Found problems: 106
2023 MMATHS, 11
A knight is on an infinite chessboard. After exactly $100$ legal moves, how many different possible squares can it end on? A knight can move to any of the $8$ closest squares not on the same row, column, or diagonal, as illustrated in the figure below.
[center][img]https://cdn.artofproblemsolving.com/attachments/0/7/144226144fb3ead533e7b517f5f65d8a70da5a.png[/img]
[/center]
2022 Girls in Math at Yale, R3
[b]p7[/b] Cindy cuts regular hexagon $ABCDEF$ out of a sheet of paper. She folds $B$ over $AC$, resulting in a pentagon. Then, she folds $A$ over $CF$, resulting in a quadrilateral. The area of $ABCDEF$ is $k$ times the area of the resulting folded shape. Find $k$.
[b]p8[/b] Call a sequence $\{a_n\} = a_1, a_2, a_3, . . .$ of positive integers [i]Fib-o’nacci[/i] if it satisfies $a_n = a_{n-1}+a_{n-2}$ for all $n \ge 3$. Suppose that $m$ is the largest even positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = m$, and suppose that $n$ is the largest odd positive integer such that exactly one [i]Fib-o’nacci[/i] sequence satisfies $a_5 = n$. Find $mn$.
[b]p9[/b] Compute the number of ways there are to pick three non-empty subsets $A$, $B$, and $C$ of $\{1, 2, 3, 4, 5, 6\}$, such that $|A| = |B| = |C|$ and the following property holds:
$$A \cap B \cap C = A \cap B = B \cap C = C \cap A.$$
2021 Girls in Math at Yale, R1
1. If $5x+3y-z=4$, $x=y$, and $z=4$, find $x+y+z$.
2. How many ways are there to pick three distinct vertices of a regular hexagon such that the triangle with those three points as its vertices shares exactly one side with the hexagon?
3. Sirena picks five distinct positive primes, $p_1 < p_2 < p_3 < p_4 < p_5$, and finds that they sum to $192$. If the product $p_1p_2p_3p_4p_5$ is as large as possible, what is $p_1 - p_2 + p_3 - p_4 + p_5$?
2024 MMATHS, 10
In acute $\triangle{ABC},$ $AB=11$ and $CB=10.$ Points $E$ and $D$ are constructed such that $\angle{CBE}$ and $\angle{ABD}$ are right, and $ACEBD$ is a non-degenerate pentagon. Additionally, $\angle{AEB} \cong \angle{DCB}, AE=CD,$ and $ED=20.$ Given that $EA$ and $CD$ intersect at $P$ and $AP=4,$ find $CP^2.$
2023 MMATHS, 5
$\omega_A, \omega_B, \omega_C$ are three concentric circles with radii $2,3,$ and $7,$ respectively. We say that a point $P$ in the plane is [i]nice[/i] if there are points $A, B,$ and $C$ on $\omega_A, \omega_B,$ and $\omega_C,$ respectively, such that $P$ is the centroid of $\triangle{ABC}.$ If the area of the smallest region of the plane containing all nice points can be expressed as $\tfrac{a\pi}{b},$ where $a$ and $b$ are relatively prime positive integers , what is $a+b$?
2023 MMATHS, 8
$30$ people sit around a table, some of which are Yale students. Each person is asked if the person to their right is a Yale student. Yale students will always answer correctly, but non-Yale students will answer randomly. Find the smallest possible number of Yale students such that, after hearing everyone’s answers and knowing the number of Yale students, it is possible to identify for certain at least one Yale student.
2024 MMATHS, 8
Let circle $A$ have radius $9,$ and let circle $B$ have radius $5$ and be internally tangent to circle $A.$ The largest radius $r$ such that there are two circles with radius $r$ that lie inside circle $A,$ are externally tangent to each other, and externally tangent with circle $B$ can be expressed as a fraction $\tfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2023 MMATHS, 6
Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$
2022 Girls in Math at Yale, 4
Kara rolls a six-sided die, and if on that first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled?
[i]Proposed by Andrew Wu[/i]
2021 MMATHS, 7
Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked?
[i]Proposed by Andrew Wu[/i]
2024 MMATHS, 11
Define a sequence $a_{m,n}$ where $a_{m,0}=1,$ and for all other $m,n$ (assuming $m \ge 1$): $$a_{m,n}=\begin{cases}
0 & n<0 \\
1 & n \equiv 0 \mod{m} \\
a_{m,n-1}+a_{m, n-m} & \text{else}
\end{cases}$$ If $\tfrac{a_{2025, (2025^2-1)}}{a_{2025, (2024^2-1)}} = \tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, then what is $a+b$?
2024 MMATHS, 6
How many $7$ digit numbers are there that satisfy the following?
[list]
[*] All digits are distinct from $1-7.$
[*] The first digit (from the left) is divisible by $1.$
[*] The two-digit number formed by the first two digits is divisible by $2.$
[*] The three-digit number formed by the first three digits is divisible by $3.$
[*] The four-digit number formed by the first four digits is divisible by $4.$
[*] The five-digit number formed by the first five digits is divisible by $5.$
[*] The six-digit number formed by the first six digits is divisible by $6.$
[/list]
2021 Girls in Math at Yale, 12
Let $\Gamma_1$ and $\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that they are tangent to and lie on the same side of line $\ell$. Points $A$ and $B$ are selected on $\ell$ such that $\Gamma_1$ and $\Gamma_2$ are internally tangent to the circle with diameter $AB$. If $AB = a + b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, then find $a + b + c$.
[i]Proposed by Andrew Wu, Deyuan Li, and Andrew Milas[/i]
2021 Girls in Math at Yale, R2
4. Suppose that $\overline{A2021B}$ is a six-digit integer divisible by $9$. Find the maximum possible value of $A \cdot B$.
5. In an arbitrary triangle, two distinct segments are drawn from each vertex to the opposite side. What is the minimum possible number of intersection points between these segments?
6. Suppose that $a$ and $b$ are positive integers such that $\frac{a}{b-20}$ and $\frac{b+21}{a}$ are positive integers. Find the maximum possible value of $a + b$.
2024 MMATHS, 5
Two subsets are called [i]disjoint[/i] if they do not share any common elements. Compute the number of ordered tuples $(A,B,C),$ where $A,B,$ and $C$ are subsets (not necessarily distinct or non-empty) of $\{1, 2, 3,4,5\}$ such that $A$ and $B$ are disjoint and $B$ and $C$ are disjoint.
2023 MMATHS, 5
We call $\triangle{ABC}$ with centroid $G$ [i]balanced[/i] on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B.$ $\triangle{XYZ},$ with $XY=2023$ and $\angle{ZXY}=120^\circ,$ is balanced on $XY.$ What is the maximum value of $XZ$?
2021 Girls in Math at Yale, R3
7. Peggy picks three positive integers between $1$ and $25$, inclusive, and tells us the following information about those numbers:
[list]
[*] Exactly one of them is a multiple of $2$;
[*] Exactly one of them is a multiple of $3$;
[*] Exactly one of them is a multiple of $5$;
[*] Exactly one of them is a multiple of $7$;
[*] Exactly one of them is a multiple of $11$.
[/list]
What is the maximum possible sum of the integers that Peggy picked?
8. What is the largest positive integer $k$ such that $2^k$ divides $2^{4^8}+8^{2^4}+4^{8^2}$?
9. Find the smallest integer $n$ such that $n$ is the sum of $7$ consecutive positive integers and the sum of $12$ consecutive positive integers.
2023 MMATHS, 12
Let $ABC$ be a triangle with incenter $I.$ The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA,$ and $AB$ at points $D, E,$ and $F,$ respectively. Let $D'$ be the reflection of $D$ over $I.$ Let $P$ be a point on $\omega$ such that $\angle{ADP}=90^\circ.$ $\mathcal{H}$ is a hyperbola passing through $D', E, F, I,$ and $P.$ Given that $\angle{BAD}=45^\circ$ and $\angle{CAD}=30^\circ,$ the acute angle between the asymptotes of $\mathcal{H}$ can be expressed as $\left(\tfrac{m}{n}\right)^\circ,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$
2021 MMATHS, 5
Suppose that $a_1 = 1$, and that for all $n \ge 2$, $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$. If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$, find $k$.
[i]Proposed by Andrew Wu[/i]
2022 Girls in Math at Yale, 5
Cat and Claire are having a conversation about Cat's favorite number.
Cat says, "My favorite number is a two-digit positive integer with distinct nonzero digits, $\overline{AB}$, such that $A$ and $B$ are both factors of $\overline{AB}$."
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, and my favorite number is among those four numbers! Also, the square of my number is the product of two of the other numbers among the four you mentioned!"
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, 11
A right rectangular prism has integer side lengths $a$, $b$, and $c$. If $\text{lcm}(a,b)=72$, $\text{lcm}(a,c)=24$, and $\text{lcm}(b,c)=18$, what is the sum of the minimum and maximum possible volumes of the prism?
[i]Proposed by Deyuan Li and Andrew Milas[/i]
2024 MMATHS, 4
Consider a pattern of squares and triangles. The first move of the pattern is to place an isosceles right triangle with side lengths $1, 1, \sqrt{2}.$ For each subsequent move, you need to attach a square to every non-hypotenuse side of a triangle and attach the same isosceles right triangle to every side of a square. After $2024$ moves, what is smallest possible area of the resulting shape?
2023 MMATHS, 4
How many distinct real numbers $x$ satisfy the equation $4\cos^3(x)+\sqrt{x}=3\sin(x)+\cos(3x)$?
2021 Girls in Math at Yale, 6
Kara rolls a six-sided die six times, and notices that the results satisfy the following conditions:
[list]
[*] She rolled a $6$ exactly three times;
[*] The product of her first three rolls is the same as the product of her last three rolls.
[/list]
How many distinct sequences of six rolls could Kara have rolled?
[i]Proposed by Andrew Wu[/i]
2024 MMATHS, 6
Cat and Claire are having a discussion about their favorite positive two-digit numbers.
[b]Cat:[/b] My number has a $1$ in its tens digit. Is it possible that your number is a multiple of my number?
[b]Claire:[/b] No, however, my number is not prime. Additionally, if I told you the two digits of my number, you still wouldn't know my number.
[b]Cat:[/b] Aha, my number and your number aren't relatively prime!
[b]Claire:[/b] Then our numbers must share the same ones digit!
What is the product of Cat and Claire's numbers?