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Evaluate the sum of reciprocals of double factorial of even nonnegative integers.
x^3 + y^3 + z^3 = w^3.
The smallest set of positive integers x,y,z,w qualifying as a solution of the bolded equation above is:
3,4,5,6
a.Find some other sets of distinct integers, not necessarily positive.
b. Provide some comments on "The sum of 3 cubes is a cube" equation.
Find all solutions of (x+y^2)*(x^2+y)=(xy)^3 . x and y have to be nonzero integers.
One Martian, one Venusian and one Human reside on Pluto. One day they make the following conversation:
Martian: I have spent 1/12 of my life in Pluto.
Human: I also have.
Venusian: Me too.
Martian: But Venusian and I have spent much more time here than you, Human.
Human: However Venusian and I are of the same age.
Venusian: I have lived 300 Earth years.
Martian: Venusian and I have been on Pluto for the last 13 years.
It is known that Human and Martian together have lived for 104 Earth years.
Find their respective ages.
HINT:
The numbers are not necessarily expressed in base 10 . Source: 1971 IMO longlisted problem.
How many ones are used in writing all of the numbers from 0 to 2018 (inclusive) in binary form?
This problem was inspired by the infamous problem 6 in the 1988 Math Olympiad:
When is a^{2}+b^{2} divisible by (ab+1)^{2} where a and b are nonnegative integers?
Let S = a^5 + b^5 + c^5 + d^5, and a,b,c,d are integers fulfilling a+b+c+d=0
Prove that S must be divisible by 10.
Consider this game:
You have a 1% chance of winning on the first try. If you win, great!
If you don't, win on the first try, you have a 2% chance of winning on the second try.
If needed you have a 3% chance on the third try and so on until you eventually win.
What is the expected number of tries to win this game?
For each positive integer n, let Mn be the square matrix (nxn) where each diagonal entry is 2018, and every other entry is 1. Determine the smallest positive integer n (if any) for which the value of det(Mn) is a perfect square.
Below is a 13letter phrase with no repeated letters and its space removed, and below it is placed the remaining 13 letters of the alphabet in alphabetic order:
PUBLICSERVANT
DFGHJKMOQWXYZ
It can be used to make a code so that vertically touching letters substitute, mutually, for one another, such as coding GROVEL as BQEWOH. The phrase PUBLICSERVANT was the key used in this encoding.
Below are the plain text and encoded text of a silly 3word phrase, encoded using the same type of scheme, but using a different 13letter phrase with no repeated letters rather than PUBLICSERVANT.
JOYFUL NUDIST BAKERY
CFMOKH PKIDZQ WXUVGM
What was the 13letter phrase that formed the key to this encoding?
From Mensa Puzzle Calendar 2018 by Fraser Simpson, Workman Publishing, New York. Puzzle for June 12.
Find a sixdigit number ABCDEF such that
AB^{3}+CD^{3}+EF^{3}=ABCDEF.
Rem: VWXYZ denotes the value of the string (not necessarily distinct digits).
If a and b are positive integers, find the probability that (a^2+b^2)/11 is a positive integer.
3^{2701} = 3 mod 2701
With neither calculator nor direct evaluation prove the above statement.
Generalize.
Any quadrilateral can tessellate.
Let Q be a cyclic quadrilateral whose inradius is r and whose incenter lies on the interior of Q.
Tessellate Q then connect the incenters of all the neighboring copies of Q.
Prove that the resulting quadrilaterals (which form another tessellation) are also cyclic with inradius r.
