MathDB

2022 CMIMC

Part of CMIMC Problems

Subcontests

(34)
2.6 1.2
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2.3
1
2.7 1.3
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2.6
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2.4
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2022 Geo Div 2 P4

Circle Γ\Gamma has diameter AB‾\overline{AB} with AB=6AB = 6. Point CC is constructed on line ABAB so that AB=BCAB = BC and A≠CA \neq C. Let DD be on Γ\Gamma so that CD↔\overleftrightarrow{CD} is tangent to Γ\Gamma. Compute the distance from line AD↔\overleftrightarrow{AD} to the circumcenter of △ADC\triangle ADC.
Proposed by Justin Hsieh

2022 Combo Div 2 P4

Dilhan is running around a track for 1212 laps. If halfway through a lap, Dilhan has his phone on him, he has a 13\frac{1}{3} chance to drop it there. If Dilhan runs past his phone on the ground, he will attempt to pick it up with a 23\frac{2}{3} chance of success, and won't drop it for the rest of the lap. He starts with his phone at the start of the 5K, what is the chance he still has it when he finished the 5K?
Proposed by Zack Lee, Daniel Li, Dilhan Salgado
2.3 1.2
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2.2 1.1
2

2022 Geo Div 1 P1 (Div 2 P2)

A circle has radius 5252 and center OO. Points AA is on the circle, and point PP on OA‾\overline{OA} satisfies OP=28OP = 28. Point QQ is constructed such that QA=QP=15QA = QP = 15, and point BB is constructed on the circle so that QQ is on OB‾\overline{OB}. Find QBQB.
Proposed by Justin Hsieh

2022 Combo Div 1 P1 (Div 2 P2)

Starting with a 5×55 \times 5 grid, choose a 4×44 \times 4 square in it. Then, choose a 3×33 \times 3 square in the 4×44 \times 4 square, and a 2×22 \times 2 square in the 3×33 \times 3 square, and a 1×11 \times 1 square in the 2×22 \times 2 square. Assuming all squares chosen are made of unit squares inside the grid. In how many ways can the squares be chosen so that the final 1×11 \times 1 square is the center of the original 5×55 \times 5 grid?
Proposed by Nancy Kuang
1.8
3

2022 Alg/NT Div 1 P8

Find the largest c>0c > 0 such that for all n≥1n \ge 1 and a1,…,an,b1,…,bn>0a_1,\dots,a_n, b_1,\dots, b_n > 0 we have ∑j=1naj4≥c∑k=1n(∑j=1kajbk+1−j)4(∑j=1kbj2j!)2\sum_{j=1}^n a_j^4 \ge c\sum_{k = 1}^n \frac{\left(\sum_{j=1}^k a_jb_{k+1-j}\right)^4}{\left(\sum_{j=1}^k b_j^2j!\right)^2}
Proposed by Grant Yu

2022 Geo Div 1 P8

Let ABCDABCD be a cyclic quadrilateral with circumcenter OO. Rays OB→\displaystyle \overrightarrow{OB} and DC→\displaystyle \overrightarrow{DC} intersect at EE, and rays OC→\displaystyle \overrightarrow{OC} and AB→\displaystyle \overrightarrow{AB} intersect at FF. Suppose that AE=EC=CF=4AE = EC = CF = 4, and the circumcircle of ODEODE bisects BF‾\overline{BF}. Find the area of triangle ADFADF.
Proposed by Howard Halim

2022 Combo Div 1 P8

Daniel has a (mostly) standard deck of 54 cards, consisting of 4 suits each containing the ranks 1 to 13 as well as 2 jokers. Daniel plays the following game: He shuffles the deck uniformly randomly and then takes all of the cards that end up strictly between the two jokers. He then sums up the ranks of all the cards he has taken and calls that his score. Let pp be the probability that his score is a multiple of 13. There exists relatively prime positive integers aa and b,b, with bb as small as possible, such that ∣p−a/b∣<10−10.|p - a/b| < 10^{-10}. What is a/b?a/b?
Proposed by Dilhan Salgado, Daniel Li
1.7
3

2022 Alg/NT Div 1 P7

Let f(n)f(n) count the number of values 0≤k≤n20\le k\le n^2 such that 43∤(n2k)43\nmid\binom{n^2}{k}. Find the least positive value of nn such that 4343∣f(43n−142)43^{43}\mid f\left(\frac{43^{n}-1}{42}\right)
Proposed by Adam Bertelli

2022 Geo Div 1 P7

In acute △ABC,\triangle ABC, let II denote the incenter and suppose that line AIAI intersects segment BCBC at a point D.D. Given that AI=3,ID=2,AI=3, ID=2, and BI2+CI2=64,BI^2+CI^2=64, compute BC2.BC^2.
Proposed by Kyle Lee

2022 Combo Div 1 P7

In a class of 1212 students, no two people are the same height. Compute the total number of ways for the students to arrange themselves in a line such that:
[*] for all 1<i<121 < i < 12, the person in the ii-th position (with the leftmost position being 11) is taller than exactly i(mod3)i\pmod 3 of their adjacent neighbors, and [*] the students standing at positions which are multiples of 33 are strictly increasing in height from left to right.
Proposed by Nancy Kuang
1.6
3

2022 Alg/NT Div 1 P6

Find the probability such that when a polynomial in Z2027[x]\mathbb Z_{2027}[x] having degree at most 20262026 is chosen uniformly at random, x2027−x∣Pk(x)−x  ⟺  2021∣kx^{2027}-x | P^{k}(x) - x \iff 2021 | k (note that 20272027 is prime). Here Pk(x)P^k(x) denotes PP composed with itself kk times.
Proposed by Grant Yu

2022 Geo Div 1 P6

Let Γ1\Gamma_1 and Γ2\Gamma_2 be two circles with radii r1r_1 and r2,r_2, respectively, where r1>r2.r_1>r_2. Suppose Γ1\Gamma_1 and Γ2\Gamma_2 intersect at two distinct points AA and B.B. A point CC is selected on ray AB→,\overrightarrow{AB}, past B,B, and the tangents to Γ1\Gamma_1 and Γ2\Gamma_2 from CC are marked as points PP and Q,Q, respectively. Suppose that Γ2\Gamma_2 passes through the center of Γ1\Gamma_1 and that points P,B,QP, B, Q are collinear in that order, with PB=3PB=3 and QB=2.QB=2. What is the length of AB?AB?
Proposed by Kyle Lee

2022 Combo Div 1 P6

Barry has a standard die containing the numbers 1-6 on its faces.
He rolls the die continuously, keeping track of the sum of the numbers he has rolled so far, starting from 0. Let EnE_n be the expected number of time he needs to until his recorded sum is at least nn.
It turns out that there exist positive reals a,ba, b such that lim⁡n→∞En−(an+b)=0\lim_{n \rightarrow \infty} E_n - (an + b) = 0
Find (a,b)(a,b).
Proposed by Dilhan Salgado
1.5
3

2022 Alg/NT DIv 1 P5

Grant is standing at the beginning of a hallway with infinitely many lockers, numbered, 1,2,3,…1, 2, 3, \ldots All of the lockers are initially closed. Initially, he has some set S={1,2,3,…}S = \{1, 2, 3, \ldots\}.
Every step, for each element ss of SS, Grant goes through the hallway and opens each locker divisible by ss that is closed, and closes each locker divisible by ss that is open. Once he does this for all ss, he then replaces SS with the set of labels of the currently open lockers, and then closes every door again.
After 20222022 steps, SS has nn integers that divide 102022{10}^{2022}. Find nn.
Proposed by Oliver Hayman

2022 Geo Div 1 P5

In triangle ABCABC, let I,O,HI, O, H be the incenter, circumcenter and orthocenter, respectively. Suppose that AI=11AI = 11 and AO=AH=13AO = AH = 13. Find OHOH.
Proposed by Kevin You

2022 Combo Div 1 P5

At CMIMC headquarters, there is a row of nn lightbulbs, each of which is connected to a light switch. Daniel the electrician knows that exactly one of the switches doesn't work, and needs to find out which one. Every second, he can do exactly one of 3 things:
[*] Flip a switch, changing the lightbulb from off/on or on/off (unless the switch is broken). [*] Check if a given lightbulb is on or off. [*] Measure the total electricity usage of all the lightbulbs, which tells him exactly how many are currently on. Initially, all the lightbulbs are off. Daniel was given the very difficult task of finding the broken switch in at most nn seconds, but fortunately he showed up to work 10 seconds early today. What is the largest possible value nn such that he can complete his task on time?
Proposed by Adam Bertelli
2.8 1.4
3

2022 Alg/NT Div 1 P4 (Div 2 P8)

Let zz be a complex number that satisfies the equation z−4z2−5z+1+2z−42z2−5z+1+z−2z2−3z+1=3z.\frac{z-4}{z^2-5z+1} + \frac{2z-4}{2z^2-5z+1} + \frac{z-2}{z^2-3z+1} = \frac{3}{z}. Over all possible values of zz, find the sum of the values of ∣1z2−5z+1+12z2−5z+1+1z2−3z+1∣.\left| \frac{1}{z^2-5z+1} + \frac{1}{2z^2-5z+1} + \frac{1}{z^2-3z+1} \right|.
Proposed by Justin Hsieh

2022 Geo Div 1 P4 (Div 2 P8)

Let AA and BB be points on circle Γ\Gamma such that AB=10.AB=\sqrt{10}. Point CC is outside Γ\Gamma such that △ABC\triangle ABC is equilateral. Let DD be a point on Γ\Gamma and suppose the line through CC and DD intersects ABAB and Γ\Gamma again at points EE and F≠D.F \neq D. It is given that points C,D,E,FC, D, E, F are collinear in that order and that CD=DE=EF.CD=DE=EF. What is the area of Γ?\Gamma?
Proposed by Kyle Lee

2022 Combo Div 1 P4 (Div 2 P8)

The CMU Kiltie Band is attempting to crash a helicopter via grappling hook. The helicopter starts parallel (angle 00 degrees) to the ground. Each time the band members pull the hook, they tilt the helicopter forward by either xx or x+1x+1 degrees, with equal probability, if the helicopter is currently at an angle xx degrees with the ground. Causing the helicopter to tilt to 9090 degrees or beyond will crash the helicopter. Find the expected number of times the band must pull the hook in order to crash the helicopter.
Proposed by Justin Hsieh
2.7
1

2022 Alg/NT Div 2 P7

For polynomials P(x)=anxn+⋯+a0P(x) = a_nx^n + \cdots + a_0, let f(P)=an⋯a0f(P) = a_n\cdots a_0 be the product of the coefficients of PP. The polynomials P1,P2,P3,QP_1,P_2,P_3,Q satisfy P1(x)=(x−a)(x−b)P_1(x) = (x-a)(x-b), P2(x)=(x−a)(x−c)P_2(x) = (x-a)(x-c), P3(x)=(x−b)(x−c)P_3(x) = (x-b)(x-c), Q(x)=(x−a)(x−b)(x−c)Q(x) = (x-a)(x-b)(x-c) for some complex numbers a,b,ca,b,c. Given f(Q)=8f(Q) = 8, f(P1)+f(P2)+f(P3)=10f(P_1) + f(P_2) + f(P_3) = 10, and abc>0abc>0, find the value of f(P1)f(P2)f(P3)f(P_1)f(P_2)f(P_3).
Proposed by Justin Hsieh
2.6 1.3
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2.5
3

2022 Alg/NT Div 2 P5

Alan is assigning values to lattice points on the 3d coordinate plane. First, Alan computes the roots of the cubic 20x3−22x2+2x+120x^3-22x^2+2x+1 and finds that they are α\alpha, β\beta, and γ\gamma. He finds out that each of these roots satisfy ∣α∣,∣β∣,∣γ∣≤1|\alpha|,|\beta|,|\gamma|\leq 1 On each point (x,y,z)(x,y,z) where x,y,x,y, and zz are all nonnegative integers, Alan writes down αxβyγz\alpha^x\beta^y\gamma^z. What is the value of the sum of all numbers he writes down?
Proposed by Alan Abraham

2022 Geo Div 2 P5

Let ABCABC be an equilateral triangle of unit side length and suppose DD is a point on segment BC‾\overline{BC} such that DB<DC.DB<DC. Let MM and NN denote the midpoints of AB‾\overline{AB} and AC‾,\overline{AC}, respectively. Suppose XX and YY are the intersections of lines ABAB and ND,ND, and lines ACAC and MD,MD, respectively. Given that XY=4,XY=4, what is the value of DBDC?\frac{DB}{DC}?
Proposed by Kyle Lee

2022 Combo Div 2 P5

Daniel, Ethan, and Zack are playing a multi-round game of Tetris. Whoever wins 1111 rounds first is crowned the champion. However Zack is trying to pull off a "reverse-sweep", where (at-least) one of the other two players first hits 1010 wins while Zack is still at 00, but Zack still ends up being the first to reach 1111. How many possible sequences of round wins can lead to Zack pulling off a reverse sweep?
Proposed by Dilhan Salgado
2.4 1.2
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2.3 1.1
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2.2
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2.1
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2022 Alg/NT Div 2 P1

Alice and Bob live on the same road. At time tt, they both decide to walk to each other's houses at constant speed. However, they were busy thinking about math so that they didn't realize passing each other. Alice arrived at Bob's house at 3:19pm3:19\text{pm}, and Bob arrived at Alice's house at 3:29pm3:29\text{pm}. Charlie, who was driving by, noted that Alice and Bob passed each other at 3:11pm3:11\text{pm}. Find the difference in minutes between the time Alice and Bob left their own houses and noon on that day.
Proposed by Kevin You

2022 Geo Div 2 P1

An equilateral 1212-gon has side length 1010 and interior angle measures that alternate between 90∘90^\circ, 90∘90^\circ, and 270∘270^\circ. Compute the area of this 1212-gon.
Proposed by Connor Gordon

2022 Combo Div 2 P1

A particle starts at (0,0,0)(0,0,0) in three-dimensional space. Each second, it randomly selects one of the eight lattice points a distance of 3\sqrt{3} from its current location and moves to that point. What is the probability that, after two seconds, the particle is a distance of 222\sqrt{2} from its original location?
Proposed by Connor Gordon
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2022 Team P14

Let a tree on mn+1mn + 1 vertices be (m,n)(m,n)-nice if the following conditions hold:
[*] m+1m + 1 colors are assigned to the nodes of the tree [*] for the first mm colors, there will be exactly nn nodes of color ii (1≤i≤m)(1\le i \le m) [*] the root node of the tree will be the unique node of color m+1m+1. \item the (m,n)(m,n)-nice trees must also satisfy the condition that for any two non-root nodes i,ji, j, if the color of ii equals the color of jj, then the color of the parent of ii equals the color of the parent of jj. [*] Nodes of the same color are considered indistinguishable (swapping any two of them results in the same tree).
Let N(u,v,l)N(u,v,l) denote the number of (u,v)(u,v)-nice trees with ll leaves. Note that N(2,2,2)=2,N(2,2,3)=4,N(2,2,4)=6N(2,2,2) = 2, N(2,2,3) = 4, N(2,2,4) = 6. Compute the remainder when ∑l=123789N(8,101,l)\sum_{l = 123}^{789} N(8,101,l) is divided by 101101.
Definition: Any rooted, ordered tree consists of some set of nodes, each of which has a (possibly empty) ordered list of children. Each node is the child of exactly one other node, with the exception of the root, which has not parent. There also cannot be any cycles of nodes which are all linearly children of each other.
Proposed by Advait Nene
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2022 Team P11

Let {εi}i≥1,{θi}i≥0\{\varepsilon_i\}_{i\ge 1}, \{\theta_i\}_{i\ge 0} be two infinite sequences of real numbers, such that εi∈{−1,1}\varepsilon_i \in \{-1,1\} for all ii, and the numbers θi\theta_i obeytan⁡θn+1=tan⁡θn+εnsec⁡(θn)−tan⁡θn−1,n≥1\tan \theta_{n+1} = \tan \theta_{n}+\varepsilon_n \sec(\theta_{n})-\tan \theta_{n-1} , \qquad n \ge 1and θ0=π4,θ1=2π3\theta_0 = \frac{\pi}{4}, \theta_1 = \frac{2\pi}{3}. Compute the sum of all possible values of lim⁡m→∞(∑n=1m1tan⁡θn+1+tan⁡θn−1+tan⁡θm−tan⁡θm+1)\lim_{m \to \infty} \left(\sum_{n=1}^m \frac{1}{\tan \theta_{n+1} + \tan \theta_{n-1}} + \tan \theta_m - \tan \theta_{m+1}\right)
Proposed by Grant Yu
10
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2022 Team P8

There are 36 contestants in the CMU Puyo-Puyo Tournament, each with distinct skill levels. The tournament works as follows: First, all (362)\binom{36}{2} pairings of players are written down on slips of paper and are placed in a hat. Next, a slip of paper is drawn from the hat, and those two players play a match. It is guaranteed that the player with a higher skill level will always win the match. We continue drawing slips (without replacement) and playing matches until the results of the match completely determine the order of skill levels of all 36 contestants (i.e. there is only one possible ordering of skill levels consistent with the match results), at which point the tournament immediately finishes. What is the expected value of the number of matches played before the stopping point is reached?
Proposed by Dilhan Salgado
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