# Quintic Nightmare

Here is another challenging problem from the 2004 Pi in the Sky Canadian magazine for high school students.

Problem 4. Find the real solutions of the system
________________ (x + y)^5 = z,
________________ (y + z)^5 = x,
________________ (z + x)^5 = y.”

See the Quintic Nightmare

# Circle Tangent Chord Problem

This is another problem from the Math Challenges section of the 2000 Pi in the Sky Canadian math magazine for high school students.

Problem 4. From a point P on the circumference of a circle, a distance PT of 10 meters is laid out along the tangent. The shortest distance from T to the circle is 5 meters. A straight line is drawn through T cutting the circle at X and Y. The length of TX is 15/2 meters.

(a) Determine the radius of the circle,
(b) Determine the length of XY.”

See the Circle Tangent Chord Problem

# Mysterious Dopplegänger Problem

I found this problem from the Math Challenges section of the 2002 Pi in the Sky Canadian math magazine for high school students to be truly astonishing.

Problem 4. Inside of the square ABCD, take any point P. Prove that the perpendiculars from A on BP, from B on CP, from C on DP, and from D on AP are concurrent (i.e. they meet at one point).”

How could such a complicated arrangement produce such an amazing result? I didn’t know where to begin to try to prove it. My wandering path to discovery produced one of my most satisfying “aha!” moments.

Update (12/27/2019) I goofed.  I had plotted the original figure incorrectly. (No figure was given in the Pi in the Sky statement of the problem.) Fortunately, the original solution idea still worked.