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AD and AE are respectively the altitude and angle bisector of ΔABC (D and E are on side BC). If DB - DC = 1029 and EB - EC = 189 then what is the value of AB - AC?

Find the minimum of √(58 - 42x) + √(149 - 140√(1 - x²)), where -1 <= x <= 1.

Put three congruent triangles inside a unit square so that they don't overlap one another.

What is the maximum possible area of one of the triangles?

66, 78, 93, 105, 111, 114...
Consider the 5 numbers above.

Evaluate the next 3 numbers.

8,9,2,0,1,5 ...

To get all ten digits you have to add 3,4,6 and 7.

Just so, but in what order?

Prove that all numbers of the form 12008, 120308, 1203308, ... are divisible by 19.

The sum of all divisors of this semiprime number is 2696.

What is the number?

Bonus: What if the word "semiprime" was erased from the puzzle?

A cube on a table has edge length 24. A plane intersects the cube's four vertical edges at points A, B, C, and D such that point A is a vertex of the cube lying on the table. The heights of points B and C from the table floor are 7 and 12, respectively.

Calculate the volume of the portion of the cube that lies underneath the cutting plane.

A mathematician wanted to teach his children the value of cooperation, so he told them the following

"I chose a secret triangle for which the lengths of its sides are all integers.

To you my dear son Charlie, I am giving the triangle's perimeter. And to you, my beloved daughter Ariella, I am giving its area.

Since you are both such talented mathematicians, I'm sure that together you can find the lengths of the triangle's sides."

Instead of working together, Charlie and Ariella had the following conversation after their father gave each of them the information he promised.

Charlie: "Alas, I cannot deduce the lengths of the sides from my knowledge of the perimeter."

Ariella: "I do not know the perimeter, but I cannot deduce the lengths of the sides from just knowing the area. Maybe our father is right and we should cooperate after all."

Charlie: "Oh no, no need. Now I know the lengths of the sides."

Ariella: "Well, now I know them as well."

Find the lengths of the triangle's sides and explain the dialogue above.

Alex and Bert have the same walking speed and the same running speed. They both decide to take a lap around the same track.

Alex walks to a point and then runs such that one half of the distance is spent walking and the other half is spent running.

Bert walks to a point and then runs such that one half of his time is spent walking and the other half is spent running.

Who finishes first?

Just insert 2 missing numbers, so it makes some sense: 101,112,131, X ,161, 718, Y

Can a number consisting of 600 sixes and some zeros be a square?