MathDB

4

Part of 2022 Stanford Mathematics Tournament

Problems(4)

SMT 2022 Algebra #4

Source:

3/27/2023
Let the roots of x2022−7x2021+8x2+4x+2x^{2022}-7x^{2021}+8x^2+4x+2 be r1,r2,…,r2022r_1,r_2,\dots,r_{2022}, the roots of x2022−8x2021+27x2+9x+3x^{2022}-8x^{2021}+27x^2+9x+3 be s1,s2,…,s2022s_1,s_2,\dots,s_{2022}, and the roots of x2022−9x2021+64x2+16x+4x^{2022}-9x^{2021}+64x^2+16x+4 be t1,t2,…,t2022t_1,t_2,\dots,t_{2022}. Compute the value of ∑1≤i,j≤2022risj+∑1≤i,j≤2022sitj+∑1≤i,j≤2022tirj.\sum_{1\le i,j\le2022}r_is_j+\sum_{1\le i,j\le2022}s_it_j+\sum_{1\le i,j\le2022}t_ir_j.
SMT 2022 Calculus #4

Source:

3/29/2023
Evaluate the integral: ∫π244π2sin⁡(x)dx.\int_{\frac{\pi^2}{4}}^{4\pi^2}\sin(\sqrt{x})dx.
SMT 2022 Discrete #4

Source:

4/1/2023
Frank mistakenly believes that the number 10111011 is prime and for some integer xx writes down (x+1)1011≡x1011+1(mod1011)(x+1)^{1011}\equiv x^{1011}+1\pmod{1011}. However, it turns out that for Frank's choice of xx, this statement is actually true. If xx is positive and less than 10111011, what is the sum of the possible values of xx?
SMT 2022 Geometry #4

Source:

4/1/2023
Let ABCABC be a triangle with ∠A=1352∘\angle A=\tfrac{135}{2}^\circ and BC‾=15\overline{BC}=15. Square WXYZWXYZ is drawn inside ABCABC such that WW is on ABAB, XX is on ACAC, ZZ is on BCBC, and triangle ZBWZBW is similar to triangle ABCABC, but WZWZ is not parallel to ACAC. Over all possible triangles ABCABC, find the maximum area of WXYZWXYZ.