Subcontests
(11)2022-23 Winter CHMMC Individual - Caltech Harvey Mudd Mathematics Competition
p1. Given any four digit number X=ABCD, consider the quantity Y(X)=2⋅AB+CD. For example, if X=1234, then Y(X)=2⋅12+34=58. Find the sum of all natural numbers n≤10000 such that over all four digit numbers X, the number n divides X if and only if it also divides Y(X).
p2. A sink has a red faucet, a blue faucet, and a drain. The two faucets release water into the sink at constant but different rates when turned on, and the drain removes water from the sink at a constant rate when opened. It takes 5 minutes to fill the sink (from empty to full) when the drain is open and only the red faucet is on, it takes 10 minutes to fill the sink when the drain is open and only the blue faucet is on, and it takes 15 seconds to fill the sink when both faucets are on and the drain is closed. Suppose that the sink is currently one-thirds full of water, and the drain is opened. Rounded to the nearest integer, how many seconds will elapse before the sink is emptied (keeping the two faucets closed)?
p3. One of the bases of a right triangular prism is a triangle XYZ with side lengths XY=13, YZ=14, ZX=15. Suppose that a sphere may be positioned to touch each of the five faces of the prism at exactly one point. A plane parallel to the rectangular face of the prism containing YZ cuts the prism and the sphere, giving rise to a cross-section of area A for the prism and area 15π for the sphere. Find the sum of all possible values of A.
p4. Albert, Brian, and Christine are hanging out by a magical tree. This tree gives each of them a stick, each of which have a non-negative real length. Say that Albert gets a branch of length x, Brian a branch of length y, and Christine a branch of length z, and the lengths follow the condition that x+y+z=2. Let m and n be the minimum and maximum possible values of xy+yz+xz−xyz, respectively. What is m+n?
p5. Let S:=MATHEMATICSMATHEMATICSMATHE... be the sequence where 7 copies of the word MATHEMATICS are concatenated together. How many ways are there to delete all but five letters of S such that the resulting subsequence is CHMMC?
p6. Consider two sequences of integers an and bn such that a1=a2=1, b1=b2=1 and that the following recursive relations are satisfied for integers n>2:
an=an−1an−2−bn−1bn−2,
bn=bn−1an−2+an−1bn−2.
Determine the value of 1≤n≤2023,bn=0∑bnan.
p7. Suppose ABC is a triangle with circumcenter O. Let A′ be the reflection of A across BC. If BC=12, ∠BAC=60o, and the perimeter of ABC is 30, then find A′O.
p8. A class of 10 students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a 21 probability of putting a slip of paper with their own name into the box and a 21 probability of not doing so. Later, one slip is randomly selected from the box. Given that Bob’s slip is selected, find the expected number of slips of paper in the box before the slip is selected.
p9. Let a and b be positive integers, a>b, such that 6!⋅11 divides xa−xb for all positive integers x. What is the minimum possible value of a+b?
p10. Find the number of pairs of positive integers (m,n) such that n<m≤100 and the polynomial xm+xn+1 has a root on the unit circle.
p11. Let ABC be a triangle and let ω be the circle passing through A, B, C with center O. Lines ℓA, ℓB, ℓC are drawn tangent to ω at A, B, C respectively. The intersections of these lines form a triangle XYZ where X is the intersection of ℓB and ℓC, Y is the intersection of ℓC and ℓA, and Z is the intersection of ℓA and ℓB. Let P be the intersection of lines OX and YZ. Given ∠ACB=23∠ABC and ABAC=1615 , find YPZP.
p12. Compute the remainder when 1≤a,k≤2021∑ak is divided by 2022 (in the above summation a,k are integers).
p13. Consider a 7×2 grid of squares, each of which is equally likely to be colored either red or blue. Madeline would like to visit every square on the grid exactly once, starting on one of the top two squares and ending on one of the bottom two squares. She can move between two squares if they are adjacent or diagonally adjacent. What is the probability that Madeline may visit the squares of the grid in this way such that the sequence of colors she visits is alternating (i.e., red, blue, red,... or blue, red, blue,... )?
p14. Let ABC be a triangle with AB=8, BC=10, and CA=12. Denote by ΩA the A-excircle of ABC, and suppose that ΩA is tangent to AB and AC at F and E, respectively. Line ℓ=BC is tangent to ΩA and passes through the midpoint of BC. Let T be the intersection of EF and ℓ. Compute the area of triangle ATB.
p15. For any positive integer n, let Dn be the set of ordered pairs of positive integers (m,d) such that d divides n and gcd(m,n)=1, 1≤m≤n. For any positive integers a, b, let r(a,b) be the non-negative remainder when a is divided by b. Denote by Sn the sum Sn=(m,d)∈Dn∑r(m,d). Determine the value of S396.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here. CHMMC 2022 Winter / 2022-23 Team #7
Let N0 be the set of all non-negative integers. Let f:N0×N0→N0 be a function such that for all non-negative integers a,b:
f(a,b)=f(b,a),
f(a,0)=0,
f(a+b,b)=f(a,b)+b.
Compute i=0∑30j=0∑2i−1f(2i,j) CHMMC 2022 Winter / 2022-23 Team #6
Let A be a set of 8 elements, and B:=(B1,...,B7) be an ordered 7-tuple of subsets of A. Let N be the number of such 7-tuples B such that there exists a unique 4-element subset I⊆{1,2,...,7} for which the intersection ∩i∈IBi is nonempty. Find the remainder when N is divided by 67. CHMMC 2022 Winter / 2022-23 Team #4
Gus is an inhabitant on an 11 by 11 grid of squares. He can walk from one square to an adjacent square (vertically or horizontally) in 1 unit of time. There are also two vents on the grid, one at the top left and one at the bottom right. If Gus is at one vent, he can teleport to the other vent in 0.5 units of time. Let an ordered pair of squares (a,b) on the grid be sus if the fastest path from a to b requires Gus to teleport between vents. Walking on top of a vent does not count as teleporting between vents.
What is the total number of ordered pairs of squares that are sus?
Note that the pairs (a1,b1) and (a2,b2) are considered distinct if and only if a1=a2 or b1=b2. CHMMC 2022 Winter / 2022-23 Team #3
Let a1,a2,... be a strictly increasing sequence of positive real numbers such that a1=1,a2=4, and that for every positive integer k, the subsequence a4k−3,a4k−2,a4k−1,a4k is geometric and the subsequence a4k−1,a4k,a4k+1,a4k+2 is arithmetic. For each positive integer k, let rk be the common ratio of the geometric sequence a4k−3,a4k−2,a4k−1,a4k. Compute
k=1∑∞(rk−1)(rk+1−1) 2022 W CHMMC Tiebreaker 3 - max |-a^6+b^6+c^6|, complex
Suppose that a,b,c are complex numbers with a+b+c=0, ∣abc∣=1, ∣b∣=∣c∣, and 489−33≤cos2(arg(ab))≤489+33.
Find the maximum possible value of ∣−a6+b6+c6∣. CHMMC 2022 Winter / 2022-23 Team # 2
Select a number X from the set of all 3-digit natural numbers uniformly at random. Let A∈[0,1] be the probability that X is divisible by 11, given that it is palindromic. Let B∈[0,1] be the probability that X is palindromic, given that it is divisible by 11. Compute B−A.
Recall that a 3-digit number is a palindrome if it reads the same left to right as right to left. For instance, 484 is a palindrome, but 603 is not a palindrome.