Olympiad problems

2021 AMC 10 Fall, 14

Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6{ }$ numbers obtained. What is the probability that the product is divisible by $4?$ $\textbf{(A)}\: \frac34\qquad\textbf{(B)} \: \frac{57}{64}\qquad\textbf{(C)} \: \frac{59}{64}\qquad\textbf{(D)} \: \frac{187}{192}\qquad\textbf{(E)} \: \frac{63}{64}$

2021 AMC 10 Fall, 9

Tags: AMC , AMC 10 , AMC 10 B

The knights in a certain kingdom come in two colors. $\frac{2}{7}$ of them are red, and the rest are blue. Furthermore, $\frac{1}{6}$ of the knights are magical, and the fraction of red knights who are magical is $2$ times the fraction of blue knights who are magical. What fraction of red knights are magical? $\textbf{(A) }\frac{2}{9}\qquad\textbf{(B) }\frac{3}{13}\qquad\textbf{(C) }\frac{7}{27}\qquad\textbf{(D) }\frac{2}{7}\qquad\textbf{(E) }\frac{1}{3}$

2021 AMC 12/AHSME Fall, 7

Tags: AMC , AMC 10 , AMC 10 B , AMC 12 , AMC 12 B , Important Announcement

Which of the following conditions is sufficient to guarantee that integers $x$, $y$, and $z$ satisfy the equation $$x(x-y)+y(y-z)+z(z-x) = 1?$$$\textbf{(A)}\: x>y$ and $y=z$ $\textbf{(B)}\: x=y-1$ and $y=z-1$ $\textbf{(C)} \: x=z+1$ and $y=x+1$ $\textbf{(D)} \: x=z$ and $y-1=x$ $\textbf{(E)} \: x+y+z=1$

2016 AMC 12/AHSME, 3

Tags: absolute value , AMC , AMC 10 , AMC 10 B , 2016 AMC 10B , 2016 AMC 12B

Let $x=-2016$. What is the value of $\left| \ \bigl \lvert { \ \lvert x\rvert -x }\bigr\rvert -|x|{\frac{}{}}^{}_{}\right|-x$? $\textbf{(A)}\ -2016\qquad\textbf{(B)}\ 0\qquad\textbf{(C)}\ 2016\qquad\textbf{(D)}\ 4032\qquad\textbf{(E)}\ 6048$

2019 AMC 10, 15

Tags: AMC , AMC 10 , AMC 10 B

Two right triangles, $T_1$ and $T_2$, have areas of 1 and 2, respectively. One side length of one triangle is congruent to a different side length in the other, and another side length of the first triangle is congruent to yet another side length in the other. What is the square of the product of the third side lengths of $T_1$ and $T_2$? $\textbf{(A) }\frac{28}3\qquad\textbf{(B) }10\qquad\textbf{(C) }\frac{32}3\qquad\textbf{(D) }\frac{34}3\qquad\textbf{(E) }12$

2017 AMC 10, 16

Tags: AMC , AMC 10 , AMC 10 B , 2017 AMC 10B , counting

How many of the base-ten numerals for the positive integers less than or equal to 2017 contain the digit 0? $\textbf{(A)} \text{ 469} \qquad \textbf{(B)} \text{ 471} \qquad \textbf{(C)} \text{ 475} \qquad \textbf{(D)} \text{ 478} \qquad \textbf{(E)} \text{ 481}$

2017 AMC 10, 23

Tags: AMC , AMC 10 , AMC 10 B , 2017 AMC 10B , 2017 AMC 12B

Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained that is formed by writing the integers from $1$ to $44$ in order, one after the other. What is the remainder when $N$ is divided by $45$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 18 \qquad\textbf{(E)}\ 44$

2019 AMC 10, 25

Tags: AMC , AMC 10 , AMC 10 B , AMC 12 , AMC 12 B , cool solution rusczyk

How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? $\textbf{(A) }55\qquad\textbf{(B) }60\qquad\textbf{(C) }65\qquad\textbf{(D) }70\qquad\textbf{(E) }75$

2015 AMC 12/AHSME, 23

Tags: geometry , AMC , AMC 10 B , AMC 12

A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? $ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $

2017 AMC 10, 6

Tags: AMC , AMC 10 , AMC 10 B , 2017 AMC 10B

What is the largest number of solid $2\text{-in}\times 2\text{-in}\times 1\text{-in}$ blocks that can fit in a $3\text{-in}\times 2\text{-in}\times 3\text{-in}$ box? $\textbf{(A) } 3\qquad \textbf{(B) } 4\qquad \textbf{(C) } 5\qquad \textbf{(D) } 6\qquad \textbf{(E) } 7$

1972 Canada National Olympiad, 1

Tags: trigonometry , AMC , AMC 10 , AMC 10 B , ratio

Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.

2021 AMC 10 Fall, 11

Tags: AMC , AMC 10 , AMC 10 B

A regular hexagon of side length $1{ }$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. What is the area of the region bounded by these $6$ reflected arcs? $(\textbf{A})\: \frac{5\sqrt{3}}{2} - \pi\qquad(\textbf{B}) \: 3\sqrt{3}-\pi\qquad(\textbf{C}) \: 4\sqrt{3}-\frac{3\pi}{2}\qquad(\textbf{D}) \: \pi - \frac{\sqrt{3}}{2}\qquad(\textbf{E}) \: \frac{\pi + \sqrt{3}}{2}$

2018 AMC 10, 3

Tags: AMC , AMC 10 , AMC 10 B , 2018 AMC 10B , 2018 AMC

In the expression $\left(\underline{\qquad}\times\underline{\qquad}\right)+\left(\underline{\qquad}\times\underline{\qquad}\right)$ each blank is to be filled in with one of the digits $1,2,3,$ or $4,$ with each digit being used once. How many different values can be obtained? $ \textbf{(A) }2 \qquad \textbf{(B) }3\qquad \textbf{(C) }4 \qquad \textbf{(D) }6 \qquad \textbf{(E) }24 \qquad $

2023 AMC 10, 22

Tags: AMC , AMC 10 , AMC 10 B

How many distinct values of $x$ satisfy $\lfloor x \rfloor ^2 – 3x + 2 = 0$ where $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$? $\textbf{(A) } \text{an infinite number} \qquad \textbf{(B) } 4 \qquad \textbf{(C) } 2 \qquad \textbf{(D) } 3 \qquad \textbf{(E) } 0$

2019 AMC 12/AHSME, 6

Tags: conics , ellipse , AMC 10 , AMC , AMC 10 B , AMC 12 , AMC 12 B

In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? $\textbf{(A) }0\qquad\textbf{(B) }2\qquad\textbf{(C) }4\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{infinitely many}$

2021 AMC 10 Fall, 23

Tags: AMC 10 , AMC , AMC 10 B

Each of the $5{ }$ sides and the $5{ }$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color? $(\textbf{A})\: \frac23\qquad(\textbf{B}) \: \frac{105}{128}\qquad(\textbf{C}) \: \frac{125}{128}\qquad(\textbf{D}) \: \frac{253}{256}\qquad(\textbf{E}) \: 1$

2013 AMC 10, 23

Tags: AMC 12 B , analytic geometry , geometry , ratio , cyclic quadrilateral , AMC 10 B , similar triangles

In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? ${ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 21\qquad\textbf{(C)}\ 24\qquad\textbf{(D}}\ 27\qquad\textbf{(E)}\ 30 $

2020 AMC 10, 8

Tags: geometry , AMC , AMC 10 , AMC 10 B

Points $P$ and $Q$ lie in a plane with $PQ=8$. How many locations for point $R$ in this plane are there such that the triangle with vertices $P,$ $Q,$ and $R$ is a right triangle with area $12$ square units? $\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) }8 \qquad\textbf{(E) } 12$

2016 AMC 10, 21

Tags: geometry , AMC , AMC 10 , AMC 10 B

What is the area of the region enclosed by the graph of the equation $x^2+y^2=|x|+|y|?$ $\textbf{(A)}\ \pi+\sqrt{2} \qquad\textbf{(B)}\ \pi+2 \qquad\textbf{(C)}\ \pi+2\sqrt{2} \qquad\textbf{(D)}\ 2\pi+\sqrt{2} \qquad\textbf{(E)}\ 2\pi+2\sqrt{2}$

2015 AMC 10, 25

Tags: geometry , AMC , AMC 10 B , AMC 12

A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? $ \textbf{(A) }4\qquad\textbf{(B) }10\qquad\textbf{(C) }12\qquad\textbf{(D) }21\qquad\textbf{(E) }26 $

2021 AMC 12/AHSME Fall, 4

Tags: AMC , AMC 10 , AMC 10 B , AMC 12 , AMC 12 B

Let $n = 8^{2022}$. Which of the following is equal to $\frac{n}{4}$? $\textbf{(A) }4^{1010}\qquad\textbf{(B) }2^{2022}\qquad\textbf{(C) }8^{2018}\qquad\textbf{(D) }4^{3031}\qquad\textbf{(E) }4^{3032}$

2016 AMC 10, 12

Tags: AMC10 , AMC , AMC 10 , AMC 10 B

Two different numbers are selected at random from $( 1, 2, 3, 4, 5)$ and multiplied together. What is the probability that the product is even? $\textbf{(A)}\ 0.2\qquad\textbf{(B)}\ 0.4\qquad\textbf{(C)}\ 0.5\qquad\textbf{(D)}\ 0.7\qquad\textbf{(E)}\ 0.8$

2016 AMC 10, 13

Tags: AMC , AMC 10 , 2016 AMC 10B , AMC 10 B

At Megapolis Hospital one year, multiple-birth statistics were as follows: Sets of twins, triplets, and quadruplets accounted for $1000$ of the babies born. There were four times as many sets of triplets as sets of quadruplets, and there was three times as many sets of twins as sets of triplets. How many of these $1000$ babies were in sets of quadruplets? $\textbf{(A)}\ 25\qquad\textbf{(B)}\ 40\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 100\qquad\textbf{(E)}\ 160$

2016 AMC 10, 16

Tags: AMC , AMC 10 , AMC 10 B , 2015 AMC 8 , geometric series

The sum of an infinite geometric series is a positive number $S$, and the second term in the series is $1$. What is the smallest possible value of $S?$ $\textbf{(A)}\ \frac{1+\sqrt{5}}{2} \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \sqrt{5} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$

2022 AMC 10, 11

Tags: AMC , AMC 10 , AMC 10 B

All the high schools in a large school district are involved in a fundraiser selling T-shirts. Which of the choices below is logically equivalent to the statement “No school bigger than Euclid HS sold more T-shirts than Euclid HS”? $(\textbf{A})$ All schools smaller than Euclid HS sold fewer T-shirts than Euclid HS. $(\textbf{B})$ No school that sold more T-shirts than Euclid HS is bigger than Euclid HS. $(\textbf{C})$ All schools bigger than Euclid HS sold fewer T-shirts than Euclid HS. $(\textbf{D})$ All schools that sold fewer T-shirts than Euclid HS are smaller than Euclid HS. $(\textbf{E})$ All schools smaller than Euclid HS sold more T-shirts than Euclid HS.