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2018-10-22 Intermediate

         

This block oscillates harmonically with amplitude A.A.

What is the probability of finding the block's center of mass between positions A2-\frac{A}{2} and A2?\frac{A}{2}?

According to the exponent tower rule, x(xx)=(xx)x x^{(x^x)} = (x^x)^x is generally not true. However, this equality holds for some x>0.x>0.

What is the sum of all such xx's?

The equation x2ax+1=0x^2 - ax + 1 = 0 has real solutions, the greater of which is r.r.

Now, let bb be a positive real number such that the greater solution of the equation x2bx+1=0x^2 - bx + 1 = 0 is r.\sqrt{r}.

If a=79,a = 79, then what is b?b?

Three friends, Slim, Jim, and Tim, each hold three cards—an ace, a two, a three—in a random order.

There are three turns and, on each turn, they simultaneously flip over a card face up, as illustrated below (without returning the card to their hands).

The probability that there's a three-way match on at least one of the flips is ab,\frac{a}{b}, where aa and bb are coprime positive integers.

What is a+b?a+b?

The three-way match can be on any of the flips, and can be on more than one flip.  In this example, there is a three-way match on the third flip. The three-way match can be on any of the flips, and can be on more than one flip. In this example, there is a three-way match on the third flip.

The sides of a triangle are in an arithmetic progression, and the greatest angle of the triangle is double the smallest angle.

The ratio of the three sides can be expressed as a:b:c,a:b:c, where a,b,ca,b,c are coprime positive integers.

What is a+b+c?a + b + c ?

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