OlympiadBench

2231

Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \ldots$. Turbo successively crawls distances $c_{1}, c_{2}, c_{3}, \ldots$ around the circle, each time choosing to crawl either clockwise or counterclockwise. For example, if the sequence $c_{1}, c_{2}, c_{3}, \ldots$ is $0.4,0.6,0.3, \ldots$, then Turbo may start crawling as follows: <image_1> Determine the largest constant $C>0$ with the following property: for every sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \ldots$ with $c_{i}<C$ for al …

The largest possible $C$ is $C=\frac{1}{2}$. For $0<C \leqslant \frac{1}{2}$, Turbo can simply choose an arbitrary point $P$ (different from its starting point) to avoid. When Turbo is at an arbitrary point $A$ different from $P$, the two arcs $A P$ have total length 1; therefore, the larger of the two the arcs (or either arc in case $A$ is diametrically opposite to $P$ ) must have length $\geqslant \frac{1}{2}$. By always choosing this larger arc (or either arc in case $A$ is diametrically opposite to $P$ ), Turbo will manage to avoid the point $P$ forever. For $C>\frac{1}{2}$, we write $C=\frac{1}{2}+a$ with $a>0$, and we choose the sequence $$ \frac{1}{2}, \quad \frac{1+a}{2}, \quad \f … To show that $C\le \frac12$ We consider the following related problem: We assume instead that the snail Chet is moving left and right on the real line. Find the size $M$ of the smallest (closed) interval, that we cannot force Chet out of, using a sequence of real numbers $d_{i}$ with $0<d_{i}<1$ for all $i$. Then $C=1 / M$. Indeed if for every sequence $c_{1}, c_{2}, \ldots$, with $c_{i}<C$ there exists a point that Turbo can avoid, then the circle can be cut open at the avoided point and mapped to an interval of size $M$ such that Chet can stay inside this interval for any sequence of the from $c_{1} / C, c_{2} / C, \ldots$, see Figure 5 . However, all sequences $d_{1}, d_{2}, \ldots$ wi …

$\frac{1}{2}$

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Multimodal

Competition

false

Numerical

Open-ended

Geometry

Math

English

2237

In the diagram, $\angle A B F=41^{\circ}, \angle C B F=59^{\circ}, D E$ is parallel to $B F$, and $E F=25$. If $A E=E C$, determine the length of $A E$, to 2 decimal places. <image_1>

Let the length of $A E=E C$ be $x$. Then $A F=x-25$. In, $\triangle B C F, \frac{x+25}{B F}=\tan \left(59^{\circ}\right)$. In $\triangle A B F, \frac{x-25}{B F}=\tan \left(41^{\circ}\right)$. Solving for $B F$ in these two equations and equating, $$ B F=\frac{x+25}{\tan 59^{\circ}}=\frac{x-25}{\tan 41^{\circ}} $$ so $\quad\left(\tan 41^{\circ}\right)(x+25)=\left(\tan 59^{\circ}\right)(x-25)$ $$ \begin{aligned} 25\left(\tan 59^{\circ}+\tan 41^{\circ}\right) & =x\left(\tan 59^{\circ}-\tan 41^{\circ}\right) \\ x & =\frac{25\left(\tan 59^{\circ}+\tan 41^{\circ}\right)}{\tan 59^{\circ}-\tan 41^{\circ}} \\ x & \doteq 79.67 . \end{aligned} $$ <img_3420> Therefore the length of $A E$ is 79. …

79.67

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Multimodal

Competition

false

Numerical

1e-1

Open-ended

Geometry

Math

English