Unnamed: 0
int64 0
40.3k
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stringlengths 10
5.15k
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stringlengths 1
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100
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|---|---|---|---|
1,100 | Let $n$ denote the smallest positive integer that is divisible by both $4$ and $9,$ and whose base-$10$ representation consists of only $4$'s and $9$'s, with at least one of each. What are the last four digits of $n?$ | 4944 | 59.375 |
1,101 | A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (2020, 0), (2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\frac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth? | 0.4 | 44.53125 |
1,102 | A $9 \times 9 \times 9$ cube is composed of twenty-seven $3 \times 3 \times 3$ cubes. The big cube is ‘tunneled’ as follows: First, the six $3 \times 3 \times 3$ cubes which make up the center of each face as well as the center $3 \times 3 \times 3$ cube are removed. Second, each of the twenty remaining $3 \times 3 \times 3$ cubes is diminished in the same way. That is, the center facial unit cubes as well as each center cube are removed. The surface area of the final figure is: | 1056 | 0 |
1,103 | King Middle School has $1200$ students. Each student takes $5$ classes a day. Each teacher teaches $4$ classes. Each class has $30$ students and $1$ teacher. How many teachers are there at King Middle School? | 50 | 75.78125 |
1,104 | A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? | 26 | 88.28125 |
1,105 | The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? | 10 | 72.65625 |
1,106 | Mr. Earl E. Bird gets up every day at 8:00 AM to go to work. If he drives at an average speed of 40 miles per hour, he will be late by 3 minutes. If he drives at an average speed of 60 miles per hour, he will be early by 3 minutes. How many miles per hour does Mr. Bird need to drive to get to work exactly on time? | 48 | 42.1875 |
1,107 | As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles? | 6\sqrt{3} - 3\pi | 94.53125 |
1,108 | A regular 15-gon has $L$ lines of symmetry, and the smallest positive angle for which it has rotational symmetry is $R$ degrees. What is $L+R$? | 39 | 64.84375 |
1,109 | On a checkerboard composed of 64 unit squares, what is the probability that a randomly chosen unit square does not touch the outer edge of the board? | \frac{9}{16} | 57.03125 |
1,110 | In $\triangle ABC$, side $a = \sqrt{3}$, side $b = \sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals: | 120^{\circ} | 77.34375 |
1,111 | The price of an article is cut $10 \%$. To restore it to its former value, the new price must be increased by: | $11\frac{1}{9} \%$ | 0 |
1,112 | For any real number a and positive integer k, define
$\binom{a}{k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}$
What is
$\binom{-\frac{1}{2}}{100} \div \binom{\frac{1}{2}}{100}$? | -199 | 92.96875 |
1,113 | Josanna's test scores to date are $90, 80, 70, 60,$ and $85.$ Her goal is to raise her test average at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal? | 95 | 95.3125 |
1,114 | A certain calculator has only two keys [+1] and [x2]. When you press one of the keys, the calculator automatically displays the result. For instance, if the calculator originally displayed "9" and you pressed [+1], it would display "10." If you then pressed [x2], it would display "20." Starting with the display "1," what is the fewest number of keystrokes you would need to reach "200"? | 9 | 96.09375 |
1,115 | Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \frac{1}{2}$? | \frac{7}{16} | 10.9375 |
1,116 | Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\frac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. Then $r$ is: | 10 | 46.09375 |
1,117 | At a gathering of $30$ people, there are $20$ people who all know each other and $10$ people who know no one. People who know each other hug, and people who do not know each other shake hands. How many handshakes occur within the group? | 245 | 85.9375 |
1,118 | Bridget bought a bag of apples at the grocery store. She gave half of the apples to Ann. Then she gave Cassie 3 apples, keeping 4 apples for herself. How many apples did Bridget buy? | 14 | 82.03125 |
1,119 | Let $n$ be the least positive integer greater than $1000$ for which
\[\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.\]What is the sum of the digits of $n$? | 18 | 69.53125 |
1,120 | LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally? | \frac{B - A}{2} | 92.1875 |
1,121 | Consider the statements:
$\textbf{(1)}\ p\wedge \sim q\wedge r \qquad\textbf{(2)}\ \sim p\wedge \sim q\wedge r \qquad\textbf{(3)}\ p\wedge \sim q\wedge \sim r \qquad\textbf{(4)}\ \sim p\wedge q\wedge r$
where $p,q$, and $r$ are propositions. How many of these imply the truth of $(p\rightarrow q)\rightarrow r$? | 4 | 71.875 |
1,122 | Kiana has two older twin brothers. The product of their three ages is 128. What is the sum of their three ages? | 18 | 32.8125 |
1,123 | A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, with blanks indicating the missing numbers. Based on her observations, the scientist was able to reconstruct the lost data. What was the average height of the trees, in meters?
\begin{tabular}{|c|c|} \hline Tree 1 & meters \\ Tree 2 & 11 meters \\ Tree 3 & meters \\ Tree 4 & meters \\ Tree 5 & meters \\ \hline Average height & .2 meters \\ \hline \end{tabular}
| 24.2 | 0 |
1,124 | Al gets the disease algebritis and must take one green pill and one pink pill each day for two weeks. A green pill costs $1 more than a pink pill, and Al's pills cost a total of $546 for the two weeks. How much does one green pill cost? | $19 | 0 |
1,125 | The simplest form of $1 - \frac{1}{1 + \frac{a}{1 - a}}$ is: | a | 67.96875 |
1,126 | In a jar of red, green, and blue marbles, all but 6 are red marbles, all but 8 are green, and all but 4 are blue. How many marbles are in the jar? | 9 | 92.1875 |
1,127 | [asy]
draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle, black+linewidth(.75));
draw((0,-1)--(0,1), black+linewidth(.75));
draw((-1,0)--(1,0), black+linewidth(.75));
draw((-1,-1/sqrt(3))--(1,1/sqrt(3)), black+linewidth(.75));
draw((-1,1/sqrt(3))--(1,-1/sqrt(3)), black+linewidth(.75));
draw((-1/sqrt(3),-1)--(1/sqrt(3),1), black+linewidth(.75));
draw((1/sqrt(3),-1)--(-1/sqrt(3),1), black+linewidth(.75));
[/asy]
Amy painted a dartboard over a square clock face using the "hour positions" as boundaries. If $t$ is the area of one of the eight triangular regions such as that between 12 o'clock and 1 o'clock, and $q$ is the area of one of the four corner quadrilaterals such as that between 1 o'clock and 2 o'clock, then $\frac{q}{t}=$ | 2\sqrt{3}-2 | 0 |
1,128 | The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? | 15 | 3.90625 |
1,129 | The slope of the line $\frac{x}{3} + \frac{y}{2} = 1$ is | -\frac{2}{3} | 99.21875 |
1,130 | Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly $7.$ Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? | \frac{7}{36} | 0.78125 |
1,131 | The number $(2^{48}-1)$ is exactly divisible by two numbers between $60$ and $70$. These numbers are | 63,65 | 30.46875 |
1,132 | Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$. Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$. Let $M$ be the midpoint of $\overline{AE}$, and $N$ be the midpoint of $\overline{CD}$. The area of $\triangle BMN$ is $x$. Find $x^2$. | 64 | 0.78125 |
1,133 | A train, an hour after starting, meets with an accident which detains it a half hour, after which it proceeds at $\frac{3}{4}$ of its former rate and arrives $3\tfrac{1}{2}$ hours late. Had the accident happened $90$ miles farther along the line, it would have arrived only $3$ hours late. The length of the trip in miles was: | 600 | 21.875 |
1,134 | The numbers $1, 2, 3, 4, 5$ are to be arranged in a circle. An arrangement is $\textit{bad}$ if it is not true that for every $n$ from $1$ to $15$ one can find a subset of the numbers that appear consecutively on the circle that sum to $n$. Arrangements that differ only by a rotation or a reflection are considered the same. How many different bad arrangements are there? | 2 | 17.1875 |
1,135 | Trickster Rabbit agrees with Foolish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays $40$ coins in toll to Rabbit after each crossing. The payment is made after the doubling. Fox is excited about his good fortune until he discovers that all his money is gone after crossing the bridge three times. How many coins did Fox have at the beginning? | 35 | 58.59375 |
1,136 | Suppose $a$ is $150\%$ of $b$. What percent of $a$ is $3b$? | 200 | 88.28125 |
1,137 | The number of distinct pairs of integers $(x, y)$ such that $0<x<y$ and $\sqrt{1984}=\sqrt{x}+\sqrt{y}$ is | 3 | 70.3125 |
1,138 | The number of diagonals that can be drawn in a polygon of 100 sides is: | 4850 | 92.96875 |
1,139 | Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline {CD}$ of length $21$. The other two sides are of lengths $10$ and $14$. The angles $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$? | $25$ | 0 |
1,140 | The sum of $n$ terms of an arithmetic progression is $153$, and the common difference is $2$. If the first term is an integer, and $n>1$, then the number of possible values for $n$ is: | 5 | 73.4375 |
1,141 | The numbers $\log(a^3b^7)$, $\log(a^5b^{12})$, and $\log(a^8b^{15})$ are the first three terms of an arithmetic sequence, and the $12^{\text{th}}$ term of the sequence is $\log{b^n}$. What is $n$? | 112 | 29.6875 |
1,142 | At Rachelle's school, an A counts $4$ points, a B $3$ points, a C $2$ points, and a D $1$ point. Her GPA in the four classes she is taking is computed as the total sum of points divided by 4. She is certain that she will get A's in both Mathematics and Science and at least a C in each of English and History. She thinks she has a $\frac{1}{6}$ chance of getting an A in English, and a $\frac{1}{4}$ chance of getting a B. In History, she has a $\frac{1}{4}$ chance of getting an A, and a $\frac{1}{3}$ chance of getting a B, independently of what she gets in English. What is the probability that Rachelle will get a GPA of at least $3.5$? | \frac{11}{24} | 29.6875 |
1,143 | The entries in a $3 \times 3$ array include all the digits from $1$ through $9$, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? | 42 | 79.6875 |
1,144 | The length of rectangle $ABCD$ is 5 inches and its width is 3 inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F$. The area of triangle $BEF$, expressed in square inches, is: | \frac{5}{2} | 78.90625 |
1,145 | A plastic snap-together cube has a protruding snap on one side and receptacle holes on the other five sides as shown. What is the smallest number of these cubes that can be snapped together so that only receptacle holes are showing?
[asy] draw((0,0)--(4,0)--(4,4)--(0,4)--cycle); draw(circle((2,2),1)); draw((4,0)--(6,1)--(6,5)--(4,4)); draw((6,5)--(2,5)--(0,4)); draw(ellipse((5,2.5),0.5,1)); fill(ellipse((3,4.5),1,0.25),black); fill((2,4.5)--(2,5.25)--(4,5.25)--(4,4.5)--cycle,black); fill(ellipse((3,5.25),1,0.25),black); [/asy] | 4 | 17.96875 |
1,146 | If $M$ is $30 \%$ of $Q$, $Q$ is $20 \%$ of $P$, and $N$ is $50 \%$ of $P$, then $\frac {M}{N} =$ | \frac {3}{25} | 78.90625 |
1,147 | A man travels $m$ feet due north at $2$ minutes per mile. He returns due south to his starting point at $2$ miles per minute. The average rate in miles per hour for the entire trip is: | 48 | 41.40625 |
1,148 | A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible? | 729 | 57.8125 |
1,149 | In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of quadrilateral $ABCD$? | 36 | 35.9375 |
1,150 | For $x$ real, the inequality $1 \le |x-2| \le 7$ is equivalent to | $-5 \le x \le 1$ or $3 \le x \le 9$ | 0 |
1,151 | It is given that one root of $2x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is $3+2i (i = \sqrt{-1})$. The value of $s$ is: | 26 | 100 |
1,152 | Star lists the whole numbers $1$ through $30$ once. Emilio copies Star's numbers, replacing each occurrence of the digit $2$ by the digit $1$. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's? | 103 | 51.5625 |
1,153 | Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hour, what was her greatest possible average speed? | 70 \frac{1}{3} | 0 |
1,154 | Johann has $64$ fair coins. He flips all the coins. Any coin that lands on tails is tossed again. Coins that land on tails on the second toss are tossed a third time. What is the expected number of coins that are now heads? | 56 | 60.9375 |
1,155 | When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.
What is the remainder when $x+2uy$ is divided by $y$? | v | 100 |
1,156 | In rectangle $ABCD$, $AB=1$, $BC=2$, and points $E$, $F$, and $G$ are midpoints of $\overline{BC}$, $\overline{CD}$, and $\overline{AD}$, respectively. Point $H$ is the midpoint of $\overline{GE}$. What is the area of the shaded region? | \frac{1}{6} | 3.125 |
1,157 | The expression $a^3-a^{-3}$ equals: | \left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right) | 2.34375 |
1,158 | The remainder can be defined for all real numbers $x$ and $y$ with $y \neq 0$ by $\text{rem} (x ,y)=x-y\left \lfloor \frac{x}{y} \right \rfloor$ where $\left \lfloor \tfrac{x}{y} \right \rfloor$ denotes the greatest integer less than or equal to $\tfrac{x}{y}$. What is the value of $\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )$? | -\frac{1}{40} | 76.5625 |
1,159 | Segment $BD$ and $AE$ intersect at $C$, as shown, $AB=BC=CD=CE$, and $\angle A = \frac{5}{2} \angle B$. What is the degree measure of $\angle D$? | 52.5 | 5.46875 |
1,160 | A frog sitting at the point $(1, 2)$ begins a sequence of jumps, where each jump is parallel to one of the coordinate axes and has length $1$, and the direction of each jump (up, down, right, or left) is chosen independently at random. The sequence ends when the frog reaches a side of the square with vertices $(0,0), (0,4), (4,4),$ and $(4,0)$. What is the probability that the sequence of jumps ends on a vertical side of the square? | \frac{5}{8} | 38.28125 |
1,161 | The circle having $(0,0)$ and $(8,6)$ as the endpoints of a diameter intersects the $x$-axis at a second point. What is the $x$-coordinate of this point? | 8 | 79.6875 |
1,162 | The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? | 10 | 92.1875 |
1,163 | For real numbers $x$ and $y$, define $x\spadesuit y = (x + y)(x - y)$. What is $3\spadesuit(4\spadesuit 5)$? | -72 | 91.40625 |
1,164 | $\frac{2}{25}=$ | .08 | 32.03125 |
1,165 | If $x^4 + 4x^3 + 6px^2 + 4qx + r$ is exactly divisible by $x^3 + 3x^2 + 9x + 3$, the value of $(p + q)r$ is: | 15 | 89.0625 |
1,166 | Debra flips a fair coin repeatedly, keeping track of how many heads and how many tails she has seen in total, until she gets either two heads in a row or two tails in a row, at which point she stops flipping. What is the probability that she gets two heads in a row but she sees a second tail before she sees a second head？ | \frac{1}{24} | 0.78125 |
1,167 | The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex? | 12 | 0 |
1,168 | The first three terms of a geometric progression are $\sqrt 3$, $\sqrt[3]3$, and $\sqrt[6]3$. What is the fourth term? | 1 | 87.5 |
1,169 | In the expression $\frac{x + 1}{x - 1}$ each $x$ is replaced by $\frac{x + 1}{x - 1}$. The resulting expression, evaluated for $x = \frac{1}{2}$, equals: | 3 | 0 |
1,170 | A piece of graph paper is folded once so that (0,2) is matched with (4,0), and (7,3) is matched with $(m,n)$. Find $m+n$. | 6.8 | 0 |
1,171 | If $x$ is a number satisfying the equation $\sqrt[3]{x+9}-\sqrt[3]{x-9}=3$, then $x^2$ is between: | 75 \text{ and } 85 | 6.25 |
1,172 | A rectangle with diagonal length $x$ is twice as long as it is wide. What is the area of the rectangle? | \frac{2}{5} x^2 | 0 |
1,173 | The table below displays the grade distribution of the $30$ students in a mathematics class on the last two tests. For example, exactly one student received a 'D' on Test 1 and a 'C' on Test 2. What percent of the students received the same grade on both tests? | 40\% | 9.375 |
1,174 | The Blue Bird High School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible? | 12 | 98.4375 |
1,175 | How many whole numbers from $1$ through $46$ are divisible by either $3$ or $5$ or both? | 21 | 100 |
1,176 | Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is | 24 | 44.53125 |
1,177 | Which terms must be removed from the sum
$\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+\frac{1}{8}+\frac{1}{10}+\frac{1}{12}$
if the sum of the remaining terms is to equal $1$? | \frac{1}{8} \text{ and } \frac{1}{10} | 29.6875 |
1,178 | At Clover View Junior High, one half of the students go home on the school bus. One fourth go home by automobile. One tenth go home on their bicycles. The rest walk home. What fractional part of the students walk home? | \frac{3}{20} | 69.53125 |
1,179 | In a particular game, each of $4$ players rolls a standard $6$-sided die. The winner is the player who rolls the highest number. If there is a tie for the highest roll, those involved in the tie will roll again and this process will continue until one player wins. Hugo is one of the players in this game. What is the probability that Hugo's first roll was a $5,$ given that he won the game? | \frac{41}{144} | 0 |
1,180 | A merchant placed on display some dresses, each with a marked price. He then posted a sign "$1/3$ off on these dresses." The cost of the dresses was $3/4$ of the price at which he actually sold them. Then the ratio of the cost to the marked price was: | \frac{1}{2} | 97.65625 |
1,181 | The digit-sum of $998$ is $9+9+8=26$. How many 3-digit whole numbers, whose digit-sum is $26$, are even? | 1 | 53.125 |
1,182 | In the sequence
\[..., a, b, c, d, 0, 1, 1, 2, 3, 5, 8,...\]
each term is the sum of the two terms to its left. Find $a$. | -3 | 67.1875 |
1,183 | The number $6545$ can be written as a product of a pair of positive two-digit numbers. What is the sum of this pair of numbers? | 162 | 81.25 |
1,184 | Francesca uses 100 grams of lemon juice, 100 grams of sugar, and 400 grams of water to make lemonade. There are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar. Water contains no calories. How many calories are in 200 grams of her lemonade? | 137 | 57.8125 |
1,185 | Lines in the $xy$-plane are drawn through the point $(3,4)$ and the trisection points of the line segment joining the points $(-4,5)$ and $(5,-1)$. One of these lines has the equation | x-4y+13=0 | 17.1875 |
1,186 | The largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$, is: | 6 | 96.09375 |
1,187 | The arithmetic mean of the nine numbers in the set $\{9, 99, 999, 9999, \ldots, 999999999\}$ is a $9$-digit number $M$, all of whose digits are distinct. The number $M$ doesn't contain the digit | 0 | 92.1875 |
1,188 | I'm thinking of two whole numbers. Their product is 24 and their sum is 11. What is the larger number? | 8 | 99.21875 |
1,189 | The sum of $25$ consecutive even integers is $10,000$. What is the largest of these $25$ consecutive integers? | 424 | 97.65625 |
1,190 | If five geometric means are inserted between $8$ and $5832$, the fifth term in the geometric series: | $648$ | 0 |
1,191 | A regular polygon of $n$ sides is inscribed in a circle of radius $R$. The area of the polygon is $3R^2$. Then $n$ equals: | 12 | 92.96875 |
1,192 | How many ways can a student schedule $3$ mathematics courses -- algebra, geometry, and number theory -- in a $6$-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other $3$ periods is of no concern here.) | 24 | 98.4375 |
1,193 | Kaashish has written down one integer two times and another integer three times. The sum of the five numbers is $100$, and one of the numbers is $28$. What is the other number? | 8 | 64.0625 |
1,194 | Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$. What is the area of $S_3$? | 4 | 96.09375 |
1,195 | Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than $5$ steps left). Suppose Dash takes $19$ fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? | 13 | 38.28125 |
1,196 | Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line that is parallel to side $AB$, how many of the four quantities listed below change?
(a) the length of the segment $MN$
(b) the perimeter of $\triangle PAB$
(c) the area of $\triangle PAB$
(d) the area of trapezoid $ABNM$ | 1 | 42.1875 |
1,197 | A $\text{palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$? | 24 | 57.8125 |
1,198 | In an arcade game, the "monster" is the shaded sector of a circle of radius $1$ cm, as shown in the figure. The missing piece (the mouth) has central angle $60^\circ$. What is the perimeter of the monster in cm? | \frac{5}{3}\pi + 2 | 0 |
1,199 | A man has part of $4500 invested at 4% and the rest at 6%. If his annual return on each investment is the same, the average rate of interest which he realizes of the $4500 is: | 4.8\% | 95.3125 |
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