Found problems: 146
2022 MMATHS, 2
Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.
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]
MMATHS Mathathon Rounds, 2020
[u]Round 1[/u]
[b]p1.[/b] Let $n$ be a two-digit positive integer. What is the maximum possible sum of the prime factors of $n^2 - 25$ ?
[b]p2.[/b] Angela has ten numbers $a_1, a_2, a_3, ... , a_{10}$. She wants them to be a permutation of the numbers $\{1, 2, 3, ... , 10\}$ such that for each $1 \le i \le 5$, $a_i \le 2i$, and for each $6 \le i \le 10$, $a_i \le - 10$. How many ways can Angela choose $a_1$ through $a_{10}$?
[b]p3.[/b] Find the number of three-by-three grids such that
$\bullet$ the sum of the entries in each row, column, and diagonal passing through the center square is the same, and
$\bullet$ the entries in the nine squares are the integers between $1$ and $9$ inclusive, each integer appearing in exactly one square.
[u]Round 2 [/u]
[b]p4.[/b] Suppose that $P(x)$ is a quadratic polynomial such that the sum and product of its two roots are equal to each other. There is a real number $a$ that $P(1)$ can never be equal to. Find $a$.
[b]p5.[/b] Find the number of ordered pairs $(x, y)$ of positive integers such that $\frac{1}{x} +\frac{1}{y} =\frac{1}{k}$ and k is a factor of $60$.
[b]p6.[/b] Let $ABC$ be a triangle with $AB = 5$, $AC = 4$, and $BC = 3$. With $B = B_0$ and $C = C_0$, define the infinite sequences of points $\{B_i\}$ and $\{C_i\}$ as follows: for all $i \ge 1$, let $B_i$ be the foot of the perpendicular from $C_{i-1}$ to $AB$, and let $C_i$ be the foot of the perpendicular from $B_i$ to $AC$. Find $C_0C_1(AC_0 + AC_1 + AC_2 + AC_3 + ...)$.
[u]Round 3 [/u]
[b]p7.[/b] If $\ell_1, \ell_2, ... ,\ell_{10}$ are distinct lines in the plane and $p_1, ... , p_{100}$ are distinct points in the plane, then what is the maximum possible number of ordered pairs $(\ell_i, p_j )$ such that $p_j$ lies on $\ell_i$?
[b]p8.[/b] Before Andres goes to school each day, he has to put on a shirt, a jacket, pants, socks, and shoes. He can put these clothes on in any order obeying the following restrictions: socks come before shoes, and the shirt comes before the jacket. How many distinct orders are there for Andres to put his clothes on?
[b]p9. [/b]There are ten towns, numbered $1$ through $10$, and each pair of towns is connected by a road. Define a backwards move to be taking a road from some town $a$ to another town $b$ such that $a > b$, and define a forwards move to be taking a road from some town $a$ to another town $b$ such that $a < b$. How many distinct paths can Ali take from town $1$ to town $10$ under the conditions that
$\bullet$ she takes exactly one backwards move and the rest of her moves are forward moves, and
$\bullet$ the only time she visits town $10$ is at the very end?
One possible path is $1 \to 3 \to 8 \to 6 \to 7 \to 8 \to 10$.
[u]Round 4[/u]
[b]p10.[/b] How many prime numbers $p$ less than $100$ have the properties that $p^5 - 1$ is divisible by $6$ and $p^6 - 1$ is divisible by $5$?
[b]p11.[/b] Call a four-digit integer $\overline{d_1d_2d_3d_4}$ [i]primed [/i] if
1) $d_1$, $d_2$, $d_3$, and $d_4$ are all prime numbers, and
2) the two-digit numbers $\overline{d_1d_2}$ and $\overline{d_3d_4}$ are both prime numbers.
Find the sum of all primed integers.
[b]p12.[/b] Suppose that $ABC$ is an isosceles triangle with $AB = AC$, and suppose that $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, with $\overline{DE} \parallel \overline{BC}$. Let $r$ be the length of the inradius of triangle $ADE$. Suppose that it is possible to construct two circles of radius $r$, each tangent to one another and internally tangent to three sides of the trapezoid $BDEC$. If $\frac{BC}{r} = a + \sqrt{b}$ forpositive integers $a$ and $b$ with $b$ squarefree, then find $a + b$.
PS. You should use hide for answers. Rounds 5-7 have been posted [url=https://artofproblemsolving.com/community/c4h2800986p24675177]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
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 $a+b$?
2019 MMATHS, 2
In the trapezoid $ABCD$, both $\angle B$ and $\angle C$ are right angles, and all four sides of the trapezoid are tangent to the same circle. If $\overline{AB} = x$ and $\overline{CD} = y$, find the area of $ABCD$ (with proof).
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 $mn.$
MMATHS Mathathon Rounds, 2020
[u]Round 5 [/u]
[b]p13.[/b] A palindrome is a number that reads the same forward as backwards; for example, $121$ and $36463$ are palindromes. Suppose that $N$ is the maximal possible difference between two consecutive three-digit palindromes. Find the number of pairs of consecutive palindromes $(A, B)$ satisfying $A < B$ and $B - A = N$.
[b]p14.[/b] Suppose that $x, y$, and $z$ are complex numbers satisfying $x +\frac{1}{yz} = 5$, $y +\frac{1}{zx} = 8$, and $z +\frac{1}{xy} = 6$. Find the sum of all possible values of $xyz$.
[b]p15.[/b] Let $\Omega$ be a circle with radius $25\sqrt2$ centered at $O$, and let $C$ and $J$ be points on $\Omega$ such that the circle with diameter $\overline{CJ}$ passes through $O$. Let $Q$ be a point on the circle with diameter $\overline{CJ}$ satisfying $OQ = 5\sqrt2$. If the area of the region bounded by $\overline{CQ}$, $\overline{QJ}$, and minor arc $JC$ on $\Omega$ can be expressed as $\frac{a\pi-b}{c}$ for integers $a, b$, and $c$ with $gcd \,\,(a, c) = 1$, then find $a + b + c$.
[u]Round 6[/u]
[b]p16.[/b] Veronica writes $N$ integers between $2$ and $2020$ (inclusive) on a blackboard, and she notices that no number on the board is an integer power of another number on the board. What is the largest possible value of $N$?
[b]p17.[/b] Let $ABC$ be a triangle with $AB = 12$, $AC = 16$, and $BC = 20$. Let $D$ be a point on $AC$, and suppose that $I$ and $J$ are the incenters of triangles $ABD$ and $CBD$, respectively. Suppose that $DI = DJ$. Find $IJ^2$.
[b]p18.[/b] For each positive integer $a$, let $P_a = \{2a, 3a, 5a, 7a, . . .\}$ be the set of all prime multiples of $a$. Let $f(m, n) = 1$ if $P_m$ and $P_n$ have elements in common, and let $f(m, n) = 0$ if $P_m$ and $P_n$ have no elements in common. Compute $$\sum_{1\le i<j\le 50} f(i, j)$$ (i.e. compute $f(1, 2) + f(1, 3) + ,,, + f(1, 50) + f(2, 3) + f(2, 4) + ,,, + f(49, 50)$.)
[u]Round 7[/u]
[b]p19.[/b] How many ways are there to put the six letters in “$MMATHS$” in a two-by-three grid such that the two “$M$”s do not occupy adjacent squares and such that the letter “$A$” is not directly above the letter “$T$” in the grid? (Squares are said to be adjacent if they share a side.)
[b]p20.[/b] Luke is shooting basketballs into a hoop. He makes any given shot with fixed probability $p$ with $p < 1$, and he shoots n shots in total with $n \ge 2$. Miraculously, in $n$ shots, the probability that Luke makes exactly two shots in is twice the probability that Luke makes exactly one shot in! If $p$ can be expressed as $\frac{k}{100}$ for some integer $k$ (not necessarily in lowest terms), find the sum of all possible values for $k$.
[b]p21.[/b] Let $ABCD$ be a rectangle with $AB = 24$ and $BC = 72$. Call a point $P$ [i]goofy [/i] if it satisfies the following conditions:
$\bullet$ $P$ lies within $ABCD$,
$\bullet$ for some points $F$ and $G$ lying on sides $BC$ and $DA$ such that the circles with diameter $BF$ and $DG$ are tangent to one another, $P$ lies on their common internal tangent.
Find the smallest possible area of a polygon that contains every single goofy point inside it.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2800971p24674988]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
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 $a+b+c.$
2018 MMATHS, 1
Daniel has an unlimited supply of tiles labeled “$2$” and “$n$” where $n$ is an integer. Find (with proof) all the values of $n$ that allow Daniel to fill an $8 \times 10$ grid with these tiles such that the sum of the values of the tiles in each row or column is divisible by $11$.
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 $41.$
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]
2017 MMATHS, Mixer Round
[b]p1.[/b] Suppose Mitchell has a fair die. He is about to roll it six times. The probability that he rolls $1$, $2$, $3$, $4$, $5$, and then $6$ in that order is $p$. The probability that he rolls $2$, $2$, $4$, $4$, $6$, and then $6$ in that order is $q$. What is $p - q$?
[b]p2.[/b] What is the smallest positive integer $x$ such that $x \equiv 2017$ (mod $2016$) and $x \equiv 2016$ (mod $2017$) ?
[b]p3.[/b] The vertices of triangle $ABC$ lie on a circle with center $O$. Suppose the measure of angle $ACB$ is $45^o$. If $|AB| = 10$, then what is the distance between $O$ and the line $AB$?
[b]p4.[/b] A “word“ is a sequence of letters such as $YALE$ and $AELY$. How many distinct $3$-letter words can be made from the letters in $BOOLABOOLA$ where each letter is used no more times than the number of times it appears in $BOOLABOOLA$?
[b]p5.[/b] How many distinct complex roots does the polynomial $p(x) = x^{12} - x^8 - x^4 + 1$ have?
[b]p6.[/b] Notice that $1 = \frac12 + \frac13 + \frac16$ , that is, $1$ can be expressed as the sum of the three fractions $\frac12 $, $\frac13$ , and $\frac16$ , where each fraction is in the form $\frac{1}{n}$, with each $n$ different. Give a $6$-tuple of distinct positive integers $(a, b, c, d, e, f)$ where $a < b < c < d < e < f$ such that $\frac{1}{a} +\frac{1}{b} + \frac{1}{c} + \frac{1}{d} + \frac{1}{e} + \frac{1}{f} = 1$ and explain how you arrived at your $6$-tuple. Multiple answers will be accepted.
[b]p7.[/b] You have a Monopoly board, an $11 \times 11$ square grid with the $9 \times 9$ internal square grid removed, where every square is blank except for Go, which is the square in the bottom right corner. During your turn, you determine how many steps forward (which is in the counterclockwise direction) to move by rolling two standard $6$-sided dice. Let $S$ be the set of squares on the board such that if you are initially on a square in $S$, no matter what you roll with the dice, you will always either land on Go (move forward enough squares such that you end up on Go) or you pass Go (you move forward enough squares such that you step on Go during your move and then you advance past Go). You randomly and uniformly select one square in $S$ as your starting position. What is the probability that you land on Go?
[b]p8.[/b] Using $L$-shaped triominos, and dominos, where each square of a triomino and a domino covers one unit, what is the minimum number of tiles needed to cover a $3$-by-$2017$ rectangle without any gaps?
[b]p9.[/b] Does there exist a pair of positive integers $(x, y)$, where $x < y$, such that $x^2 + y^2 = 1009^3$? If so, give a pair $(x, y)$ and explain how you found that pair. If not, explain why.
[b]p10.[/b] Triangle $ABC$ has inradius $8$ and circumradius $20$. Let $M$ be the midpoint of side $BC$, and let $N$ be the midpoint of arc $BC$ on the circumcircle not containing $A$. Let $s_A$ denote the length of segment $MN$, and define $s_B$ and $s_C$ similarly with respect to sides $CA$ and $AB$. Evaluate the product $s_As_Bs_C$.
[b]p11.[/b] Julia and Dan want to divide up $256$ dollars in the following way: in the first round, Julia will offer Dan some amount of money, and Dan can choose to accept or reject the offer. If Dan accepts, the game is over. Otherwise, if Dan rejects, half of the money disappears. In the second round, Dan can offer Julia part of the remaining money. Julia can then choose to accept or reject the offer. This process goes on until an offer is accepted or until $4$ rejections have been made; once $4$ rejections are made, all of the money will disappear, and the bargaining process ends. If Julia or Dan is indifferent between accepting and rejecting an offer, they will accept the offer. Given that Julia and Dan are both rational and both have the goal of maximizing the amount of money they get, how much will Julia offer Dan in the first round?
[b]p12.[/b] A perfect partition of a positive integer $N$ is an unordered set of numbers (where numbers can be repeated) that sum to $N$ with the property that there is a unique way to express each positive integer less than $N$ as a sum of elements of the set. Repetitions of elements of the set are considered identical for the purpose of uniqueness. For example, the only perfect partitions of $3$ are $\{1, 1, 1\}$ and $\{1, 2\}$. $\{1, 1, 3, 4\}$ is NOT a perfect partition of $9$ because the sum $4$ can be achieved in two different ways: $4$ and $1 + 3$. How many integers $1 \le N \le 40$ each have exactly one perfect partition?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2018 MMATHS, Mixer Round
[b]p1.[/b] Suppose $\frac{x}{y} = 0.\overline{ab}$ where $x$ and $y$ are relatively prime positive integers and $ab + a + b + 1$ is a multiple of $12$. Find the sum of all possible values of $y$.
[b]p2.[/b] Let $A$ be the set of points $\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}$. How many distinct circles pass through at least three points in $A$?
[b]p3.[/b] Jack and Jill need to bring pails of water home. The river is the $x$-axis, Jack is initially at the point $(-5, 3)$, Jill is initially at the point $(6, 1)$, and their home is at the point $(0, h)$ where $h > 0$. If they take the shortest paths home given that each of them must make a stop at the river, they walk exactly the same total distance. What is $h$?
[b]p4.[/b] What is the largest perfect square which is not a multiple of $10$ and which remains a perfect square if the ones and tens digits are replaced with zeroes?
[b]p5.[/b] In convex polygon $P$, each internal angle measure (in degrees) is a distinct integer. What is the maximum possible number of sides $P$ could have?
[b]p6.[/b] How many polynomials $p(x)$ of degree exactly $3$ with real coefficients satisfy $$p(0), p(1), p(2), p(3) \in \{0, 1, 2\}?$$
[b]p7.[/b] Six spheres, each with radius $4$, are resting on the ground. Their centers form a regular hexagon, and adjacent spheres are tangent. A seventh sphere, with radius $13$, rests on top of and is tangent to all six of these spheres. How high above the ground is the center of the seventh sphere?
[b]p8.[/b] You have a paper square. You may fold it along any line of symmetry. (That is, the layers of paper must line up perfectly.) You then repeat this process using the folded piece of paper. If the direction of the folds does not matter, how many ways can you make exactly eight folds while following these rules?
[b]p9.[/b] Quadrilateral $ABCD$ has $\overline{AB} = 40$, $\overline{CD} = 10$, $\overline{AD} = \overline{BC}$, $m\angle BAD = 20^o$, and $m \angle ABC = 70^o$. What is the area of quadrilateral $ABCD$?
[b]p10.[/b] We say that a permutation $\sigma$ of the set $\{1, 2,..., n\}$ preserves divisibilty if $\sigma (a)$ divides $\sigma (b)$ whenever $a$ divides $b$. How many permutations of $\{1, 2,..., 40\}$ preserve divisibility? (A permutation of $\{1, 2,..., n\}$ is a function $\sigma$ from $\{1, 2,..., n\}$ to itself such that for any $b \in \{1, 2,..., n\}$, there exists some $a \in \{1, 2,..., n\}$ satisfying $\sigma (a) = b$.)
[b]p11.[/b] In the diagram shown at right, how many ways are there to remove at least one edge so that some circle with an “A” and some circle with a “B” remain connected?
[img]https://cdn.artofproblemsolving.com/attachments/8/7/fde209c63cc23f6d3482009cc6016c7cefc868.png[/img]
[b]p12.[/b] Let $S$ be the set of the $125$ points in three-dimension space of the form $(x, y, z)$ where $x$, $y$, and $z$ are integers between $1$ and $5$, inclusive. A family of snakes lives at the point $(1, 1, 1)$, and one day they decide to move to the point $(5, 5, 5)$. Snakes may slither only in increments of $(1,0,0)$, $(0, 1, 0)$, and $(0, 0, 1)$. Given that at least one snake has slithered through each point of $S$ by the time the entire family has reached $(5, 5, 5)$, what is the smallest number of snakes that could be in the family?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
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 $a+b$?
2022 MMATHS, 3
There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet?
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 $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$
MMATHS Mathathon Rounds, 2021
[u]Round 1 [/u]
[b]p1.[/b] Ben the bear has an algorithm he runs on positive integers- each second, if the integer is even, he divides it by $2$, and if the integer is odd, he adds $1$. The algorithm terminates after he reaches $1$. What is the least positive integer n such that Ben's algorithm performed on n will terminate after seven seconds? (For example, if Ben performed his algorithm on $3$, the algorithm would terminate after $3$ seconds: $3 \to 4 \to 2 \to 1$.)
[b]p2.[/b] Suppose that a rectangle $R$ has length $p$ and width $q$, for prime integers $p$ and $q$. Rectangle $S$ has length $p + 1$ and width $q + 1$. The absolute difference in area between $S$ and $R$ is $21$. Find the sum of all possible values of $p$.
[b]p3.[/b] Owen the origamian takes a rectangular $12 \times 16$ sheet of paper and folds it in half, along the diagonal, to form a shape. Find the area of this shape.
[u]Round 2[/u]
[b]p4.[/b] How many subsets of the set $\{G, O, Y, A, L, E\}$ contain the same number of consonants as vowels? (Assume that $Y$ is a consonant and not a vowel.)
[b]p5.[/b] Suppose that trapezoid $ABCD$ satisfies $AB = BC = 5$, $CD = 12$, and $\angle ABC = \angle BCD = 90^o$. Let $AC$ and $BD$ intersect at $E$. The area of triangle $BEC$ can be expressed as $\frac{a}{b}$, for positive integers $a$ and $b$ with $gcd(a, b) = 1$. Find $a + b$.
[b]p6.[/b] Find the largest integer $n$ for which $\frac{101^n + 103^n}{101^{n-1} + 103^{n-1}}$ is an integer.
[u]Round 3[/u]
[b]p7.[/b] For each positive integer n between $1$ and $1000$ (inclusive), Ben writes down a list of $n$'s factors, and then computes the median of that list. He notices that for some $n$, that median is actually a factor of $n$. Find the largest $n$ for which this is true.
[b]p8.[/b] ([color=#f00]voided[/color]) Suppose triangle $ABC$ has $AB = 9$, $BC = 10$, and $CA = 17$. Let $x$ be the maximal possible area of a rectangle inscribed in $ABC$, such that two of its vertices lie on one side and the other two vertices lie on the other two sides, respectively. There exist three rectangles $R_1$, $R_2$, and $R_3$ such that each has an area of $x$. Find the area of the smallest region containing the set of points that lie in at least two of the rectangles $R_1$, $R_2$, and $R_3$.
[b]p9.[/b] Let $a, b,$ and $c$ be the three smallest distinct positive values of $\theta$ satisfying $$\cos \theta + \cos 3\theta + ... + \cos 2021\theta = \sin \theta+ \sin 3 \theta+ ... + \sin 2021\theta. $$
What is $\frac{4044}{\pi}(a + b + c)$?
[color=#f00]Problem 8 is voided. [/color]
PS. You should use hide for answers.Rounds 4-5 have been posted [url=https://artofproblemsolving.com/community/c4h3131422p28368457]here [/url] and 6-7 [url=https://artofproblemsolving.com/community/c4h3131434p28368604]here [/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
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 $a+b$?