AMC 10 Exercise – 1

Posted on September 20, 2024 by kevin
  1. Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2:1$?
  2. Two right circular cylinders have the same volume. The radius of the second cylinder is $10%$ more than the radius of the first. What is the ratio between the height of the first cylinder and that of the second one?
  3. How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $cd$.
  4. The diagram below shows the circular face of a clock with radius $20$cm, and a circular disk with radius $12$cm externally tangent to the clock face at $12$ o’clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
  5. Consider the set of all fractions $\dfrac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10%$?
  6. If $y+4=(x-2)^2$, $x+4=(y-2)^2$,and $x\ne y$, what is the value of $x^2+y^2$?
  7. A line that passes through the origin intersects both the line $x=1$ and the line $y=1+\dfrac{\sqrt{3}}{3}x$. The three lines create an equilateral triangle. What is the perimeter of the triangle?
  8. Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$as well as the letters $A$ through $F$ to represent $10$ through 15. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$?
  9. A rectangle with positive integer side lengths in $\text{cm}$ has area $A$ ${cm}^2$ and perimeter $P$ $cm$. Which of the following numbers cannot equal to $A+P$? (A) 100 (B) 102 (C) 104 (D) 106 (E) 108
  10. Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\dfrac{12}{5}\sqrt{2}$. What is the volume of the tetrahedron?
  11. Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
  12. The zeroes of the function $f(x)=x^2-ax+2a$are integers. What is the sum of the possible values of $a$?
  13. For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p\lt 2015$ are possible?
  14. Let $S$ be a square of side length $1$. Two points are chosen at random on the sides of $S$ The probability that the straight-line distance between the points is at least $\dfrac{1}{2}$ is $\dfrac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers with $gcd(a,b,c)=1$. What is $a+b+c$?
  15. The town of Hamlet has $3$ people for each horse, $4$ sheep for each cow, and $3$ ducks for each person. Which of the following could not possibly be the total number of people, horses, sheep, cows, and ducks in Hamlet? $(A) 41 (B) 47 (C) 59 (D) 61 (E) 66$
  16. When the centers of the faces of the right rectangular prism, with three side length as $3$, $4$, and $5$, are joined to create an octahedron, what is the volume of the octahedron?
  17. In $\triangle{ABC}$, $\angle{C}=90^\circ$, and $AB=12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$ and $W$lie on a circle. What is the perimeter of the triangle?
  18. Erin the ant starts at a given corner of a cube and crawls along exactly $7$ edges in such a way that she visits every corner exactly once and then finds that she is unable to return along an edge to her starting point. How many paths are there meeting these conditions?
  19. Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
  20. In the figure shown below, $ABCDE$ is a regular pentagon and $AG=1$. What is $FG+JH+CD$?
  21. A rectangular box measures $a\times b\times c$, where $a$,$b$, and $c$ are integers and $1\le a \le b \le c$. The volume and surface area of the box are numerically equal. How many ordered triples $(a, b, c)$ are possible?
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