MathDB

7

Part of 2022 Stanford Mathematics Tournament

Problems(4)

SMT 2022 Algebra #7

Source:

3/27/2023
Let M={0,1,2,…,2022}M=\{0,1,2,\dots,2022\} and let f:M×M→Mf:M\times M\to M such that for any a,b∈Ma,b\in M, f(a,f(b,a))=bf(a,f(b,a))=b and f(x,x)≠xf(x,x)\neq x for each x∈Mx\in M. How many possible functions ff are there (mod1000)\pmod{1000}?
SMT 2022 Calculus #7

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3/29/2023
Let Aj={(x,y):0≤xsin⁡(jπ3)+ycos⁡(jπ3)≤6−(xcos⁡(jπ3)−ysin⁡(jπ3))2}A_j=\left\{(x,y):0\le x\sin\left(\frac{j\pi}{3}\right)+y\cos\left(\frac{j\pi}{3}\right)\le6-\left(x\cos\left(\frac{j\pi}{3}\right)-y\sin\left(\frac{j\pi}{3}\right)\right)^2\right\} The area of ∪j=05Aj\cup_{j=0}^5A_j can be expressed as mnm\sqrt{n}. What is the area?
SMT 2022 Discrete #7

Source:

4/1/2023
Let n0n_0 be the product of the first 2525 primes. Now, choose a random divisor n1n_1 of n0n_0, where a choice n1n_1 is taken with probability proportional to ϕ(n1)\phi(n_1). (ϕ(m)\phi(m) is the number of integers less than mm which are relatively prime to mm.) Given this n1n_1, we let n2n_2 be a random divisor of n1n_1, again chosen with probability proportional to ϕ(n2)\phi(n_2). Compute the probability that n2≡0(mod2310)n_2\equiv0\pmod{2310}.
SMT 2022 Geometry #7

Source:

4/1/2023
△ABC\triangle ABC has side lengths AB=20AB=20, BC=15BC=15, and CA=7CA=7. Let the altitudes of △ABC\triangle ABC be ADAD, BEBE, and CFCF. What is the distance between the orthocenter (intersection of the altitudes) of △ABC\triangle ABC and the incenter of △DEF\triangle DEF?