MathDB

2020 CHMMC Winter (2020-21)

Part of CHMMC problems

Subcontests

(15)
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2020-2021 Indiv 15

For an integer n≥2n \ge 2, let GnG_n be an n×nn \times n grid of unit cells. A subset of cells H⊆GnH \subseteq G_n is considered quasi-complete if and only if each row of GnG_n has at least one cell in HH and each column of GnG_n has at least one cell in HH. A subset of cells K⊆GnK \subseteq G_n is considered quasi-perfect if and only if there is a proper subset L⊂KL \subset K such that ∣L∣=n|L| = n and no two elements in LL are in the same row or column. Let ϑ(n)\vartheta(n) be the smallest positive integer such that every quasi-complete subset H⊆GnH \subseteq G_n with ∣H∣≥ϑ(n)|H| \ge \vartheta(n) is also quasi-perfect. Moreover, let ϱ(n)\varrho(n) be the number of quasi-complete subsets H⊆GnH \subseteq G_n with ∣H∣=ϑ(n)−1|H| = \vartheta(n) - 1 such that HH is not quasi-perfect. Compute ϑ(20)+ϱ(20)\vartheta(20) + \varrho(20).
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2020-2021 Indiv 14

Let aa be a positive real number. Collinear points Z1,Z2,Z3,Z4Z_1, Z_2, Z_3, Z_4 (in that order) are plotted on the (x,y)(x, y) Cartesian plane. Suppose that the graph of the equation x2+(y+a)2+x2+(y−a)2=4a2+(x2+(y+a)2)(x2+(y−a)2) x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 + \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} passes through points Z1Z_1 and Z4Z_4, and the graph of the equation x2+(y+a)2+x2+(y−a)2=4a2−(x2+(y+a)2)(x2+(y−a)2) x^2 + (y+a)^2 + x^2 + (y-a)^2 = 4a^2 - \sqrt{(x^2 + (y+a)^2)(x^2 + (y-a)^2)} passes through points Z2Z_2 and Z3Z_3. If Z1Z2=5Z_1Z_2 = 5, Z2Z3=1Z_2Z_3 = 1, and Z3Z4=3Z_3Z_4 = 3, then a2a^2 can be written as m+npq\frac{m + n\sqrt{p}}{q}, where mm, nn, pp, and qq are positive integers, mm, nn, and qq are relatively prime, and pp is squarefree. Find m+n+p+qm + n + p + q.
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2020-2021 Indiv 10

A research facility has 6060 rooms, numbered 1,2,…601, 2, \dots 60, arranged in a circle. The entrance is in room 11 and the exit is in room 6060, and there are no other ways in and out of the facility. Each room, except for room 6060, has a teleporter equipped with an integer instruction 1≤i<601 \leq i < 60 such that it teleports a passenger exactly ii rooms clockwise. On Monday, a researcher generates a random permutation of 1,2,…,601, 2, \dots, 60 such that 11 is the first integer in the permutation and 6060 is the last. Then, she configures the teleporters in the facility such that the rooms will be visited in the order of the permutation. On Tuesday, however, a cyber criminal hacks into a randomly chosen teleporter, and he reconfigures its instruction by choosing a random integer 1≤j′<601 \leq j' < 60 such that the hacked teleporter now teleports a passenger exactly j′j' rooms clockwise (note that it is possible, albeit unlikely, that the hacked teleporter's instruction remains unchanged from Monday). This is a problem, since it is possible for the researcher, if she were to enter the facility, to be trapped in an endless \say{cycle} of rooms. The probability that the researcher will be unable to exit the facility after entering in room 11 can be written as mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.

2020-2021 Team 10

Let ω\omega be a nonreal 4747th root of unity. Suppose that S\mathcal{S} is the set of polynomials of degree at most 4646 and coefficients equal to either 00 or 11. Let NN be the number of polynomials Q∈SQ \in \mathcal{S} such that ∑j=046Q(ω2j)−Q(ωj)ω4j+ω3j+ω2j+ωj+1=47. \sum_{j = 0}^{46} \frac{Q(\omega^{2j}) - Q(\omega^{j})}{\omega^{4j} + \omega^{3j} + \omega^{2j} + \omega^j + 1} = 47. The prime factorization of NN is p1α1p2α2…psαsp_1^{\alpha_1}p_2^{\alpha_2} \dots p_s^{\alpha_s} where p1,…,psp_1, \ldots, p_s are distinct primes and α1,α2,…,αs\alpha_1, \alpha_2, \ldots, \alpha_s are positive integers. Compute ∑j=1spjαj\sum_{j = 1}^s p_j\alpha_j.
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2020-2021 Indiv 9

For a positive integer mm, let φ(m)\varphi(m) be the number of positive integers k≤mk \le m such that kk and mm are relatively prime, and let σ(m)\sigma(m) be the sum of the positive divisors of mm. Find the sum of all even positive integers nn such that n5σ(n)−2φ(n) \frac{n^5\sigma(n) - 2}{\varphi(n)} is an integer.

2020-2021 Team 9

Triangle ABCABC has circumcenter OO and circumcircle ω\omega. Let AωA_{\omega} be the point diametrically opposite AA on ω\omega, and let HH be the foot of the altitude from AA onto BCBC. Let HBH_B and HCH_C be the reflections of HH over BB and CC, respectively. Point PP is the intersection of line AωBA_{\omega}B and the perpendicular of BCBC at point HBH_B, and point QQ is the intersection of line AωCA_{\omega}C and the perpendicular of CBCB at point HCH_C. The circles ω1\omega_1 and ω2\omega_2 have the respective centers PP and QQ and respective radii PAPA and QAQA. Suppose that ω\omega, ω1\omega_1, and ω2\omega_2 intersect at another common point XX. If AO=1055AO = \frac{\sqrt{105}}{5} and AX=4AX = 4, then ∣AB−CA∣2|AB - CA|^2 can be written as m−npm - n\sqrt{p} for positive integers mm and nn and squarefree positive integer pp. Find m+n+pm + n + p. Note: the reflection of a point PP over another point Q≠PQ \neq P is the point P′P' such that QQ is the midpoint of PP and P′P'.
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2020-2021 Indiv 8

Define S=tan⁡−1(2020)+∑j=02020tan⁡−1(j2−j+1). S = \tan^{-1}(2020) + \sum_{j = 0}^{2020} \tan^{-1}(j^2 - j + 1). Then SS can be written as mπn\frac{m \pi}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.

2020-2021 Team 8

1515 ladies and 3030 gentlemen attend a luxurious party. At the start of the party, each one of the ladies shakes hands with a random gentleman. At the end of the party, each of the ladies shakes hands with another random gentleman. A lady may shake hands with the same gentleman twice (first at the start and then at the end of the party), and no two ladies shake hands with the same gentleman at the same time. Let mm and nn be relatively prime positive integers such that mn\frac{m}{n} is the probability that the collection of ladies and gentlemen that shook hands at least once can be arranged in a single circle such that each lady is directly adjacent to someone if and only if she shook hands with that person. Find the remainder when mm is divided by 1000010000.
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2020-2021 Indiv 7

Given 1010 points on a plane such that no three are collinear, we connect each pair of points with a segment and color each segment either red or blue. Assume that there exists some point AA among the 1010 points such that: 1. There is an odd number of red segments connected to AA} 2. The number of red segments connected to each of the other points are all different Find the number of red triangles (i.e, a triangle whose three sides are all red segments) on the plane.

2020-2021 Team 7

For any positive integer nn, let f(n)f(n) denote the sum of the positive integers k≤nk \le n such that kk and nn are relatively prime. Let SS be the sum of 1f(m)\frac{1}{f(m)} over all positive integers mm that are divisible by at least one of 22, 33, or 55, and whose prime factors are only 22, 33, or 55. Then S=pqS = \frac{p}{q} for relatively prime positive integers pp and qq. Find p+qp+q.

2020-2021 Tiebreak 7

Consider the polynomial x3−3x2+10x^3-3x^2+10. Let a,b,ca, b, c be its roots. Compute a2b2c2+a2b2+b2c2+c2a2+a2+b2+c2a^2b^2c^2 + a^2b^2 + b^2c^2 + c^2a^2 + a^2 + b^2 + c^2.
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