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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?

5 is in 5²=25, but 5 is at the end of 25. 10 is in 10²=100, but 100 ends in 0. What is the smallest positive integer n such that n is in n², and there is at least one nonzero digit after n in n²?

Does there exist a hexagon whose 6 vertices lie on a circle, and whose internal angles in degrees are

70, 90, 110, 130, 150, 170

in some order?

Given :
a is either a square or a factorial
b is a factorial

Prove that there is only one one pair of integers (a ,b) satisfying the equation
a-b=2019.

As in the first Blob's Borders:

Divide the below grid into ten areas (blobs) so that each area:

• ... is contiguous (all cells connect via edges, not just vertices).
• ... has exactly one of the ten numbers shown in the grid.
• ... contains as many cells as the number that it contains.
• ... borders (along edges, not vertices) the same number of other areas as the number that it contains; for example, a 6-cell area will border six other areas, again based on the digit 6 that is contained within its bounds.

2	4
					3
			4
			5
6	2			3
5			2

An example of a completed puzzle of this type is

3	4
		4	4		3
					6
					4
3	3				2

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Puzzle by Thinh Van Duc Lai, from N.Y.Times Magazine, March 31, 2019.

How many positive integers less than or equal to 100 cannot be written as the sum of distinct powers of 3?

How many integer solutions has the following equation:
sqrt(x)+sqrt(y)=sqrt(3888)?

Assume x>y.

F_n is the n-th Fibonacci number defined by the recurrence relation F_n = F_n-1 + F_n-2 with F₁ = F₂ = 1. If n is a perfect square and n > 4, then find the value of the determinant below

| F₁          F₂          ... F_√n  |
| F_√n+1     F_√n+2    ... F_2√n |
|    .           .                .    |
|    .           .                .    |
| F_n-√n+1   F_n-√n+2  ... F_n   |

The grid below has two of each digit from 1 to 8. Connect each matching pair of digits with a path that connects vertically and/or horizontally (no diagonals), without going through any cell occupied by any other path (no crossing paths).

					1				2
	3						4
5									2
6				6
7	5							4
	1							7
								8
8
3

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From Daily Brain Games calendar 2019, by HAPPYneuron, Andrews McMeel Publishing, Kansas City, MO. Puzzle for March 27.

9 distinct positive integers are given, such that the sum of their reciprocals is equal to 1. If 5 of these integers are 3, 7, 9, 11 and 33, and the other 4 integers all have a units digit of 5, what is the sum of these integers?

The equation (x-2^1/9)(x-3^1/9)(x-4^1/9)...(x-10^1/9)=1 has 9 distinct complex solutions x₁, x₂, x₃, ..., x₉. Find the value of x₁⁹ + x₂⁹ + x₃⁹ + ... + x₉⁹.

There is a collection of points inside a unit cube that are closer to the center of the cube than to any of the cube’s vertices.

What is the volume of this 3D region?

Mobile screen brightness easily auto-adjusts in a variety of conditions so that it remains easily visible. However, if the screen faces the sun, the objects on the screen become difficult to discern with the human eye.

What could be the reason that best explains this phenomenon?