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Problem of the Day #246: Sums of Powers of 7 November 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 Centers November 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 Building November 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 Strings November 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 Ducks November 16, 2011

Posted by Alex in : potd , 2 comments

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 Root November 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 Area November 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 Circle November 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 Quartic November 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 Eleven November 11, 2011

Posted by Saketh in : potd , add a comment

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.