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math competition problems starter pack | /r/starterpacks
![math competition problems starter pack lots of Markov chains Let A1 A2 A3 A12 be a dodecagon (12-gon). Three frogs initially sit at A4, A8, and A12. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is where m and n are relatively prime positive integers. Find m + n. m n ' combinatorial identity spam Consider all 1000-element subsets of the set {1, 2, 3, ..., 2015}. From each such subset choose the least element. The arithmetic mean of all of these least elements is where P and q are relatively prime positive integers. Find p+q. q Vieta's Formulas/Newton sums bash Let s denote the sum of the kth powers of the roots of the polynomial x³ - 5x²+8x . In particular, so = 3, 8₁ = 5, and 82 = 9. Let a, b, and c be real numbers such that Sk+1 = ask + b Sk-1+cSk-2 for k = 2, 3, What is a+b+c? very rudimentary complex number/calculus problems just for the sake of including those topics ==== Find the small- For a positive integer n, let fn(x) cos(x) cos(2x) cos(3x)...cos(nx). est n such that f" (0)|> 2023. geometry, but only Euclidean geometry. and pulled from a small set of common configurations ― 13 non-sentient creatures with no free will to speak of at least 3 problems featuring the number of the current year For each positive integer n, let k(n) be the number of ones in the binary representation of 2023. n. What is the minimum value of k(n)? multiplicative orders/FLT can be Find all primes p > 5 for which there exists an integer a and an integer r satisfying 1 ≤r≤p-1 with the fol- lowing property: the sequence 1, a, a²,..., rearranged to form a sequence bo,b1, b2,...,bp–5 such that bn — bn−1 — r is divisible by p for 1 ≤ n ≤ p −5. find the determinant of [insert goofy ahh matrix] Let S1, S2,..., S2n-1 be the nonempty subsets of {1,2,………,n} in some order, and let M be the (2" - 1) × (2n-1) matrix whose (i, j) entry is mij = So if sins; = 0; 0 1 otherwise. Calculate the determinant of M. D X L E M G N B roots of unity filter plug and chug ... m For every subset T of U = {1, 2, 3, 18}, let s(T) be the sum of the elements of T, with s(Ø) defined to be 0. If I is chosen at random among all subsets of U, the probability that s(T) is divisible by 3 is, where m and n are relatively prime positive integers. Find m. n](https://i.kym-cdn.com/photos/images/masonry/003/051/831/567.png)
![math competition problems starter pack lots of Markov chains Let A1 A2 A3 A12 be a dodecagon (12-gon). Three frogs initially sit at A4, A8, and A12. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is where m and n are relatively prime positive integers. Find m + n. m n ' combinatorial identity spam Consider all 1000-element subsets of the set {1, 2, 3, ..., 2015}. From each such subset choose the least element. The arithmetic mean of all of these least elements is where P and q are relatively prime positive integers. Find p+q. q Vieta's Formulas/Newton sums bash Let s denote the sum of the kth powers of the roots of the polynomial x³ - 5x²+8x . In particular, so = 3, 8₁ = 5, and 82 = 9. Let a, b, and c be real numbers such that Sk+1 = ask + b Sk-1+cSk-2 for k = 2, 3, What is a+b+c? very rudimentary complex number/calculus problems just for the sake of including those topics ==== Find the small- For a positive integer n, let fn(x) cos(x) cos(2x) cos(3x)...cos(nx). est n such that f" (0)|> 2023. geometry, but only Euclidean geometry. and pulled from a small set of common configurations ― 13 non-sentient creatures with no free will to speak of at least 3 problems featuring the number of the current year For each positive integer n, let k(n) be the number of ones in the binary representation of 2023. n. What is the minimum value of k(n)? multiplicative orders/FLT can be Find all primes p > 5 for which there exists an integer a and an integer r satisfying 1 ≤r≤p-1 with the fol- lowing property: the sequence 1, a, a²,..., rearranged to form a sequence bo,b1, b2,...,bp–5 such that bn — bn−1 — r is divisible by p for 1 ≤ n ≤ p −5. find the determinant of [insert goofy ahh matrix] Let S1, S2,..., S2n-1 be the nonempty subsets of {1,2,………,n} in some order, and let M be the (2" - 1) × (2n-1) matrix whose (i, j) entry is mij = So if sins; = 0; 0 1 otherwise. Calculate the determinant of M. D X L E M G N B roots of unity filter plug and chug ... m For every subset T of U = {1, 2, 3, 18}, let s(T) be the sum of the elements of T, with s(Ø) defined to be 0. If I is chosen at random among all subsets of U, the probability that s(T) is divisible by 3 is, where m and n are relatively prime positive integers. Find m. n](https://i.kym-cdn.com/photos/images/original/003/051/831/567.png)
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