BMC Quiz 1

What is the value of

( 2² - 2 ) - ( 3² - 3 ) + ( 4² - 4 ) ?

( Problem 1 in AMC 10 A 2021)

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The sum of two natural numbers is 17402. One of the two numbers is divisible by 10. If the units digit of that number is erased, the other number is obtained. What is the difference of these two numbers?

( 2021 AMC 10 A Problem 3 )

• 10272

• 11700

• 13362

• 14238

• 15426

In a cube ABCDEFGH, all side are 1 unit long. What is the shortest distance between A and G?

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• $\sqrt{2}$

• $\sqrt{3}$

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Tom has a collection of snakes, of which are purple and of which are happy. He observes that

all of his happy snakes can add

none of his purple snakes can subtract

all of his snakes that can't subtract also can't add

Which of these conclusions can be drawn about Tom's snakes?

( 2021 AMC 12 A Problem 4 )

• Purple snakes can add.

• Purple snakes are happy.

• Snakes that can add are purple.

• Happy snakes are not purple.

• Happy snakes can't subtract.

What non-zero real value for x satisfies  $\left(7x\right)^{14}=\left(14x\right)^7$ ?

( 2006 AMC 10 A Problem 6 )

•  $\frac{1}{7}$  

•  $\frac{2}{7}$  

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When a student multiplied the number 66 by the repeating decimal,  $1.ababab...\ =\ 1.\overline{ab}$ , where a and b are digits, he did not notice the notation and just multiplied 66 times $1.ab$  . Later he found that his answer is 0.5 less than the correct answer. What is the 2-digit integer $ab$  ?

( 2021 AMC 12 A Problem 5 )

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• 75

Which of the following is equal to $\left(2+3\right)\left(2^2+3^2\right)\left(2^4+3^4\right)\left(2^8+3^8\right)\left(2^{16}+3^{16}\right)\left(2^{32}+3^{32}\right)\left(2^{64}+3^{64}\right)$   ( 2021 AMC 12 A Problem 9 )

•  $3^{127}+2^{127}$  

•   $3^{128}-2^{128}$  

•  $5^{127}$  

•  $3^{128}+2^{128}$  

•  $3^{127}+2^{127}$  
   $+2\cdot3^{63}+3\cdot2^{63}$  

The two legs of a right triangle, which are altitudes, have lengths $2\sqrt{3}$ and 6. How long is the third altitude of the triangle?

( 2014 AMC 10 A Problem 9 )

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Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?

( 2015 AMC 10 Problem 13 )

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What is the value of

 $\left(\sum_{k=1}^{20}\log_{5^k}\left(3^{k^2}\right)\right)\left(\sum_{k=1}^{100}\log_{9^k}25^k\right)$ ?
( 2021 AMC 12 A Problem 14 )

• I haven't learned log. 

•  $100\log_53$  

•  $200\log_35$  

• 2200
  

• 21000