Found problems: 106
2023 MMATHS, 12
Let $ABC$ be a triangle with incenter $I,$ circumcenter $O,$ and $A$-excenter $J_A.$ The incircle of $\triangle{ABC}$ touches side $BC$ at a point $D.$ Lines $OI$ and $J_AD$ meet at a point $K.$ Line $AK$ meets the circumcircle of $\triangle{ABC}$ again at a point $L \neq A.$ If $BD=11, CD=5,$ and $AO=10,$ the length of $DL$ can be expressed as $\tfrac{m\sqrt{p}}{n},$ where $m,n,p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p.$
2024 MMATHS, 9
$2048$ frogs are sitting in a circle and each have a $\$1$ bill. After each minute, each frog will independently give away each of their $\$1$ bills to either the closest frog to their left or the closest frog to their right with equal probability. If a frog has $\$0$ at the end of any given minute, then they will not give any money but may receive money. The expected number of frogs to have at least $\$1$ after $3$ minutes can be denoted as a common fraction in the form $\tfrac{a}{b}.$ Find $a+b.$
2022 Girls in Math at Yale, 11
Georgina calls a $992$-element subset $A$ of the set $S = \{1, 2, 3, \ldots , 1984\}$ a [b]halfthink set[/b] if
[list]
[*] the sum of the elements in $A$ is equal to half of the sum of the elements in $S$, and
[*] exactly one pair of elements in $A$ differs by $1$.
[/list]
She notices that for some values of $n$, with $n$ a positive integer between $1$ and $1983$, inclusive, there are no halfthink sets containing both $n$ and $n+1$. Find the last three digits of the product of all possible values of $n$.
[i]Proposed by Andrew Wu and Jason Wang[/i]
(Note: wording changed from original to specify what $n$ can be.)
2021 MMATHS, 1
Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$. Find $\frac{100bdf}{ace}$.
[i]Proposed by Andrew Wu[/i]
2023 MMATHS, 2
$20$ players enter a chess tournament in which each player will play every other player exactly once. Some competitors are cheaters and will cheat in every game they play, but the rest of the competitors are not cheaters. A game is cheating if both players cheat, and a game is half-cheating if one player cheats and one player does not. If there were $68$ more half-cheating games than cheating games, how many of the players are cheaters?
2024 MMATHS, 1
Let $f$ be a function over the domain of all positive real numbers such that $$f(x)=\frac{1-\sqrt{x}}{1+\sqrt{x}}$$ Now, let $g(x)$ be a function given by $$g(x)=f(x)^{\tfrac{2f\left(\tfrac{1}{x}\right)}{f(x)}}$$ $g(100)$ can be expressed as a fraction $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime integers. What is the sum of $a$ and $b$?
2024 MMATHS, 9
Grant and Stephen are playing Square-Tac-Toe. In this game, players alternate placing $X$'s and $O$'s on a $3 \times 3$ board, and the first person to complete a $2 \times 2$ square with their respective symbols wins the game. If all tiles are filled and no such square exists, the game is a tie. Grant moves first. Given that Stephen plays randomly and Grant plays optimally (knowing that Stephen is playing randomly), the probability that Grant wins is $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
([i]Note: Grant playing "optimally" means he is maximizing his win probability[/i])
2023 MMATHS, 1
Let $n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i},$ where $p_1<p_2<\dots<p_k$ are primes and $e_1, e_2, \dots, e_k$ are positive integers, and let $f(n) = \prod_{i=1}^k e_i^{p_i}.$ Find the number of integers $n$ such that $2 \le n \le 2023$ and $f(n)=128.$
2024 MMATHS, 10
Find the sum of all prime numbers $p$ such that $\binom{20242024p}{p}\equiv 2024\pmod{p}.$
2023 MMATHS, 6
$10$ points are drawn on each of two parallel lines. What is the largest number of acute triangles of positive area that can be formed using three of these $20$ points as vertices?
2023 MMATHS, 10
Find the number of ordered pairs of integers $(m,n)$ with $0 \le m,n \le 22$ such that $k^2+mk+n$ is not a multiple of $23$ for all integers $k.$
2024 MMATHS, 8
Triangle $ABC$ is an acute triangle with $BC=6$ and $AC=7.$ Let $D, E,$ and $F$ be the feet of the altitudes from $A, B,$ and $C$ respectively. $\overline{AD}$ bisects angle $FDE.$ Let $m$ be the maximum possible value of $FD+ED.$ Find $m^2.$
2023 MMATHS, 7
$ABCD$ is a regular tetrahedron of side length $4.$ Four congruent spheres are inside $ABCD$ such that each sphere is tangent to exactly three of the faces, the spheres have distinct centers, and the four spheres are concurrent at one point. Let $v$ be the volume of one of the spheres. If $v^2$ can be written as $\tfrac{a}{b}\pi^2,$ where $a$ and $b$ are relatively prime positive integers, find $a+b.$
2022 Girls in Math at Yale, R5
[b]p13[/b] Let $ABCD$ be a square. Points $E$ and $F$ lie outside of $ABCD$ such that $ABE$ and $CBF$ are equilateral triangles. If $G$ is the centroid of triangle $DEF$, then find $\angle AGC$, in degrees.
[b]p14 [/b]The silent reading $s(n)$ of a positive integer $n$ is the number obtained by dropping the zeros not at the end of the number. For example, $s(1070030) = 1730$. Find the largest $n < 10000$ such that $s(n)$ divides $n$ and $n\ne s(n)$.
[b]p15[/b] Let $ABCDEFGH$ be a regular octagon with side length $12$. There exists a region $R$ inside the octagon such that for each point $X$ in $R$, exactly three of the rays $AX$, $BX$, $CX$, $DX$, $GX$, and $HX$ intersect segment $EF$. If the area of region $R$ can be expressed as $a -b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.
2021 MMATHS, 2
In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's [i]score[/i] is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$.
[i]Proposed by Andrew Wu[/i]
2021 Girls in Math at Yale, 4
Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer that is the product of three distinct prime numbers!"
Claire says, "I don't know your favorite number yet, but I do know that among four of the numbers that might be your favorite number, you could start with any one of them, add a second, subtract a third, and get the fourth!"
Cat says, "That's cool! My favorite number is not among those four numbers, though."
Claire says, "Now I know your favorite number!"
What is Cat's favorite number?
[i]Proposed by Andrew Wu and Andrew Milas[/i]
2021 Girls in Math at Yale, 3
Suppose that $a_1 = 1,$ $a_2 = 2$, and for any $n \ge 3$, $a_n = a_1 + a_2 + \cdots + a_{n-1}$. Find $\frac{a_{2021}}{a_{2020}}$.
[i]Proposed by Andrew Wu[/i]
2021 Girls in Math at Yale, 7
Suppose two circles $\Omega_1$ and $\Omega_2$ with centers $O_1$ and $O_2$ have radii $3$ and $4$, respectively. Suppose that points $A$ and $B$ lie on circles $\Omega_1$ and $\Omega_2$, respectively, such that segments $AB$ and $O_1O_2$ intersect and that $AB$ is tangent to $\Omega_1$ and $\Omega_2$. If $O_1O_2=25$, find the area of quadrilateral $O_1AO_2B$.
[asy]
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
import graph; size(12cm);
real labelscalefactor = 0.5; /* changes label-to-point distance */
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
pen dotstyle = black; /* point style */
real xmin = -12.81977592804657, xmax = 32.13023014338037, ymin = -14.185056097058798, ymax = 12.56855801985179; /* image dimensions */
/* draw figures */
draw(circle((-3.4277328104418046,-1.4524996726688195), 3), linewidth(1.2));
draw(circle((21.572267189558197,-1.4524996726688195), 4), linewidth(1.2));
draw((-2.5877328104418034,1.4275003273311748)--(20.452267189558192,-5.2924996726687885), linewidth(1.2));
/* dots and labels */
dot((-3.4277328104418046,-1.4524996726688195),linewidth(3pt) + dotstyle);
label("$O_1$", (-4.252707018231291,-1.545940604327141), N * labelscalefactor);
dot((21.572267189558197,-1.4524996726688195),linewidth(3pt) + dotstyle);
label("$O_2$", (21.704189347819636,-1.250863978037686), NE * labelscalefactor);
dot((-2.5877328104418034,1.4275003273311748),linewidth(3pt) + dotstyle);
label("$A$", (-2.3937351324858342,1.6999022848568643), NE * labelscalefactor);
dot((20.452267189558192,-5.2924996726687885),linewidth(3pt) + dotstyle);
label("$B$", (20.671421155806545,-4.9885012443707835), NE * labelscalefactor);
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
/* end of picture */
[/asy]
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2021 Girls in Math at Yale, R5
13. The triangle with vertices $(0,0), (a,b)$, and $(a,-b)$ has area $10$. Find the sum of all possible positive integer values of $a$, given that $b$ is a positive integer.
14. Elsa is venturing into the unknown. She stands on $(0,0)$ in the coordinate plane, and each second, she moves to one of the four lattice points nearest her, chosen at random and with equal probability. If she ever moves to a lattice point she has stood on before, she has ventured back into the known, and thus stops venturing into the unknown from then on. After four seconds have passed, the probability that Elsa is still venturing into the unknown can be expressed as $\frac{a}{b}$ in simplest terms. Find $a+b$.
(A lattice point is a point with integer coordinates.)
15. Let $ABCD$ be a square with side length $4$. Points $X, Y,$ and $Z$, distinct from points $A, B, C,$ and $D$, are selected on sides $AD, AB,$ and $CD$, respectively, such that $XY = 3, XZ = 4$, and $\angle YXZ = 90^{\circ}$. If $AX = \frac{a}{b}$ in simplest terms, then find $a + b$.
2021 Girls in Math at Yale, Tiebreaker
[b]p1.[/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 a strictly 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]p2.[/b] Suppose that $A$ and $B$ are digits between $1$ and $9$ such that
$$0.\overline{ABABAB...}+ B \cdot (0.\overline{AAA...}) = A \cdot (0.\overline{B1B1B1...}) + 1$$
Find the sum of all possible values of $10A + B$.
[b]p3.[/b] Let $ABC$ be an isosceles right triangle with $m\angle ABC = 90^o$. Let $D$ and $E$ lie on segments $\overline{AC}$ and $\overline{BC}$, 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]p4.[/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 $\frac{a}{b}$ 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]p5.[/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 coeffcients equal to $1011$?
[b]p6.[/b] Suppose $f(x)$ is a monic quadratic polynomial with distinct nonzero roots $p$ and $q$, and suppose $g(x)$ is a monic quadratic polynomial with roots $p + \frac{1}{q}$ and $q + \frac{1}{p}$ . If we are given that $g(-1) = 1$ and $f(0)\ne -1$, then there exists some real number $r$ that must be a root of $f(x)$. Find $r$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2022 Girls in Math at Yale, 12
Let $ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $D$ be a point on arc $\widehat{BC}$ of its circumcircle $\Omega$. Suppose that the angle bisectors of $\angle ADB$ and $\angle ADC$ meet $AB$ and $AC$ at $E$ and $F$, respectively, and that $EF$ and $BC$ meet at $G$. Line $GD$ meets $\Omega$ at $T$. If the maximum possible value of $AT^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd (a,b) = 1$, find $a + b$.
[i]Proposed by Andrew Wu[/i]
2024 MMATHS, 1
On a planet, far, far away, the Yaliens have defined: $x$ "equals" $y$ if and only if $|x-y| \le 3.$ Let $S$ be a set of positive integers. What is the smallest possible number of elements in $S$ such that, for any positive integer $r,$ where $1 \le r \le 2024,$ $r$ "equals" some element in $S$?
2021 MMATHS, 3
Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod_{n=2}^x \log_{n^n}(n+1)^{n+2}$$ is an integer.
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2022 Girls in Math at Yale, Tiebreaker
[b]p1.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$.
[b]p2.[/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]p3.[/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]p4.[/b] In triangle $ABC$, points $D$ 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]p5.[/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]p6.[/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].
2023 MMATHS, 3
There are $360$ permutations of the letters in $MMATHS.$ When ordered alphabetically, starting from $AHMMST,$ $MMATHS$ is in the $n$th permutation. What is $n$?