AoPS Community 2023 MMATHS

Math Majors of America Tournament for High Schools 2023

www.artofproblemsolving.com/community/c4026055
by parmenides51, lpieleanu

– Individual

1 Cat and Claire are having a conversation about Cat’s favorite number. Cat says, ”My favorite
number is a two-digit multiple of 7.”
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 100 minus my number,
you could determine my favorite number.”
Claire says, ”Now I know your favorite number!”
What is Cat’s favorite number?

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 ab , where a and
b are relatively prime positive integers, find a + b.

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?

4 Let A and B be unit hexagons that share a center. Then, let P be the set of points contained in at
least one of the hexagons. If the maximum possible area of P √is X and the minimum possible
area of P is Y, then the value of Y − X can be expressed as a db−c , where a, b, c, d are positive
integers such that b is square-free and gcd(a, c, d) = 1. Find a + b + c + d.

5 We call △ABC with centroid G balanced on side AB if the foot of the altitude from G onto line
AB lies between A and B. △XY Z, with XY = 2023 and ∠ZXY = 120◦ , is balanced on XY.
What is the maximum value of XZ?
P2022 P2022  
ijπ
6 Compute i=1 j=1 cos 2023 .

7 A 2023 × 2023 grid of lights begins with every light off. Each light is assigned a coordinate (x, y).
For every distinct pair of lights (x1 , y1 ), (x2 , y2 ), with x1 < x2 and y1 > y2 , all lights strictly

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AoPS Community 2023 MMATHS

between them (i.e. x1 < x < x2 and y2 < y < y1 ) are toggled. After this procedure is done, how
many lights are on?

8 Find the number of ordered pairs of integers (m, n) such that 0 ≤ m, n ≤ 2023 and
X
m2 ≡ nd (mod 2024).
d|2023

9 In △ABC with ∠BAC = 60◦ , points D, E, and F lie on BC, AC, and AB respectively, such that
D is the midpoint of BC and √ △DEF is equilateral. If BF = 1 and EC = 13, then the area of
△DEF can be written as c , where a and c are relatively prime positive integers and b is not
a b

divisible by a square of a prime. Compute a + b + c.

10 Consider the recurrence relation xn+2 = 2xn+1 + xn , with x0 = 0, x1 = 1. What is the greatest
common divisor of x2023 and x721 ?

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.

https://cdn.artofproblemsolving.com/attachments/0/7/
144226144fb3ead533e7b517f5f65d8a70da5a.png

12 Let ABC be a triangle with incenter I. The incircle ω 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 ω such
that ∠ADP = 90◦ . H is a hyperbola passing through D′ , E, F, I, and P. Given that ∠BAD = 45 ◦
m ◦
and ∠CAD = 30 , the acute angle between the asymptotes of H can be expressed as n ,
◦

where m and n are relatively prime positive integers. Find m + n.

– Team

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.

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?

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AoPS Community 2023 MMATHS

3 There are 360 permutations of the letters in M M AT HS. When ordered alphabetically, starting
from AHM M ST, M M AT HS is in the nth permutation. What is n?
√
4 How many distinct real numbers x satisfy the equation 4 cos3 (x) + x = 3 sin(x) + cos(3x)?

5 ωA , ωB , ωC are three concentric circles with radii 2, 3, and 7, respectively. We say that a point P
in the plane is nice if there are points A, B, and C on ωA , ωB , and ωC , respectively, such that P is
the centroid of △ABC. If the area of the smallest region of the plane containing all nice points
can be expressed as aπ b , where a and b are relatively prime positive integers , what is a + b?

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?

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 ab π 2 , where a and b are relatively prime positive integers, find a + b.

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 hear-
ing everyone’s answers and knowing the number of Yale students, it is possible to identify for
certain at least one Yale student.
P40
Let (x + x−1 + 1)40 = i Find the remainder when is divided by 41.
P
9 i=−40 ai x . p prime ap

10 Find the number of ordered pairs of integers (m, n) with 0 ≤ m, n ≤ 22 such that k 2 + mk + n is
not a multiple of 23 for all integers k.

11 Suppose we have sequences (an )n≥0 and (bn )n≥0 and the function f (x) = 1
x such that for all n
we have:
-an+1 = f (f (an + bn ) − f (f (an ) + f (bn ))
-an+2 = f (1 − an ) − f (1 + an )
-bn+2 = f (1 − bn ) − f (1 + bn )
Given that a0 = 16 and b0 = 17 , then b5 = m
n , where m and n are relatively prime positive integers.
Find the sum of the prime factors of mn.

12 Let ABC be a triangle with incenter I, circumcenter O, and A-excenter JA . The incircle of △ABC
touches side BC at a point D. Lines OI and JA D meet at a point K. Line AK meets the circum-
circle of △ABC again at a point L ̸= A. If BD = 11, CD = 5, and AO = 10, the length of DL

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AoPS Community 2023 MMATHS
√
m p
can be expressed as 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.

– Tiebreaker

Let n = pe11 pe22 . . . pekk = ki=1 pei i , where p1 < p2 < · · · < pk are primes and e1 , e2 , . . . , ek are
Q
1
positive integers, and let f (n) = ki=1 epi i . Find the number of integers n such that 2 ≤ n ≤ 2023
Q
and f (n) = 128.

2 The lengths of the altitudes of △ABC are the roots of the polynomial x3 − 34x2 + 360x − 1200.
Find the area of △ABC.

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