## Problem of the Day #246: Sums of Powers of 7November 20, 2011

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For how many nonnegative integers $n < 1111$ is $\displaystyle\sum_{i=0}^n 7^i \equiv 1 \pmod{4}$?

## Problem of the Day #245: Distance between Two CentersNovember 19, 2011

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Albert draws a random isosceles triangle with perimeter $6$. What is the expected distance between the circumcenter and the incenter of the triangle?

## Problem of the Day #244: Sequential Shell BuildingNovember 18, 2011

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Arjun is constructing a shell pattern for art as seen below, for some monotonically increasing sequence $\{a_k\}$ with $a_0 = 1$.

The right angles for all triangles are fixed at point $P$. The addition of the last triangle (with leg lengths of $a_{n+3}$ and $a_{n+4}$, $n \ge 1$) increases the shape’s area by a factor of:

$$\frac{8 a_n^2 + 10 a_n a_{n-1} + 3 a_{n-1}^2}{3 a_n^2 + 4 a_n a_{n-1} + a_{n-1}^2}$$

Find $a_{15}$.

## Problem of the Day #243: Sets of Binary StringsNovember 17, 2011

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Alex wants to find a set of binary strings of length $n$ such that they all differ by at least $3$ bits. Find, in terms of $n$, the size of the largest set he can make.

## Problem of the Day #242: Lone DucksNovember 16, 2011

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Saketh wants to organize his ducks into groups. He has somewhere between $1$ and $1000$ ducks. When he tries to organize his ducks into groups of size $2$ each, he ends up with $1$ duck left over. When he tries to organize his ducks into groups of size $5$ each, he ends up with $2$ ducks left over. When he tries to organize his ducks into groups of size $9$, he ends up with $5$ ducks left over. Let $x$ be the number of ducks Saketh has. Find the number of possible values of $x$.

## Problem of the Day #241: Approximating a Square RootNovember 15, 2011

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Suppose $r$ is an estimate of the exact value of $\sqrt{240}$. Find an expression, in terms of $r$, for a better estimate. How quickly converging an expression can you find?

## Problem of the Day #240: Minimum Triangle AreaNovember 15, 2011

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Triangle $ABC$’s bisectors all have length $1$. What is the least possible area $ABC$ could have?

## Problem of the Day #239: Connecting Points on a CircleNovember 13, 2011

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Albert has $20$ distinct points evenly spaced on the circumference of a circle. He draws $10$ line segments that connect each point to exactly one other point. What is the expected value of the number of intersections there will be? Note that this is not the same as asking how many intersection points there will be.

## Problem of the Day #238: Perfect Square QuarticNovember 12, 2011

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Given a number $x$, Albert will tell you the value of $$x(x+1)(x+2)(x+3)$$ For how many integers $x$ will Albert give you a perfect square?

## Problem of the Day #237: Eleven Eleven ElevenNovember 11, 2011

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Alex is thinking of $111111$ numbers $a_1, a_2, a_3, …, a_{111111}$ such that

$$a_1+a_2+a_3+…+a_{111111} = 111111$$ $$a_1^3 + a_2^3 + a_3^3+…+a_{111111}^3 = a_1^4 + a_2^4 + a_3^4+…+a_{111111}^4$$

Find all distinct sequences of numbers that he could be thinking of.