MathDB

2023 MMATHS

Part of MMATHS problems

Subcontests

(12)
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MMATHS 2023 Individual Problem 12: Hyperbola through 5 points

Let ABCABC be a triangle with incenter I.I. The incircle ω\omega of ABCABC is tangent to sides BC,CA,BC, CA, and ABAB at points D,E,D, E, and F,F, respectively. Let D′D' be the reflection of DD over I.I. Let PP be a point on ω\omega such that ∠ADP=90∘.\angle{ADP}=90^\circ. H\mathcal{H} is a hyperbola passing through D′,E,F,I,D', E, F, I, and P.P. Given that ∠BAD=45∘\angle{BAD}=45^\circ and ∠CAD=30∘,\angle{CAD}=30^\circ, the acute angle between the asymptotes of H\mathcal{H} can be expressed as (mn)∘,\left(\tfrac{m}{n}\right)^\circ, where mm and nn are relatively prime positive integers. Find m+n.m+n.

MMATHS 2023 Team Problem 12: An incenter and excenter configuration

Let ABCABC be a triangle with incenter I,I, circumcenter O,O, and AA-excenter JA.J_A. The incircle of △ABC\triangle{ABC} touches side BCBC at a point D.D. Lines OIOI and JADJ_AD meet at a point K.K. Line AKAK meets the circumcircle of △ABC\triangle{ABC} again at a point L≠A.L \neq A. If BD=11,CD=5,BD=11, CD=5, and AO=10,AO=10, the length of DLDL can be expressed as mpn,\tfrac{m\sqrt{p}}{n}, where m,n,pm,n,p are positive integers, mm and nn are relatively prime, and pp is not divisible by the square of any prime. Find m+n+p.m+n+p.
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MMATHS 2023 Individual Problem 11: Knight on infinite chessboard

A knight is on an infinite chessboard. After exactly 100100 legal moves, how many different possible squares can it end on? A knight can move to any of the 88 closest squares not on the same row, column, or diagonal, as illustrated in the figure below. https://cdn.artofproblemsolving.com/attachments/0/7/144226144fb3ead533e7b517f5f65d8a70da5a.png

MMATHS 2023 Team Problem 11: Two sequences and a function

Suppose we have sequences (an)n≥0(a_n)_{n \ge 0} and (bn)n≥0(b_n)_{n \ge 0} and the function f(x)=1xf(x)=\tfrac{1}{x} such that for all nn we have:
[*]an+1=f(f(an+bn)−f(f(an)+f(bn))a_{n+1} = f(f(a_n+b_n)-f(f(a_n)+f(b_n)) [*]an+2=f(1−an)−f(1+an)a_{n+2} = f(1-a_n) - f(1+a_n) [*]bn+2=f(1−bn)−f(1+bn)b_{n+2} = f(1-b_n) - f(1+b_n)
Given that a0=16a_0=\tfrac{1}{6} and b0=17,b_0=\tfrac{1}{7}, then b5=mn,b_5=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find the sum of the prime factors of mn.mn.
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MMATHS 2023 Individual Problem 9: Equilateral triangle inscribed in a triangle

In △ABC\triangle{ABC} with ∠BAC=60∘,\angle{BAC}=60^\circ, points D,E,D, E, and FF lie on BC,AC,BC, AC, and ABAB respectively, such that DD is the midpoint of BCBC and △DEF\triangle{DEF} is equilateral. If BF=1BF=1 and EC=13,EC=13, then the area of △DEF\triangle{DEF} can be written as abc,\tfrac{a\sqrt{b}}{c}, where aa and cc are relatively prime positive integers and bb is not divisible by a square of a prime. Compute a+b+c.a+b+c.

MMATHS 2023 Team Problem 9: Sum of prime-indexed coefficients

Let (x+x−1+1)40=∑i=−4040aixi.(x+x^{-1}+1)^{40} = \sum_{i=-40}^{40} a_ix^i. Find the remainder when ∑p primeap\sum_{p \text{ prime}} a_p is divided by 41.41.
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MMATHS 2023 Individual Problem 7: Lights in a grid toggled

A 2023×20232023 \times 2023 grid of lights begins with every light off. Each light is assigned a coordinate (x,y).(x,y). For every distinct pair of lights (x1,y1),(x2,y2),(x_1, y_1), (x_2, y_2), with x1<x2x_1<x_2 and y1>y2,y_1>y_2, all lights strictly between them (i.e. x1<x<x2x_1<x<x_2 and y2<y<y1y_2<y<y_1) are toggled. After this procedure is done, how many lights are on?

MMATHS 2023 Team Problem 7: Spheres inside a tetrahedron concurrent at a point

ABCDABCD is a regular tetrahedron of side length 4.4. Four congruent spheres are inside ABCDABCD 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 vv be the volume of one of the spheres. If v2v^2 can be written as abπ2,\tfrac{a}{b}\pi^2, where aa and bb are relatively prime positive integers, find a+b.a+b.
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MMATHS 2023 Individual Problem 5: Maximizing XZ given XY and &lt;ZXY

We call △ABC\triangle{ABC} with centroid GG balanced on side ABAB if the foot of the altitude from GG onto line AB‾\overline{AB} lies between AA and B.B. △XYZ,\triangle{XYZ}, with XY=2023XY=2023 and ∠ZXY=120∘,\angle{ZXY}=120^\circ, is balanced on XY.XY. What is the maximum value of XZXZ?

MMATHS 2023 Team Problem 5: Locus of centroids

ωA,ωB,ωC\omega_A, \omega_B, \omega_C are three concentric circles with radii 2,3,2,3, and 7,7, respectively. We say that a point PP in the plane is nice if there are points A,B,A, B, and CC on ωA,ωB,\omega_A, \omega_B, and ωC,\omega_C, respectively, such that PP is the centroid of △ABC.\triangle{ABC}. If the area of the smallest region of the plane containing all nice points can be expressed as aπb,\tfrac{a\pi}{b}, where aa and bb are relatively prime positive integers , what is a+ba+b?
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MMATHS 2023 Individual Problem 4: Common area of unit hexagons

Let AA and BB be unit hexagons that share a center. Then, let P\mathcal{P} be the set of points contained in at least one of the hexagons. If the maximum possible area of P\mathcal{P} is XX and the minimum possible area of P\mathcal{P} is Y,Y, then the value of Y−XY-X can be expressed as ab−cd,\tfrac{a\sqrt{b}-c}{d}, where a,b,c,da,b,c,d are positive integers such that bb is square-free and gcd⁡(a,c,d)=1.\gcd(a,c,d)=1. Find a+b+c+d.a+b+c+d.

MMATHS 2023 Team Problem 4: Real number solutions to a trigonometric equation

How many distinct real numbers xx satisfy the equation 4cos⁡3(x)+x=3sin⁡(x)+cos⁡(3x)4\cos^3(x)+\sqrt{x}=3\sin(x)+\cos(3x)?
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MMATHS 2023 Individual Problem 3: SFFT polynomials

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),(m - a)(n - b), where a,b,m,na, 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 mm and n,n, he realizes that the value of ChatSFFT’s result is the same as the real result in terms of aa and bb. How many such pairs are there?

MMATHS 2023 Team Problem 3: Ordering the permutations of MMATHS

There are 360360 permutations of the letters in MMATHS.MMATHS. When ordered alphabetically, starting from AHMMST,AHMMST, MMATHSMMATHS is in the nnth permutation. What is nn?
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MMATHS 2023 Individual Problem 2: Game of Life on infinite grid

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 11, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is ab,\tfrac{a}{b}, where aa and bb are relatively prime positive integers, find a+b.a + b.

MMATHS 2023 Team Problem 2: Cheaters at a chess tournament

2020 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 6868 more half-cheating games than cheating games, how many of the players are cheaters?

MMATHS 2023 Tiebreaker Problem 2: Area of triangle given altitudes

The lengths of the altitudes of △ABC\triangle{ABC} are the roots of the polynomial x3−34x2+360x−1200.x^3-34x^2+360x-1200. Find the area of △ABC.\triangle{ABC}.
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MMATHS 2023 Team Problem 1: The favorite numbers of Lucy&#039;s children

Lucy has 88 children, each of whom has a distinct favorite integer from 11 to 10,10, inclusive. The smallest number that is a perfect multiple of all of these favorite numbers is 1260,1260, and the average of these favorite numbers is at most 5.5. Find the sum of the four largest numbers.

MMATHS 2023 Individual Problem 1: Cat and Claire are back again

Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit multiple of 77.” 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 100100 minus my number, you could determine my favorite number.” Claire says, “Now I know your favorite number!" What is Cat’s favorite number?

MMATHS 2023 Tiebreaker Problem 1: Tranformations of prime factorizations

Let n=p1e1p2e2…pkek=∏i=1kpiei,n=p_1^{e_1}p_2^{e_2}\dots p_k^{e_k}=\prod_{i=1}^k p_i^{e_i}, where p1<p2<⋯<pkp_1<p_2<\dots<p_k are primes and e1,e2,…,eke_1, e_2, \dots, e_k are positive integers, and let f(n)=∏i=1keipi.f(n) = \prod_{i=1}^k e_i^{p_i}. Find the number of integers nn such that 2≤n≤20232 \le n \le 2023 and f(n)=128.f(n)=128.