MathDB

30

Part of 2018 Online Math Open Problems

Problems(2)

2017-2018 Spring OMO Problem 30

Source:

4/3/2018
Let p=2017p = 2017. Given a positive integer nn, an n×nn\times n matrix AA is formed with each element aija_{ij} randomly selected, with equal probability, from {0,1,…,p−1}\{0,1,\ldots,p - 1\}. Let qnq_n be probability that det⁡A≡1(modp)\det A\equiv 1\pmod{p}. Let q=lim⁡n→∞qnq=\displaystyle\lim_{n\rightarrow\infty} q_n. If d1,d2,d3,…d_1, d_2, d_3, \ldots are the digits after the decimal point in the base pp expansion of qq, then compute the remainder when ∑k=1p2dk\displaystyle\sum_{k = 1}^{p^2} d_k is divided by 10910^9.
Proposed by Ashwin Sah
2018-2019 Fall OMO Problem 30

Source:

11/7/2018
Let ABCABC be an acute triangle with cos⁡B=13,cos⁡C=14\cos B =\frac{1}{3}, \cos C =\frac{1}{4}, and circumradius 7272. Let ABCABC have circumcenter OO, symmedian point KK, and nine-point center NN. Consider all non-degenerate hyperbolas H\mathcal H with perpendicular asymptotes passing through A,B,CA,B,C. Of these H\mathcal H, exactly one has the property that there exists a point P∈HP\in \mathcal H such that NPNP is tangent to H\mathcal H and P∈OKP\in OK. Let N′N' be the reflection of NN over BCBC. If AKAK meets PN′PN' at QQ, then the length of PQPQ can be expressed in the form a+bca+b\sqrt{c}, where a,b,ca,b,c are positive integers such that cc is not divisible by the square of any prime. Compute 100a+b+c100a+b+c.
Proposed by Vincent Huang