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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?

Before trying the problem "note your opinion as to whether the observed pattern is known to continue, known not to continue, or not known at all."

Lets factor polynomials of the form x^n-1. Starting with 1 the following list can be generated:
n=1: (x-1)
n=2: (x-1)*(x+1)
n=3: (x-1)*(x^2+x+1)
n=4: (x-1)*(x+1)*(x^2+1)
n=5: (x-1)*(x^4+x^3+x^2+x+1)
n=6: (x-1)*(x+1)*(x^2+x+1)*(x^2-x+1)

One thing to notice is that each line has exactly one polynomial factor not seen earlier in the list:
n=1: x-1
n=2: x+1
n=3: x^2+x+1
n=4: x^2+1
n=5: x^4+x^3+x^2+x+1
n=6: x^2-x+1

Does each new factorization always produce exactly one new polynomial factor?

Looking more closely you may see that all the coefficients are -1, 0, or 1. Does this continue to be the case for all factors?

Try to solve it without software:

SOMETHING+NOTHING=STONEFISH

ABC is a right triangle with legs a and b and hypotenuse c. Two circles of radius r are placed inside the triangle, the first tangent to a and c, the second tangent to b and c, and both circles externally tangent to each other. Draw a third circle of radius s tangent externally to the first two circles, and to the hypotenuse. What is the smallest possible radius of the third circle if a, b, c, r and s are distinct integers?

A man set out at noon to walk from Appleminster to Boneyham, and a friend of his started at two P.M. on the same day to walk from Boneyham to Appleminster. They met on the road at five minutes past four o'clock, and each man reached his destination at exactly the same time. Can you say at what time they both arrived?

There are 4 solid spheres arranged so that each one is touching all of the others. The 3 bottom spheres touch the flat floor at points A, B and C. The top sphere has a radius of 12 centimeters. If it were replaced by a sphere with radius 25 cm, then its center would be 14, 15 and 16 cm further from from points A, B and C, respectively.

What is the radius of each sphere?

In an election among three candidates, Charles came in last and Bob received 24.8% of the votes. After counting two additional votes, he overtook Bob with 25.1% of the votes. Assuming there were no ties and all the results are rounded to the nearest one-tenth of a percent, how many votes did Alice get?

The following English words have something in common:

we're, dying, nice, slight

Don't give away the common factor, but instead list another word on the list in your answer.

Doing her homework Jane evaluated xy as a concatenation of two 3-digit numbers instead of a product x*y, thus obtaining a result n times bigger than the correct one.

Knowing that n is an integer find all possible solutions fitting the above fact.

The time between seeing a lightning flash and hearing the resulting thunder gives an accurate estimate of the distance to its location. During one storm a meteorologist observes 29 such flashes, and calculates the distances in miles as follows:

6.02, 3.01, 0.69, 3.38, 2.01, 4.69, 3.54, 3.67, 3.67, 4.55, 5.79, 3.72, 1.05, 3.73, 3.98, 7.28, 2.10, 2.90, 6.95, 6.18, 7.20, 4.89, 6.60, 2.53, 2.09, 1.30, 1.81, 4.75, 6.91

Assume that the cloud is circular, is not moving, and that the lightning strikes are uniformly distributed through the cloud. Determine the diameter of the cloud.

Polynomial p(x) has integer coefficients and p(3)=−2.

For what values n may it be possible for (x-n) to be a factor of p(x)?

Inspired by 16 choices

M is the smallest possible sum for a set of four distinct primes such that the sum of any three is prime - (p1,p2,p3,p4}.
N is the smallest possible sum for a set of six distinct primes such that the sum of any five is prime - (q1,q2,q3,q4,q5,q6}.

Find M & N and the corresponding sets.

Find 3 distinct coprime numbers, all below 50, such that
the square of the first is an average of the cubes of the others.