Olympiad problems

2023 LMT Fall, 1C

Tags: theme , geo

How many distinct triangles are there with prime side lengths and perimeter $100$? [i]Proposed by Muztaba Syed[/i] [hide=Solution][i]Solution.[/i] $\boxed{0}$ As the perimeter is even, $1$ of the sides must be $2$. Thus, the other $2$ sides are congruent by Triangle Inequality. Thus, for the perimeter to be $100$, both of the other sides must be $49$, but as $49$ is obviously composite, the answer is thus $\boxed{0}$.[/hide]

2024 LMT Fall, B5

Tags: theme

Tnag is repeating the phrase "sigma sigma on the wall'' an infinite number times. Between each word, there is exactly one second of pause. Adam has heard the phrase so many times that he has come up with a game using two numbers $x$ and $y$: Start with a score of 0. [list] [*] At a random time, Adam will hear the word $a$ (each of the 5 words are equally likely to be heard). [*] Then [list] [*] if $a$ is "sigma'', Adam will multiply his score by $x$, and [*] if $a$ is any of the other words, Adam will add $y$ to his score. [/list] [/list] Let $f(x,y)$ be Adam's expected score after infinitely many steps. Find \[ \sum_{n=2}^{\infty}f\left(\frac{1}{n}, \frac{1}{n^2}\right). \]

2020 MOAA, TO2

Tags: algebra , theme

The Den has two deals on chicken wings. The first deal is $4$ chicken wings for $3$ dollars, and the second deal is $11$ chicken wings for $ 8$ dollars. If Jeremy has $18$ dollars, what is the largest number of chicken wings he can buy?