Olympiad problems

2014 Contests, 3

Tags: MATHCOUNTS , geometry , perimeter , ratio , 3D geometry , sphere , Alcumus

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

2021 AMC 10 Spring, 12

Tags: FTW , AMC , AMC 10 , AMC 10 B , Alcumus

Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? $\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$

2014 NIMO Problems, 3

Tags: MATHCOUNTS , geometry , perimeter , ratio , 3D geometry , sphere , Alcumus

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

2021 AMC 12/AHSME Spring, 7

Tags: FTW , AMC , AMC 10 , AMC 10 B , Alcumus

Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? $\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$

2014 NIMO Summer Contest, 3

Tags: MATHCOUNTS , geometry , perimeter , ratio , 3D geometry , sphere , Alcumus

A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square? [i]Proposed by Evan Chen[/i]

2003 AMC 10, 15

Tags: probability , geometry , rectangle , Alcumus , videos , floor function

What is the probability that an integer in the set $ \{1,2,3,\ldots,100\}$ is divisible by $ 2$ and not divisible by $ 3$? $ \textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{33}{100} \qquad \textbf{(C)}\ \frac{17}{50} \qquad \textbf{(D)}\ \frac{1}{2} \qquad \textbf{(E)}\ \frac{18}{25}$