Unnamed: 0
int64 0
40.3k
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5.15k
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1.22k
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100
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|---|---|---|---|
1,500 | If $f(x)=4^x$ then $f(x+1)-f(x)$ equals: | 3f(x) | 0 |
1,501 | $R$ varies directly as $S$ and inversely as $T$. When $R = \frac{4}{3}$ and $T = \frac{9}{14}$, $S = \frac{3}{7}$. Find $S$ when $R = \sqrt{48}$ and $T = \sqrt{75}$. | 30 | 97.65625 |
1,502 | Tom has twelve slips of paper which he wants to put into five cups labeled $A$, $B$, $C$, $D$, $E$. He wants the sum of the numbers on the slips in each cup to be an integer. Furthermore, he wants the five integers to be consecutive and increasing from $A$ to $E$. The numbers on the papers are $2, 2, 2, 2.5, 2.5, 3, 3, 3, 3, 3.5, 4,$ and $4.5$. If a slip with $2$ goes into cup $E$ and a slip with $3$ goes into cup $B$, then the slip with $3.5$ must go into what cup? | D | 21.09375 |
1,503 | In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is traced in the picture. | 24 | 23.4375 |
1,504 | Camilla had twice as many blueberry jelly beans as cherry jelly beans. After eating 10 pieces of each kind, she now has three times as many blueberry jelly beans as cherry jelly beans. How many blueberry jelly beans did she originally have? | 40 | 70.3125 |
1,505 | For how many positive integers $n$ is $\frac{n}{30-n}$ also a positive integer? | 7 | 86.71875 |
1,506 | The amount $2.5$ is split into two nonnegative real numbers uniformly at random, for instance, into $2.143$ and $.357$, or into $\sqrt{3}$ and $2.5-\sqrt{3}$. Then each number is rounded to its nearest integer, for instance, $2$ and $0$ in the first case above, $2$ and $1$ in the second. What is the probability that the two integers sum to $3$? | \frac{3}{5} | 14.84375 |
1,507 | Pablo buys popsicles for his friends. The store sells single popsicles for $1 each, 3-popsicle boxes for $2, and 5-popsicle boxes for $3. What is the greatest number of popsicles that Pablo can buy with $8? | 13 | 86.71875 |
1,508 | The sum of the distances from one vertex of a square with sides of length $2$ to the midpoints of each of the sides of the square is | $2+2\sqrt{5}$ | 0 |
1,509 | Let $N$ be the positive integer $7777\ldots777$, a $313$-digit number where each digit is a $7$. Let $f(r)$ be the leading digit of the $r^{\text{th}}$ root of $N$. What is $f(2) + f(3) + f(4) + f(5)+ f(6)$? | 8 | 23.4375 |
1,510 | Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then
\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\]
For every $m\ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$s in the range $1\le N\le 400$ is the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ unbounded?
Note: A sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$. | 18 | 85.9375 |
1,511 | Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book? | 250 | 100 |
1,512 | Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard? | 481 | 38.28125 |
1,513 | The letters $A$, $B$, $C$ and $D$ represent digits. If $\begin{array}{ccc}&A&B\\ +&C&A\\ \hline &D&A\end{array}$and $\begin{array}{ccc}&A&B\\ -&C&A\\ \hline &&A\end{array}$,what digit does $D$ represent? | 9 | 40.625 |
1,514 | If you walk for $45$ minutes at a rate of $4 \text{ mph}$ and then run for $30$ minutes at a rate of $10\text{ mph}$, how many miles will you have gone at the end of one hour and $15$ minutes? | 8 | 89.84375 |
1,515 | In the product shown, $\text{B}$ is a digit. The value of $\text{B}$ is
$\begin{array}{rr} &\text{B}2 \\ \times& 7\text{B} \\ \hline &6396 \\ \end{array}$ | 8 | 92.1875 |
1,516 | A box contains a collection of triangular and square tiles. There are $25$ tiles in the box, containing $84$ edges total. How many square tiles are there in the box? | 9 | 100 |
1,517 | A piece of paper containing six joined squares labeled as shown in the diagram is folded along the edges of the squares to form a cube. The label of the face opposite the face labeled $\text{X}$ is | Y | 0 |
1,518 | Suppose that the euro is worth 1.3 dollars. If Diana has 500 dollars and Etienne has 400 euros, by what percent is the value of Etienne's money greater that the value of Diana's money? | 4 | 32.03125 |
1,519 | The area of the shaded region $\text{BEDC}$ in parallelogram $\text{ABCD}$ is | 64 | 0 |
1,520 | Sandwiches at Joe's Fast Food cost $3 each and sodas cost $2 each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? | 31 | 93.75 |
1,521 | A circle passes through the vertices of a triangle with side-lengths $7\tfrac{1}{2},10,12\tfrac{1}{2}.$ The radius of the circle is: | \frac{25}{4} | 53.125 |
1,522 | The lines $x = \frac{1}{4}y + a$ and $y = \frac{1}{4}x + b$ intersect at the point $(1,2)$. What is $a + b$? | \frac{9}{4} | 89.84375 |
1,523 | A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? | 9 | 67.96875 |
1,524 | Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? | 20\pi + 40 | 0 |
1,525 | The faces of each of $7$ standard dice are labeled with the integers from $1$ to $6$. Let $p$ be the probabilities that when all $7$ dice are rolled, the sum of the numbers on the top faces is $10$. What other sum occurs with the same probability as $p$? | 39 | 56.25 |
1,526 | Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$? | 45 | 80.46875 |
1,527 | The horizontal and vertical distances between adjacent points equal 1 unit. What is the area of triangle $ABC$? | \frac{1}{2} | 0 |
1,528 | Tamara has three rows of two $6$-feet by $2$-feet flower beds in her garden. The beds are separated and also surrounded by $1$-foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet? | 78 | 6.25 |
1,529 | Handy Aaron helped a neighbor $1 \frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\textdollar 3$ per hour. How much did he earn for the week? | \$15 | 68.75 |
1,530 | Let $a_1, a_2, \dots, a_{2018}$ be a strictly increasing sequence of positive integers such that $a_1 + a_2 + \cdots + a_{2018} = 2018^{2018}$. What is the remainder when $a_1^3 + a_2^3 + \cdots + a_{2018}^3$ is divided by $6$? | 2 | 0 |
1,531 | For real numbers $x$ and $y$, define $x \spadesuit y = (x+y)(x-y)$. What is $3 \spadesuit (4 \spadesuit 5)$? | -72 | 91.40625 |
1,532 | How many distinguishable rearrangements of the letters in $CONTEST$ have both the vowels first? (For instance, $OETCNST$ is one such arrangement but $OTETSNC$ is not.) | 120 | 89.0625 |
1,533 | Cindy was asked by her teacher to subtract 3 from a certain number and then divide the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the problem correctly? | 15 | 94.53125 |
1,534 | The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people? | 33 | 98.4375 |
1,535 | Soda is sold in packs of 6, 12 and 24 cans. What is the minimum number of packs needed to buy exactly 90 cans of soda? | 5 | 87.5 |
1,536 | On each horizontal line in the figure below, the five large dots indicate the populations of cities $A, B, C, D$ and $E$ in the year indicated.
Which city had the greatest percentage increase in population from $1970$ to $1980$? | C | 17.96875 |
1,537 | If $a, b, c$ are real numbers such that $a^2 + 2b = 7$, $b^2 + 4c = -7$, and $c^2 + 6a = -14$, find $a^2 + b^2 + c^2$. | 14 | 88.28125 |
1,538 | For real numbers $a$ and $b$, define $a\textdollar b = (a - b)^2$. What is $(x - y)^2\textdollar(y - x)^2$? | 0 | 91.40625 |
1,539 | A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?
[asy]
unitsize(2mm); defaultpen(linewidth(.8pt));
fill((0,0)--(0,5)--(5,5)--cycle,gray);
fill((25,0)--(25,5)--(20,5)--cycle,gray);
draw((0,0)--(0,5)--(25,5)--(25,0)--cycle);
draw((0,0)--(5,5));
draw((20,5)--(25,0));
[/asy] | \frac{1}{5} | 61.71875 |
1,540 | If $4^x - 4^{x - 1} = 24$, then $(2x)^x$ equals: | 25\sqrt{5} | 66.40625 |
1,541 | In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(110);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy] | 12 | 11.71875 |
1,542 | Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? | 5 | 91.40625 |
1,543 | Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Green expect from his garden? | 600 | 90.625 |
1,544 | A privateer discovers a merchantman $10$ miles to leeward at 11:45 a.m. and with a good breeze bears down upon her at $11$ mph, while the merchantman can only make $8$ mph in her attempt to escape. After a two hour chase, the top sail of the privateer is carried away; she can now make only $17$ miles while the merchantman makes $15$. The privateer will overtake the merchantman at: | $5\text{:}30\text{ p.m.}$ | 0 |
1,545 | Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$ | 360 | 1.5625 |
1,546 | In rectangle $ABCD$, $DC = 2 \cdot CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD$? | \frac{3\sqrt{3}}{16} | 0 |
1,547 | A point $(x, y)$ is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line $x+y=2$. Then $x$ is | 1 | 2.34375 |
1,548 | The values of $y$ which will satisfy the equations $2x^{2}+6x+5y+1=0$, $2x+y+3=0$ may be found by solving: | $y^{2}+10y-7=0$ | 0 |
1,549 | An American traveling in Italy wishes to exchange American money (dollars) for Italian money (lire). If 3000 lire = 1.60, how much lire will the traveler receive in exchange for 1.00? | 1875 | 81.25 |
1,550 | What is the probability that a randomly drawn positive factor of $60$ is less than $7$? | \frac{1}{2} | 85.9375 |
1,551 | All three vertices of $\triangle ABC$ lie on the parabola defined by $y=x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis. The area of the triangle is $64$. What is the length of $BC$? | 8 | 95.3125 |
1,552 | Consider the set of all equations $x^3 + a_2x^2 + a_1x + a_0 = 0$, where $a_2$, $a_1$, $a_0$ are real constants and $|a_i| < 2$ for $i = 0,1,2$. Let $r$ be the largest positive real number which satisfies at least one of these equations. Then | \frac{5}{2} < r < 3 | 0 |
1,553 | 2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k} is equal to | -2^{-(2k+1)} | 49.21875 |
1,554 | How many integers $x$ satisfy the equation $(x^2-x-1)^{x+2}=1?$ | 4 | 97.65625 |
1,555 | Let $x_1$ and $x_2$ be such that $x_1 \not= x_2$ and $3x_i^2-hx_i=b$, $i=1, 2$. Then $x_1+x_2$ equals | -\frac{h}{3} | 0 |
1,556 | In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.
\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list? | 142 | 97.65625 |
1,557 | $\sqrt{\frac{1}{9} + \frac{1}{16}} = $ | \frac{5}{12} | 96.09375 |
1,558 | The altitude drawn to the base of an isosceles triangle is $8$, and the perimeter $32$. The area of the triangle is: | 48 | 59.375 |
1,559 | How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? | 4 | 98.4375 |
1,560 | There are twenty-four $4$-digit numbers that use each of the four digits $2$, $4$, $5$, and $7$ exactly once. Listed in numerical order from smallest to largest, the number in the $17\text{th}$ position in the list is | 5724 | 89.84375 |
1,561 | The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is: | 2 | 91.40625 |
1,562 | Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. What is the sum of the minimum values of $P(x)$ and $Q(x)$? | -100 | 0 |
1,563 | 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 | 78.125 |
1,564 | If $r$ and $s$ are the roots of $x^2-px+q=0$, then $r^2+s^2$ equals: | p^2-2q | 97.65625 |
1,565 | The degree of $(x^2+1)^4 (x^3+1)^3$ as a polynomial in $x$ is | 17 | 82.8125 |
1,566 | The ratio of the interior angles of two regular polygons with sides of unit length is $3: 2$. How many such pairs are there? | 3 | 15.625 |
1,567 | Homer began peeling a pile of 44 potatoes at the rate of 3 potatoes per minute. Four minutes later Christen joined him and peeled at the rate of 5 potatoes per minute. When they finished, how many potatoes had Christen peeled? | 20 | 77.34375 |
1,568 | The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6 \times 5$. What is the missing number in the top row? | 4 | 0.78125 |
1,569 | 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 | 23.4375 |
1,570 | Each piece of candy in a store costs a whole number of cents. Casper has exactly enough money to buy either 12 pieces of red candy, 14 pieces of green candy, 15 pieces of blue candy, or $n$ pieces of purple candy. A piece of purple candy costs 20 cents. What is the smallest possible value of $n$? | 21 | 97.65625 |
1,571 | Find $i + 2i^2 +3i^3 + ... + 2002i^{2002}.$ | -1001 + 1000i | 0 |
1,572 | Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:57 and 36 seconds. Assuming that her watch loses time at a constant rate, what will be the actual time when her watch first reads 10:00 PM? | 10:25 PM | 0.78125 |
1,573 | The annual incomes of $1,000$ families range from $8200$ dollars to $98,000$ dollars. In error, the largest income was entered on the computer as $980,000$ dollars. The difference between the mean of the incorrect data and the mean of the actual data is | 882 | 58.59375 |
1,574 | The minimum value of the quotient of a (base ten) number of three different non-zero digits divided by the sum of its digits is | 10.5 | 67.96875 |
1,575 | Assume the adjoining chart shows the $1980$ U.S. population, in millions, for each region by ethnic group. To the nearest percent, what percent of the U.S. Black population lived in the South?
\begin{tabular}{|c|cccc|} \hline & NE & MW & South & West \\ \hline White & 42 & 52 & 57 & 35 \\ Black & 5 & 5 & 15 & 2 \\ Asian & 1 & 1 & 1 & 3 \\ Other & 1 & 1 & 2 & 4 \\ \hline \end{tabular} | 56\% | 95.3125 |
1,576 | In rectangle $ABCD$, we have $A=(6,-22)$, $B=(2006,178)$, $D=(8,y)$, for some integer $y$. What is the area of rectangle $ABCD$? | 40400 | 7.8125 |
1,577 | If $a$ and $b$ are positive numbers such that $a^b=b^a$ and $b=9a$, then the value of $a$ is | \sqrt[4]{3} | 27.34375 |
1,578 | The remainder when the product $1492 \cdot 1776 \cdot 1812 \cdot 1996$ is divided by 5 is | 4 | 96.09375 |
1,579 | The arithmetic mean between $\frac {x + a}{x}$ and $\frac {x - a}{x}$, when $x \not = 0$, is: | 1 | 98.4375 |
1,580 | Two right circular cones with vertices facing down as shown in the figure below contain the same amount of liquid. The radii of the tops of the liquid surfaces are $3$ cm and $6$ cm. Into each cone is dropped a spherical marble of radius $1$ cm, which sinks to the bottom and is completely submerged without spilling any liquid. What is the ratio of the rise of the liquid level in the narrow cone to the rise of the liquid level in the wide cone? | 4:1 | 0 |
1,581 | The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. How many integers less than $2013$ but greater than $1000$ have this property? | 46 | 92.1875 |
1,582 | A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon? | \sqrt{6} | 95.3125 |
1,583 | Given that $i^2=-1$, for how many integers $n$ is $(n+i)^4$ an integer? | 3 | 82.03125 |
1,584 | In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is | \frac{7}{9} | 2.34375 |
1,585 | Abby, Bridget, and four of their classmates will be seated in two rows of three for a group picture, as shown.
\begin{tabular}{ccc} X & X & X \\ X & X & X \end{tabular}
If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column? | \frac{7}{15} | 9.375 |
1,586 | The roots of $64x^3-144x^2+92x-15=0$ are in arithmetic progression. The difference between the largest and smallest roots is: | 1 | 82.8125 |
1,587 | A circle of radius $10$ inches has its center at the vertex $C$ of an equilateral triangle $ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. The number of degrees of angle $ADB$ is: | 90 | 0 |
1,588 | Danica drove her new car on a trip for a whole number of hours, averaging $55$ miles per hour. At the beginning of the trip, $abc$ miles was displayed on the odometer, where $abc$ is a $3$-digit number with $a\ge1$ and $a+b+c\le7$. At the end of the trip, the odometer showed $cba$ miles. What is $a^2+b^2+c^2$? | 37 | 72.65625 |
1,589 | Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^3 - 22x^2 + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that \begin{equation*} \frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r} \end{equation*} for all $s\not\in\{p,q,r\}$. What is $\tfrac1A+\tfrac1B+\tfrac1C$? | 244 | 74.21875 |
1,590 | A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A = (-3, 2)$ and $B = (3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths? | 195 | 6.25 |
1,591 | A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so that the inner corners each touch an inner corner of an adjacent mat. What is $x$? | $\frac{3\sqrt{7}-\sqrt{3}}{2}$ | 0 |
1,592 | The price of an article was increased $p\%$. Later the new price was decreased $p\%$. If the last price was one dollar, the original price was: | \frac{10000}{10000-p^2} | 95.3125 |
1,593 | What is $\frac{2+4+6}{1+3+5} - \frac{1+3+5}{2+4+6}$ ? | \frac{7}{12} | 94.53125 |
1,594 | Riders on a Ferris wheel travel in a circle in a vertical plane. A particular wheel has radius $20$ feet and revolves at the constant rate of one revolution per minute. How many seconds does it take a rider to travel from the bottom of the wheel to a point $10$ vertical feet above the bottom? | 10 | 43.75 |
1,595 | A semicircle is inscribed in an isosceles triangle with base 16 and height 15 so that the diameter of the semicircle is contained in the base of the triangle. What is the radius of the semicircle? | \frac{120}{17} | 6.25 |
1,596 | The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make? | 13 | 85.15625 |
1,597 | Given a geometric progression of five terms, each a positive integer less than $100$. The sum of the five terms is $211$. If $S$ is the sum of those terms in the progression which are squares of integers, then $S$ is: | 133 | 0 |
1,598 | The area of a circle whose circumference is $24\pi$ is $k\pi$. What is the value of $k$? | 144 | 91.40625 |
1,599 | Jacob uses the following procedure to write down a sequence of numbers. First he chooses the first term to be 6. To generate each succeeding term, he flips a fair coin. If it comes up heads, he doubles the previous term and subtracts 1. If it comes up tails, he takes half of the previous term and subtracts 1. What is the probability that the fourth term in Jacob's sequence is an integer? | \frac{5}{8} | 16.40625 |
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