NRP Math Challenge Club Level Two Countdown Round

Question 1 A potato synthesizing firm hires workers to make potatoes. If 5 workers can complete an order of potatoes in 7 days, what is the minimum number of whole days necessary for 3 workers to complete the same order?

Question 1 A potato synthesizing firm hires workers to make potatoes. If 5 workers can complete an order of potatoes in 7 days, what is the minimum number of whole days necessary for 3 workers to complete the same order? Answer: 12 days

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Question 2 If 𝑥 cubed plus 3 equals 11, what is the value of 7𝑥 + 2𝑥 − 8?

Question 2 If 𝑥 cubed plus 3 equals 11, what is the value of 7𝑥 + 2𝑥 − 8? Answer : 10

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Question 3 How long does it take for the hour hand of a clock to move 15 degrees in minutes?

Question 3 How long does it take for the hour hand of a clock to move 15 degrees in minutes? Answer : 30 minutes

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Question 4 What is the sum of prime factors of 2015?

Question 4 What is the sum of prime factors of 2015? Answer : 49

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Question 5 Joe and Phil’s math teacher assigned them a project to do. If Joe can do it by himself in 6 hours, and Phil can do it by himself in 4 hours, then if they split the work 50-50, and start working on it at the same time, how long will it take them to finish?

Question 5 Joe and Phil’s math teacher assigned them a project to do. If Joe can do it by himself in 6 hours, and Phil can do it by himself in 4 hours, then if they split the work 50-50, and start working on it at the same time, how long will it take them to finish? Answer : 3 hours

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Question 6 What percent of number less than 100 are divisible by 6?

Question 6 What percent of number less than 100 are divisible by 6? Answer : 16%

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Question 7 What is the product of the least common multiple and the greatest common factor of 25 and 78?

Question 7 What is the product of the least common multiple and the greatest common factor of 25 and 78? Answer : 1,950

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Question 8 Evaluate: 1 + 3 + 5 … + 17 + 19 − 2 − 4 − 6 … − 18 – 20.

Question 8 Evaluate: 1 + 3 + 5 … + 17 + 19 − 2 − 4 − 6 … − 18 – 20. Answer : -10

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Question 9 The ratio of 10 to what number is equivalent to 20% of 250?

Question 9 The ratio of 10 to what number is equivalent to 20% of 250? Answer : 1 5

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Question 10 25 two-legged monsters and four-legged monsters are trapped in a cage. If there is a total of 68 legs in the cage, how many four-legged monsters are there?

Question 10 25 two-legged monsters and four-legged monsters are trapped in a cage. If there is a total of 68 legs in the cage, how many four-legged monsters are there? Answer : 9 four-legged monsters

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Question 11 If two numbers are relatively prime, what is their greatest common factor?

Question 11 If two numbers are relatively prime, what is their greatest common factor? Answer : 1

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Question 12 What is the height of an equilateral triangle with a side length of 4?

Question 12 What is the height of an equilateral triangle with a side length of 4? Answer : 2 3

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Question 13 Matthew and Irving start running from the same point around a circular track in opposite directions. Matthew runs at a speed of 11 meters per second and Irving at a speed of 13 meters per second. If the length of the track is 144 meters, after how many seconds will Matthew and Irving meet?

Question 13 Matthew and Irving start running from the same point around a circular track in opposite directions. Matthew runs at a speed of 11 meters per second and Irving at a speed of 13 meters per second. If the length of the track is 144 meters, after how many seconds will Matthew and Irving meet? Answer : 6 seconds

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Question 14 The positive square root of 𝑛 is 4. What is 𝑛2 ?

Question 14 The positive square root of 𝑛 is 4. What is 𝑛2 ? Answer : 256

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Question 15 The sum of 3 consecutive integers is 99. What is the median?

Question 15 The sum of 3 consecutive integers is 99. What is the median? Answer : 33

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Question 16 12! is divisible by 10n , where 𝑛 is a positive integer. Find the maximum value of 𝑛.

Question 16 12! is divisible by 10n , where 𝑛 is a positive integer. Find the maximum value of 𝑛. Answer : 2

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Question 17 𝑋 has a remainder of 1 when divided by 3. If 𝑋 is less than 100, how many such possible positive integers are exist?

Question 17 𝑋 has a remainder of 1 when divided by 3. If 𝑋 is less than 100, how many such possible positive integers are exist? Answer : 33 integers

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Question 18 What is the sum of the units digit of 20142014 and the units digit of 55 × 55 ?

Question 18 What is the sum of the units digit of 20142014 and the units digit of 55 × 55 ? Answer : 11

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Question 19 How many positive factors does 37 have?

Question 19 How many positive factors does 37 have? Answer : 8 factors

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Question 20 Matthew walks 12 miles east and 4 miles south. He then walks 3 miles east and 10 miles north. How far is Matthew now located from his starting location?

Question 20 Matthew walks 12 miles east and 4 miles south. He then walks 3 miles east and 10 miles north. How far is Matthew now located from his starting location? Answer : 3 29 miles

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Question 21 Complete the statement: 29 is the ___th prime number .

Question 21 Complete the statement: 29 is the ___th prime number . Answer : 10th

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Question 22 Let 𝑚 = 200 and 𝑛 = 1 (((𝑚 2 ) 0 ) 6 ) −5 What is 1000 − 𝑛?

Question 22 Let 𝑚 = 200 and 𝑛 = 1 (((𝑚 2 ) 0 ) 6 ) −5 What is 1000 − 𝑛? Answer : 999

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Question 23 Find the sum of the first 20 positive odd integers.

Question 23 Find the sum of the first 20 positive odd integers. Answer : 400

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Question 24 If 52 + 𝑥2 = 132 , 72 + 𝑦2 = 252 , and 32 + 𝑧2= 52 , and 𝑥, 𝑦, and 𝑧 are positive, what is 𝑥 + 𝑦 + 𝑧?

Question 24 If 52 + 𝑥2 = 132 , 72 + 𝑦2 = 252 , and 32 + 𝑧2= 52 , and 𝑥, 𝑦, and 𝑧 are positive, what is 𝑥 + 𝑦 + 𝑧? Answer : 40

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Question 25 In rhombus 𝐴𝐵𝐶𝐷, diagonal 𝐴𝐶 = 5 and 𝐶𝐷 = 6. What is area of the rhombus?

Question 25 In rhombus 𝐴𝐵𝐶𝐷, diagonal 𝐴𝐶 = 5 and 𝐶𝐷 = 6. What is area of the rhombus? Answer : 15

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Question 26 Chris jogs at 6 miles per hour to get to school on time, then runs home along the same path at 8 miles per hour to get the homework he forgot. He then sprints at 9 miles per hour back to school along the same path to avoid being late. What is the average speed, in miles per hour, of Chris’s round-trip?

Question 26 Chris jogs at 6 miles per hour to get to school on time, then runs home along the same path at 8 miles per hour to get the homework he forgot. He then sprints at 9 miles per hour back to school along the same path to avoid being late. What is the average speed, in miles per hour, of Chris’s round-trip? Answer : 216 29 miles/hour

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Question 27 What is the smallest number greater than 1 that is both a perfect cube and a square of a perfect square?

Question 27 What is the smallest number greater than 1 that is both a perfect cube and a square of a perfect square? Answer : 4096

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Question 28 How many palindromes exist between 10 and 200?

Question 28 How many palindromes exist between 10 and 200? Answer : 19

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Question 29 How many composite numbers less than 40 exist?

Question 29 How many composite numbers less than 40 exist? Answer : 26 numbers

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Question 30 400 people are shaking hands with each other. If only one handshake can occur per two participants, what is the total number of handshakes that can occur?

Question 30 400 people are shaking hands with each other. If only one handshake can occur per two participants, what is the total number of handshakes that can occur? Answer : 79,800 handshakes

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Question 31 Matthew the Magician is entertaining some students after the competition. For one of his tricks, he asks Cathy to select a number between 1 and 100. He then asks her to add 7 to her number, multiply the result by 2, subtract 4, divide by 2, and then subtract the original number. What should Matthew say is her final result?

Question 31 Matthew the Magician is entertaining some students after the competition. For one of his tricks, he asks Cathy to select a number between 1 and 100. He then asks her to add 7 to her number, multiply the result by 2, subtract 4, divide by 2, and then subtract the original number. What should Matthew say is her final result? Answer : 5

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Question 32 Find the remainder when the result of the following operation is divided by 5: 2014 × 2014 × 2014 − (2013 × 2013 × 2013).

Question 32 Find the remainder when the result of the following operation is divided by 5: 2014 × 2014 × 2014 − (2013 × 2013 × 2013). Answer : 2

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Question 33 Find the 10th term of the sequence of which the initial term is 1 and the difference between the reciprocals of two consecutive terms is 2.

Question 33 Find the 10th term of the sequence of which the initial term is 1 and the difference between the reciprocals of two consecutive terms is 2. Answer : 1 19

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Question 34 Let E be the intersection of the diagonals of square ABCD. How many triangles of any size are in the resulting figure? ABE is one such triangle.

Question 34 Let E be the intersection of the diagonals of square ABCD. How many triangles of any size are in the resulting figure? ABE is one such triangle. Answer : 8 triangles

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Question 35 Twice the positive integer n is a perfect cube. If n is less than 5, what is the cube root of 2n?

Question 35 Twice the positive integer n is a perfect cube. If n is less than 5, what is the cube root of 2n? Answer : 2

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Question 36 In the land of BLAB, 4 BLARBs and 2 BLALBs cost $620 and 3 BLARBs and 1 BLALB cost $440. How much does 1 BLARB cost?

Question 36 In the land of BLAB, 4 BLARBs and 2 BLALBs cost $620 and 3 BLARBs and 1 BLALB cost $440. How much does 1 BLARB cost? Answer : $130

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Question 37 If it takes 3 people of equal strength to move a cart that weighs 0.4 tons, how many people are required to move a cart that weighs 6 tons?

Question 37 If it takes 3 people of equal strength to move a cart that weighs 0.4 tons, how many people are required to move a cart that weighs 6 tons? Answer : 45 people

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Question 38 Tree A is 5 meters taller than tree B, and tree B is 4 meters taller than tree C. If tree A is 19 meters tall, what is the ratio of the height of tree A to that of tree C?

Question 38 Tree A is 5 meters taller than tree B, and tree B is 4 meters taller than tree C. If tree A is 19 meters tall, what is the ratio of the height of tree A to that of tree C? Answer : 19 10

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Question 39 The sum of five consecutive even integers is 100. What is the smallest of these five integers?

Question 39 The sum of five consecutive even integers is 100. What is the smallest of these five integers? Answer : 16

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Question 40 Beaker A is 3/5 filled with water and beaker B is 1/3 filled with orange juice. If half of the amount of water present in beaker A is transferred to beaker B, Beaker B will be 1/2 filled. If the maximum capacity of beaker A is 400ml, find the maximum capacity of beaker B.

Question 40 Beaker A is 3/5 filled with water and beaker B is 1/3 filled with orange juice. If half of the amount of water present in beaker A is transferred to beaker B, Beaker B will be 1/2 filled. If the maximum capacity of beaker A is 400ml, find the maximum capacity of beaker B. Answer : 720 mL

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Question 41 The sum of the first n consecutive positive integers is 78. What is n?

Question 41 The sum of the first n consecutive positive integers is 78. What is n? Answer : 12

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Question 42 Gary is guessing 2 multiple choice questions with 5 answer choices. If only one choice per question is correct, what is the probability that he’ll get at least one question right?

Question 42 Gary is guessing 2 multiple choice questions with 5 answer choices. If only one choice per question is correct, what is the probability that he’ll get at least one question right? Answer : 9 25

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Question 43 Evaluate: (7! × 6!)÷(5! × 4!)

Question 43 Evaluate: (7! × 6!)÷(5! × 4!) Answer : 1,260

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Question 44 What is the smaller angle, in degrees, formed by the hour hand and the minute hand of a clock face at 4:00am?

Question 44 What is the smaller angle, in degrees, formed by the hour hand and the minute hand of a clock face at 4:00am? Answer : 120 degrees

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Question 45 The denominator of a certain reduced fraction is double the numerator. What is this fraction?

Question 45 The denominator of a certain reduced fraction is double the numerator. What is this fraction? Answer : 1 2

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Question 46 If X is an integer and X squared plus 2X is divisible by 2, what is the remainder when X is divided by 2?

Question 46 If X is an integer and X squared plus 2X is divisible by 2, what is the remainder when X is divided by 2? Answer : 0

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Question 47 If February tenth in the year 2017 is a Friday, what day is March eleventh in the same year?

Question 47 If February tenth in the year 2017 is a Friday, what day is March eleventh in the same year? Answer : Saturday

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Question 48 How many diagonals does an octagon have?

Question 48 How many diagonals does an octagon have? Answer : 20 diagonals

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Question 49 How many integers less than or equal to 90 have an odd number of positive factors?

Question 49 How many integers less than or equal to 90 have an odd number of positive factors? Answer : 9 integers

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Question 50 What is the largest integer that does not exceed its reciprocal?

Question 50 What is the largest integer that does not exceed its reciprocal? Answer : 1

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Question 51 What is the circumference of a circle with an area of 25𝜋?

Question 51 What is the circumference of a circle with an area of 25𝜋? Answer :10𝜋

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Question 52 How many space diagonals are in a cube?

Question 52 How many space diagonals are in a cube? Answer : 4 space diagonals

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Question 53 The length of a diagonal of a certain square is the square root of 2. What is its area?

Question 53 The length of a diagonal of a certain square is the square root of 2. What is its area? Answer : 1

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Question 54 What is the smallest positive integer that has 3 positive integer factors?

Question 54 What is the smallest positive integer that has 3 positive integer factors? Answer : 4

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Question 55 How many digits does 2 to the power of 10 have?

Question 55 How many digits does 2 to the power of 10 have? Answer : 4 digits

The End