MathDB
SMT 2023 Team

Source:

May 3, 2023

Problem Statement

In the spirit of parmenides51, I guess.
p1. We call a time on a 1212 hour digital clock nice if the sum of the minutes digits is equal to the hour. For example, 10:5510:55, 3:123:12 and 5:055:05 are nice times. How many nice times occur during the course of one day? (We do not consider times of the form 00:XX00:\text{XX}.)
p2. Along Stanford’s University Avenue are 20232023 palm trees which are either red, green, or blue. Let the positive integers RR, GG, BB be the number of red, green, and blue palm trees respectively. Given that R3+2B+G=12345,R^3+2B+G=12345, compute RR.
p3. 55 integers are each selected uniformly at random from the range 11 to 55 inclusive and put into a set SS. Each integer is selected independently of the others. What is the expected value of the minimum element of SS?
p4. Cornelius chooses three complex numbers a,b,ca,b,c uniformly at random from the complex unit circle. Given that real parts of a⋅c‾a\cdot\overline{c} and b⋅c‾b\cdot\overline{c} are 110\tfrac{1}{10}, compute the expected value of the real part of a⋅b‾a\cdot\overline{b}.
p5. A computer virus starts off infecting a single device. Every second an infected computer has a 7/307/30 chance to stay infected and not do anything else, a 7/157/15 chance to infect a new computer, and a 1/61/6 chance to infect two new computers. Otherwise (a 2/152/15 chance), the virus gets exterminated, but other copies of it on other computers are unaffected. Compute the probability that a single infected computer produces an infinite chain of infections.
p6. In the language of Blah, there is a unique word for every integer between 00 and 9898 inclusive. A team of students has an unordered list of these 9999 words, but do not know what integer each word corresponds to. However, the team is given access to a machine that, given two, not necessarily distinct, words in Blah, outputs the word in Blah corresponding to the sum modulo 9999 of their corresponding integers. What is the minimum NN such that the team can narrow down the possible translations of "11" to a list of NN Blah words, using the machine as many times as they want?
p7. Compute 6∑t=1∞(1+∑k=1∞(∑j=1∞(1+k)−j)2)−t.\sqrt{6\sum_{t=1}^\infty\left(1+\sum_{k=1}^\infty\left(\sum_{j=1}^\infty(1+k)^{-j}\right)^2\right)^{-t}}.
p8. What is the area that is swept out by a regular hexagon of side length 11 as it rotates 30∘30^\circ about its center?
p9. Let AA be the the area enclosed by the relation x2+y2≤2023.x^2+y^2\le2023. Let BB be the area enclosed by the relation x2n+y2n≤(A⋅716π)n/2.x^{2n}+y^{2n}\le\left(A\cdot\frac{7}{16\pi}\right)^{n/2}. Compute the limit of BB as n→∞n\rightarrow\infty for n∈Nn\in\mathbb{N}.
p10. Let S={1,6,10,… }\mathcal{S}=\{1,6,10,\dots\} be the set of positive integers which are the product of an even number of distinct primes, including 11. Let T={2,3,…,}\mathcal{T}=\{2,3,\dots,\} be the set of positive integers which are the product of an odd number of distinct primes. Compute ∑n∈S⌊2023n⌋−∑n∈T⌊2023n⌋.\sum_{n\in\mathcal{S}}\left\lfloor\frac{2023}{n}\right\rfloor-\sum_{n\in\mathcal{T}}\left\lfloor\frac{2023}{n}\right\rfloor.
p11. Define the Fibonacci sequence by F0=0F_0=0, F1=1F_1=1, and Fi=Fi−1+Fi−2F_i=F_{i-1}+F_{i-2} for i≥2i\ge2. Compute lim⁡n→∞FFn+1+1FFn⋅FFn−1−1.\lim_{n\rightarrow\infty}\frac{F_{F_{n+1}+1}}{F_{F_n}\cdot F_{F_{n-1}-1}}.
p12. Let AA, BB, CC, and DD be points in the plane with integer coordinates such that no three of them are collinear, and where the distances ABAB, ACAC, ADAD, BCBC, BDBD, and CDCD are all integers. Compute the smallest possible length of a side of a convex quadrilateral formed by such points.
p13. Suppose the real roots of p(x)=x9+16x8+60x7+1920x2+2048x+512p(x)=x^9+16x^8+60x^7+1920x^2+2048x+512 are r1,r2,…,rkr_1,r_2,\dots,r_k (roots may be repeated). Compute ∑i=1k12−ri.\sum_{i=1}^k\frac{1}{2-r_i}.
p14. A teacher stands at (0,10)(0,10) and has some students, where there is exactly one student at each integer position in the following triangle: https://cdn.artofproblemsolving.com/attachments/2/2/0cddcedf318d7b53bd33bd14353ece9614ec44.png Here, the circle denotes the teacher at (0,10)(0,10) and the triangle extends until and includes the column (21,y)(21,y). A teacher can see a student (i,j)(i,j) if there is no student in the direct line of sight between the teacher and the position (i,j)(i,j). Compute the number of students the teacher can see (assume that each student has no width—that is, each student is a point).
p15. Suppose we have a right triangle △ABC\triangle ABC where AA is the right angle and lengths AB=AC=2AB=AC=2. Suppose we have points DD, EE, and FF on ABAB, ACAC, and BCBC respectively with DE⊥EFDE\perp EF. What is the minimum possible length of DFDF?
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