MathDB

2022 CMWMC

Part of CMWMC

Subcontests

(9)
1

2022 CMWMC Individual Round 1-25 Carnegie Mellon University Womens'; Competition

p1. Let x=1521315−213.x =\frac{15213}{15 - 213}. Find the integer nearest to x.
p2. A grocery store sells oranges for either $1\$1 each or five for $4\$4. If Theo wants to buy 4040 oranges, they would save $k\$k by buying all five-packs instead of all single oranges. What is kk?
p3. Let ABCDABCD be a square. If ABAB and CDCD were increased in length by 20%20\% and ADAD and BCBC were decreased in length by 20%20\% while keeping ABCDABCD a rectangle, the area of ABCDABCD would change by k%k\%. Find kk.
p4. Polly writes down all nonnegative integers that contain at most one 00, at most three 22s, and no other digits. What is the median of all numbers that Polly writes down?
p5. Let PP be a point. 77 circles of distinct radii all pass through PP. Let nn be the total number of intersection points, including PP. What is the ratio of the maximum possible value of nn to the minimum possible value of nn?
p6. Define the sequence {an}\{a_n\} recursively with a0=2a_0 = 2, a1=3a_1 = 3, and an=a0+...+an−1a_n = a_0 +...+ a_{n-1} for n≥2n \ge 2. What is a2022a_{2022}?
p7. Define the sequence {an}\{a_n\} recursively with a0=1a_0 = 1, a1=0a_1 = 0, and an=2an−1+9an−2a_n = 2a_{n-1} + 9a_{n-2} for all n≥2n \ge 2. What is the units digit of a2022a_{2022}?
p8. Suppose that x satisfies ∣2x−2∣−2≥x|2x - 2| -2 \ge x. Find the sum of the minimum and maximum possible value of xx.
p9. Clarabelle has 50005000 cards numbered 11 to 50005000. They pick five at random and then place them face down such that their numbers are increasing from left to right. They then turn over the third card to reveal the number 20222022. What is the probability that the first card is a 11?
p10. Rays r1r_1 and r2r_2 share a common endpoint. Three squares have sides on one of the rays and vertices on the other, as shown in the diagram. If the side lengths of the smallest two squares are 2020 and 2222, find the side length of the largest square. https://cdn.artofproblemsolving.com/attachments/1/2/3717aa86a65e20b94a0d8161f89bea603411e9.png
p11. There exists a rectangle ABCDABCD and a point PP inside ABCDABCD such that AP=20AP = 20, BP=21BP = 21, and CP=22CP = 22. In such a setup, find the square of the length DPDP.
p12. Compute the smallest integer NN such that 56=156255^6 = 15625 appears as the last five digits of 5N5^N , where N>6N > 6.
p13. There exist two complex numbers z1z_1, z2z_2 such that ∣z1+z2∣2+∣z1−z2∣2=338.|z_1 + z_2| ^2 + |z_1 - z_2| ^2 = 338. Find the length of the hypotenuse of the right triangle formed with legs of length ∣z1∣|z_1|,∣z2∣ |z_2|.
p14. Blahaj has two rays with a common endpoint A0 that form an angle of 1o1^o. They construct a sequence of points A0A_0, .... . . , AnA_n such that for all 1≤i≤n1 \le i \le n, ∣Ai−1Ai∣=1|A_{i-1}A_i | = 1, and ∣AiA0∣>∣Ai−1A0∣|A_iA_0| > |A_{i-1}A_0|. Find the largest possible value of nn. https://cdn.artofproblemsolving.com/attachments/d/b/99c1adbdcf18e7b62ebfc1786cd6ad4bc2253e.png
p15. Consider the sequence 1,1,2,1,2,3,1,2,3,4,..1, 1, 2, 1, 2, 3, 1, 2, 3, 4, . . . Find the sum of the first 100100 terms of the sequence.
p16. Suppose Annie the Ant is walking on a regular icosahedron (as shown). She starts on point AA and will randomly create a path to go to point ZZ which is the point directly opposite to AA. Every move she makes never moves further from Z, and she has equal probability to go down every valid move. What is the expected number of moves she can make? https://cdn.artofproblemsolving.com/attachments/6/1/38b92c54a5f01cdb948ff565843cb08407e6db.png
p17. Suppose that zz is a complex number, where the expression z−2iz+1\frac{z -2i}{z + 1} is real. Find min∣z−1∣min |z - 1|.
p18. Scotty has a circular sheet of paper with radius 11. They split this paper into nn congruent sectors, and with each sector, tape the two straight edges together to form a cone. Let VV be the combined volume of all n cones. What is the maximum value of VV ?
p19. Let P(x)=(x−3)m(x−13)nP(x) = (x- 3)^m \left(x -\frac13\right)^n where m,nm, n are positive integers. How many ordered pairs (m,n)(m, n) for m,n≤100m, n \le 100 result in P(x)P(x) having integer coefficients for its first three terms and last term? Assume P(x)P(x) is depicted from greatest to least exponent of xx.
p20. Let f(x)=∣x∣−1f(x) = |x| - 1 and g(x)=∣x−1∣g(x) = |x - 1|. Define fn(x)=f(f(f(...f⏟n  times(x))),f^n(x) = \underbrace{f(f(f(...f}{n\,\, times}(x))), and define gn(x)g^n(x) similarly. Let the number of solutions to f20(x)=0f^{20}(x) = 0 and g20(x)=0g^{20}(x) = 0 be a,ba, b,respectively. Find a−ba - b.
p21. (Estimation) Let MM be the mean absolute deviation of all submissions to this question. In other words, if the submissions to this question are x1x_1, x−2x-2, .... . . , xnx_n, with mean xx, then M=1n∑i=1n∣xi−x∣.M =\frac{1}{n} \sum^n_{i=1} |x_i - x|. Estimate MM. Your answer must an integer between 00 and 999999, inclusive.
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R8
1

2022 CMWMC Guts Round 8/8 - Carnegie Mellon University Womens' Competition

Set 8
p22. For monic quadratic polynomials P=x2+ax+bP = x^2 + ax + b and Q=x2+cx+dQ = x^2 + cx + d, where 1≤a,b,c,d≤101 \le a, b, c, d \le 10 are integers, we say that PP and QQ are friends if there exists an integer 1≤n≤101 \le n \le 10 such that P(n)=Q(n)P(n) = Q(n). Find the total number of ordered pairs (P,Q)(P, Q) of such quadratic polynomials that are friends.
p23. A three-dimensional solid has six vertices and eight faces. Two of these faces are parallel equilateral triangles with side length 11, △A1A2A3\vartriangle A_1A_2A_3 and △B1B2B3\vartriangle B_1B_2B_3. The other six faces are isosceles right triangles — △A1B2A3\vartriangle A_1B_2A_3, △A2B3A1\vartriangle A_2B_3A_1, △A3B1A2\vartriangle A_3B_1A_2, △B1A2B3\vartriangle B_1A_2B_3, △B2A3B1\vartriangle B_2A_3B_1, △B3A1B2\vartriangle B_3A_1B_2 — each with a right angle at the second vertex listed (so for instace △A1B2A3\vartriangle A_1B_2A_3 has a right angle at B2B_2). Find the volume of this solid.
p24. The digits 0,1,2,3,4,5,6,7,8,90, 1, 2, 3, 4, 5, 6, 7, 8, 9 are each colored red, blue, or green. Find the number of colorings such that any integer n≥2 n \ge 2 has that (a) If nn is prime, then at least one digit of nn is not blue. (b) If nn is composite, then at least one digit of nn is not green.
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R7
1

2022 CMWMC Guts Round 7/8 - Carnegie Mellon University Womens' Competition

Set 7
p19. The polynomial x4+ax3+bx2−32xx^4 + ax^3 + bx^2 - 32x, wherea a and bb are real numbers, has roots that form a square in the complex plane. Compute the area of this square.
p20. Tetrahedron ABCDABCD has equilateral triangle base ABCABC and apex DD such that the altitude from DD to ABCABC intersects the midpoint of BC‾\overline{BC}. Let MM be the midpoint of AC‾\overline{AC}. If the measure of ∠DBA\angle DBA is 67o67^o, find the measure of ∠MDC\angle MDC in degrees.
p21. Last year’s high school graduates started high school in year n−4=2017n- 4 = 2017, a prime year. They graduated high school and started college in year n=2021n = 2021, a product of two consecutive primes. They will graduate college in year n+4=2025n + 4 = 2025, a square number. Find the sum of all n<2021n < 2021 for which these three properties hold. That is, find the sum of those n<2021n < 2021 such that n−4n -4 is prime, n is a product of two consecutive primes, and n+4n + 4 is a square.
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R6
1

2022 CMWMC Guts Round 6/8 - Carnegie Mellon University Womens&#039; Competition

Set 6
p16. Let xx and yy be non-negative integers. We say point (x,y)(x, y) is square if x2+yx^2 + y is a perfect square. Find the sum of the coordinates of all distinct square points which also satisfy x2+y≤64x^2 + y \le 64.
p17. Two integers aa and bb are randomly chosen from the set {1,2,13,17,19,87,115,121}\{1, 2, 13, 17, 19, 87, 115, 121\}, with a>ba > b. What is the expected value of the number of factors of abab?
p18. Marnie the Magical Cello is jumping on nonnegative integers on number line. She starts at 00 and jumps following two specific rules. For each jump she can either jump forward by 11 or jump to the next multiple of 44 (the next multiple must be strictly greater than the number she is currently on). How many ways are there for her to jump to 20222022? (Two ways are considered distinct only if the sequence of numbers she lands on is different.)
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R5
1

2022 CMWMC Guts Round 5/8 - Carnegie Mellon University Womens&#039; Competition

Set 5
p13. An equiangular 1212-gon has side lengths that alternate between 22 and 3\sqrt3. Find the area of the circumscribed circle of this 1212-gon.
p14. For positive integers nn, let σ(n)\sigma(n) denote the number of positive integer factors of nn. Then σ(17280)=σ(27⋅33⋅5)=64\sigma(17280) = \sigma (2^7 \cdot 3^3 \cdot 5)= 64. Let SS be the set of factors kk of 1728017280 such that σ(k)=32\sigma(k) = 32. If pp is the product of the elements of SS, find σ(p)\sigma(p).
p15. How many odd 33-digit numbers have exactly four 11’s in their binary (base 22) representation? For example, 22510=111000012225_{10} = 11100001_2 would be valid.
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R4
2

2022 CMWMC Relay Round 4/4 - Carnegie Mellon University Womens&#039; Competition

Set 4
4.1 Quadrilateral ABCDABCD (with A,B,CA, B, C not collinear and A,D,CA, D, C not collinear) has AB=4AB = 4, BC=7BC = 7, CD=10CD = 10, and DA=5DA = 5. Compute the number of possible integer lengths ACAC. https://cdn.artofproblemsolving.com/attachments/1/6/4f43873a64bc00a0e6173002ccd80e8f1529a9.png
4.2 Let TT be the answer from the previous part. 2T2T congruent isosceles triangles with base length bb and leg length ℓ\ell are arranged to form a parallelogram as shown below (not necessarily the correct number of triangles). If the total length of all drawn line segments (not double counting overlapping sides) is exactly three times the perimeter of the parallelogram, find ℓb\frac{\ell}{b}. https://cdn.artofproblemsolving.com/attachments/5/c/744f503ed822bc43acafe2633e6108022f2c88.png
4.3 Let TT be the answer from the previous part. Rectangle RR has length TT times its width. RR is inscribed in a square SS such that the diagonals of S S are parallel to the sides of RR. What proportion of the area of SS is contained within RR? https://cdn.artofproblemsolving.com/attachments/a/1/0928dd1ffbeb4d7dee9b697fdb7696cc70c03d.png
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2022 CMWMC Guts Round 4/8 - Carnegie Mellon University Womens&#039; Competition

Set 4
p10. Eve has nine letter tiles: three CC’s, three MM’s, and three WW’s. If she arranges them in a random order, what is the probability that the string “CMWMCCMWMC” appears somewhere in the arrangement?
p11. Bethany’s Batteries sells two kinds of batteries: CC batteries for $4\$4 per package, and DD batteries for $7\$7 per package. After a busy day, Bethany looks at her ledger and sees that every customer that day spent exactly $2021\$2021, and no two of them purchased the same quantities of both types of battery. Bethany also notes that if any other customer had come, at least one of these two conditions would’ve had to fail. How many packages of batteries did Bethany sell?
p12. A deck of cards consists of 3030 cards labeled with the integers 11 to 3030, inclusive. The cards numbered 11 through 1515 are purple, and the cards numbered 1616 through 3030 are green. Lilith has an expansion pack to the deck that contains six indistinguishable copies of a green card labeled with the number 3232. Lilith wants to pick from the expanded deck a hand of two cards such that at least one card is green. Find the number of distinguishable hands Lilith can make with this deck.
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R3
2

2022 CMWMC Guts Round 3/8 - Carnegie Mellon University Womens&#039; Competition

Set 3
p7. On unit square ABCDABCD, a point PP is selected on segment CDCD such that DP=14DP =\frac14 . The segment BPBP is drawn and its intersection with diagonal ACAC is marked as EE. What is the area of triangle AEPAEP?
p8. Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance 11 apart, two pairs of points are distance 3\sqrt3 apart, and one pair of points is distance 22 apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be?
p9. The inhabitants of Mars use a base 88 system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base 88 numerals 124541581245415_8, 756326587563265_8, and 63214738 6321473_8. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is 1014133027■276620411381014133027\blacksquare 27662041138. What is the missing digit?
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2022 CMWMC Relay Round 3/4 - Carnegie Mellon University Womens&#039; Competition

Set 3
3.1 Annie has 2424 letter tiles in a bag; 88 C’s, 88 M’s, and 88 W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
3.2 Let TT be the answer from the previous problem. Charlotte is initially standing at (0,0)(0, 0) in the coordinate plane. She takes TT steps, each of which moves her by 11 unit in either the +x+x, −x-x, +y+y, or −y-y direction (e.g. her first step takes her to (1,0)(1, 0), (1,0)(1, 0), (0,1)(0, 1) or (0,−1)(0, -1)). After the T steps, how many possibilities are there for Charlotte’s location?
3.3 Let TT be the answer from the previous problem, and let SS be the sum of the digits of TT. Francesca has an unfair coin with an unknown probability pp of landing heads on a given flip. If she flips the coin SS times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability pp.
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R2
2

2022 CMWMC Relay Round 2/4 - Carnegie Mellon University Womens&#039; Competition

Set 2
2.1 What is the last digit of 2022+20222022+2022(20222022)2022 + 2022^{2022} + 2022^{(2022^{2022})}?
2.2 Let TT be the answer to the previous problem. CMIMC executive members are trying to arrange desks for CMWMC. If they arrange the desks into rows of 55 desks, they end up with 11 left over. If they instead arrange the desks into rows of 77 desks, they also end up with 11 left over. If they instead arrange the desks into rows of 1111 desks, they end up with TT left over. What is the smallest possible (non-negative) number of desks they could have?
2.3 Let TT be the answer to the previous problem. Compute the largest value of kk such that 11k11^k divides T!=T(T−1)(T−2)...(2)(1).T! = T(T - 1)(T - 2)...(2)(1).
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2022 CMWMC Guts Round 2/8 - Carnegie Mellon University Womens&amp;#039; Competition

Set 2
p4. △ABC\vartriangle ABC is an isosceles triangle with AB=BCAB = BC. Additionally, there is DD on BCBC with AC=DA=BD=1AC = DA = BD = 1. Find the perimeter of △ABC\vartriangle ABC.
p5. Let rr be the positive solution to the equation 100r2+2r−1=0100r^2 + 2r - 1 = 0. For an appropriate AA, the infinite series Ar+Ar2+Ar3+Ar4+...Ar + Ar^2 + Ar^3 + Ar^4 +... has sum 11. Find AA.
p6. Let N(k)N(k) denote the number of real solutions to the equation x4−x2=kx^4 -x^2 = k. As kk ranges from −∞-\infty to ∞\infty, the value of N(k)N(k) changes only a finite number of times. Write the sequence of values of N(k)N(k) as an ordered tuple (i.e. if N(k)N(k) went from 11 to 33 to 22, you would write (1,3,2)(1, 3, 2)).
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R1
2

2022 CMWMC Relay Round 1/4 - Carnegie Mellon University Womens&#039; Competition

Set 1
1.1 Compute the number of real numbers x such that the sequence xx, x2x^2, x3x^3,x4 x^4, x5x^5, ...... eventually repeats. (To be clear, we say a sequence “eventually repeats” if there is some block of consecutive digits that repeats past some point—for instance, the sequence 11, 22, 33, 44, 55, 66, 55, 66, 55, 66, ...... is eventually repeating with repeating block 55, 66.)
1.2 Let TT be the answer to the previous problem. Nicole has a broken calculator which, when told to multiply aa by bb, starts by multiplying aa by bb, but then multiplies that product by b again, and then adds bb to the result. Nicole inputs the computation “k×kk \times k” into the calculator for some real number kk and gets an answer of 10T10T. If she instead used a working calculator, what answer should she have gotten?
1.3 Let TT be the answer to the previous problem. Find the positive difference between the largest and smallest perfect squares that can be written as x2+y2x^2 + y^2 for integers x,yx, y satisfying T≤x≤T\sqrt{T} \le x \le T and T≤y≤T\sqrt{T} \le y \le T.
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2022 CMWMC Guts Round 1/8 - Carnegie Mellon University Womens&#039; Competition

Set 1
p1. Assume the speed of sound is 343343 m/s. Anastasia and Bananastasia are standing in a field in front of you. When they both yell at the same time, you hear Anastasia’s yell 55 seconds before Bananastasia’s yell. If Bananastasia yells first, and then Anastasia yells when she hears Bananastasia yell, you hear Anastasia’s yell 55 seconds after Bananastasia’s yell. What is the distance between Anastasia and Bananastasia in meters?
p2. Michelle picks a five digit number with distinct digits. She then reverses the digits of her number and adds that to her original number. What is the largest possible sum she can get?
p3. Twain is trying to crack a 44-digit number combination lock. They know that the second digit must be even, the third must be odd, and the fourth must be different from the previous three. If it takes Twain 1010 seconds to enter a combination, how many hours would it take them to try every possible combination that satisfies these rules?
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