP}$ have? (Y is not a vowel.)
[i]Proposed by Andrew Wu, with significant inspiration from ali cy[/i]
2022 Girls in Math at Yale, 1
Charlotte is playing the hit new web number game, Primle. In this game, the objective is to guess a two-digit positive prime integer between $10$ and $99$, called the [i]Primle[/i]. For each guess, a digit is highlighted blue if it is in the [i]Primle[/i], but not in the correct place. A digit is highlighted orange if it is in the [i]Primle[/i] and is in the correct place. Finally, a digit is left unhighlighted if it is not in the [i]Primle[/i]. If Charlotte guesses $13$ and $47$ and is left with the following game board, what is the [i]Primle[/i]?
$$\begin{array}{c}
\boxed{1} \,\, \boxed{3} \\[\smallskipamount]
\boxed{4}\,\, \fcolorbox{black}{blue}{\color{white}7}
\end{array}$$
[i]Proposed by Andrew Wu and Jason Wang[/i]
2023 MMATHS, 4
Let $A$ and $B$ be unit hexagons that share a center. Then, let $\mathcal{P}$ be the set of points contained in at least one of the hexagons. If the maximum possible area of $\mathcal{P}$ is $X$ and the minimum possible area of $\mathcal{P}$ is $Y,$ then the value of $Y-X$ can be expressed as $\tfrac{a\sqrt{b}-c}{d},$ where $a,b,c,d$ are positive integers such that $b$ is square-free and $\gcd(a,c,d)=1.$ Find $a+b+c+d.$
2021 MMATHS, 10
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, and suppose that $OI$ meets $AB$ and $AC$ at $P$ and $Q$, respectively. There exists a point $R$ on arc $\widehat{BAC}$ such that the circumcircles of triangles $PQR$ and $ABC$ are tangent. Given that $AB = 14$, $BC = 20$, and $CA = 26$, find $\frac{RC}{RB}$.
[i]Proposed by Andrew Wu[/i]
2022 Girls in Math at Yale, Mixer Round
[b]p1.[/b] Find the smallest positive integer $N$ such that $2N -1$ and $2N +1$ are both composite.
[b]p2.[/b] Compute the number of ordered pairs of integers $(a, b)$ with $1 \le a, b \le 5$ such that $ab - a - b$ is prime.
[b]p3.[/b] Given a semicircle $\Omega$ with diameter $AB$, point $C$ is chosen on $\Omega$ such that $\angle CAB = 60^o$. Point $D$ lies on ray $BA$ such that $DC$ is tangent to $\Omega$. Find $\left(\frac{BD}{BC} \right)^2$.
[b]p4.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If $f(x)$ is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$?
[b]p5.[/b] Regular hexagon $ABCDEF$ has side length $3$. Circle $\omega$ is drawn with $AC$ as its diameter. $BC$ is extended to intersect $\omega$ at point $G$. If the area of triangle $BEG$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree and $gcd(a, c) = 1$, find $a + b + c$.
[b]p6.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$.
[b]p7.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers.
[b]p8.[/b] Let $\vartriangle ABC$ be an equilateral triangle. Isosceles triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, not overlapping $\vartriangle ABC$, are constructed such that each has area seven times the area of $\vartriangle ABC$. Compute the ratio of the area of $\vartriangle DEF$ to the area of $\vartriangle ABC$.
[b]p9.[/b] Consider the sequence of polynomials an(x) with $a_0(x) = 0$, $a_1(x) = 1$, and $a_n(x) = a_{n-1}(x) + xa_{n-2}(x)$ for all $n \ge 2$. Suppose that $p_k = a_k(-1) \cdot a_k(1)$ for all nonnegative integers $k$. Find the number of positive integers $k$ between $10$ and $50$, inclusive, such that $p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}$.
[b]p10.[/b] In triangle $ABC$, point $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle BEC has area 45 and triangle $ADC$ has area $72$, and lines CH and AB meet at F. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with c squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$.
[b]p11.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer x such that $x^2 + 18x + y$ is a perfect fourth power.
[b]p12.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a$, $b$ with $b$ squarefree, then find $a + b$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 Girls in Math at Yale, 3
The [b]Collaptz function[/b] is defined as $$C(n) = \begin{cases} 3n - 1 & n\textrm{~odd}, \\ \frac{n}{2} & n\textrm{~even}.\end{cases}$$
We obtain the [b]Collaptz sequence[/b] of a number by repeatedly applying the Collaptz function to that number. For example, the Collaptz sequence of $13$ begins with $13, 38, 19, 56, 28, \cdots$ and so on. Find the sum of the three smallest positive integers $n$ whose Collaptz sequences do not contain $1,$ or in other words, do not [b]collaptzse[/b].
[i]Proposed by Andrew Wu and Jason Wang[/i]
2021 Girls in Math at Yale, 10
Suppose that $a_1, a_2, a_3, \ldots$ is an infinite geometric sequence such that for all $i \ge 1$, $a_i$ is a positive integer. Suppose furthermore that $a_{20} + a_{21} = 20^{21}$. If the minimum possible value of $a_1$ can be expressed as $2^a 5^b$ for positive integers $a$ and $b$, find $a + b$.
[i]Proposed by Andrew Wu[/i]
2021 Girls in Math at Yale, Mixer Round
[b]p1.[/b] Find the number of ordered triples $(a, b, c)$ satisfying
$\bullet$ $a, b, c$ are are single-digit positive integers, and
$\bullet$ $a \cdot b + c = a + b \cdot c$.
[b]p2.[/b] In their class Introduction to Ladders at Greendale Community College, Jan takes four tests. They realize that their test scores in chronological order form an increasing arithmetic progression with integer terms, and that the average of those scores is an integer greater than or equal to $94$. How many possible combinations of test scores could they have had? (Test scores at Greendale range between $0$ and $100$, inclusive.)
[b]p3.[/b] Suppose that $a + \frac{1}{b} = 2$ and $b +\frac{1}{a} = 3$. If$ \frac{a}{b} + \frac{b}{a}$ can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$.
[b]p4.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that $$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B_1B_1B_1...}) + 1$$
Find the sum of all possible values of $10A + B$.
[b]p5.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $AC$ and $BC$, respectively, such that triangles $\vartriangle ADB$ and $\vartriangle CDE$ are similar and $DE = EB$. If $\frac{AC}{AD} = 1 +\frac{\sqrt{a}}{b}$ with $a, b$ positive integers and a squarefree, then find $a + b$.
[b]p6.[/b] Five bowling pins $P_1$, $P_2$,..., $P_5$ are lined up in a row. Each turn, Jemma picks a pin at random from the standing pins, and throws a bowling ball at that pin; that pin and each pin directly adjacent to it are knocked down. If the expected value of the number of turns Jemma will take to knock down all the pins is a b where a and b are relatively prime, find $a + b$. (Pins $P_i$ and $P_j$ are adjacent if and only if $|i -j| = 1$.)
[b]p7.[/b] Let triangle $ABC$ have side lengths $AB = 10$, $BC = 24$, and $AC = 26$. Let $I$ be the incenter of $ABC$. If the maximum possible distance between $I$ and a point on the circumcircle of $ABC$ can be expressed as $a +\sqrt{b}$ for integers $a$ and $b$ with $b$ squarefree, find $a + b$.
(The incenter of any triangle $XY Z$ is the intersection of the angle bisectors of $\angle Y XZ$, $\angle XZY$, and $\angle ZY X$.)
[b]p8.[/b] How many terms in the expansion of
$$(1 + x + x^2 + x^3 +... + x^{2021})(1 + x^2 + x^4 + x^6 + ... + x^{4042})$$
have coefficients equal to $1011$?
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2024 MMATHS, 12
$S_1,S_2,\ldots,S_n$ are subsets of $\{1,2,\ldots,10000\}$ which satisfy that, whenever $|S_i| > |S_j|$, the sum of all elements in $S_i$ is less than the sum of all elements in $S_j$. Let $m$ be the maximum number of distinct values among $|S_1|,\ldots,|S_n|$. Find $\left\lfloor\frac{m}{100}\right\rfloor$.
2022 Girls in Math at Yale, R4
[b]p10 [/b]Kathy has two positive real numbers, $a$ and $b$. She mistakenly writes
$$\log (a + b) = \log (a) + \log( b),$$
but miraculously, she finds that for her combination of $a$ and $b$, the equality holds. If $a = 2022b$, then $b = \frac{p}{q}$ , for positive integers $p, q$ where $gcd(p, q) = 1$. Find $p + q$.
[b]p11[/b] Points $X$ and $Y$ lie on sides $AB$ and $BC$ of triangle $ABC$, respectively. Ray $\overrightarrow{XY}$ is extended to point $Z$ such that $A, C$, and $Z$ are collinear, in that order. If triangle$ ABZ$ is isosceles and triangle $CYZ$ is equilateral, then the possible values of $\angle ZXB$ lie in the interval $I = (a^o, b^o)$, such that $0 \le a, b \le 360$ and such that $a$ is as large as possible and $b$ is as small as possible. Find $a + b$.
[b]p12[/b] Let $S = \{(a, b) | 0 \le a, b \le 3, a$ and $b$ are integers $\}$. In other words, $S$ is the set of points in the coordinate plane with integer coordinates between $0$ and $3$, inclusive. Prair selects four distinct points in $S$, for each selected point, she draws lines with slope $1$ and slope $-1$ passing through that point. Given that each point in $S$ lies on at least one line Prair drew, how many ways could she have selected those four points?
2023 MMATHS, 1
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit multiple of $7$.”
Claire asks, “If you just told me the tens digit of the number, would I know your number?”
Cat says, “No. However, without knowing that, if I told you the tens digit of $100$ minus my number, you could determine my favorite number.”
Claire says, “Now I know your favorite number!"
What is Cat’s favorite number?
2021 MMATHS, 4
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a,b,c$ with $c$ squarefree, find $a + b + c$.
[i]Proposed by Andrew Wu[/i]
2024 MMATHS, 2
Grant has a box with $6$ red balls, $5$ blue balls, $4$ green balls, $3$ yellow balls, $2$ orange balls, and $1$ purple ball. Grant selects $6$ balls at random, without replacement. Let $P$ be the probability that Grant selects six balls of different colors, and let $Q$ be the probability that Grant selects six balls of the same color. What is $\tfrac{P}{Q}$?
2021 MMATHS, 12
$ABCD$ is a regular tetrahedron with side length 1. Points $X,$ $Y,$ and $Z,$ distinct from $A,$ $B,$ and $C,$ respectively, are drawn such that $BCDX,$ $ACDY,$ and $ABDZ$ are also regular tetrahedra. If the volume of the polyhedron with faces $ABC,$ $XYZ,$ $BXC,$ $XCY,$ $CYA,$ $YAZ,$ $AZB,$ and $ZBX$ can be written as $\frac{a\sqrt{b}}{c}$ for positive integers $a,b,c$ with $\gcd(a,c) = 1$ and $b$ squarefree, find $a+b+c.$
[i]Proposed by Jason Wang[/i]
2024 MMATHS, 3
Let $f(x)$ be a function, where if $q$ is an integer, then $f(\tfrac{1}{q})=q,$ and if $m$ and $n$ are real numbers, $f(m\cdot n)$ $ =$ $ f(m)\cdot f(n).$ If $f(\sqrt{2})$ can be written as $\tfrac{\sqrt{a}}{b}$ where $a$ is not divisible by the square of any prime and $b$ is a positive integer, then what is $a+b$?
2023 MMATHS, 11
Suppose we have sequences $(a_n)_{n \ge 0}$ and $(b_n)_{n \ge 0}$ and the function $f(x)=\tfrac{1}{x}$ such that for all $n$ we have:
[list]
[*]$a_{n+1} = f(f(a_n+b_n)-f(f(a_n)+f(b_n))$
[*]$a_{n+2} = f(1-a_n) - f(1+a_n)$
[*]$b_{n+2} = f(1-b_n) - f(1+b_n)$
[/list]
Given that $a_0=\tfrac{1}{6}$ and $b_0=\tfrac{1}{7},$ then $b_5=\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find the sum of the prime factors of $mn.$
2022 Girls in Math at Yale, R1
[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that
1) neither digit is $0$, and
2) the units digit is a multiple of the tens digit?
[b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as possible, she has some candies left over. Find the smallest integer $k$ such that Mirabel could have had $k$ leftover candies.
[b]p3[/b] Callie picks two distinct numbers from $\{1, 2, 3, 4, 5\}$ at random. The probability that the sum of the numbers she picked is greater than the sum of the numbers she didn’t pick is $p$. $p$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$. Find $a + b$.
2024 MMATHS, 12
Let $ABC$ be a triangle with $\angle{A}=60^\circ$ and orthocenter $H.$ Let $B'$ be the reflection of $B$ over $AC,$ $C'$ be the reflection of $C$ over $AB,$ and $A'$ be the intersection of $BC'$ and $B'C.$ Let $D$ be the intersection of $A'H$ and $BC.$ If $BC=5$ and $A'D=4,$ then the area of $\triangle{ABC}$ can be expressed as $a\sqrt{b}+\sqrt{c},$ where $a,b,$ and $c$ are positive integers, and $b$ and $c$ are not divisible by the square of any prime. Find $a+b+c.$
2023 MMATHS, 9
In $\triangle{ABC}$ with $\angle{BAC}=60^\circ,$ points $D, E,$ and $F$ lie on $BC, AC,$ and $AB$ respectively, such that $D$ is the midpoint of $BC$ and $\triangle{DEF}$ is equilateral. If $BF=1$ and $EC=13,$ then the area of $\triangle{DEF}$ can be written as $\tfrac{a\sqrt{b}}{c},$ where $a$ and $c$ are relatively prime positive integers and $b$ is not divisible by a square of a prime. Compute $a+b+c.$
2023 MMATHS, 10
Consider the recurrence relation $x_{n+2}=2x_{n+1}+x_n,$ with $x_0=0, x_1=1.$ What is the greatest common divisor of $x_{2023}$ and $x_{721}$?
2021 MMATHS, 6
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater than the first can be expressed as $\frac{a}{b}$ for positive integers $a,b$ with $\gcd (a,b) = 1$, then find $a + b$.
[i]Proposed by Vismay Sharan[/i]
2023 MMATHS, 9
Let $(x+x^{-1}+1)^{40} = \sum_{i=-40}^{40} a_ix^i.$ Find the remainder when $\sum_{p \text{ prime}} a_p$ is divided by $41.$