MATHCOUNTS 2022 Exercises – 1

Posted on November 24, 2021 by kevin
  1. Six cocker spaniels have a total weight of 192 pounds. Five golden retrievers have an
    average weight of 71 pounds. What is the average weight of all 11 dogs? Express your
    answer as a decimal to the nearest tenth.
  2. How many different assortments of pennies, nickels, dimes and quarters can Ashley’s coin holder contain if it has 15 coins total?
  3. Carla selects 5 fruit-flavored candies from a bowl containing 6 apple, 5 banana and 4 cherry candies. How many possible combinations of candies can Carla select?
  4. Lynn has 3 cats and a row of 4 cat beds. Each cat bed can hold one or two cats, and each cat is in a bed. Listing the number of cats in each bed from left to right, how many unique sequences are there?
  5. Sadako has a 6-inch by 8-inch rectangle of paper. She folds it in half from left to right, and then in half from top to bottom, so that the folds are on the top and left of the resulting rectangle, as shown in steps 1 and 2. Then she cuts the paper along a straight diagonal line from the bottom right corner to the top left corner, as shown in step 3. When she unfolds the paper, how many separate pieces are there?
  1. Let $a↑b=a^2+b^2$, what is the value of $[(3↑1)↑2]-[(3↑(1↑2)]$?
  2. A triangle has vertices $A(-1,2)$, $B(5,8)$ and $C(-1,-7)$. Point $D(x,y)$ is on side $BC$, and the area of triangle $ACD$ is half the area of triangle $ABC$. What is the value of $x+y$?
  3. A pair of six-sided dice are rolled. What is the probability that the product of the numbers shown is a multiple of 3? Express your answer as a common fraction.
  4. How many more quadrilaterals than triangles are in the figure shown?
  1. What is the value of the expression $\dfrac{22^2-17^2}{20^2-19^2}$?
  2. The graph of the line $y=6x-5$ intersects the graph of the parabola $y=x^2$ at two points $(x,y)$. What is the distance between those two points? Express your answer in simplest radical form.
  3. A set of double-six dominoes consists of 28 dominoes. Each domino has two ends, with 0 to 6 dots on each end and every possible combination occurring on exactly one domino. For example, there is exactly one of the domino shown, which has 3 dots on one side and one dot on the other. How many total dots are there on all 28 dominoes?
  4. Micah is at a donut shop and wants to choose his own dozen. There are ample glazed, chocolate and jelly donuts. If Micah wants at least two jelly donuts, how many different combinations of donuts can Micah use to create a box of a dozen donuts?
  5. For a school project, Yun records the weather at his house every day for a 20-day period in September. On what percent of the days did it rain? Express your answer to the nearest percent.
  1. Michelle charges \$45 per hour for tutoring. She is saving up to buy a \$14,000 car. How many full hours must she tutor to earn enough money to purchase the car?
  2. A certain burger chain sells, on average, 75 burgers every second. How many burgers are sold each hour? Express your answer in scientific notation to two significant figures.
  3. A specialty candy is packaged in a box holding only 1 candy, in a box holding 5 candies or in a box holding 16 candies. The company receives an order for 238 candies. What is the least number of boxes needed to exactly fill this order?
  4. At Bruno’s shipping warehouse, any item weighing 5 pounds or less costs \$3.50 to ship. For items over 5 pounds, the item costs \$3.50 plus \$1.25 for each additional pound. If Rachel paid \$14.75 to ship a package, how many pounds did it weigh?
  5. The bar graph shows the quiz scores received by the students in Ms. Novak’s Civics class. What was the mean quiz score? Express your answer as a decimal to the nearest hundredth.
  1. The sum of four positive integers is 7. What is the least possible sum of their reciprocals? Express your answer as a mixed number.
  2. The great pyramid of Giza is a pyramid with a square base that is 230 meters on each side, and it is 145 meters tall. It is built mostly of limestone that has a density of 2900 $kg/m^3$. Assume that the entire pyramid is solid. What is the pyramid’s mass, in kilograms? Express your answer in scientific notation to three significant digits.
  3. The three-digit number ABC is 675 more than the three-digit number DEF. If the letters A through F represent distinct positive digits other than 6, 7 or 5, what is the value of ABC?
  4. Roy juggles three balls—one red, one orange and one yellow. He starts by throwing the yellow ball into the air while the red ball is in his left hand and the orange ball is in his right hand. Every second, the yellow ball takes the place of the red ball, the red ball takes the place of the orange ball, and the orange ball takes the place of the yellow ball. How many seconds after starting will he take the red ball in his right hand for the fifth time?
  5. A duck quacks every 3 minutes, a cow moos every 7 minutes, and a horse neighs every 12 minutes. At noon, the animals all sound off together. What is the next time that all three sound off together again?
  6. There are 30 balls, each a different color, and 30 boxes in the same 30 colors. One ball is placed in each box. How many ways can the balls be placed so that exactly 3 of the balls do not match the color of the box they are in?
  7. Nine fair coins are flipped. What is the probability that the total number of coins that land heads up is a multiple of 3? Express your answer as a common fraction.
  8. A certain cube has volume $n\ ft^3$ and surface area $n\ in^2$. If the integer $n$ is written in the form $n=2^b\cdot 3^b$ for integers $a$ and $b$, what is the value of $a+b$?
  9. How many rectangles of any size appear in the figure shown?
  1. In the figure shown, lines AB and CD are parallel, line AB passes B through the center of circle O, AB = 12 cm, and the distance between lines AB and CD is 3 cm. What is the area of the shaded region? Express your answer as a decimal to the nearest tenth.
  1. The numbers 1 through 10 are marked with points on the number line as shown. Two distinct points are chosen at random from these points. What is the expected value of the distance between the two points? Express your answer as a decimal to the nearest tenth.
  1. Madison creates a batch of orange paint by mixing 1 pint of red paint with 1.5 quarts of yellow paint. How many more gallons of yellow paint does she need to add to the mixture to produce a 2-gallon batch of orange paint? Express your answer as a common fraction.
  2. Convex quadrilateral ABCD has AB = 4, BC = 5, AC = 5, AD = 3 and CD = 4. What is the area of quadrilateral ABCD? Express your answer as a decimal to the nearest tenth.
  3. At the end of Day 1, Larry has \$40 and Moe has \$9. Every day, Larry spends \$1.50 while Moe saves \$1.50. What is the first day Moe will end with more money than Larry?
  4. A positive number $n$ is $25\%$ of $\dfrac{6}{5}$ of its square. What is the value of $n$? Express your answer as a decimal to the nearest tenth.

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Storing Cans on Shelf

Posted on November 8, 2021 by kevin

If cans can be placed on top of the one another straight up, how many cylindrical cans $4$ inches in diameter and $6$ inches high can be stored on a shelf $2$ feet wide and $6$ feet long if the shelf is $1$ feet down from the ceiling? Courtney CML Questions Grade 4-6: Problem 362

The standard answer given by CML is $$\dfrac{(2\times 12)\times (6\times 12)}{4\times 4}\times\dfrac{1\times 12}{6}=216$$ That is to store $108$ cans at the bottom, and another $108$ cans on the top.

However, if the cans are stored in the following pattern on the shelf, there will be $$(6\times 10+5\times 10)\times 2=220$$ with $2$ more cans stored on the shelf:

The above packing is more space efficient, as the required length $L$ for placing $n$ columns of cans with diameter as $d$ is $$L=d+(n-1)\cdot\dfrac{\sqrt{3}}{2}\cdot d$$ Given the shelf length as $l$, we have $$n=1+\Big\lfloor \dfrac{2\cdot\sqrt{3}\cdot(l-d)}{3\cdot d}\Big\rfloor$$

If $l=72$ and $d=4$, then $n$=20, with $$L=4+38\sqrt{3}\approx69.816\lt 72$$ leaving more than 2 inches to spare.

If $l=32$ or $l=60$, can you verify that more tighter packing can be done?

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Math Olympiad for 3rd Grade – 6

Posted on October 29, 2021 by kevin
  1. Is the following a square number? $$1!+2!+3!+4!+5!+6!+7!$$
  2. There are $100$ cups labeled from $1, 2, 3, … 100$. You distribute $1$ candy in the first cup, $2$ candies in the second cup, and $3$ candies in the third cup, and so on. The cup labeled with number __________ holds the $2020^{th}$ candy distributed.
  3. There are $10000$ small boxes drawn on a piece of paper as the following and each box is filled in a number as the following. The sum of all $10000$ numbers is __________. $$\begin{array}{|r|r|r|r|r|r|r|r|r|r|c|r|} \hline 1&2&3&4&5&6&7&8&9&10&……&100\\ \hline 2&3&4&5&6&7&8&9&10&11&……&101\\ \hline 3&4&5&6&7&8&9&10&11&12&……&102\\ \hline 4&5&6&7&8&9&10&11&12&13&……&103\\ \hline …&…&&&&&&&&&……&…\\ \hline …&…&&&&&&&&&……&…\\ \hline 99&100&101&102&103&104&105&106&107&108&……&198\\ \hline 100&101&102&103&104&105&106&107&108&109&……&199\\ \hline \end{array} $$
  4. 12 students in Mrs. King’s class took a math exam and their scores are: $86, 82, 71, 88, 90, 78, 83, 81, 85, 76, 87, 77$. The average of their scores is __________.
  5. A group of students voted to select $5$ students to form a committee. The top $5$ students received a total of $168$ votes. Among them, Amy received $2$ more votes than Bob, $5$ more votes than Charlie, $10$ more votes than Danny and $15$ more votes than Emily. The number of votes received by Amy is __________.
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Geometry Challenge – 8 ⭐

Posted on October 16, 2021 by kevin

In unit square $ABCD$, point $E$ and $F$ are located on edge $CD$, with $E$ closer to $D$ and $F$ closer to $C$. Line $BE$ and $AF$ intersect at $G$, forming two triangles: $\triangle{ABG}$ and $\triangle{EFG}$. Find the minimum value of the total area formed by these two triangles. Click here for the solution.

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Math Olympiad for 3rd Grade – 5

Posted on October 15, 2021 by kevin
  1. Amy, Bob, Charlie, David and Emily took part in a math exam. Their average score was 85. After checking his answer sheet against the answer sheet, Amy found that her score was wrong and reported it to her teacher. After the teacher corrected Amy’s score, their average score increased to 88. And Amy’s final score is ________.
  2. Two numbers on Kevin’s report card got smudged and unreadable, one is for English, the other for Math. Kevin’s score for English is ________ and his score for Math is __________. $$ \begin{array}{|c|c|c|c|c|}\hline \texttt{Subject}&English&Math&Science&\texttt{Average} \\ \hline \texttt{Score}&8\blacksquare & \blacksquare 6&75&\texttt{86} \\ \hline \end{array}$$
  3. Adam, Bill, Carl and Dan took a math quiz. The teacher told Adam that the sum of other 3 students’ score is $9$; told Bill that the sum of the other 3’s scores is $9$; told Carl that the sum of other 3’s scores is $11$; and told Dan that the sum of other 3’s scores is $10$. Adam’s score is __________, Bill’s __________, Carl’s __________, and Dan’s __________.
  4. Five students went to a gym to get their weight measured. However, the scale was malfunctioning that it could not measure weight below 100 pounds. So each student paired with another student to weight together on the scale and took a total of 10 measurements in pounds as: $$153, 156, 159, 162, 159, 162, 165, 168, 171$$, What is the weight of each student? __________, __________, __________, __________, __________.
  5. Will has two clocks, one is always accurate and the other would be slower and lose 20 seconds every hour. At 8 AM in the morning, Will reset the slower clock. How many days later will both clocks point to 8 AM in the morning at the same time again?
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