AMC 8 Exercises 2 – 02/07/2022

Posted on February 7, 2022 by kevin

  1. From a point $P$ inside a unit equilateral $\triangle{ABC}$, draw line $PQ$, $PS$, and $PT$ pendicular to $AB$, $BC$, and $CA$ respectively. The total length of $PQ$, $PS$, and $PT$ is $$ $$ $\textbf{(A) } \dfrac{1}{2} \qquad\textbf{(B) } \dfrac{\sqrt{6}}{4} \qquad\textbf{(C) } \dfrac{\sqrt{3}}{3} \qquad\textbf{(D) } \dfrac{\sqrt{3}}{2} \qquad\textbf{(E) } \dfrac{2}{3}$ $$ $$
  2. A full circle $R$ is inscribed inside a quarter-circle $Q$. The area ratio of circle $R$ to $Q$ is $$ $$ $\textbf{(A) } \sqrt{2}-1 \qquad\textbf{(B) } 2\sqrt{2}-2 \qquad\textbf{(C) } \dfrac{\sqrt{2}-1}{2} \qquad\textbf{(D) } 3\sqrt{2}-3 \qquad\textbf{(E) } 3-2\sqrt{2}$ $$ $$
  3. For unit square $ABCD$, side $AB$ and $AD$ are folded along the diagonal $AC$, forming a rhombus. The area of the rhombus is $$ $$ $\textbf{(A) } \sqrt{2}-1 \qquad\textbf{(B) } 2\sqrt{2}-2 \qquad\textbf{(C) } 2-\sqrt{2} \qquad\textbf{(D) } \dfrac{\sqrt{2}}{2} \qquad\textbf{(E) } \dfrac{1}{2}$ $$ $$
  4. 3 congrunt circles are inscribed inside a unit circle, tangent with each other and the unit circle. The radius of the circles is $$ $$ $\textbf{(A) } \dfrac{2\sqrt{3}-3}{2} \qquad\textbf{(B) } \dfrac{\sqrt{3}}{8} \qquad\textbf{(C) } \dfrac{3\sqrt{3}-4}{4} \qquad\textbf{(D) } \dfrac{1}{3} \qquad\textbf{(E) } \dfrac{3\sqrt{3}-3}{8}$ $$ $$
  5. A square is inscribed inside a right triangle which is also isosceles. The four corners of the square are on the side of the triangle. If the area of the right triangle is 1, the area sum of the largest and smallest squares is $$ $$ $\textbf{(A) } \dfrac{8}{9} \qquad\textbf{(B) } \dfrac{17}{18} \qquad\textbf{(C) } 1 \qquad\textbf{(D) } \dfrac{19}{18} \qquad\textbf{(E) } \dfrac{10}{9}$ $$ $$
  6. The sum of all $x$ values satisfying $|x+2|+|3x|=14$ is $$ $$ $\textbf{(A) } -7 \qquad\textbf{(B) } -1 \qquad\textbf{(C) } 0 \qquad\textbf{(D) } 1 \qquad\textbf{(E) } 7$ $$ $$
  7. Ant $Z$ can only move diagonally on a Descartes plane, with each step as $\sqrt{2}$ unit long. After each move, it changes direction by turning $90^\circ$ counter-clock wise, if the next stop has not been touched before; otherwise it keeps the same direction for the next move. Its first step is to move from $(0, 0)$ to $(1,1)$. After 2020 steps, it moves to $(x,y)$. The value of $x+y$ is $$ $$ $\textbf{(A) } 34 \qquad\textbf{(B) } 35 \qquad\textbf{(C) } 36 \qquad\textbf{(D) } 37 \qquad\textbf{(E) } \text{None of the above values}$ $$ $$
  8. $\boxed{a}-\dfrac{\boxed{b}}{\boxed{c}}\times\boxed{d}$ Four single digit number 3, 5, 7, and 8 are used to replace $a$, $b$, $c$, and $d$, and every number must be used. The difference between the maximum possible value and the minimum possible value is $$ $$ $\textbf{(A) } 18\dfrac{19}{24}\qquad\textbf{(B) } 13\dfrac{13}{40} \qquad\textbf{(C) } 14\dfrac{2}{35} \qquad\textbf{(D) } 19\dfrac{7}{15} \qquad\textbf{(E) } 19\dfrac{11}{21}$ $$ $$
  9. In $\triangle{ABC}$, $D$ is a point on $AC$, and $E$ is a point on $AB$. Point $O$ is the intersection of line $BD$ and $CE$. If the area of $\triangle{BOC}$, $\triangle{BOE}$ and $\triangle{COD}$ are $10$, $5$ and $8$ respectively, the area of quadrilateral $ADOE$ is $$ $$ $\textbf{(A) } 16 \qquad\textbf{(B) } 18 \qquad\textbf{(C) } 20 \qquad\textbf{(D) } 22 \qquad\textbf{(E) } 24$ $$ $$
  10. Jack bought two used cars for resale. He made $30$ percent profit on the first car, but suffered a loss of $20$ percent for the second car. Both cars are sold with the same price. Jack’s overall percentage of profit or loss is $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } \dfrac{10}{21} \qquad\textbf{(C) } -1\dfrac{20}{21} \qquad\textbf{(D) } -1\dfrac{1}{21} \qquad\textbf{(E) } \text{None of the above values}$ $$ $$
  11. Kevin is walking to the park to meet his friends. After walking 5 minutes, he slows down his pace for 20% and spends another 5 minutes to reach the park. The ratio of the time spent on the first half of the trip against that of the second half is: $$ $$ $\textbf{(A) } \dfrac{2}{3} \qquad\textbf{(B) } \dfrac{3}{4} \qquad\textbf{(C) } \dfrac{5}{6} \qquad\textbf{(D) } \dfrac{9}{11} \qquad\textbf{(E) } \dfrac{6}{7}$ $$ $$
  12. William was walking to the school from home. He walked the first 2 minutes at a speed of 150 feet per minutes. Then he realized that at this speed, he would be 3 minutes late to be in the school at 8:00 AM. So he picked up his pace at 200 feet per minute and arrived at the school with 2 minutes to spare. The distance William walked from home to the school is closest to: $$ $$ $\textbf{(A) } 0.4\ mile \qquad\textbf{(B) } 0.5\ mile \qquad\textbf{(C) } 0.6\ mile \qquad\textbf{(D) } 0.7\ mile \qquad\textbf{(E) } 0.8\ mile $ $$ $$
  13. The unit cube has its bottom face as $ABCD$, and the top face as $EFGH$, the front face as $ABFE$. $P$ and $Q$ are midpoints of $AE$ and $CG$. The area of quadlateral $PFQD$ is $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } \dfrac{\sqrt{5}}{2} \qquad\textbf{(C) } \dfrac{\sqrt{6}}{2} \qquad\textbf{(D) } \dfrac{\sqrt{3}}{3} \qquad\textbf{(E) } \dfrac{\sqrt{2}}{2}$ $$ $$
  14. The minimum number of toothpicks required to construct 26 equilateral triangles is: $$ $$ $\textbf{(A) } 24 \qquad\textbf{(B) } 30 \qquad\textbf{(C) } 39 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 53 $ $$ $$
  15. The maximum score of a math exam is 100. The average score of 6 students is 91, and each of them got a different score. One of students got 65. The minimum score of the student who ranks 3rd is $$ $$ $\textbf{(A) } 91 \qquad\textbf{(B) } 92 \qquad\textbf{(C) } 93 \qquad\textbf{(D) } 94 \qquad\textbf{(E) } 95$ $$ $$
  16. In the following arrangement, all the odd numbers were placed in such a way that in the $n^{th}$ row there are $n$ consecutive odd numbers:
     1
     3  5
     7  9 11
    13 15 17 19
    21 23 25 27 29
    .. .. .. .. ..
    The first number of $100^{th}$ row is $$ $$ $\textbf{(A) } 8,997 \qquad\textbf{(B) } 8,999\qquad\textbf{(C) } 9,901 \qquad\textbf{(D) } 9,999 \qquad\textbf{(E) } 10,001$ $$ $$
  17. If $x+\dfrac{1}{y}=1$ and $y+\dfrac{1}{z}=1$, then the value of $z+\dfrac{1}{x}$ is $$ $$ $\textbf{(A) } -1 \qquad\textbf{(B) } 0 \qquad\textbf{(C) } 1 \qquad\textbf{(D) } 2 \qquad\textbf{(E) } \text{None of the above values}$ $$ $$
  18. Alan and Bob travel with Chuck from City A to City B which is 72 miles away. Because Chuck’s motorcycle can only carry one passenger, so Alan first rides with Chuck while Bob walks toward City B. Chuck drops Alan before reaching to City B, and turns back to pick up Bob, while Alan continues to walk toward City B. In the end, Alan, Bob and Chuck arrive at City B at the same time. Assume the walking speed of Alan and Bob is 4 miles per hour, and the speed of the motorcycle is 24 miles per hour, the number of miles does Alan ride with Chuck is $$ $$ $\textbf{(A) } 40 \qquad\textbf{(B) } 48 \qquad\textbf{(C) } 50 \qquad\textbf{(D) } 56 \qquad\textbf{(E) } 60$ $$ $$
  19. A triangle has its side lengths as positive integers. The length of one side, which is not the longest one, is 4. The number of unique triangles fitting the description is $$ $$ $\textbf{(A) } 6 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qquad\textbf{(E) } 10$ $$ $$
  20. The remainder of $2^{1000}$ divided by $13$ is $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 7$ $$ $$
  21. An arithmetic sequence consists of consuctive natural numbers. The sum of the sequence is 200. The number of different sequences meeting the description is $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } 0 $ $$ $$
  22. 27 regular dices are used to form a $3\times 3\times 3$ cube. The minimum value of the sum of all numbers exposed on the surface of the cube $$ $$ $\textbf{(A) } 78 \qquad\textbf{(B) } 81 \qquad\textbf{(C) } 90 \qquad\textbf{(D) } 108 \qquad\textbf{(E) } 120 $ $$ $$
  23. Let $\pi=3.1415926535…$, and $f(n)$ is the value of $n^{th}$ digit of the decimal part of $\pi$. The value of $f(f(f(f(f(f(f(f(f(f(10))))))))))$ is equal to $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } 3 \qquad\textbf{(C) } 4 \qquad\textbf{(D) } 5 \qquad\textbf{(E) } 6 $ $$ $$
  24. Andy, Ben, Charlie, Dan and Eddy are playing a dot-and-cat game. Each kid can be a dog or a cat, but not both. Dogs are always honest, while cats are always lying.
    • Andy said that Ben is a dog.
    • Charlie said that Dan is a cat.
    • Eddy said that Andy is not a cat.
    • Ben said that Charlie is not a dog.
    • Dan said that Eddy and Andy are different animals.

      The number of cats in this game is $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{Cannot be determined}$ $$ $$

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