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Solve in integers for x and y:
6(x! + 3) = y^2 + 5
N=sum of primes between smallest and largest prime factor of N (inclusive).
Find all possible values for the composite number N below 300.
The square ABCD has side length 2*sqrt2.
A circle with centre A and radius 1 is drawn. A second circle with centre C is drawn so that it just
touches the 1st circle at point P on AC . Determine the total area of the
regions inside the square but outside the two circles. Source: one of Mayhem problems
The number 1987 can be written as a three digit number xyz in some base
b. If x + y + z = 1 + 9 + 8 + 7=25, determine all possible values of x, y, z, b.
Source: 1987 Canadian Mathematical Olympiad.
A box contains p white balls and q black balls. Beside the box
there is a pile of black balls. Two balls are taken out from the
box. If they are of the same colour, a black ball from the pile is put into
the box. If they are of different colours, the white ball is put back into the box. This procedure is repeated until the last pair of balls are removed from the box and one last ball is put in. What is the probability that this last ball is white? Source: Australian Olympiad 1983
Dinner would have been splendid … if the wine had been as cold as the XXXX, the beef as rare as the XXXXXXX, the brandy as old as the XXXX, and the maid as willing as the XXXXXXX.
Attributed to Winston Churchill.
Try to insert the missing words, taking into account the immense wit of W.C.
A sphenic number S.N. is a product of three distinct prime numbers.
a. Clearly each S.N. has 8 divisors. Show that not only S.N.s claim this feature.
b. Find the smallest consecutive pair n,n+1 of sphenic numbers.
c. Same for the smallest triplet.
d. Prove that sphenic quadruplet n,n+1,n+2,n+3 is "mission impossible".
Prove: If the sides a, b, c of a triangle satisfy a^2 + b^2 = kc^2, then k >1/2.
An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. 13, 17, 31, 37, 71... are
emirps.
List values of q(n) (the number of emirps of length n) for n=2 to 6.
Within the light gray area form eight common 6letter English words vertically. Then form an 8letter word in the yellow spaces at the top using only one letter from the three letters just below each one in the top half of the gray area. Then form a second 8letter word in the yellow spaces at the bottom, and in this case for each position use only one letter from among the three immediately above each in the bottom half of the gray area.
The title is a clue to the two 8letter words.








F

B


F

E

P


U



D


N


F


B

R


Z


R


P

R


B




I







C

C

C


U

E

E

B












From Mensa Puzzle Calendar 2018 by Fraser Simpson, Workman Publishing, New York. Puzzle for October 1.
Let Sp(n) be a sum of all primes from p(1) to p(n) inclusive. Let m(n) be the average value of all those primes, i.e. m(n)= Sp(n)/n.
Find the nth prime N such that m(n) equals the reversal of N.
Example for n=6: Sp(6)=2+3+5+7+11+13= 41; m(6)=41/6; not the reversal of 13,
that is 31 . So 13 is not our prime. Another 2digit number is.
Find it.
Are there any additional i.e. "numbers over 100" solutions?
Some currencies, such as the German Papiermark, the Hungarian pengő, the first Turkish lira, and the second and third Zimbabwean dollars, had banknotes with this number in the denomination.
Place a 1digit number in each of the 16 white cells below so that the product of the numbers in each row will be the number shown to the right of that row, and the product of the numbers in each column will be the number at the bottom of that column.
There will be a digit 1 in exactly one cell in each row and exactly one cell in each column.
From Mensa Puzzle Calendar 2018 by Fraser Simpson, Workman Publishing, New York. Puzzle for October 3.
S is the smallest sum of the squares of the first prime numbers that is a palindrome.
What else can be said about S?
