MathDB

2022 MMATHS

Part of MMATHS problems

Subcontests

(12)
8
2

2022 MMATHS Team Online p8 - number puzzle

In the number puzzle below, each cell contains a digit, each cell in the same bolded region has the same digit, and cells in different bolded regions have different digits. The answers to the clues are to be read as three-, four-, or five-digit numbers. Find the unique solution to the puzzle, given that no answer to any clue has a leading 00. https://cdn.artofproblemsolving.com/attachments/b/a/23514673819aea46c30fd2947f8c82710a1fb3.png

2022 MMATHS Individual p8 - sum of gcd of 64 pairs

Let S={1,2,3,5,6,10,15,30}S = \{1, 2, 3, 5, 6, 10, 15, 30\}. For each of the 6464 ordered pairs (a,b)(a, b) of elements of SS, AJ computes gcd(a,b)gcd(a, b). They then sum all of the numbers they computed. What is AJ’s sum?
12
2

2022 MMATHS Team Online p12 - 5th and final geo on the test, long live geo

Let triangle ABCABC with incenter II satisfy AB=3AB = 3, AC=4AC = 4, and BC=5BC = 5. Suppose that DD and EE lie on ABAB and ACAC, respectively, such that DD, II, and EE are collinear and DE⊥AIDE \perp AI. Points PP and QQ lie on side BCBC such that IP=BPIP = BP and IQ=CQIQ = CQ, and lines DPDP and EQEQ meet at SS. Compute SI2SI^2.

2022 MMATHS Individual p12 - Geometry, where art thou?

Let triangle ABCABC with incenter II satisfy AB=10AB = 10, BC=21BC = 21, and CA=17CA = 17. Points DD and E lie on side BCBC such that BD=4BD = 4, DE=6DE = 6, and EC=11EC = 11. The circumcircles of triangles BIEBIE and CIDCID meet again at point PP, and line IPIP meets the altitude from AA to BCBC at XX. Find (DX⋅EX)2(DX \cdot EX)^2.
11
2

2022 MMATHS Team Online p11 - 3 emojis: barbecue, bacon, or burger

Every time Josh and Ron tap their screens, one of three emojis appears, each with equal probability: barbecue, bacon, or burger. Josh taps his screen until he gets a sequence of barbecue, bacon, and burger consecutively (in that specific order.) Ron taps his screen until he gets a sequence of three bacons in a row. Let MM and NN be the expected number of times Josh and Ron tap their screens, respectively. What is ∣M−N∣|M-N|?

2022 MMATHS Individual p11 - Im(1/w)=Im(k/w^2)=Im(k/w^3) complex

Denote by Re(z)Re(z) and Im(z)Im(z) the real part and imaginary part, respectively, of a complex number zz; that is, if z=a+biz = a + bi, then Re(z)=aRe(z) = a and Im(z)=bIm(z) = b. Suppose that there exists some real number kk such that Im(1w)=Im(kw2)=Im(kw3)Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) for some complex number ww with ∣∣w∣∣=32||w||=\frac{\sqrt3}{2} , Re(w)>0Re(w) > 0, and Im(w)≠0Im(w) \ne 0. If kk can be expressed as a−bc\frac{\sqrt{a}-b}{c} for integers aa, bb, cc with aa squarefree, find a+b+ca + b + c.
10
2

2022 MMATHS Team Online p10 - sequence of circumcenters, fixed sum ZA^2_i

Suppose that A1A2A3A_1A_2A_3 is a triangle with A1A2=16A_1A_2 = 16 and A1A3=A2A3=10A_1A_3 = A_2A_3 = 10. For each integer n≥4n \ge 4, set An to be the circumcenter of triangle An−1An−2An−3A_{n-1}A_{n-2}A_{n-3}. There exists a unique point ZZ lying in the interiors of the circumcircles of triangles AkAk+1Ak+2A_kA_{k+1}A_{k+2} for all integers k≥1k \ge 1. If ZA12+ZA22+ZA32+ZA42ZA^2_1+ ZA^2_2+ ZA^2_3+ ZA^2_4 can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd(a,b)=1gcd(a, b) = 1, find a+ba + b.

2022 MMATHS Individual p10 - sum of f(d) when d|M

Define a function ff on the positive integers as follows: f(n)=mf(n) = m, where mm is the least positive integer such that nn is a factor of m2m^2. Find the smallest integer MM such that M\sqrt{M} is both a product of prime numbers, of which there are at least 33, and a factor of ∑d∣Mf(d),\sum_{ d|M} f(d), the sum of f(d)f(d) for all positive integers dd that divide MM.
9
2

2022 MMATHS Team Online p9 - sum of roots of monic cubic polynomial =1

Let ff be a monic cubic polynomial such that the sum of the coefficients of ff is 55 and such that the sum of the roots of ff is 11. Find the absolute value of the sum of the cubes of the roots of ff.

2022 MMATHS Individual p9 - 2 sequences

Suppose sequence {ai}=a1,a2,a3,....\{a_i\} = a_1, a_2, a_3, .... satisfies an+1=1an+1a_{n+1} = \frac{1}{a_n+1} for all positive integers nn. Define bkb_k for positive integers k≥2k \ge 2 to be the minimum real number such that the product a1⋅a2⋅...⋅aka_1 \cdot a_2 \cdot ...\cdot a_k does not exceed bkb_k for any positive integer choice of a1a_1. Find 1b2+1b3+1b4+...+1b10.\frac{1}{b_2}+\frac{1}{b_3}+\frac{1}{b_4}+...+\frac{1}{b_{10}}. .
7
2

2022 MMATHS Team Online p7 - 3 right angles and midpoint arc

△ABC\vartriangle ABC satisfies AB=16AB = 16, BC=30BC = 30, and ∠ABC=90o\angle ABC = 90^o. On the circumcircle of △ABC\vartriangle ABC, let PP be the midpoint of arc ACAC not containing BB, and let XX and YY lie on lines ABAB and BCBC, respectively, with PX⊥ABPX \perp AB and PY⊥BCPY \perp BC. Find XY2XY^2.

2022 MMATHS Individual p7 - percentages and dollars

Katherine makes Benj play a game called 5050 Cent. Benj starts with $0.50\$0.50, and every century thereafter has a 50%50\% chance of doubling his money and a 50%50\% chance of having his money reset to $0.50\$0.50. What is the expected value of the amount of money Benj will have, in dollars, after 5050 centuries?
6
2

2022 MMATHS Team Online p6 - word by 5 different letters

Prair writes the letters A,B,C,DA,B,C,D, and EE such that neither vowel are written first, and they are not adjacent; such that there exists at least one pair of adjacent consonants; and such that exactly five pairs of letters are in alphabetical order. How many possible ways could Prair have ordered the letters?

2022 MMATHS Individual p6 - flipping a coin to get + or -

Siva has the following expression, which is missing operations: 12  _  14  _  18  _  116  _  132.\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}. For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, he computes the value of the expression. He then repeats the entire process with a new set of coinflips and operations. If the probability that the positive difference between his computed values is greater than 12\frac12 can be expressed as ab\frac{a}{b} for relatively prime positive integers aa, bb, then find a+ba + b.
5
2

2022 MMATHS Team Online p5 - min sum of 2 areas, starting with equilateral

Equilateral triangle △ABC\vartriangle ABC has side length 66. Points DD and EE lie on BC‾\overline{BC} such that BD=CEBD = CE and BB, DD, EE, CC are collinear in that order. Points FF and GG lie on AB‾\overline{AB} such that FD‾⊥BC‾\overline{FD} \perp \overline{BC}, and GF=GAGF = GA. If the minimum possible value of the sum of the areas of △BFD\vartriangle BFD and △DGE\vartriangle DGE can be expressed as abc\frac{a\sqrt{b}}{c} for positive integers a,b,ca, b, c with gcd(a,c)=1gcd (a, c) = 1 and bb squarefree, find a+b+ca + b + c.

2022 MMATHS Individual p5 - folding a rectange

Holding a rectangular sheet of paper ABCDABCD, Prair folds triangle ABDABD over diagonal BDBD, so that the new location of point AA is A′A'. She notices that A′C=13BDA'C =\frac13 BD. If the area of ABCDABCD is 27227\sqrt2, find BDBD.
4
2

2022 MMATHS Team Online p4 - even 3 digit numbrt in base b >=10

How many ways are there to choose three digits A,B,CA,B,C with 1≤A≤91 \le A \le 9 and 0≤B,C≤90 \le B,C \le 9 such that ABC‾b\overline{ABC}_b is even for all choices of base bb with b≥10b \ge 10?

2022 MMATHS Individual p4 - 2-digit favorite number

Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!” Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?” Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.” Claire says, “Now I know your favorite number!” What is Cat’s favorite number?
3
2
2
2
1
2