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Problem of the Day #318: Pascal Paths January 31, 2012

Posted by Alex in : potd , 1 comment so far

Albert starts at the top corner of Pascal’s triangle. At each step, Albert either goes down-left or down-right, eating each number he comes across. By the time Albert is at the $9^\text{th}$ row, $4^\text{th}$ column ($0$-indexed), what is the maximum possible total of the numbers Albert has eaten?

Problem of the Day #317: Fiboscal January 30, 2012

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Express the following in terms of the Fibonacci function, $F(n)$:

$$\sum\limits_{k=0}^{n} \binom{n+k}{n-k}$$

Problem of the Day #316: Möbius $\mu$ Sums January 29, 2012

Posted by Albert in : potd , add a comment

Let $\mu$ be the Möbius function. Find the smallest value of $n \geq 1$ such that:

$$\sum\limits_{i=1}^n \; i \cdot \mu(i) = 0$$

Can you find the next smallest value of $n$?

Problem of the Day #315: Divisibility into $99 \ldots 99900 \ldots 000$ January 28, 2012

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Thanks to my dad for this problem.

Prove (or find a counterexample) that given a positive integer $n$, $\exists \quad i,j: \quad n \mid (10^i – 1)(10^j)$.

Problem of the Day #314: $\pi$ January 27, 2012

Posted by Albert in : potd , add a comment

Saketh graphs $f(x) = \frac{1}{1+x^2}$ for $x \in [0,1]$. He cuts out the square $0 \leq x \leq 1$, $0 \leq y \leq 1$, then cuts along the line $f(x)$, producing piece 1 (below $f(x)$) and piece 2 (above $f(x)$). He finds that the ratio of the weights of piece 1 to piece 2 is $R$. Given this, help him find an expression for $\pi$.

Problem of the Day #313: A System of Four Equations January 26, 2012

Posted by Saketh in : potd , add a comment

Find all real solutions:

\begin{array}{lcl}
w^4 = 313x – 1 \\
x^4 = 313y – 1 \\
y^4 = 313z – 1 \\
z^4 = 313w – 1
\end{array}

Problem of the Day #312: Count the Possiblities January 25, 2012

Posted by Saketh in : potd , add a comment

Steve is thinking of two numbers, $a$ and $b$. If their least common multiple is $100!$ and their greatest common divisor is $24$, how many possible unordered pairs of integers could he be thinking of?

Problem of the Day #311: Making Powers January 24, 2012

Posted by Alex in : potd , add a comment

Albert starts with a list, containing only $1$ and $2$. At any step, he is allowed to take any two elements from his list (not necessarily distinct) and either multiply or divide them, adding the new result to the list. After some set of operations, the number $2^{17}$ is in the list. What is the smallest possible size of the list?

Problem of the Day #310: The volume of a simply defined region January 23, 2012

Posted by Mitchell in : potd , add a comment

Let $\theta$ be a real number with $0 < \theta < \pi$. Let $A$ and $O$ be distinct points in three-dimensional space. Let $S$ be the set of all points $P$ in space with both $OP < 1$ and $\angle AOP <\theta$. Find the volume of $S$ in terms of $\theta$.

(This problem is possible to solve with calculus, but there’s also a very nice solution that avoids it.)

Problem of the Day #309: Reflecting a Triangle January 22, 2012

Posted by Alex in : potd , add a comment

$\triangle ABC$ has coordinates located at $(1, 1)$, $(-2, 1)$, and $(2, 0)$. When $\triangle ABC$ is reflected across the line $y=kx$, the total area covered by $\triangle ABC$ and its image is $\frac{3[ABC]}{2}$. Find $k$.