Olympiad problems

2024 AMC 12/AHSME, 13

There are real numbers $x,y,h$ and $k$ that satisfy the system of equations $$x^2 + y^2 - 6x - 8y = h$$ $$x^2 + y^2 - 10x + 4y = k$$ What is the minimum possible value of $h+k$? $ \textbf{(A) }-54 \qquad \textbf{(B) }-46 \qquad \textbf{(C) }-34 \qquad \textbf{(D) }-16 \qquad \textbf{(E) }16 \qquad $

2024 AMC 10, 3

Tags: AMC , AMC 10 , AMC 12 , AMC 12 B , 2024 AMC , 2024 AMC 10b , 2024 AMC 12B

For how many integer values of $x$ is $|2x|\leq 7\pi?$ $\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

2024 AMC 12/AHSME, 6

Tags: AMC , AMC 12 , AMC 12 B , 2024 AMC , 2024 AMC 12B

The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.) $ \textbf{(A) }18 \qquad \textbf{(B) }20 \qquad \textbf{(C) }22 \qquad \textbf{(D) }24 \qquad \textbf{(E) }26 \qquad $

2024 AMC 12/AHSME, 3

Tags: AMC , AMC 10 , AMC 12 , AMC 12 B , 2024 AMC , 2024 AMC 10b , 2024 AMC 12B

For how many integer values of $x$ is $|2x|\leq 7\pi?$ $\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

2024 AMC 12/AHSME, 12

Tags: complex numbers , geometry , AMC , AMC 12 , AMC 12 B , 2024 AMC , 2024 AMC 12B

Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$? $\textbf{(A)}~\displaystyle\frac{3}{4}\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\displaystyle\frac{4}{3}\qquad\textbf{(D)}~\displaystyle\frac{3}{2}\qquad\textbf{(E)}~\displaystyle\frac{5}{3}$

2024 AMC 12/AHSME, 8

Tags: AMC , AMC 12 , AMC 12 B , 2024 AMC , 2024 AMC 12B , logarithms

What value of $x$ satisfies \[\frac{\log_2x\cdot\log_3x}{\log_2x+\log_3x}=2?\] $ \textbf{(A) }25\qquad \textbf{(B) }32\qquad \textbf{(C) }36\qquad \textbf{(D) }42\qquad \textbf{(E) }48\qquad $