MathDB

Mixer Round

Part of 2022 Girls in Math at Yale

Problems(1)

2022 Girls in Math at Yale - Mixer Round

Source:

11/9/2023
p1. Find the smallest positive integer NN such that 2N−12N -1 and 2N+12N +1 are both composite.
p2. Compute the number of ordered pairs of integers (a,b)(a, b) with 1≤a,b≤51 \le a, b \le 5 such that ab−a−bab - a - b is prime.
p3. Given a semicircle Ω\Omega with diameter ABAB, point CC is chosen on Ω\Omega such that ∠CAB=60o\angle CAB = 60^o. Point DD lies on ray BABA such that DCDC is tangent to Ω\Omega. Find (BDBC)2\left(\frac{BD}{BC} \right)^2.
p4. Let the roots of x2+7x+11x^2 + 7x + 11 be rr and ss. If f(x)f(x) is the monic polynomial with roots rs+r+srs + r + s and r2+s2r^2 + s^2, what is f(3)f(3)?
p5. Regular hexagon ABCDEFABCDEF has side length 33. Circle ω\omega is drawn with ACAC as its diameter. BCBC is extended to intersect ω\omega at point GG. If the area of triangle BEGBEG can be expressed as abc\frac{a\sqrt{b}}{c} for positive integers a,b,ca, b, c with bb squarefree and gcd(a,c)=1gcd(a, c) = 1, find a+b+ca + b + c.
p6. Suppose that xx and yy are positive real numbers such that log⁡2x=log⁡xy=log⁡y256\log_2 x = \log_x y = \log_y 256. Find xyxy.
p7. Call a positive three digit integer ABC‾\overline{ABC} fancy if ABC‾=(AB‾)2−11⋅C‾\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}. Find the sum of all fancy integers.
p8. Let △ABC\vartriangle ABC be an equilateral triangle. Isosceles triangles △DBC\vartriangle DBC, △ECA\vartriangle ECA, and △FAB\vartriangle FAB, not overlapping △ABC\vartriangle ABC, are constructed such that each has area seven times the area of △ABC\vartriangle ABC. Compute the ratio of the area of △DEF\vartriangle DEF to the area of △ABC\vartriangle ABC.
p9. Consider the sequence of polynomials an(x) with a0(x)=0a_0(x) = 0, a1(x)=1a_1(x) = 1, and an(x)=an−1(x)+xan−2(x)a_n(x) = a_{n-1}(x) + xa_{n-2}(x) for all n≥2n \ge 2. Suppose that pk=ak(−1)⋅ak(1)p_k = a_k(-1) \cdot a_k(1) for all nonnegative integers kk. Find the number of positive integers kk between 1010 and 5050, inclusive, such that pk−2+pk−1=pk+1−pk+2p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}.
p10. In triangle ABCABC, point DD and EE are on line segments BCBC and ACAC, respectively, such that ADAD and BEBE intersect at HH. Suppose that AC=12AC = 12, BC=30BC = 30, and EC=6EC = 6. Triangle BEC has area 45 and triangle ADCADC has area 7272, and lines CH and AB meet at F. If BF2BF^2 can be expressed as a−bcd\frac{a-b\sqrt{c}}{d} for positive integers aa, bb, cc, dd with c squarefree and gcd(a,b,d)=1gcd(a, b, d) = 1, then find a+b+c+da + b + c + d.
p11. Find the minimum possible integer yy such that y>100y > 100 and there exists a positive integer x such that x2+18x+yx^2 + 18x + y is a perfect fourth power.
p12. Let ABCDABCD be a quadrilateral such that AB=2AB = 2, CD=4CD = 4, BC=ADBC = AD, and ∠ADC+∠BCD=120o\angle ADC + \angle BCD = 120^o. If the sum of the maximum and minimum possible areas of quadrilateral ABCDABCD can be expressed as aba\sqrt{b} for positive integers aa, bb with bb squarefree, then find a+ba + b.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
Yalealgebrageometrycombinatoricsnumber theory