The 58th William Lowell Putnam Mathematical Competition
Saturday, December 6, 1997
A–1 A rectangle, HOMF, has sides HO = 11 and OM = 5. B–1 Let {x} denote the distance between the real number
A triangle ABC has H as the intersection of the altitudes, x and the nearest integer. For each positive integer n,
O the center of the circumscribed circle, M the midpoint evaluate
of BC, and F the foot of the altitude from A. What is
6n−1
the length of BC? m m
Fn = ∑ min({ }, { }).
m=1 6n 3n
A–2 Players 1, 2, 3, . . . , n are seated around a table, and each
has a single penny. Player 1 passes a penny to player (Here min(a, b) denotes the minimum of a and b.)
2, who then passes two pennies to player 3. Player 3 B–2 Let f be a twice-differentiable real-valued function sat-
then passes one penny to Player 4, who passes two pen- isfying
nies to Player 5, and so on, players alternately passing
one penny or two to the next player who still has some f (x) + f 00 (x) = −xg(x) f 0 (x),
pennies. A player who runs out of pennies drops out
of the game and leaves the table. Find an infinite set where g(x) ≥ 0 for all real x. Prove that | f (x)| is
of numbers n for which some player ends up with all n bounded.
pennies.
B–3 For each positive integer n, write the sum ∑nm=1 1/m in
A–3 Evaluate the form pn /qn , where pn and qn are relatively prime
positive integers. Determine all n such that 5 does not
x3 x5 x7
Z ∞
x− + − +··· divide qn .
0 2 2·4 2·4·6
2 B–4 Let am,n denote the coefficient of xn in the expansion of
x4 x6
x
1 + 2 + 2 2 + 2 2 2 + · · · dx. (1 + x + x2 )m . Prove that for all [integers] k ≥ 0,
2 2 ·4 2 ·4 ·6
b 2k
3 c
A–4 Let G be a group with identity e and φ : G → G a func-
tion such that
0≤ ∑ (−1)i ak−i,i ≤ 1.
i=0
φ (g1 )φ (g2 )φ (g3 ) = φ (h1 )φ (h2 )φ (h3 ) B–5 Prove that for n ≥ 2,
whenever g1 g2 g3 = e = h1 h2 h3 . Prove that there exists n terms n − 1 terms
an element a ∈ G such that ψ(x) = aφ (x) is a homo- z}|{
2
z}|{
2··· ···2
morphism (i.e. ψ(xy) = ψ(x)ψ(y) for all x, y ∈ G). 2 ≡ 22 (mod n).
A–5 Let Nn denote the number of ordered n-tuples of pos-
itive integers (a1 , a2 , . . . , an ) such that 1/a1 + 1/a2 + B–6 The dissection of the 3–4–5 triangle shown below (into
. . . + 1/an = 1. Determine whether N10 is even or odd. four congruent right triangles similar to the original) has
diameter 5/2. Find the least diameter of a dissection of
A–6 For a positive integer n and any real number c, define xk this triangle into four parts. (The diameter of a dissec-
recursively by x0 = 0, x1 = 1, and for k ≥ 0, tion is the least upper bound of the distances between
pairs of points belonging to the same part.)
cxk+1 − (n − k)xk
xk+2 = .
k+1
Fix n and then take c to be the largest value for which
xn+1 = 0. Find xk in terms of n and k, 1 ≤ k ≤ n.
