The 85th William Lowell Putnam Mathematical Competition

Saturday, December 7, 2024

A1 Determine all positive integers n for which there exist n squares from the grid, no two in the same row or col-
positive integers a, b, and c satisfying umn, such that the numbers contained in the selected
squares are exactly 1, 2, . . . , n?
2an + 3bn = 4cn .
B2 Two convex quadrilaterals are called partners if they
A2 For which real polynomials p is there a real polynomial have three vertices in common and they can be labeled
q such that ABCD and ABCE so that E is the reflection of D across
the perpendicular bisector of the diagonal AC. Is there
p(p(x)) − x = (p(x) − x)2 q(x) an infinite sequence of convex quadrilaterals such that
each quadrilateral is a partner of its successor and no
for all real x? two elements of the sequence are congruent? [A dia-
gram has been omitted.]
A3 Let S be the set of bijections
B3 Let rn be the nth smallest positive solution to tan x = x,
T : {1, 2, 3} × {1, 2, . . . , 2024} → {1, 2, . . . , 6072} where the argument of tangent is in radians. Prove that
such that T (1, j) < T (2, j) < T (3, j) for all j ∈ 1
{1, 2, . . . , 2024} and T (i, j) < T (i, j + 1) for all i ∈ 0 < rn+1 − rn − π <
(n2 + n)π
{1, 2, 3} and j ∈ {1, 2, . . . , 2023}. Do there exist a and c
in {1, 2, 3} and b and d in {1, 2, . . . , 2024} such that the for n ≥ 1.
fraction of elements T in S for which T (a, b) < T (c, d)
is at least 1/3 and at most 2/3? B4 Let n be a positive integer. Set an,0 = 1. For k ≥ 0,
choose an integer mn,k uniformly at random from the
A4 Find all primes p > 5 for which there exists an integer set {1, . . . , n}, and let
a and an integer r satisfying 1 ≤ r ≤ p − 1 with the fol-
lowing property: the sequence 1, a, a2 , . . . , a p−5 can be

an,k + 1, if mn,k > an,k ;

rearranged to form a sequence b0 , b1 , b2 , . . . , b p−5 such
an,k+1 = an,k , if mn,k = an,k ;
that bn − bn−1 − r is divisible by p for 1 ≤ n ≤ p − 5. 
a − 1, if m < a .
n,k n,k n,k
A5 Consider a circle Ω with radius 9 and center at the ori-
gin (0, 0), and a disc ∆ with radius 1 and center at (r, 0), Let E(n) be the expected value of an,n . Determine
where 0 ≤ r ≤ 8. Two points P and Q are chosen in- limn→∞ E(n)/n.
dependently and uniformly at random on Ω. Which
value(s) of r minimize the probability that the chord PQ B5 Let k and m be positive integers. For a positive in-
intersects ∆? teger n, let f (n) be the number of integer sequences
x1 , . . . , xk , y1 , . . . , ym , z satisfying 1 ≤ x1 ≤ · · · ≤ xk ≤ z ≤
A6 Let c0 , c1 , c2 , . . . be the sequence defined so that n and 1 ≤ y1 ≤ · · · ≤ ym ≤ z ≤ n. Show that f (n) can be
√ expressed as a polynomial in n with nonnegative coeffi-
1 − 3x − 1 − 14x + 9x2 ∞
cients.
= ∑ ck xk
4 k=0 2
B6 For a real number a, let Fa (x) = ∑n≥1 na e2n xn for 0 ≤
for sufficiently small x. For a positive integer n, let A x < 1. Find a real number c such that
be the n-by-n matrix with i, j-entry ci+ j−1 for i and j in
{1, . . . , n}. Find the determinant of A. lim Fa (x)e−1/(1−x) = 0 for all a < c, and
x→1−

B1 Let n and k be positive integers. The square in the ith lim Fa (x)e−1/(1−x) = ∞ for all a > c.
x→1−
row and jth column of an n-by-n grid contains the num-
ber i + j − k. For which n and k is it possible to select