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2018-10-22 Intermediate

         

This block oscillates harmonically with amplitude \(A.\)

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

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

What is the sum of all such \(x\)'s?

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

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

If \(a = 79,\) then what is \(b?\)

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 \(\frac{a}{b},\) where \(a\) and \(b\) are coprime positive integers.

What is \(a+b?\)

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 \(a:b:c,\) where \(a,b,c\) are coprime positive integers.

What is \(a + b + c ?\)

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