MathDB
2016 LMT Team Round - Potpourri - Lexington Math Tournament

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January 12, 2022
algebrageometrycombinatoricsnumber theoryLMT

Problem Statement

p1. Let X,Y,ZX,Y ,Z be nonzero real numbers such that the quadratic function Xt2−Yt+Z=0X t^2 - Y t + Z = 0 has the unique root t=Yt = Y . Find XX.
p2. Let ABCDABCD be a kite with AB=BC=1AB = BC = 1 and CD=AD=2CD = AD =\sqrt2. Given that BD=5BD =\sqrt5, find ACAC.
p3. Find the number of integers nn such that n−2016n -2016 divides n2−2016n^2 -2016. An integer aa divides an integer bb if there exists a unique integer kk such that ak=bak = b.
p4. The points A(−16,256)A(-16, 256) and B(20,400)B(20, 400) lie on the parabola y=x2y = x^2 . There exists a point C(a,a2)C(a,a^2) on the parabola y=x2y = x^2 such that there exists a point DD on the parabola y=−x2y = -x^2 so that ACBDACBD is a parallelogram. Find aa.
p5. Figure F0F_0 is a unit square. To create figure F1F_1, divide each side of the square into equal fifths and add two new squares with sidelength 15\frac15 to each side, with one of their sides on one of the sides of the larger square. To create figure Fk+1F_{k+1} from FkF_k , repeat this same process for each open side of the smallest squares created in FnF_n. Let AnA_n be the area of FnF_n. Find lim⁡n→∞An\lim_{n\to \infty} A_n. https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png
p6. For a prime pp, let SpS_p be the set of nonnegative integers nn less than pp for which there exists a nonnegative integer kk such that 2016k−n2016^k -n is divisible by pp. Find the sum of all pp for which pp does not divide the sum of the elements of SpS_p .
p7. Trapezoid ABCDABCD has AB∥CDAB \parallel CD and AD=AB=BCAD = AB = BC. Unit circles γ\gamma and ω\omega are inscribed in the trapezoid such that circle γ\gamma is tangent to CDCD, ABAB, and ADAD, and circle ω\omega is tangent to CDCD, ABAB, and BCBC. If circles γ\gamma and ω\omega are externally tangent to each other, find ABAB.
p8. Let x,y,zx, y, z be real numbers such that (x+y)2+(y+z)2+(z+x)2=1(x+y)^2+(y+z)^2+(z+x)^2 = 1. Over all triples (x,y,z)(x, y, z), find the maximum possible value of y−zy -z.
p9. Triangle △ABC\vartriangle ABC has sidelengths AB=13AB = 13, BC=14BC = 14, and CA=15CA = 15. Let PP be a point on segment BCBC such that BPCP=3\frac{BP}{CP} = 3, and let I1I_1 and I2I_2 be the incenters of triangles △ABP\vartriangle ABP and △ACP\vartriangle ACP. Suppose that the circumcircle of △I1PI2\vartriangle I_1PI_2 intersects segment APAP for a second time at a point X≠PX \ne P. Find the length of segment AXAX.
p10. For 1≤i≤91 \le i \le 9, let Ai be the answer to problem i from this section. Let (i1,i2,...,i9)(i_1,i_2,... ,i_9) be a permutation of (1,2,...,9)(1, 2,... , 9) such that Ai1<Ai2<...<Ai9A_{i_1} < A_{i_2} < ... < A_{i_9}. For each iji_j , put the number iji_j in the box which is in the jjth row from the top and the jjth column from the left of the 9×99\times 9 grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits 1,2,...,91, 2,... , 9 such that ∙\bullet each bolded 3×3 3\times 3 grid contains exactly one of each digit from 1 1 to 99, ∙\bullet each row of the 9×99\times 9 grid contains exactly one of each digit from 1 1 to 99, and ∙\bullet each column of the 9×99\times 9 grid contains exactly one of each digit from 1 1 to 99.
PS. You had better use hide for answers.
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