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I have a positive integer X.

When I add all the positive integers from 1 to X, I get a result of YYY, where Y is a positive integer from 0 to 9.

What is X?

For all positive integers k, define f(k)=k²+k+1. Compute the largest positive integer n such that

2019f(1²)f(2²)...f(n²)>=(f(1)f(2)...f(n))²

The famous Kohinoor diamond has been put on display in a certain museum. The museum authorities have installed an electronic lock which has N buttons, each with a different symbol. To open the lock one must select the correct combination of the symbols, irrespective of the order in which the buttons are pressed. When each button is pressed it lights up. If the correct combination is lit, the lock immediately opens. On the side of the lock is a reset button which will turn off all the lights.

The Jewel Thief visited the museum and noted down all the symbols. That night, he returned to steal the precious gem. Due to shortage of time, he must open the lock as fast as possible.

What is the minimum number of presses needed to guarantee that the lock will open (including both symbols and reset)? Assume that all the lights are off when he arrives.

Three families make a remarkable discovery. The sum of the ages of their members are all the same, the sum of the squares of the ages of their members are all the same, and the sum of the cubes of the ages of their members are all the same. Everyone in all 3 families has a different age, and nobody is more than 100 years old.

What is the smallest possible sum of their ages? Can this be done with 4 families?

Four safety engineers set out to inspect a newly cut tunnel through Mt. Popocaterpillar in the Andes. Each person walks at a different constant integer speed measured in meters per minute. In the tunnel there is a mine car which travels along a fixed track, automatically going from end to end at a fixed integer speed. When people board the car they may reverse its direction, but cannot change its speed.

At noon on Monday all four engineers start at the south end, while the mine car starts at the north end. The first (fastest) engineer meets the car, and takes it some distance north. The engineer gets out and continues going north, while the car resumes heading south. Then the second engineer meets the car and also takes it some distance north. Likewise for the third and fourth engineers. All the people, and the mine car, travel continuously with no pauses. The inspectors always go north. Each person enters and exits the car at an integral number of minutes.

All four engineers reach the end of the tunnel simultaneously. What is the earliest time this could happen?

Find the sum of the digits of the number 999...999³. The number has 2000 repeated 9s.

Solve this logic number sequence puzzle

f(2202)=1
f(1999)=3
f(7351)=0
f(6666)=4
f(8080)=6
f(9068)=5
f(2386)=?

What is the minimum value of p(2) if the following 4 conditions are followed?

1) p(x) is a polynomial of degree 17.

2) All roots of p(x) are real.

3) All coefficients are positive.

4) The coefficient of x is 1.

5) The product of roots of p(x) is -1.

In right angled ΔABC, ∠ABC=90 and ∠BCA=60. The incircle of ΔABC touches sides BC and AB at points D and E respectively. Let F be the midpoint of DE. P, Q, R are the feet of perpendiculars from F on BC, CA, AB respectively.

Find (AB+BC+CA)/(PQ+QR+RP).

Find a quadratic polynomial which is a factor of x¹⁰¹+x⁹⁴+x⁵⁷+x³³-1

For some positive integer k both 4^k and 5^k start with the same digit x.

What are the possible values of x?