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Algebra
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1
Algebra
오답
How many integer values of \(x\) satisfy \(\| x \| < 3 \pi\)?
A
\( 9\)
B
\( 10\)
C
\( 18\)
\( 19\)
정답
E
\( 20\)
해설 없음
2
Algebra
오답
What is the value of
\(\sqrt{(3 - 2 \sqrt{3})^2} + \sqrt{(3 + 2 \sqrt{3})^2}\)?
A
\( 0\)
B
\( 4 \sqrt{3} - 6\)
C
\( 6\)
\( 4 \sqrt{3}\)
정답
E
\( 4 \sqrt{3} + 6\)
해설 없음
3
Algebra
오답
In an after-school program for juniors and seniors, there is a debate
team with an equal number of students from each class on the team. Among
the \(28\) students in the program, \(25 %\) of the juniors and \(10 %\) of
the seniors are on the debate team. How many juniors are in the program?
A
\( 5\)
B
\( 6\)
\( 8\)
정답
D
\( 11\)
E
\( 20\)
해설 없음
4
Counting and Probability
오답
At a math contest, \(57\) students are wearing blue shirts, and another
\(75\) students are wearing yellow shirts. The \(132\) students are assigned
into \(66\) pairs. In exactly \(23\) of these pairs, both students are
wearing blue shirts. In how many pairs are both students wearing yellow
shirts?
A
\( 23\)
\( 32\)
정답
C
\( 37\)
D
\( 41\)
E
\( 64\)
해설 없음
5
Number Theory
오답
The ages of Jonie's four cousins are distinct single-digit positive
integers. Two of the cousins' ages multiplied together give \(24\), while
the other two multiply to \(30\). What is the sum of the ages of Jonie's
four cousins?
A
\( 21\)
\( 22\)
정답
C
\( 23\)
D
\( 24\)
E
\( 25\)
해설 없음
6
Algebra
오답
Ms.~Blackwell gives an exam to two classes. The mean of the scores of
the students in the morning class is \(84\), and the afternoon class's
mean score is \(70\). The ratio of the number of students in the morning
class to the number of students in the afternoon class is \(\dfrac{3}{4}\). What
is the mean of the scores of all the students?
A
\( 74\)
B
\( 75\)
\( 76\)
정답
D
\( 77\)
E
\( 78\)
해설 없음
7
Geometry
오답
In a plane, four circles with radii \(1 , 3 , 5 ,\) and \(7\) are tangent to
line \(\ell\) at the same point \(A ,\) but they may be on either side of
\(\ell\). Region \(S\) consists of all the points that lie inside exactly one
of the four circles. What is the maximum possible area of region \(S\)?
A
\(24 \pi\)
B
\(32 \pi\)
C
\(64 \pi\)
\(65 \pi\)
정답
E
\(84 \pi\)
해설 없음
8
Competition Math
오답
Mr.~Zhou places all the integers from \(1\) to \(225\) into a \(15\) by \(15\)
grid. He places \(1\) in the middle square (eighth row and eighth column)
and places other numbers one by one clockwise, as shown in part in the
diagram below. What is the sum of the greatest number and the least
number that appear in the second row from the top?
\( 367\)
정답
B
\( 368\)
C
\( 369\)
D
\( 379\)
E
\( 380\)
해설 없음
9
Geometry
오답
The point \(P ( a , b )\) in the \(x y\)-plane is first rotated
counterclockwise by \(90^\circ\) around the point \(( 1 , 5 )\) and then
reflected about the line \(y = - x\). The image of \(P\) after these two
transformations is at \(( - 6 , 3 )\). What is \(b - a ?\)
A
\( 1\)
B
\( 3\)
C
\( 5\)
\( 7\)
정답
E
\( 9\)
해설 없음
10
Geometry
오답
An inverted cone with base radius \(12 \text{cm}\) and height
\(18 \text{cm}\) is full of water. The water is poured into a tall
cylinder whose horizontal base has a radius of \(24 \text{cm}\). What
is the height in centimeters of the water in the cylinder?
\(1.5\)
정답
B
\(3\)
C
\(4\)
D
\(4.5\)
E
\(6\)
해설 없음
11
Algebra
오답
Grandma has just finished baking a large rectangular pan of brownies.
She is planning to make rectangular pieces of equal size and shape, with
straight cuts parallel to the sides of the pan. Each cut must be made
entirely across the pan. Grandma wants to make the same number of
interior pieces as pieces along the perimeter of the pan. What is the
greatest possible number of brownies she can produce?
A
\( 24\)
B
\( 30\)
C
\( 48\)
\( 60\)
정답
E
\( 64\)
해설 없음
12
Number Theory
오답
Let \(N = 34 \cdot 34 \cdot 63 \cdot 270\). What is the ratio of the
sum of the odd divisors of \(N\) to the sum of the even divisors of \(N\)?
A
\( 1 : 16\)
B
\( 1 : 15\)
\( 1 : 14\)
정답
D
\( 1 : 8\)
E
\( 1 : 3\)
해설 없음
13
Number Theory
오답
Let \(n\) be a positive integer and \(d\) be a digit such that the value of
the numeral \(\underline{32 d}\) in base \(n\) equals \(263\), and the value of
the numeral \(\underline{324}\) in base \(n\) equals the value of the numeral
\(\underline{11 d 1}\) in base six. What is \(n + d ?\)
A
\( 10\)
\( 11\)
정답
C
\( 13\)
D
\( 15\)
E
\( 16\)
해설 없음
14
Geometry
오답
Three equally spaced parallel lines intersect a circle, creating three
chords of lengths \(38 , 38 ,\) and \(34\). What is the distance between two
adjacent parallel lines?
A
\( 5 \dfrac{1}{2}\)
\( 6\)
정답
C
\( 6 \dfrac{1}{2}\)
D
\( 7\)
E
\( 7 \dfrac{1}{2}\)
해설 없음
15
Algebra
오답
The real number \(x\) satisfies the equation \(x + \dfrac{1}{x} = \sqrt{5}\). What
is the value of \(x^11 - 7 x^7 + x^3 ?\)
A
\( - 1\)
\( 0\)
정답
C
\( 1\)
D
\( 2\)
E
\( \sqrt{5}\)
해설 없음
16
Competition Math
오답
Call a positive integer an uphill integer if every digit is strictly
greater than the previous digit. For example, \(1357\), \(89\), and \(5\) are
all uphill integers, but \(32\), \(1240\), and \(466\) are not. How many
uphill integers are divisible by \(15\)?
A
\( 4\)
B
\( 5\)
\( 6\)
정답
D
\( 7\)
E
\( 8\)
해설 없음
17
Competition Math
오답
Ravon, Oscar, Aditi, Tyrone, and Kim play a card game. Each person is
given \(2\) cards out of a set of \(10\) cards numbered
\(1 , 2 , 3 , \cdots , 10 .\) The score of a player is the sum of the
numbers of their cards. The scores of the players are as follows:
Ravon--\(11 ,\) Oscar--\(4 ,\) Aditi--\(7 ,\) Tyrone--\(16 ,\) Kim--\(17 .\) Which
of the following statements is true?
\(\mathbf{\text{(B) }} \text{Aditi was given card 3.} \) bold("(C) ") "Ravon was given card 4." \( \mathbf{\text{(D) }} \text{Aditi was given card 4.}\)\(\mathbf{\text{(E) }} \text{Tyrone was given card 7.}\)
A
\(\text{Ravon was given card 3.}\)
해설 없음
18
Counting and Probability
오답
A fair \(6\)-sided die is repeatedly rolled until an odd number appears.
What is the probability that every even number appears at least once
before the first occurrence of an odd number?
A
\( \dfrac{1}{120}\)
B
\( \dfrac{1}{32}\)
\( \dfrac{1}{20}\)
정답
D
\( \dfrac{3}{20}\)
E
\( \dfrac{1}{6}\)
해설 없음
19
Algebra
오답
Suppose that \(S\) is a finite set of positive integers. If the greatest
integer in \(S\) is removed from \(S\), then the average value (arithmetic
mean) of the integers remaining is \(32\). If the least integer in \(S\) is
also removed, then the average value of the integers remaining is \(35\).
If the greatest integer is then returned to the set, the average value
of the integers rises to \(40\). The greatest integer in the original set
\(S\) is \(72\) greater than the least integer in \(S\). What is the average
value of all the integers in the set \(S\)?
A
\(36.2\)
B
\(36.4\)
C
\(36.6\)
\(36.8\)
정답
E
\(37\)
해설 없음
20
Geometry
오답
The figure is constructed from \(11\) line segments, each of which has
length \(2\). The area of pentagon \(A B C D E\) can be written as
\(\sqrt{m} + \sqrt{n}\), where \(m\) and \(n\) are positive integers. What is
\(m + n ?\)
A
\( 20\)
B
\( 21\)
C
\( 22\)
\( 23\)
정답
E
\( 24\)
해설 없음
21
Geometry
오답
A square piece of paper has side length \(1\) and vertices \(A , B , C ,\)
and \(D\) in that order. As shown in the figure, the paper is folded so
that vertex \(C\) meets edge \(\overline{A D}\) at point \(C'\), and edge
\(\overline{A B}\) at point \(E\). Suppose that \(C' D = \dfrac{1}{3}\). What is the
perimeter of triangle \(\triangle A E C' ?\)
\( 2\)
정답
B
\( 1 + \dfrac{2}{3} \sqrt{3}\)
C
\( \dfrac{13}{6}\)
D
\( 1 + \dfrac{3}{4} \sqrt{3}\)
E
\( \dfrac{7}{3}\)
해설 없음
22
Counting and Probability
오답
Ang, Ben, and Jasmin each have \(5\) blocks, colored red, blue, yellow,
white, and green; and there are \(5\) empty boxes. Each of the people
randomly and independently of the other two people places one of their
blocks into each box. The probability that at least one box receives \(3\)
blocks all of the same color is \(\dfrac{m}{n}\), where \(m\) and \(n\) are
relatively prime positive integers. What is \(m + n ?\)
A
\( 47\)
B
\( 94\)
C
\( 227\)
\( 471\)
정답
E
\( 542\)
해설 없음
23
Counting and Probability
오답
A square with side length \(8\) is colored white except for \(4\) black
isosceles right triangular regions with legs of length \(2\) in each
corner of the square and a black diamond with side length \(2 \sqrt{2}\) in
the center of the square, as shown in the diagram. A circular coin with
diameter \(1\) is dropped onto the square and lands in a random location
where the coin is completely contained within the square. The
probability that the coin will cover part of the black region of the
square can be written as \(\dfrac{1}{196} (a + b \sqrt{2} + \pi)\), where \(a\) and
\(b\) are positive integers. What is \(a + b\)?
A
\( 64\)
B
\( 66\)
\( 68\)
정답
D
\( 70\)
E
\( 72\)
해설 없음
24
Counting and Probability
오답
Arjun and Beth play a game in which they take turns removing one brick
or two adjacent bricks from one "wall" among a set of several walls of
bricks, with gaps possibly creating new walls. The walls are one brick
tall. For example, a set of walls of sizes \(4\) and \(2\) can be changed
into any of the following by one move:
\(( 3 , 2 ) , ( 2 , 1 , 2 ) , ( 4 ) , ( 4 , 1 ) , ( 2 , 2 ) ,\)
or \(( 1 , 1 , 2 ) .\) Arjun plays first, and the player who
removes the last brick wins. For which starting configuration is there a
strategy that guarantees a win for Beth?
A
\(( 6 , 1 , 1 )\)
\(( 6 , 2 , 1 )\)
정답
C
\(( 6 , 2 , 2 )\)
D
\(( 6 , 3 , 1 )\)
E
\(( 6 , 3 , 2 )\)
해설 없음
25
Counting and Probability
오답
Let \(S\) be the set of lattice points in the coordinate plane, both of
whose coordinates are integers between \(1\) and \(30\), inclusive. Exactly
\(300\) points in \(S\) lie on or below a line with equation \(y = m x\). The
possible values of \(m\) lie in an interval of length \(\dfrac{a}{b}\), where \(a\)
and \(b\) are relatively prime positive integers. What is \(a + b ?\)
A
\( 31\)
B
\( 47\)
C
\( 62\)
D
\( 72\)
\( 85\)
정답
해설 없음
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