This is a nice Brainteaser from the *Quantum* math magazine.

“Line segment MN is the projection of a circle inscribed in a right triangle ABC onto its hypotenuse AB. Prove that angle MCN is 45°.”

See the Circle Projection Problem.

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This is a nice Brainteaser from the *Quantum* math magazine.

“Line segment MN is the projection of a circle inscribed in a right triangle ABC onto its hypotenuse AB. Prove that angle MCN is 45°.”

See the Circle Projection Problem.

Here is another problem from the *Quantum* magazine, only this time from the “Challenges” section (these are expected to be a bit more difficult than the Brainteasers).

“A quadrangle is inscribed in a parallelogram whose area is twice that of the quadrangle. Prove that at least one of the quadrangle’s diagonals is parallel to one of the parallelogram’s sides. (E. Sallinen)”

Here is another Brainteaser from the *Quantum* magazine.

“Prove that the area of the red portion of the star is exactly half the area of the whole star. (N. Avilov)”

This is a relatively simple problem, but I wanted to include it because of its cartoon. Its implied gentle post-Soviet humor reminded me of that strange decade in US-Russian affairs between the end of the Cold War and the rise of Putin in the 21st century. The strangeness was brought home when we had our annual security checks of our classified document storage. Being mostly anti-submarine warfare (ASW) material the main concern was that it would not fall into the hands of the Soviets. But with the “demise” of the Soviet Union in 1989 no one cared any more about the classification. After decades of painfully securing these documents we could not suddenly turn them loose and throw them into the public trash. So we kept them secure anyway. You can imagine how we old cold-warriors feel about the current regime.

That is not to say that I didn’t welcome the thaw. Russian literature, both classical and even “Soviet realism”, as well as Russian cinema, is some of the world’s best. And Russian mathematicians have always been superior, and especially adept at communicating with novices. The collaboration of the American mathematicians and *Kvant* contributors in *Quantum* produced excellent results during the thaw. It is unfortunate that it could not survive the rise of Putin and his oligarchs.

See the Red Star

In looking through some old files I came across a math magazine I had bought in 1998. It was called *Quantum* and was published by the National Science Teachers Association in collaboration with the Russian magazine *Kvant* during the period 1990 to 2001 (coinciding with the Russian thaw, which in the following age of Putin seems eons ago). Fortunately, they are all online now. Besides some fascinating math articles the magazine contains a column of “Brainteasers.” Here is one of them:

“Alice used to walk to school every morning, and it took 20 minutes for her from door to door. Once on her way she remembered she was going to show the latest issue of *Quantum* to her classmates but had forgotten it at home. She knew that if she continued walking to school at the same speed, she’d be there 8 minutes before the bell, and if she went back home for the magazine she’d arrive at school 10 minutes late. What fraction of the way to school had she walked at that moment in time? (S. Dvorianinov)”

This is fairly straight-forward, but other problems in the magazine are a bit more challenging.