Olympiad problems

2021 AMC 10 Spring, 23

Tags: AMC , AMC 10 , AMC 10 A , AMC 12 , AMC 12 A , 2021 AMC 10A , probs

Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around'' and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up'', the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? $\textbf{(A) }\frac{9}{16}\qquad\textbf{(B) }\frac{5}{8}\qquad\textbf{(C) }\frac{3}{4}\qquad\textbf{(D) }\frac{25}{32}\qquad\textbf{(E) }\frac{13}{16}$

2021 AMC 12/AHSME Spring, 1

Tags: AMC , AMC 12 , AMC 12 A

What is the value of $$2^{1+2+3}-(2^1+2^2+2^3)?$$ $\textbf{(A) }0 \qquad \textbf{(B) }50 \qquad \textbf{(C) }52 \qquad \textbf{(D) }54 \qquad \textbf{(E) }57$ Proposed by [b]djmathman[/b]

2020 AMC 12/AHSME, 12

Tags: AMC 12 , AMC 12 A , analytic geometry , rotation , geometry , 2020 AMC 12A , 2020 AMC

Line $\ell$ in the coordinate plane has the equation $3x - 5y + 40 = 0$. This line is rotated $45^{\circ}$ counterclockwise about the point $(20, 20)$ to obtain line $k$. What is the $x$-coordinate of the $x$-intercept of line $k?$ $\textbf{(A) } 10 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 25 \qquad \textbf{(E) } 30$

2018 AMC 12/AHSME, 1

Tags: AMC , AMC 12 , AMC 12 A

A large urn contains $100$ balls, of which $36\%$ are red and the rest are blue. How many of the blue balls must be removed so that the percentage of red balls in the urn will be $72\%?$ (No red balls are to be removed.) $ \textbf{(A) }28 \qquad \textbf{(B) }32 \qquad \textbf{(C) }36 \qquad \textbf{(D) }50 \qquad \textbf{(E) }64 \qquad $

2012 AMC 12/AHSME, 1

Tags: AMC , AMC 12 , AMC 12 A , AMC 10 , AMC 10 A

A bug crawls along a number line, starting at $-2$. It crawls to $-6$, then turns around and crawls to $5$. How many units does the bug crawl altogether? $ \textbf{(A)}\ 9 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 14 \qquad\textbf{(E)}\ 15 $

2016 AMC 12/AHSME, 19

Tags: AMC , AMC 12 , AMC 12 A , probability , number theory , relatively prime , 2016 AMC 12A

Jerry starts at 0 on the real number line. He tosses a fair coin 8 times. When he gets heads, he moves 1 unit in the positive direction; when he gets tails, he moves 1 unit in the negative direction. The probability that he reaches 4 at some time during this process is $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? (For example, he succeeds if his sequence of tosses is $HTHHHHHH$.) $\textbf{(A)}\ 69\qquad\textbf{(B)}\ 151\qquad\textbf{(C)}\ 257\qquad\textbf{(D)}\ 293\qquad\textbf{(E)}\ 313$

2016 AMC 12/AHSME, 5

Tags: AMC , AMC 12 , AMC 12 A , 2016 AMC 12A

Goldbach's conjecture states that every even integer greater than 2 can be written as the sum of two prime numbers (for example, $2016=13+2003$). So far, no one has been able to prove that the conjecture is true, and no one has found a counterexample to show that the conjecture is false. What would a counterexample consist of? $ \textbf{(A)}\ \text{an odd integer greater than } 2 \text{ that can be written as the sum of two prime numbers}$\\ $\textbf{(B)}\ \text{an odd integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$\\ $\textbf{(C)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two numbers that are not prime}$\\ $\textbf{(D)}\ \text{an even integer greater than } 2 \text{ that can be written as the sum of two prime numbers}$\\ $\textbf{(E)}\ \text{an even integer greater than } 2 \text{ that cannot be written as the sum of two prime numbers}$

2017 AMC 12/AHSME, 20

Tags: AMC , AMC 12 , AMC 12 A , 2017 AMC 12A

How many ordered pairs $(a, b)$ such that $a$ is a real positive number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\log_b a)^{2017} = \log_b (a^{2017})$? $ \textbf{(A) \ }198\qquad \textbf{(B) \ } 199 \qquad \textbf{(C) \ } 398 \qquad \textbf{(D) \ }399\qquad \textbf{(E) \ } 597$

2016 AMC 12/AHSME, 21

Tags: geometry , cyclic quadrilateral , AMC , AMC 12 , AMC 12 A

A quadrilateral is inscribed in a circle of radius $200\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? $\textbf{(A) } 200 \qquad\textbf{(B) } 200\sqrt{2} \qquad\textbf{(C) } 200\sqrt{3} \qquad\textbf{(D) } 300\sqrt{2} \qquad\textbf{(E) } 500$

2018 AMC 12/AHSME, 21

Tags: AMC , AMC 12 , AMC 12 A , 2018 AMC 12A

Which of the following polynomials has the greatest real root? $\textbf{(A) } x^{19}+2018x^{11}+1 \qquad \textbf{(B) } x^{17}+2018x^{11}+1 \qquad \textbf{(C) } x^{19}+2018x^{13}+1 \qquad \textbf{(D) } x^{17}+2018x^{13}+1 \qquad \textbf{(E) } 2019x+2018 $