Olympiad problems

2021 AMC 10 Spring, 7

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

2021 AMC 12/AHSME Spring, 4

Tags: 2021 AMC 10A , 2021 amc 12a

Tom has a collection of $13$ snakes, $4$ of which are purple and $5$ of which are happy. He observes that $\bullet$ all of his happy snakes can add $\bullet$ none of his purple snakes can subtract, and $\bullet$ all of his snakes that can’t subtract also can’t add Which of these conclusions can be drawn about Tom’s snakes? $\textbf{(A)}$ Purple snakes can add. $\textbf{(B)}$ Purple snakes are happy. $\textbf{(C)}$ Snakes that can add are purple. $\textbf{(D)}$ Happy snakes are not purple. $\textbf{(E)}$ Happy snakes can't subtract.

2021 AMC 10 Fall, 20

Tags: 2021 amc 12a , AMC , AMC 12 , AMC 12 A

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions? $\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

2021 AMC 12/AHSME Spring, 24

Tags: geometry , AMC , AMC 12 , 2021 amc 12a , trigonometry

Semicircle $\Gamma$ has diameter $\overline{AB}$ of length $14$. Circle $\Omega$ lies tangent to $\overline{AB}$ at a point $P$ and intersects $\Gamma$ at points $Q$ and $R$. If $QR=3\sqrt3$ and $\angle QPR=60^\circ$, then the area of $\triangle PQR$ is $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. What is $a+b+c$? $\textbf{(A) }110 \qquad \textbf{(B) }114 \qquad \textbf{(C) }118 \qquad \textbf{(D) }122\qquad \textbf{(E) }126$

2021 AMC 12/AHSME Fall, 17

Tags: 2021 amc 12a , AMC , AMC 12 , AMC 12 A

How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions? $\textbf{(A) } 4 \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 8 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 12 \qquad$

2021 AMC 12/AHSME Spring, 19

Tags: AMC , AMC 12 , AMC 12 A , 2021 amc 12a , AUKAAT

How many solutions does the equation $\sin \left( \frac{\pi}2 \cos x\right)=\cos \left( \frac{\pi}2 \sin x\right)$ have in the closed interval $[0,\pi]$? $\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3\qquad \textbf{(E) }4$