Problems and Solutions Never Ending Math Open 2017

1. [10] A team of 6 distinguishable students competed at a math competition. They each scored an
integer amount of points, and the sum of their scores was 170. Their highest score was a 29, and their
lowest score was a 27. How many possible ordered 6-tuples of scores could they have scored?
2. [10] Let ABC be a triangle with AB = 13, BC = 14, and CA = 15. Let P be a point inside triangle
ABC, and let ray AP meet segment BC at Q. Suppose the area of triangle ABP is three times the
area of triangle CP Q, and the area of triangle ACP is three times the area of triangle BP Q. Compute
the length of BQ.
3. [10] I am thinking of a geometric sequence with 9600 terms, a1 , a2 ,. . ., a9600 . The sum of the terms
1
with indices divisible by three (i.e. a3 + a6 + · · · + a9600 ) is 56 times the sum of the other terms (i.e.
a1 + a2 + a4 + a5 + · · · + a9598 + a9599 ). Given that the terms with even indices sum to 10, what is the
smallest possible sum of the whole sequence?
p
4. [15] Let ABCD be a regular tetrahedron with side length 6 2. There is a sphere centered at each of
the four vertices, with the radii of the four spheres forming a geometric series with common ratio 2
when arranged in increasing order. If the volume inside the tetrahedron but outside the second largest
sphere is 71, what is the volume inside the tetrahedron but outside all four of the spheres?
5. [20] Find the largest number of consecutive positive integers, each of which has exactly 4 positive
divisors.
6. [25] Let the (not necessarily distinct) roots of the equation x12 3x4 + 2 = 0 be a1 , a2 , . . . , a12 .
Compute
12
X
|Re(ai )|.
i=1

7. [25] Je↵rey is doing a three-step card trick with a row of seven cards labeled A through G. Before he
starts his trick, he picks a random permutation of the cards. During each step of his trick, he rearranges
the cards in the order of that permutation. For example, for the permutation (1, 3, 5, 2, 4, 7, 6), the
first card from the left remains in position, the second card is moved to the third position, the third
card is moved to the fifth position, etc. After Je↵rey completes all three steps, what is the probability
that the “A” card will be in the same position as where it started?
8. [30] In convex equilateral hexagon ABCDEF , AC = 13, CE = 14, and EA = 15. It is given that the
area of ABCDEF is twice the area of triangle ACE. Compute AB.
9. [35] Find all ordered pairs (x, y) of numbers satisfying
(1 + x2 )(1 + y 2 ) = 170
(1 + x)( 1 + y) = 10.

10. [40] Call a positive integer “pretty good” if it is divisible by the product of its digits. Call a positive
integer n “clever” if n, n + 1, and n + 2 are all pretty good. Find the number of clever positive integers
less than 102018 . Note: the only number divisible by 0 is 0.
y2 x2
11. [40] What is the area in the xy-plane bounded by x2 + 3  1 and 3 + y 2  1?
12. [45] Let S be the set of ordered triples (a, b, c) 2 { 1, 0, 1}3 \ {(0, 0, 0)}. Let n be the smallest positive
integer such that there exists a polynomial, with integer coefficients, of the form
X
a(i,j,k) xi y j z k
i+j+k=n
i,j,k 0

such that the absolute value of all the coefficients are less than 2, and the polynomial equals 1 for all
(x, y, z) 2 S. Compute the number of such polynomials for that value of n.
Problems and Solutions Never Ending Math Open 2017

13. [45] Let N 2017 be an odd positive integer. Two players, A and B, play a game on an N ⇥ N board,
taking turns placing numbers from the set {1, 2, . . . , N 2 } into cells, so that each number appears in
exactly one cell, and each cell contains exactly one number. Let the largest row sum be M , and the
smallest row sum be m. A goes first, and seeks to maximize M m , while B goes second and wishes to
minimize Mm . There exists real numbers a and 0 < x < y such that for all odd N 2017, if A and B
play optimally,
M
x · Na  1  y · N a.
m
Find a.
14. [50] Yunseo has a supercomputer, equipped with a function F that takes in a polynomial P (x) with
integer coefficients, computes the polynomial Q(x) = (P (x) 1)(P (x) 2)(P (x) 3)(P (x) 4)(P (x) 5),
and outputs Q(x). Thus, for example, if P (x) = x+3, then F (P (x)) = (x+2)(x+1)(x)(x 1)(x 2) =
x5 5x3 + 4x. Yunseo, being clumsy, plugs in P (x) = x and uses the function 2017 times, each time
using the output as the new input, thus, in e↵ect, calculating

F (F (F (. . . F (F ( x)) . . . ))).
| {z }
2017

She gets a polynomial of degree 52017 . Compute the number of coefficients in the polynomial that are
divisible by 5.