Problems

The following problems are printed in the May 1997 issue. Solutions will be printed in future issues.
Solutions for theses problems should be mailed before August 31, 1997. (Solutions sent after that date will be considered as deadlines permit.)

4572-S: Student Problem - only undergraduate and pre-college students are eligible to submit a solution.
Proposed by Stanley Rabinowitz, Westford, MA.
The numerical identity cos²12^° - cos6^°cos18^° = sin²6^° is a special case of the more general identity cos²2x - cosxcos3x = sin²x. Find and prove a general identity for each of the following numerical identities:
(a) 2sin40^°sin50^° = sin80^°
(b) 4cos24^°cos36^°cos84^° = sin18^°
(c) sin²10^° + cos²20^° - sin10^°cos20^° = 3/4.
Editors' Note: This problem appeared originally in the May, 1996 issue. Since no solutions were received, the problem remains "open" until August 31, 1997.

4620-OBG: Oldie But Goodie
Proposed by C.Z. Aughenbaugh, Woodstock, IL.
ABCD is a parallelogram, Y any point on AB between A and B, X any point on CD between C and D. Let YD intersect AX in P, and YC intersect BX in Q. Let PQ intersect AD in M and BC in N. Prove that MN divides the parallelogram into two equal parts.

4621: Proposed by Murray S. Klamkin, Edmonton, AB, Canada.
Lat a₁, a₂, ..., a_r be any sequence of digits in base 10.
(a) Prove that there are infinitely many squares which have a₁, a₂, ..., a_r as their first (lefmost) r digits.
(b) Prove that if there is one square which has a₁, a₂, ..., a_r as its last r digits then there are infinitely many squares which have a₁, a₂, ..., a_r as their last r digits.
(c) Prove that there are infinitely many squares which have a₁, a₂, ..., a_r as their middle digits.
(This problem generalizes Problem 4574 whose solution appears in this issue.)

4622: Proposed by Richard L. Francis, Cape Girardeau, MO If a, b, c, d are positive integers and a³ + b³ + c³ = d³, can aⁿ + bⁿ + cⁿ = dⁿ for integers n greater than 3?

4623: Proposed by Richard L. Francis, Cape Girardeau, MO.
A deceptive prime is a composite number p such that p divides the repunit number R_p-1. For example, 91 | R₉₀. Is it true that between any deceptive prime and its double another deceptive prime can be found?
(Note that by Bertrand's Conjecture (Chebyshev's Theorem) between any prime and its double another prime can be found.)

4624: Proposed by Heinz-Jürgen Seiffert, Berlin, Germany.

4625: Proposed by Kenichiro Kashihara, Sagarnihara, Kanagawa, Japan.