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Find the inverse function f(x)=x+[x].
Solve the equation
16x^{3}=(11x^{2}+x1)√(x^{2}x+1)
Find six distinct positive integers A, B, C, D, E, F, G satisfying: A^{3} + B^{3} = C^{3} + D^{3} = E^{3} + F^{3} = 19G^{3}.
Please submit primitive solutions only, that is, A, B, C, D, E, F, G should not have a common factor.
Out of 26 ABCâ€™s letters I have erased 15, leaving only those: a,e,i,j,k,o,q,u,v,x,y.
What were my criteria?
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} + 2x^{2} = x^{3} + 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:
x^{3}+y^{3}=7z^{3}
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.


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