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2016 LMT Spring Guts Round p13-p24 - Lexington Mathematical Tournament

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September 18, 2023
LMTalgebrageometrycombinatoricsnumber theory

Problem Statement

Round 5
p13. A 2016×20162016 \times 2016 chess board is cut into k≥1k \ge 1 rectangle(s) with positive integer sidelengths. Let pp be the sum of the perimeters of all kk rectangles. Additionally, let mm and MM be the minimum and maximum possible value of pk\frac{p}{k}, respectively. Determine the ordered pair (m,M)(m,M).
p14. For nonnegative integers nn, let f(n)f (n) be the product of the digits of nn. Compute ∑i=11000f(i)\sum^{1000}_{i=1}f (i ).
p15. How many ordered pairs of positive integers (m,n)(m,n) have the property that mnmn divides 20162016?
Round 6
p16. Let a,b,ca,b,c be distinct integers such that a+b+c=0a +b +c = 0. Find the minimum possible positive value of ∣a3+b3+c3∣|a^3 +b^3 +c^3|.
p17. Find the greatest positive integer kk such that 11k−2k11^k -2^k is a perfect square.
p18. Find all ordered triples (a,b,c)(a,b,c) with a≤b≤ca \le b \le c of nonnegative integers such that 2a+2b+2c=ab+bc+ca2a +2b +2c = ab +bc +ca.
Round 7
p19. Let f:N→Nf :N \to N be a function such that f(f(n))+f(n+1)=n+2f ( f (n))+ f (n +1) = n +2 for all positive integers nn. Find f(20)+f(16)f (20)+ f (16).
p20. Let △ABC\vartriangle ABC be a triangle with area 1010 and BC=10BC = 10. Find the minimum possible value of AB⋅ACAB \cdot AC.
p21. Let △ABC\vartriangle ABC be a triangle with sidelengths AB=19AB = 19, BC=24BC = 24, CA=23C A = 23. Let DD be a point on minor arc BCBC of the circumcircle of △ABC\vartriangle ABC such that DB=DCDB =DC. A circle with center DD that passes through BB and CC interests ACAC again at a point E≠CE \ne C. Find the length of AEAE.
Round 8
p22. Let m=122+2+...2m =\frac12 \sqrt{2+\sqrt{2+... \sqrt2}}, where there are 20142014 square roots. Let f1(x)=2x2−1f_1(x) =2x^2 -1 and let fn(x)=f1(fn−1(x))f_n(x) = f_1( f_{n-1}(x)). Find f2015(m)f_{2015}(m).
p23. How many ordered triples of integers (a,b,c)(a,b,c) are there such that 0<c≤b≤a≤20160 < c \le b \le a \le 2016, and a+b−c=2016a +b-c = 2016?
p24. In cyclic quadrilateral ABCDABCD, ∠BAD=120o\angle B AD = 120^o,∠ABC=150o\angle ABC = 150^o,CD=8CD = 8 and the area of ABCDABCD is 636\sqrt3. Find the perimeter of ABCDABCD.
PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3158461p28714996]here and 9-12 [url=https://artofproblemsolving.com/community/c3h3162282p28763571]here. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.
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