AMC 8 Exercises 1 – 02/01/2024

Posted on February 1, 2022 by kevin
  1. The sum of the first three prime number greater than 50 is $$ $$ $\textbf{(A) }169 \qquad\textbf{(B) }171 \qquad\textbf{(C) }173 \qquad\textbf{(D) }175 \qquad \textbf{(E) } 177$ $$ $$
  2. In a regular hexagon, the ratio between the shortest diagonal and the longest diagonal is $$ $$ $\textbf{(A) }\dfrac{1}{3} \qquad\textbf{(B) }\dfrac{1}{2} \qquad\textbf{(C) }\dfrac{1}{\sqrt{3}} \qquad\textbf{(D) }\dfrac{2}{3} \qquad \textbf{(E) } \dfrac{\sqrt{3}}{2}$ $$ $$
  3. In a box there are 5 red balls and 10 white balls. If two balls are taken at the same time, the chance of getting two balls of the same color is $$ $$ $\textbf{(A) }\dfrac{1}{2} \qquad\textbf{(B) }\dfrac{1}{4} \qquad\textbf{(C) }\dfrac{2}{21} \qquad\textbf{(D) }\dfrac{10}{21} \qquad \textbf{(E) } \dfrac{11}{21}$ $$ $$
  4. Let $x=\dfrac{1}{2+\dfrac{1}{2+\dfrac{1}{2}}}$, then $x$ = $$ $$ $\textbf{(A) }\dfrac{2}{9} \qquad\textbf{(B) }\dfrac{5}{12} \qquad\textbf{(C) }\dfrac{4}{9} \qquad\textbf{(D) }\dfrac{9}{4} \qquad \textbf{(E) } \dfrac{12}{5}$ $$ $$
  5. In $\triangle{ABC}$, point $F$ divides $AC$ in the ratio of $1: 2$. Suppose $G$ is the middle point of $BF$ and $E$ is the point of intersection between $BC$ and $AG$. Then point $E$ divides $BC$ in the ratio of $$ $$ $\textbf{(A) }\dfrac{1}{4} \qquad\textbf{(B) }\dfrac{1}{3} \qquad\textbf{(C) }\dfrac{2}{5} \qquad\textbf{(D) }\dfrac{4}{11} \qquad \textbf{(E) } \dfrac{3}{8}$ $$ $$
  6. In a gathering, 28 handshakes were made. Every two people shake hands at most once. The least number of people who attended the gathering was $$ $$ $\textbf{(A) }28 \qquad\textbf{(B) }27 \qquad\textbf{(C) }14 \qquad\textbf{(D) }8 \qquad \textbf{(E) }7 $ $$ $$
  7. Kevin’s salary is 20% more than William’s. After William earns a salary increase, his salary is more than $20%$ of Kevin’s salary. William’s salary increase percentage is $$ $$ $\textbf{(A) }0.44 \qquad\textbf{(B) }20 \qquad\textbf{(C) }44 \qquad\textbf{(D) }144 \qquad \textbf{(E) } \text{Cannot be determined}$ $$ $$
  8. Let $P$ be the set of all points on the $xy$-plane satisfying $|x|+|y|\le4|$. The area of $P$ is $$ $$ $\textbf{(A) }4 \qquad\textbf{(B) }8 \qquad\textbf{(C) }12 \qquad\textbf{(D) }16 \qquad \textbf{(E) } 32$ $$ $$
  9. How many 3-digit natural numbers are there with its value 30 times of the sum of its digits? $$ $$ $\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }5 \qquad \textbf{(E) }10 $ $$ $$
  10. Denote $a\oplus b=a+b+1$, for all integers $a$ and $b$. If $a\oplus p=a$ for all integers $a$, then $p=$ $$ $$ $\textbf{(A) }-1 \qquad\textbf{(B) }0 \qquad\textbf{(C) }1 \qquad\textbf{(D) }-2 \qquad \textbf{(E) } \text{No solution}$ $$ $$
  11. Ten ice hockey teams participate in a tournament. Each team meets every other team once. The winner of each match gets a 3 point, while the loser gets a score of 0. For a match that ends in a draw, both teams score 1 each. At the end of the tournament, the total score for all teams is 124. The number of matches that end in a draw is $$ $$ $\textbf{(A) }8 \qquad\textbf{(B) }9 \qquad\textbf{(C) }10 \qquad\textbf{(D) }11 \qquad \textbf{(E) } 12 $ $$ $$
  12. Each $dong$ is a $ding$. Some $dung$ are $dong$. The following statements are made $ding$, $dong$, and $dung$:
    • X: There is a $dong$ that is also a $ding$, at the same time a $dung$
    • Y: Some $ding$ are also $dung$
    • Z: There is a $dong$ that is not a $dung$ $$ $$ $\textbf{(A) }\text{Only X is correct} \qquad\textbf{(B) }\text{Only Y is correct} \qquad\textbf{(C) } \text{Only Z is correct} $ $\qquad\textbf{(D) }\text{X and Y are both correct} \qquad \textbf{(E) } \text{X, Y and Z are all wrong}\ \ \ \ \ \ \ \ \ $ $$ $$
  13. $x$ and $y$ are positive integers that satisfy $3x+5y=501$. The number of solution pair $(x,y)$ is $$ $$ $\textbf{(A) }33 \qquad\textbf{(B) }34 \qquad\textbf{(C) } 35 \qquad\textbf{(D) }36 \qquad \textbf{(E) }37 $ $$ $$
  14. The sum of connective integers no greater than $50$ is $1139$. What is the value of the smallest integer? $$ $$ $\textbf{(A) }-17 \qquad\textbf{(B) }-16 \qquad\textbf{(C) } 16 \qquad\textbf{(D) }17 \qquad \textbf{(E) }\text{None of the above values} $ $$ $$
  15. Among the five girls, Amy, Betty, Cathy, Debbie, and Emily, two wore skirts and three wore jeans. Amy and Cathy wore the same type of clothing. Betty and Cathy’s clothes were different, so were Betty and Debbie. The two girls wearing skirts were $$ $$ $\textbf{(A) }\text{Amy and Betty}\qquad\textbf{(B) }\text{Betty and Debbie} \qquad\textbf{(C) }\text{Carol and Emily}$ $ \qquad\textbf{(D) }\text{Amy and Carol} \qquad \textbf{(E) }\text{Betty and Emily} \qquad $ $$ $$
  16. Sequence 2, 3, 5, 6, 7, 10, 11, … consists of natural numbers which are not squares nor cubes. The $250^{th}$ number in the sequence is $$ $$ $\textbf{(A) }268 \qquad\textbf{(B) }269 \qquad\textbf{(C) }270 \qquad\textbf{(D) }271 \qquad \textbf{(E) }\text{None of the above} $ $$ $$
  17. If $f(x\cdot y)=f(x+y)$ and $f(7)=7$, then $f(49)=$ $$ $$ $\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }7 \qquad\textbf{(D) }8 \qquad \textbf{(E) }9 $ $$ $$
  18. In an arithmetic sequence, the value of the $25^{th}$ term is three times that of the $5^{th}$ term. If the value of the $n^{th}$ term is twice the value of the first term, then $n=$ $$ $$ $\textbf{(A) }5 \qquad\textbf{(B) }7 \qquad\textbf{(C) }9 \qquad\textbf{(D) }11 \qquad \textbf{(E) }13 $ $$ $$
  19. David buys a pencil every 5 days, while Andrew buys a pencil every 8 days. Yesterday David bought a pencil. Andrew bought a pencil today. How many days later will both of them buy a pencil on the same day? $$ $$ $\textbf{(A) }20\qquad\textbf{(B) }24 \qquad\textbf{(C) }25 \qquad\textbf{(D) }27 \qquad \textbf{(E) }30 $ $$ $$
  20. How many 4-digit integers are there that the difference between its value and the sum of the digits is 2016?$$ $$ $\textbf{(A) }8 \qquad\textbf{(B) }10 \qquad\textbf{(C) }12 \qquad\textbf{(D) }14 \qquad \textbf{(E) }16 $ $$ $$
  21. $\triangle{ABC}$ is an obtuse triangle, with $\angle{ACB}>90^\circ$. Point $M$ is the midpoint of $AB$. Through $C$, a perpendicular line is drawn on $BC$ that intersects $AB$ at point E. From $M$, draw the line perpendicular to $BC$ and intersecting $BC$ at $D$. If the area of $\triangle{ABC}$ is 54, then the area of $\triangle{BED}$ is $$ $$ $\textbf{(A) }15 \qquad\textbf{(B) }18 \qquad\textbf{(C) }24 \qquad\textbf{(D) }27 \qquad \textbf{(E) }30 $ $$ $$
  22. In trapezoid $ABCD$, $AB$ is parallel to $DC$ and the ratio of the area of $\triangle{ABC}$ to the area of $\triangle{ACD}$ is $\dfrac{1}{3}$. If $E$ and $F$ are the midpoints of $BC$ and $DA$, then the ratio of the area of $ABEF$ to the area of $EFDC$ is $$ $$ $\textbf{(A) }\dfrac{1}{3} \qquad\textbf{(B) }\dfrac{3}{5} \qquad\textbf{(C) }1 \qquad\textbf{(D) }\dfrac{5}{3} \qquad \textbf{(E) }3 $ $$ $$
  23. The unit cube has its bottom face as $ABCD$, and the top face as $EFGH$, the front face as $ABFE$. The cube is cut by the plane passing through diagnoal $HF$, forming an angle of $30^\circ$ to diagonal $EG$, and intersecting edge $AE$ at $P$. The ength of $AP$ is $$ $$ $\textbf{(A) }\dfrac{\sqrt{3}}{3} \qquad\textbf{(B) }\dfrac{\sqrt{6}}{4} \qquad\textbf{(C) }\dfrac{3}{5} \qquad\textbf{(D) }1-\dfrac{\sqrt6}{6} \qquad \textbf{(E) }\dfrac{\sqrt{6}-\sqrt{2}}{2}$ $$ $$
  24. In unit square $ABCD$, $\triangle{ABE}$ is equilateral and $E$ is inside the square. Draw diagonal $BD$, intersecting $AE$ at $F$. The area of $\triangle{BEF}$ is $$ $$ $\textbf{(A) }\dfrac{1}{8} \qquad\textbf{(B) }\dfrac{2\sqrt{3}-\sqrt{6}}{8} \qquad\textbf{(C) }\dfrac{\sqrt{3}}{12} \qquad\textbf{(D) }\dfrac{2\sqrt{3}-3}{4} \qquad \textbf{(E) }\dfrac{\sqrt{3}-\sqrt{2}}{3}$ $$ $$
  25. In Banana Republic, the license plate of cars must be a 4-digit number with the sum of the digit as an even number, such as 1234, but not 1235. How many cars can be registered? $$ $$ $\textbf{(A) }600 \qquad\textbf{(B) }1800 \qquad\textbf{(C) }2000 \qquad\textbf{(D) }4500 \qquad \textbf{(E) }5000$ $$ $$
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