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Problem of the Day #32: More Fun With Factorials April 20, 2011

Posted by Albert in : potd , add a comment

Find all ordered pairs $(x, y)$ for positive integers $x$ and $y$, such that: $$x(1! + 2! + \cdots + x!) = y!$$

Problem of the Day #31: Area of an equilateral triangle April 19, 2011

Posted by Sreenath in : potd , add a comment

The difference between the areas of the circumcircle and incircle of an equilateral triangle is $147\pi$ square units. Find the area of the triangle.

Problem of the Day #30: Area of a Triangle on a Cubic Function April 18, 2011

Posted by Alex in : potd , add a comment

Find the maximum area of $\triangle{}ABC$ given that $A$, $B$, and $C$ are lattice points $(x, y)$ on the function $y=x^3$ and no two points have $x$-coordinates differing by more than $11$. Express the answer using integers, the four basic operations, exponentiations, and square roots.

Problem of the Day #29: Sum of Root Cubes April 17, 2011

Posted by Saketh in : potd , add a comment

Determine the sum of the cubes of the roots of the polynomial $x^6 – 4x^5 + 12x^4 + 23x^2 – 2x + 125 = 0$.

Problem of the Day #28: The 7′s of Modularity April 16, 2011

Posted by Aziz in : potd , add a comment

Find the last 3 digits of $$7777^{777^{77^7}}$$

Problem of the Day #27: Solution to a nonlinear recurrence April 15, 2011

Posted by Mitchell in : potd , add a comment

Given that the sequence $\{a_{n}\}_{n=0}^\infty$ satisfies $a_0 = 1$ and $a_{n+1} – a_n =2^n a_{n}^2$ for all $n \geq 0$, find a closed-form expression for $a_n$.

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Problem of the Day #26: Repeating Function V(x) April 14, 2011

Posted by Seungln in : potd , add a comment

Albert’s function $V(x)$ be defined as $V(x) \equiv 1\cdot 1! + 3\cdot 2! + 5\cdot 3! + \cdots + (2x – 1)\cdot x! \pmod{1000}$ when $0 \le V(x) < 1000$. Find the smallest integer $n$ such that $V(n) = V(p)$ when $p < n$ and $p$ is also an integer.

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Problem of the Day #25: Building an Infinite Crane April 13, 2011

Posted by Billy in : potd , add a comment

Albert is constructing a crane. His basic building block is shown in Fig. 1.

Fig. 1

Fig. 1

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Problem of the Day #24: Product of areas of triangles in regular n-gon April 12, 2011

Posted by Mitchell in : potd , add a comment

Given a regular $n$-gon $\mathcal{P}$ with side length $1$, find the product of the areas of all the triangles whose vertices are vertices of $\mathcal{P}$.

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Problem of the Day #23: Divisibility of Factorials and Factorial Sums April 11, 2011

Posted by Albert in : potd , add a comment

Let $x$ be the highest positive integer such that $x!$ divides $$\sum\limits_{n=k}^{2011} n!$$ for $0 \leq k \leq 2011$. Find the number of integers $k$ such that $x \neq k$.

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