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A paper triangle has its vertices cut off. Each cut is along a straight line
parallel to the side opposite the vertex and tangent to the triangle's incircle.

Prove that the triangle's inradius is equal to the sum of inradii of the
three triangles cut off.
  

The number, 1406357289, is a 0 to 9 pandigital number because it is made up of each of the digits 0 to 9 in some order, but it also has a rather interesting sub-string divisibility property:

d2d3d4=406 is divisible by 2
d3d4d5=063 is divisible by 3
d4d5d6=635 is divisible by 5
d5d6d7=357 is divisible by 7
d6d7d8=572 is divisible by 11
d7d8d9=728 is divisible by 13
d8d9d10=289 is divisible by 17

Find the sum of all 0 to 9 pandigital numbers with this property.

Leonardo of Pisa, also known as Fibonacci, recounts that he was given this problem by John of Palermo as part of a mathematical tournament.

Three men possess a pile of money, their shares being 1/2, 1/3, 1/6. Each man takes some money from the pile until nothing is left. The first man returns 1/2 of what he took, the second 1/3 and third 1/6. When the total so returned is divided equally among the men it is found that each then possesses what he is entitled to. How much money was in the original pile, and how much did each man take from the pile?

Note: This is a Diophantine problem so you may just give the smallest whole number solution.

This particular problem is attributed to Alcuin. The wording does not give sufficient information to answer the question without making legal assumptions. Please share your assumptions with your solution.

A dying man left 960 shillings and a pregnant wife. He directed that if a boy was born, he should receive three-quarters of the whole and the child's mother should receive one-quarter. But if a daughter was born, she would receive seven-twelfths, and her mother five-twelfths. It happened however that twins were born - a boy and a girl. How much should the mother receive, how much the son, and how much the daughter?

How many distinct integer solution of the equation
2ⁿ-2=n!

exist?

List them.

For which bases b is it possible to make a pandigital number in base b that is divisible by 11 in base b?

For example, in base 4: 1023 / 11 = 33. (In base 10 this is 75 / 5 = 15.)

Consider numbers like 54 and 90.
In the lists of all divisors (for each of those two) 9 distinct digits appear!

54: (1,2,3,6,9,18,27,54); all digits , except 0
90: (1,2,3,5,6,9,10,15,18,30,45,90); 7 excluded

List all such numbers below 1000.

The palindromic decimal number N=abccba displays an interesting feature:
The value of abc in base 9 equals the value of cba in base 7.

What is N's prime factorization?

YOU*ME=OKAY

Q: 591 ?

A: Because 183 4067 43 90221 !

Below is a partially filled in grid of letters that form a crossword-style set of ten 5-letter words, 5 across and 5 down.

Add the following set of letters to the empty cells so that the set of ten different words is complete:

D, F, G, G, H, O, P, R, R, R, R, S, T, T, U

		A
		W	E
A		A		E
			E	E
	E	E

---

From Mensa Puzzle Calendar 2017 by Mark Danna and Fraser Simpson, Workman Publishing, New York. Puzzle for July 12.

From the year c. 850 the book Ganita-Sara-Sangraha contains the following:

Three merchants saw in the road a purse. One said, "If I secure this purse, I shall become twice as rich as both of you together."
Then the second said, "I shall become three times as rich."
Then the third said, "I shall become five times as rich."
What is the value of the money in the purse, as also the money on hand?

There are an infinite number of solutions. Find the smallest whole number amounts the merchants could have.

Consider the number 4808
Notice: 4^2+8^2=80
i.e. the sum of the squares of this number's first and last digits equals the number obtained when the first and last digits are erased.

How many numbers with such feature exist below 10000?

Clearly, no leading zeroes.

11262; 41207; 20X04; 04032; 03628; ...?..

a.What is the missing digit in the above sequence?
b. What is the next term?

Definition: "Brazilian" numbers ("les nombres brésiliens" in French)are numbers n such that exists a natural number k with 1<k< (n-1) such that the representation of n in base k has all equal digits.

1.Prove that all even numbers above 6 are Brazilian numbers.
2. How many odd Brazilian numbers are there below 100?