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Part of 2021 CHMMC Winter (2021-22)

Problems(1)

2021-22 Winter CHMMC Individual - Caltech Harvey Mudd Mathematics Competition

Source:

3/18/2024
p1. Fleming has a list of 8 mutually distinct integers between 9090 to 9999, inclusive. Suppose that the list has median 9494, and that it contains an even number of odd integers. If Fleming reads the numbers in the list from smallest to largest, then determine the sixth number he reads.
p2. Find the number of ordered pairs (x,y)(x,y) of three digit base-1010 positive integers such that x−yx-y is a positive integer, and there are no borrows in the subtraction x−yx-y. For example, the subtraction on the left has a borrow at the tens digit but not at the units digit, whereas the subtraction on the right has no borrows.
\begin{tabular}{ccccc} & 4 & 7 & 2 \\ - & 1 & 9 & 1\\ \hline & 2 & 8 & 1 \\ \end{tabular}\,\,\, \,\,\, \begin{tabular}{ccccc} & 3 & 7 & 9 \\ - & 2 & 6 & 3\\ \hline & 1 & 1 & 6 \\ \end{tabular}
p3. Evaluate 1⋅2⋅3−2⋅3⋅4+3⋅4⋅5−4⋅5⋅6+...+2017⋅2018⋅2019−2018⋅2019⋅2020+1010⋅2019⋅20211 \cdot 2 \cdot 3-2 \cdot 3 \cdot 4+3 \cdot 4 \cdot 5- 4 \cdot 5 \cdot 6+ ... +2017 \cdot 2018 \cdot 2019 -2018 \cdot 2019 \cdot 2020+1010 \cdot 2019 \cdot 2021
p4. Find the number of ordered pairs of integers (a,b)(a,b) such that ab+a+ba2+b2+1\frac{ab+a+b}{a^2+b^2+1} is an integer.
p5. Lin Lin has a 4×44\times 4 chessboard in which every square is initially empty. Every minute, she chooses a random square CC on the chessboard, and places a pawn in CC if it is empty. Then, regardless of whether CC was previously empty or not, she then immediately places pawns in all empty squares a king’s move away from CC. The expected number of minutes before the entire chessboard is occupied with pawns equals mn\frac{m}{n} for relatively prime positive integers mm,nn. Find m+nm+n. A king’s move, in chess, is one square in any direction on the chessboard: horizontally, vertically, or diagonally.
p6. Let P(x)=x5−3x4+2x3−6x2+7x+3P(x) = x^5-3x^4+2x^3-6x^2+7x+3 and a1,...,a5a_1,...,a_5 be the roots ofP(x) P(x). Compute ∑k=15(ak3−4ak2+ak+6).\sum^5_{k=1}(a^3_k -4a^2_k +a_k +6).
p7. Rectangle AXCYAXCY with a longer length of 1111 and square ABCDABCD share the same diagonal AC‾\overline{AC}. Assume BB,XX lie on the same side of AC‾\overline{AC} such that triangleBXC BXC and square ABCDABCD are non-overlapping. The maximum area of BXCBXC across all such configurations equals mn\frac{m}{n} for relatively prime positive integers mm,nn. Compute m+nm+n.
p8. Earl the electron is currently at (0,0)(0,0) on the Cartesian plane and trying to reach his house at point (4,4)(4,4). Each second, he can do one of three actions: move one unit to the right, move one unit up, or teleport to the point that is the reflection of its current position across the line y=xy=x. Earl cannot teleport in two consecutive seconds, and he stops taking actions once he reaches his house. Earl visits a chronologically ordered sequence of distinct points (0,0)(0,0), ......, (4,4)(4,4) due to his choice of actions. This is called an Earl-path. How many possible such Earl-paths are there?
p9. Let P(x)P(x) be a degree-20222022 polynomial with leading coefficient 11 and roots cos⁡(2πk2023)\cos \left( \frac{2\pi k}{2023} \right) for k=1k = 1 , ......,20222022 (note P(x)P(x) may have repeated roots). If P(1)=mnP(1) =\frac{m}{n} where mm and nn are relatively prime positive integers, then find the remainder when m+nm+n is divided by 100100.
p10. A randomly shuffled standard deck of cards has 5252 cards, 1313 of each of the four suits. There are 44 Aces and 44 Kings, one of each of the four suits. One repeatedly draws cards from the deck until one draws an Ace. Given that the first King appears before the first Ace, the expected number of cards one draws after the first King and before the first Ace is mn\frac{m}{n} where mm and nn are relatively prime positive integers. Find m+nm+n.
p11. The following picture shows a beam of light (dashed line) reflecting off a mirror (solid line). The angle of incidence is marked by the shaded angle; the angle of reflection is marked by the unshaded angle. https://cdn.artofproblemsolving.com/attachments/9/d/d58086e5cdef12fbc27d0053532bea76cc50fd.png The sides of a unit square ABCDABCD are magically distorted mirrors such that whenever a light beam hits any of the mirrors, the measure of the angle of incidence between the light beam and the mirror is a positive real constant qq degrees greater than the measure of the angle of reflection between the light beam and the mirror. A light beam emanating from AA strikes CD‾\overline{CD} at W1W_1 such that 2DW1=CW12DW_1 =CW_1, reflects off of CD‾\overline{CD} and then strikes BC‾\overline{BC} at W2W_2 such that 2CW2=BW22CW_2 = BW_2, reflects off of BC‾\overline{BC}, etc. To this end, denote WiW_i the ii-th point at which the light beam strikes ABCDABCD. As ii grows large, the area of WiWi+1Wi+2Wi+3W_iW_{i+1}W_{i+2}W_{i+3} approaches mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Compute m+nm+n.
p12. For any positive integer mm, define ϕ(m)\phi (m) the number of positive integers k≤mk \le m such that kk and mm are relatively prime. Find the smallest positive integer NN such that ϕ(n)≥22\sqrt{ \phi (n) }\ge 22 for any integer n≥Nn \ge N.
p13. Let nn be a fixed positive integer, and let {ak}\{a_k\} and {bk}\{b_k\} be sequences defined recursively by a1=b1=n−1a_1 = b_1 = n^{-1} aj=j(n−j+1)aj−1   ,   j>1a_j = j(n- j+1)a_{j-1}\,\,\, , \,\,\, j > 1 bj=nj2bj−1+aj   ,   j>1b_j = nj^2b_{j-1}+a_j\,\,\, , \,\,\, j > 1 When n=2021n = 2021, then a2021+b2021=m⋅20172a_{2021} +b_{2021} = m \cdot 2017^2 for some positive integer mm. Find the remainder when mm is divided by 20172017.
p14. Consider the quadratic polynomial g(x)=x2+x+1020100g(x) = x^2 +x+1020100. A positive odd integer nn is called gg-friendly if and only if there exists an integer mm such that nn divides 2⋅g(m)+20212 \cdot g(m)+2021. Find the number of gg-friendly positive odd integers less than 100100.
p15. Let ABCABC be a triangle with AB<ACAB < AC, inscribed in a circle with radius 11 and center OO. Let HH be the intersection of the altitudes of ABCABC. Let lines OH‾\overline{OH}, BC‾\overline{BC} intersect at TT. Suppose there is a circle passing through BB, HH, OO, CC. Given cos⁡(∠ABC−∠BCA)=1132\cos (\angle ABC-\angle BCA) = \frac{11}{32} , then TO=mpnTO = \frac{m\sqrt{p}}{n} for relatively prime positive integers mm,nn and squarefree positive integer pp. Find m+n+pm+n+ p.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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