2008 Mock ARML 1 Problems

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Set 1

Problem 1

Compute all real values of $x$ such that $\sqrt {\sqrt {x + 4} + 4} = x$.


Problem 2

A positive integer $n$ is a yo-yo if the absolute value of the difference between any two consecutive digits of $n$ is at least $7$ . Compute the number of $8$-digit yo-yos.


Set 2

Problem 3

In regular hexagon $ABCDEF$ with side length $1$, $AD$ intersects $BF$ at $G$, and $BD$ intersects $EC$ at $H$. Compute the length of $GH$.


Problem 4

There are $4$ black balls and $1$ white ball in a hat. A turn consists of picking a ball from the hat and replacing it with one of the opposite color. Compute the probability that, after a sequence of turns, there are $5$ black balls in the hat before there are $5$ white balls.


Set 3

Problem 5

The positive real numbers $x_1, x_2, \ldots, x_{10}$ are in arithmetic progression in that order. They also satisfy

\[x_1^2 - x_2^2 + x_3^2 - \cdots - x_{10}^2 = x_1 + x_2 + \cdots + x_{10}.\]

Compute the common difference of this arithmetic progression.


Problem 6

Square $ABCD$ has side length $2$. $M$ is the midpoint of $CD$, and $N$ is the midpoint of $BC$. $P$ is on $MN$ such that $N$ is between $M$ and $P$, and $m\angle MAN = m\angle NAP$. Compute the length of $AP$.


Set 4

Problem 7

Compute the number of $3$-digit base-$5$ positive integer multiples of $7$ that are also divisible by $7$ when read in base $10$ instead of base $5$.


Problem 8

For positive real numbers $a,b,c,d$,

\begin{align*}2a^2 + \sqrt {(a^2 + b^2)(a^2 + c^2)} &= 2bc\\ 2a^2 + \sqrt {(a^2 + c^2)(a^2 + d^2)} &= 2cd\\ 2a^2 + \sqrt {(a^2 + d^2)(a^2 + b^2)} &= 2db\end{align*} \[\sqrt {(a^2 + b^2)(a^2 + c^2)} + \sqrt {(a^2 + c^2)(a^2 + d^2)} + \sqrt {(a^2 + d^2)(a^2 + b^2)} = 2\]

Compute $ab + ac + ad$.