Found problems: 106
2023 MMATHS, 2
In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\tfrac{a}{b},$ where $a$ and $b$ are relatively prime positive integers, find $a + b.$
2022 Girls in Math at Yale, R2
[b]p4[/b] Define the sequence ${a_n}$ as follows:
1) $a_1 = -1$, and
2) for all $n \ge 2$, $a_n = 1 + 2 + . . . + n - (n + 1)$.
For example, $a_3 = 1+2+3-4 = 2$. Find the largest possible value of $k$ such that $a_k+a_{k+1} = a_{k+2}$.
[b]p5[/b] The taxicab distance between two points $(a, b)$ and $(c, d)$ on the coordinate plane is $|c-a|+|d-b|$. Given that the taxicab distance between points $A$ and $B$ is $8$ and that the length of $AB$ is $k$, find the minimum possible value of $k^2$.
[b]p6[/b] For any two-digit positive integer $\overline{AB}$, let $f(\overline{AB}) = \overline{AB}-A\cdot B$, or in other words, the result of subtracting the product of its digits from the integer itself. For example, $f(\overline{72}) = 72-7\cdot 2 = 58$. Find the maximum possible $n$ such that there exist distinct two-digit integers$ \overline{XY}$ and $\overline{WZ}$ such that $f(\overline{XY} ) = f(\overline{WZ}) = n$.
2022 Girls in Math at Yale, 8
Triangle $ABC$ has sidelengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D,E$, and $F$ are chosen on $AB, BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a, b) = 1$, find $a + b$. (Here $[DEF]$ denotes the area of triangle $DEF$.)
[i]Proposed by Vismay Sharan[/i]
2022 Girls in Math at Yale, 10
How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
[i]Proposed by Miles Yamner and Andrew Wu[/i]
(Note: wording changed from original to clarify)
2022 Girls in Math at Yale, 9
Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$.
[i]Proposed by Andrew Wu[/i]
(Note: wording changed from original to specify that $a \neq 20, 22$.)
2021 MMATHS, 9
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find the second-smallest possible value of $P(0).$
[i]Proposed by Andrew Wu[/i]
2023 MMATHS, 1
Lucy has $8$ children, each of whom has a distinct favorite integer from $1$ to $10,$ inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is $1260,$ and the average of these favorite numbers is at most $5.$ Find the sum of the four largest numbers.
2024 MMATHS, 7
The sum $\sum_{x=-5}^5\sum_{y=-5}^5\frac{2^x3^y}{(1+2^x)(1+3^y)}$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 MMATHS, 2
Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)
[i]Proposed by Andrew Wu[/i]
2023 MMATHS, 3
Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand $(m - a)(n - b),$ where $a, b, m, n$ are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for $m$ and $n,$ he realizes that the value of ChatSFFT’s result is the same as the real result in terms of $a$ and $b$. How many such pairs are there?
2023 MMATHS, 7
A $2023 \times 2023$ grid of lights begins with every light off. Each light is assigned a coordinate $(x,y).$ For every distinct pair of lights $(x_1, y_1), (x_2, y_2),$ with $x_1<x_2$ and $y_1>y_2,$ all lights strictly between them (i.e. $x_1<x<x_2$ and $y_2<y<y_1$) are toggled. After this procedure is done, how many lights are on?
2021 Girls in Math at Yale, 8
Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$.
[i]Proposed by Andrew Wu[/i]
2021 MMATHS, 1
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find \[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2022 Girls in Math at Yale, 2
How many ways are there to fill in a $2\times 2$ square grid with the numbers $1,2,3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$?
[i]Proposed by Andrew Wu[/i]
2024 MMATHS, 4
Let $ABC$ be an equilateral triangle with side length $1.$ Then, let $M$ be the midpoint of $\overline{BC}.$ The area of all points within $ABC$ that are closer to $M$ than either of $A, B,$ or $C$ can be expressed as the fraction $\tfrac{\sqrt{a}}{b}$ where $a$ is not divisible by the square of any prime and $b$ is a positive integer. Find $a+b.$
2024 MMATHS, 5
Amir and Bella play a game on a gameboard with $6$ spaces, labeled $0, 1, 2, 3, 4,$ and $5.$ Each turn, each player flips a coin. If it is heads, their character moves forward one space, and if it is tails, their character moves back one space, unless it was already at space $0,$ in which case it moves forward one space instead. If Amir and Bella each have a character that starts at space $0,$ the probability that they end turn $5$ on the same space can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$
2023 MMATHS, 8
Find the number of ordered pairs of integers $(m,n)$ such that $0 \le m,n \le 2023$ and $$m^2 \equiv \sum_{d \mid 2023} n^d \pmod{2024}.$$
2022 Girls in Math at Yale, R6
[b]p16[/b] Madelyn is being paid $\$50$/hour to find useful [i]Non-Functional Trios[/i], where a Non-Functional Trio is defined as an ordered triple of distinct real numbers $(a, b, c)$, and a Non- Functional Trio is [i]useful [/i] if $(a, b)$, $(b, c)$, and $(c, a)$ are collinear in the Cartesian plane. Currently, she’s working on the case $a+b+c = 2022$. Find the number of useful Non-Functional Trios $(a, b, c)$ such that $a + b + c = 2022$.
[b]p17[/b] Let $p(x) = x^2 - k$, where $k$ is an integer strictly less than $250$. Find the largest possible value of k such that there exist distinct integers $a, b$ with $p(a) = b$ and $p(b) = a$.
[b]p18[/b] Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$ such that $AB = 13$, $BC = 14$, and $CA = 15$. $BH$ and $CH$ meet $\Gamma$ again at points $D$ and $E$, respectively, and $DE$ meets $AB$ and $AC$ at $F$ and $G$, respectively. The circumcircles of triangles $ABG$ and $ACF$ meet BC again at points $P$ and $Q$. If $PQ$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd (a, b) = 1$, find $a + b$.
2022 Girls in Math at Yale, 7
Given that six-digit positive integer $\overline{ABCDEF}$ has distinct digits $A,$ $B,$ $C,$ $D,$ $E,$ $F$ between $1$ and $8$, inclusive, and that it is divisible by $99$, find the maximum possible value of $\overline{ABCDEF}$.
[i]Proposed by Andrew Milas[/i]
2021 MMATHS, 8
Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when the string he's written down contains the letters $M, A, T$, and $H$ in that order, not necessarily consecutively (for example, one valid string might be $MAMMSHTH$.) What is the expected length of the string Clayton wrote?
[i]Proposed by Andrew Milas and Andrew Wu[/i]
2024 MMATHS, 2
Consider the recursive sequence defined by $a_{n+1}=a_n^n+1,$ with $a_1=0.$ What is the last digit of $a_{2024}$?
2021 Girls in Math at Yale, R6
16. Suppose trapezoid $JANE$ is inscribed in a circle of radius $25$ such that the center of the circle lies inside the trapezoid. If the two bases of $JANE$ have side lengths $14$ and $30$ and the average of the lengths of the two legs is $\sqrt{m}$, what is $m$?
17. What is the radius of the circle tangent to the $x$-axis, the line $y=\sqrt{3}x$, and the circle $(x-10\sqrt{3})^2+(y-10)^2=25$?
18. Find the smallest positive integer $n$ such that $3n^3-9n^2+5n-15$ is divisible by $121$ but not $2$.
2024 MMATHS, 3
Alice picks a random three-digit number, from $100$ to $999,$ inclusive. The probability that her first digit is larger than the sum of her other two digits can be expressed as a common fraction $\tfrac{a}{b}.$ Find $a+b.$
2021 MMATHS, 4
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I'd know for certain what it is?"
Cat says, "Nope! However, if I now told you the units digit of my favorite number, you'd know which one it is!"
Claire says, "Now I know your favorite number!" What is Cat's favorite number?
[i]Proposed by Andrew Wu[/i]
2021 Girls in Math at Yale, R4
10. Prair picks a three-digit palindrome $n$ at random. If the probability that $2n$ is also a palindrome can be expressed as $\frac{p}{q}$ in simplest terms, find $p + q$. (A palindrome is a number that reads the same forwards as backwards; for example, $161$ and $2992$ are palindromes, but $342$ is not.)
11. If two distinct integers are picked randomly between $1$ and $50$ inclusive, the probability that their sum is divisible by $7$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
12. Ali is playing a game involving rolling standard, fair six-sided dice. She calls two consecutive die rolls such that the first is less than the second a "rocket." If, however, she ever rolls two consecutive die rolls such that the second is less than the first, the game stops. If the probability that Ali gets five rockets is $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, find $p+q$.