MathFun4Kids

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 $$ $$ $\textbf{(A) } 1 \qquad\textbf{(B) } 2 \qquad\textbf{(C) } 3 \qquad\textbf{(D) } 4 \qquad\textbf{(E) } \text{Cannot be determined}$ $$ $$

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? $$ $$ $\textbf{(A) }600 \qquad\textbf{(B) }1800 \qquad\textbf{(C) }2000 \qquad\textbf{(D) }4500 \qquad \textbf{(E) }5000$ $$ $$

1. Given $n>1$, what is the smallest positive integer $n$ whose positive divisors have a product of $n^6$?
2. What is the largest integer value of $n$ for which $8^n$ evenly divides $100!$?
3. A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a prime number. How many $3$-primable positive integers are there that are less than 1000?
4. How many whole numbers n, such that $100\le n \le1000$, have the same number of odd factors as even factors?
5. A right triangle has a hypotenuse of $10m$ and a perimeter of $22m$. In square meters, what is the area of the triangle?
6. Circle $O$ has radius $10$ units. Point $P$ is on radius $OQ$ and $OP=6$ units. How many chords containing $P$, including the diameter, have integer lengths?
7. What is the total surface area of the largest regular tetrahedron that can be inscribed inside of a cube of edge length $1 cm$. Express your answer in simplest radical form.
8. The diameter, in inches, of a sphere with twice the volume of a spbere of radius 9 inches can be expressed in the form of $a\sqrt[3]{b}$, where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.
9. A circular garden is surrounded by a sidewalk with a uniform width of $25$ foot. The total area of the sidewalk equals the total area of the garden. How many feet are in the diameter of the garden? Round your answer to the nearest whole number.
10. Pedro stood at the center of a circular field that had a radius of 120 feet. He walked due north halfway to the circle. He then turned and walked due east halfway to the circle. He turned again and walked due south hallway to the circle. Finally he turned and walked due west halfway to the circle. When he stopped, how many feet was Pedro from the center of the circle? Express your answer to the nearest foot.

Answers