Consider the sum 1^99 + 2^99 + 3^99 + ... + 99^99.
Finding the last digit of this sum was the task of an old problem Last Digit. With a clever setup finding the last two digits was just as easy.

So I present a higher challenge: find the last four digits. No computer programs!

A bag contains an unknown number of red balls and yellow balls. When N balls are drawn at random (without replacement) the probability that they are all yellow is 1/2. The number of balls in the bag is the minimum for this to happen.

If the first N balls were all yellow, what is the probability that the next ball drawn is red?

Express the probability as a function of N.

Solve the floor equation:

[x]³ + 2x² = x³ + 2[x]²

If N is a nonnegative integer, the triangular number T(N)=1+2+3+...+N is given by N(N+1)/2.

Find a prime P such that the sum of the divisors of T(P) is a cube.

****The divisors of a positive integer N includes 1 and N.

For which digits d, is it possible to add d to every digit of a square and get another square?
For example, adding 3 to each digit of 16 gives 49.However, adding zero to each digit in this manner is NOT permissible.

For which digits d are there infinitely many such squares?

*** Digit sums greater than 9 are not allowed. For example, you could not add 8 to the digits of 81 to get 169.

Let ABCDE be a convex pentagon such that AB = BC = CD and angle BDE = angle EAC = 30. Find the possible values of angle BEC.

Let A and B be two different squares of positive integers, A < B, such that the set of base ten digits of A is the same as the set of base ten digits of B.

Find the smallest and largest value of A+B, such that A+B consists of 10 distinct digits.

Each of x, y and z is a positive integer with gcd(x,y,z)=1

Determine all possible pairs (x,y,z) satisfying this equation:

x³+y³=7z³ where x+y+z < 10^10

THESE
FIFTY
HAPPY
GIGAS

The 4x5 matrix above has a certain peculiar feature, which allows you to perform a certain card trick. Although it serves ok as a mnemonic for this trick, it is not a nicely structured sentence, like "NEVER TRUST BLIND DATES" (better, but misses the needed feature).

I ask you to find the essential feature, and then to suggest a nice mnemonic, logically making sense.

Find the number of trapeziums that it can be formed with the vertices of a regular polygon.

Real constants a, b, c are such that there is exactly one square all of whose vertices lie on the cubic curve y = x³ + ax² + bx + c. Prove that the square has sides of length 72^1/4.

The digital product P(N) of an integer N is the product of its decimal digits. So P(128)=16.

Determine all sets of distinct positive integers A and B such that A = P(B)P(C) and B = P(A)P(C) for some integer C.

For each pair, A and B, give the lowest possible value for C.

How many non-negative integers are there with non-repeating digits?

To avoid ambiguity:
Smallest number: 0
Biggest: 9876543210

f(a,b) is defined by the following table:
f(2,3)=10,
f(3,7)=28,
f(4,5)=33,
f(5,8)=61,
f(6,7)=73,
f(7,8)= ???

Fill in the last value of f(a,b), and derive its formula.

What is the smallest positive integer N such that N+123456 and N+12345678 are both palindromic?

For example, the smallest whole number such that N+1234 and N+123456 are both palindromic is N=975445, since:975445+1234=976679 and 975445+123456=1098901.

A right circular cone has base of radius 1 and height 3.
A cube is inscribed in the cone so that one face of the cube is contained in the base of the cone.

What is the side's length of the cube?

Source: Putnam 1998

For how many natural numbers n<45² are there coprime natural numbers a and b such that 45-√n is the root of the equation x²-ax+b=0?

Choose randomly 3 points on a circumference of a circle.

What is the probability that the center of the circle
lies within the triangle ABC?

Source: Simplified from a 3-dimensional problem on a Putnam exam.

Find positive integers X, Y and N satifying the Pell equation X² - N*Y² = 1 such that such that the concatenation of X, Y and N contains exactly two of each digit 0 to 9.

Find the smallest distinct whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M⁷ to get N⁷, and where neither M nor M⁷ contains a 0.

KR died about 4 years ago.
He would be 85 today.

KENNY+ROGERS=CHOSEN

I miss him.

Derive an algorithm to calculate the total number of zeros in the decimal expansion of 2024²⁰²⁴.

Prove that there exists no quadratic polynomials f, g, h, such that the equation f(g(h(x)))=0 has the solutions 1, 2, 3, 4, 5, 6, 7, 8.

HERE+THERE= each 5-letter word from the list below

Malta, Porto, London, Rouen, Miami, movies, races, sales

each of the above can easily replace the sum defining my whereabouts thus creating a puzzle with a unique answer.

Please expand the list, using valid words providing a unique solution

Find the smallest distinct whole numbers, M and N such that you can rearrange the digits of M to get N, and you can rearrange the digits of M⁶ to get N⁶, and where neither M nor M⁶ contains a 0.