Download AoPS Kings - Art of Problem Solving

AoPS Kings
Level 2
Level 2 Problems
1. On a regular die, one of the dots is removed at random, with an
equally likely probability of each dot being chosen. When the die is
rolled, what is the probability that the bottom face has an odd number
of dots?
2. Let f (x) = x2 − 2x.
f (f (f (f (a)))) = 3?
How many distinct real numbers a satisfy
3. Find the smallest possible integer x such that z 10 +z 9 +z 6 +z 5 +z 4 +z+1
divides z x − 1.
p
√
√
√
√
√
4. Find the sum a+b+c+d+e if 7 + 40 = a+b 2+c 5+d 7+e 10
5. Let n = 10222 − 3. Then what is the sum of the digits of n2 ?
6. Find the largest number x such that (2004!)! is divisible by ((x!)!)!
7. A regular tetrahedron A of volume 1 is reflected through its center to
form tetrahedron B. What is the volume of the intersection of tetrahedra A and B? Express your answer as a common fraction.
8. What is the sum of the first 10 multiples of 1 multiplied by the sum of
the first 10 multiples of 2 multiplied by the sum of the first 10 multiples
of 3 multiplied by...multiplied by the sum of the first 10 multiples of
10?
9. In the expansion of (a + b)20 , what is the coeffficient of the a9 b11 term?
10. A facsum is the number of proper factors of a number multiplied by
the sum of all the factors. Find the sum of all the facsums for the
number 900 to 910, exclusive.
11. Find the value of 101108 + 302106 + 501426 + 374618 .
12. How many numbers in base 9 can be written with 4 digits in base 2?
13. A figure is a 6 × 6 square minus the 2 × 2 square in the middle. Find
the number of paths from the topleft point to the bottomright point.
14. A circle with radius 5 is inscribed
√ in a right triangle. Given that the
area of the triangle is 75 + 50 2, find the area of a regular hexagon
that has a side length equal to that of the hypotenuse.
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AoPS Kings
Level 2
15. What is the sum of the first 25 cubes minus the sum of the first 50
squares?
16. Given a rectangular piece of paper ABCD that is 8 × 11, where AB is
8, find the length of the fold when the midpoint of AB is folded over
to C.
17. Solve the following exponential equation: 7x−5 = 34. Round your
answer to the nearest thousandth.
18. Solve the following logarithmic equation: log2 (8x ) = −3. Round your
answer to the nearest thousandth.
19. Kristian has $7200 in a bank account. The bank’s interest rate is 4%.
How many years should Kristian leave his money in his account so that
he will end up with $18000? The interest is compounded continously.
Round your answer to the nearest whole number.
20. Given that f (x) = 3x − 7 and g(x) = 4 + 5x, find the value of
f (g(f (f (g(g(f (5))))))).
21. Find the number of ordered pairs of integers (x + y) that satisfy the
following: x + y = 10000, x − y < 1000, and neither x nor y has a digit
that is a prime number.
22. A covenant armada of 15 dropships wants to organize itself in a set
of 3 rectangular formations before going to battle. If each formation
must have at least one dropship, how many ways can the armada go
to battle? A 1 × 4 is different from a 2 × 2 but not different from a
4 × 1.
23. What is the LCM of the first 50 prime numbers, the first 10 squares,
and the first 5 cubes? Just answer what the calculator shows.
24. Find the first time after 7:30 where the hour and minute hand of a
clock form a 75◦ angle. Round to the nearest second if necessary.
Express your answer as hour:minute:second.
25. The Master Chief has a sniper rifle with 4 bullets left. He sees 4 enemy
carriers flying. The enemy has 3 species, elites, jackals, and grunts.
The first carrier has 7 elites, 4 jackals, and 2 grunts. The second one
has 5 elites, 7 jackals, and 1 grunt. The third one has 2 elites, 8 jackals,
and 3 grunts. The fourth one has 5 elites, 1 jackal, and 6 grunts. The
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AoPS Kings
Level 2
probability of sniping an elite is twice as much as either a jackal or a
grunt. What is the probability that the Master Chief snipes 2 elites,
1 jackal, and 1 grunt out of the carriers, not necessarily in that order.
Assume that the Master Chief never misses a shot and that he shoots
all 4 bullets. Assume also that he may shoot at any of carriers and that
the probability of selecting each carrier is based on the total enemies
in it. An elite is the same as two regular enemies. Express your answer
as a common fraction.
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