Is 2ⁿ + 3ⁿ (where n is an integer) ever the square of a rational number?

Triangle ABC is isosceles with AB = AC. Point D on AB is such that angle BCD = 15 degrees and BC = AD√6. Find, with proof, the measure of angle CAB.

Solution

(a) Is there a largest perfect square with all odd digits? If so, what is it?
(b) Is there a largest perfect square with all even digits? If so, what is it?

(a)The largest perfect square with all odd digits is 9.
Proof. A square that contains all odd digits must itself be odd, and hence must be the square of an odd integer. But the tens digit of the square of an odd integer is always even. (There are various ways to show this. For instance, consider that, modulo 20, (±1)² = (±9)² = 1, (±3)² = (±7)² = 9, and (±5)² = 5, are all less than 10.) Hence the largest perfect square with all odd digits is the largest such square with no tens digit: namely, 9.

(b)There is no largest perfect square with all even digits.
Proof. Consider f(n) = (2 * 10ⁿ)² = 4 * 10²ⁿ, where n is a non-negative integer. Clearly f(n) consists entirely of even digits, for any value of n. Since there is no largest integer n, there is no largest f(n). (Alternatively, note that for any positive integer K, we can find an integer n such that f(n) > K. Thus there is no largest f(n).)

In triangle ABC, side AB = 20, AC = 11, and BC = 13. Find the diameter of the semicircle inscribed in ABC, whose diameter lies on AB, and that is tangent to AC and BC.

Solution

Let a, b, m, and n be positive integers, with n > 1. Show that aⁿ + bⁿ = 2^m implies a = b.

The minute hand of a clock is twice as long as the hour hand. At what time, between 00:00 and when the hands are next aligned (just after 01:05), is the distance between the tips of the hands increasing at its greatest rate?

Solution

In triangle ABC, draw AD, where D is the midpoint of BC. If angle ACB = 30° and angle ADB = 45°, find angle ABC.

Solution

In how many ways, counting ties, can eight horses cross the finishing line? (For example, two horses, A and B, can finish in three ways: A wins, B wins, A and B tie.)

Solution

Let p be a polynomial of degree n with complex coefficients.
Can
p(1) = 1/1,
p(2) = 1/2,
...
p(n) = 1/n,
p(n + 1) = 1/(n + 1),
p(n + 2) = 1/(n + 2),
be satisfied simultaneously?

Solution

Let P = {p₁, ... , p_n} be the set of the first n prime numbers. Let S be an arbitrary (possibly empty) subset of P. Let A be the product of the elements of S, and B the product of the elements of S', the complement of S. (An empty product is assigned the value of 1.)

Prove that each of A + B and |A - B| is prime, provided that it is less than p_n+1² and greater than 1.

For example, if P = {2, 3, 5, 7}, the table below shows all the distinct possibilities for A + B and |A - B|. Values of A + B and |A - B| that are less than p₅² = 121 and greater than 1, shown in bold, are all prime.

The sides of two squares (not necessarily of the same size) intersect in eight points: A, B, C, D, E, F, G, and H. These eight points form an octagon.
Join opposite pairs of vertices to form two non-adjacent diagonals. (For example, diagonals AE and CG.)
Show that these two diagonals are perpendicular.
Solution

In triangle ABC, with AB not equal to AC, drop a perpendicular from A to BC, meeting at O.
Let AD be the median joining A to BC.
If angle OAB = angle CAD, show that angle CAB is a right angle.
Solution

If 2ⁿ and 5ⁿ (where n is a positive integer) have the same initial digit, what is this digit?
Numbers are written in standard decimal notation, with no leading zeroes.
Solution

Let {a_n} be a sequence of positive integers satisfying a_n+3 = a_n+2(a_n+1 + a_n), for n = 1, 2, 3, ... .
Given that a₆ = 8820, find a₇.
Solution