# Special Log Sum

Here is a fairly computationally challenging 1994 AIME problem .

“Find the positive integer n for which

⌊log2 1⌋ + ⌊log2 2⌋ + ⌊log2 3⌋ + … + ⌊log2 n⌋ = 1994.

where for real x, ⌊x⌋ is the greatest integer ≤  x.”

There is some fussy consideration of indices.

See the Special Log Sum for a solution.

# Elliptic Circles

Here is another UKMT Senior Challenge problem for 2017.

“The diagram shows a square PQRS with edges of length 1, and four arcs, each of which is a quarter of a circle. Arc TRU has centre P; arc VPW has centre R; arc UV has centre S; and arc WT has centre Q.

What is the length of the perimeter of the shaded region?

A_6___B_(2√2 – 1)π___C_(√2 – 1/2 ___D_2___E_(3√2 – 2)π”

See Elliptic Circles for a solution.

# Amazing Identity

This is a most surprising and amazing identity from the 1965 Polish Mathematical Olympiads.

“31.  Prove that if n is a natural number, then we have

(√2 – 1)n = √m – √(m – 1),

where m is a natural number.”

Here, natural numbers are 1, 2, 3, …

I found it to be quite challenging, as all the Polish Math Olympiad problems seem to be.

See the Amazing Identity

# The Tired Messenger Problem

Here is another challenging problem from the Polish Mathematical Olympiads.  Its generality will cause more thought than for a simpler, specific problem.

“A cyclist sets off from point O and rides with constant velocity v along a rectilinear highway.  A messenger, who is at a distance a from point O and at a distance b from the highway, wants to deliver a letter to the cyclist.  What is the minimum velocity with which the messenger should run in order to attain his objective?”

See the Tired Messenger Problem

# Incredible Trick Puzzle

Here is another typical sum puzzle from Presh Talwalkar.

“Solve the following sums:

_____1/(1×3) + 1/(3×5) + 1/(5×7) + 1/(7×9) + 1/(9×11) =

_____1/(4×7) + 1/(7×10) + 1/(10×13) + 1/(13×16) =

_____1/(2×7) + 1/(7×12) + 1/(12×17) + … =”

The only reason I am including this puzzle is that Talwalkar gets very excited about deriving a formula that can solve sums of this type.  This gives me an opportunity to discuss the “formula vs. procedure” way of doing math.

See the Incredible Trick Puzzle for solutions.

# Fireworks Rocket

This is another physics-based problem from Colin Hughes’s Maths Challenge website (mathschallenge.net) that may take a bit more thought.

“A firework rocket is fired vertically upwards with a constant acceleration of 4 m/s2 until the chemical fuel expires. Its ascent is then slowed by gravity until it reaches a maximum height of 138 metres.

Assuming no air resistance and taking g = 9.8 m/s2, how long does it take to reach its maximum height?”

I can never remember the formulas relating acceleration, velocity, and distance, so I always derive them via integration.

See the Fireworks Rocket for solutions.

# Stimulating Sequence

This is another stimulating little problem from the 2022 Math Calendar.

a1 = 1, a2 = 2, …, an+1 = an + 6an-1

x = lim an+1/an   as   n → ∞

Solve for x.”

As before, recall that all the answers are integer days of the month.

See Stimulating Sequence for a solution.

# Minimum Path Via Circle

James Tanton provides another imaginative problem on Twitter.

“I am at point A and want to walk to point B via some point, any point, P on the circle. What point P should I choose so that my journey A → P → B is as short as possible?”

Hint: I got ideas for a solution from two of my posts, “Square Root Minimum” and “Maximum Product”.