problem stringlengths 10 5.15k | answer dict |
|---|---|
Equilateral $\triangle ABC$ has side length $1$, and squares $ABDE$, $BCHI$, $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$? | {
"answer": "3+\\sqrt3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many integers $n$ between $1$ and $50$, inclusive, is $\frac{(n^2-1)!}{(n!)^n}$ an integer? | {
"answer": "34",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All students at Adams High School and at Baker High School take a certain exam. The average scores for boys, for girls, and for boys and girls combined, at Adams HS and Baker HS are shown in the table, as is the average for boys at the two schools combined. What is the average score for the girls at the two schools combined?
$\begin{tabular}[t]{|c|c|c|c|} \multicolumn{4}{c}{Average Scores}\\ \hline Category&Adams&Baker&Adams\&Baker\\ \hline Boys&71&81&79\\ Girls&76&90&?\\ Boys\&Girls&74&84& \\ \hline \end{tabular}$
| {
"answer": "84",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. For what value of $n$ does $a_n=1+\cos x$? | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A positive number $x$ satisfies the inequality $\sqrt{x} < 2x$ if and only if | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 $|x-y| > \tfrac{1}{2}$? | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$ | {
"answer": "16:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three cards, each with a positive integer written on it, are lying face-down on a table. Casey, Stacy, and Tracy are told that
(a) the numbers are all different,
(b) they sum to $13$, and
(c) they are in increasing order, left to right.
First, Casey looks at the number on the leftmost card and says, "I don't have enough information to determine the other two numbers." Then Tracy looks at the number on the rightmost card and says, "I don't have enough information to determine the other two numbers." Finally, Stacy looks at the number on the middle card and says, "I don't have enough information to determine the other two numbers." Assume that each person knows that the other two reason perfectly and hears their comments. What number is on the middle card? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten tiles numbered $1$ through $10$ are turned face down. One tile is turned up at random, and a die is rolled. What is the probability that the product of the numbers on the tile and the die will be a square? | {
"answer": "\\frac{11}{60}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure, $\triangle ABC$ has $\angle A =45^{\circ}$ and $\angle B =30^{\circ}$. A line $DE$, with $D$ on $AB$
and $\angle ADE =60^{\circ}$, divides $\triangle ABC$ into two pieces of equal area.
(Note: the figure may not be accurate; perhaps $E$ is on $CB$ instead of $AC.)$
The ratio $\frac{AD}{AB}$ is | {
"answer": "\\frac{1}{\\sqrt[4]{12}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Reduced to lowest terms, $\frac{a^{2}-b^{2}}{ab} - \frac{ab-b^{2}}{ab-a^{2}}$ is equal to: | {
"answer": "\\frac{a}{b}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three cubes are each formed from the pattern shown. They are then stacked on a table one on top of another so that the $13$ visible numbers have the greatest possible sum. What is that sum? | {
"answer": "164",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(\triangle ABC\) is isosceles with base \(AC\). Points \(P\) and \(Q\) are respectively in \(CB\) and \(AB\) and such that \(AC=AP=PQ=QB\).
The number of degrees in \(\angle B\) is: | {
"answer": "25\\frac{5}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four balls of radius $1$ are mutually tangent, three resting on the floor and the fourth resting on the others.
A tetrahedron, each of whose edges have length $s$, is circumscribed around the balls. Then $s$ equals | {
"answer": "2+2\\sqrt{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
[asy] size(0,50); draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle); dot((-1,1)); dot((-2,2)); dot((-3,1)); dot((-2,0)); draw((1,0){up}..{left}(0,1)); dot((1,0)); dot((0,1)); draw((0,1){right}..{up}(1,2)); dot((1,2)); draw((1,2){down}..{right}(2,1)); dot((2,1)); draw((2,1){left}..{down}(1,0));[/asy] | {
"answer": "\\frac{4-\\pi}{\\pi}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $78$ is divided into three parts which are proportional to $1, \frac{1}{3}, \frac{1}{6},$ the middle part is: | {
"answer": "17\\frac{1}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive two-digit integers are factors of $2^{24}-1$?~ pi_is_3.14 | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is | {
"answer": "2: 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular box measures $a \times b \times c$, where $a$, $b$, and $c$ are integers and $1\leq a \leq b \leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, what is the ratio $\dfrac{XQ}{QY}$? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If rose bushes are spaced about $1$ foot apart, approximately how many bushes are needed to surround a circular patio whose radius is $12$ feet? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jack had a bag of $128$ apples. He sold $25\%$ of them to Jill. Next he sold $25\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equivalent of the expression
$\left(\frac{x^2+1}{x}\right)\left(\frac{y^2+1}{y}\right)+\left(\frac{x^2-1}{y}\right)\left(\frac{y^2-1}{x}\right)$, $xy \not= 0$,
is: | {
"answer": "2xy+\\frac{2}{xy}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The product $(1.8)(40.3 + .07)$ is closest to | {
"answer": "74",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle ACB$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle CDE$? | {
"answer": "\\frac{50-25\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are | {
"answer": "$3^{1/3},\\ 2^{1/2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the plane figure shown below, $3$ of the unit squares have been shaded. What is the least number of additional unit squares that must be shaded so that the resulting figure has two lines of symmetry?
[asy] import olympiad; unitsize(25); filldraw((1,3)--(1,4)--(2,4)--(2,3)--cycle, gray(0.7)); filldraw((2,1)--(2,2)--(3,2)--(3,1)--cycle, gray(0.7)); filldraw((4,0)--(5,0)--(5,1)--(4,1)--cycle, gray(0.7)); for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { pair A = (j,i); } } for (int i = 0; i < 5; ++i) { for (int j = 0; j < 6; ++j) { if (j != 5) { draw((j,i)--(j+1,i)); } if (i != 4) { draw((j,i)--(j,i+1)); } } } [/asy] | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < \dots < a_k$ such that\[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + \dots + 2^{a_k}.\]What is $k?$ | {
"answer": "137",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Paula the painter and her two helpers each paint at constant, but different, rates. They always start at 8:00 AM, and all three always take the same amount of time to eat lunch. On Monday the three of them painted 50% of a house, quitting at 4:00 PM. On Tuesday, when Paula wasn't there, the two helpers painted only 24% of the house and quit at 2:12 PM. On Wednesday Paula worked by herself and finished the house by working until 7:12 P.M. How long, in minutes, was each day's lunch break? | {
"answer": "48",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\triangle ABC$ is $50$ units and the area of $\triangle ABC$ is $100$ square units? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many sequences of zeros and ones of length 20 have all the zeros consecutive, or all the ones consecutive, or both? | {
"answer": "382",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The symbol $|a|$ means $a$ is a positive number or zero, and $-a$ if $a$ is a negative number.
For all real values of $t$ the expression $\sqrt{t^4+t^2}$ is equal to? | {
"answer": "|t|\\sqrt{1+t^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\,?$ | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many ordered triples $(x,y,z)$ of positive integers satisfy $\text{lcm}(x,y) = 72, \text{lcm}(x,z) = 600 \text{ and lcm}(y,z)=900$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $t = \frac{1}{1 - \sqrt[4]{2}}$, then $t$ equals | {
"answer": "$-(1+\\sqrt[4]{2})(1+\\sqrt{2})$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
$\begin{array}{cccccc}&A&B&B&C&B\ +&B&C&A&D&A\ \hline &D&B&D&D&D\end{array}$ | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The cost $C$ of sending a parcel post package weighing $P$ pounds, $P$ an integer, is $10$ cents for the first pound and $3$ cents for each additional pound. The formula for the cost is: | {
"answer": "C=10+3(P-1)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six different digits from the set \(\{ 1,2,3,4,5,6,7,8,9\}\) are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is | {
"answer": "29",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid. | {
"answer": "\\frac{5}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \cdot Q(z) = R(z)$. What is the minimum possible value of $N$? | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Six chairs are evenly spaced around a circular table. One person is seated in each chair. Each person gets up and sits down in a chair that is not the same and is not adjacent to the chair he or she originally occupied, so that again one person is seated in each chair. In how many ways can this be done? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four congruent rectangles are placed as shown. The area of the outer square is 4 times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?
[asy] unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p); [/asy] | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The percent that $M$ is greater than $N$ is: | {
"answer": "\\frac{100(M-N)}{N}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The graph below shows the total accumulated dollars (in millions) spent by the Surf City government during $1988$. For example, about $.5$ million had been spent by the beginning of February and approximately $2$ million by the end of April. Approximately how many millions of dollars were spent during the summer months of June, July, and August? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mr. Garcia asked the members of his health class how many days last week they exercised for at least 30 minutes. The results are summarized in the following bar graph, where the heights of the bars represent the number of students.
What was the mean number of days of exercise last week, rounded to the nearest hundredth, reported by the students in Mr. Garcia's class? | {
"answer": "4.36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the product $\frac{3}{2} \cdot \frac{4}{3} \cdot \frac{5}{4} \cdot \frac{6}{5} \cdot \ldots \cdot \frac{a}{b} = 9$, what is the sum of $a$ and $b$? | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before the yellow house. The blue house was not next to the yellow house. How many orderings of the colored houses are possible? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x=\frac{a}{b}$, $a\neq b$ and $b\neq 0$, then $\frac{a+b}{a-b}=$ | {
"answer": "\\frac{x+1}{x-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Keiko walks once around a track at the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of $6$ meters, and it takes her $36$ seconds longer to walk around the outside edge of the track than around the inside edge. What is Keiko's speed in meters per second? | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the areas of the three squares in the figure, what is the area of the interior triangle? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider $x^2+px+q=0$, where $p$ and $q$ are positive numbers. If the roots of this equation differ by 1, then $p$ equals | {
"answer": "\\sqrt{4q+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Harry has 3 sisters and 5 brothers. His sister Harriet has $\text{S}$ sisters and $\text{B}$ brothers. What is the product of $\text{S}$ and $\text{B}$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When simplified $\sqrt{1+ \left (\frac{x^4-1}{2x^2} \right )^2}$ equals: | {
"answer": "\\frac{x^2}{2}+\\frac{1}{2x^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then: | {
"answer": "a > 1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $f(x) = x^{2}(1-x)^{2}$. What is the value of the sum
\[f \left(\frac{1}{2019} \right)-f \left(\frac{2}{2019} \right)+f \left(\frac{3}{2019} \right)-f \left(\frac{4}{2019} \right)+\cdots + f \left(\frac{2017}{2019} \right) - f \left(\frac{2018}{2019} \right)?\] | {
"answer": "0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "729",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$ is equal to: | {
"answer": "\\frac{\\sqrt{3}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(x)$ is a real valued function of the real variable $x$, and $f(x)$ is not identically zero, and for all $a$ and $b$, $f(a+b)+f(a-b)=2f(a)+2f(b)$, then for all $x$ and $y$ | {
"answer": "$f(-x)=f(x)$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A trapezoid has side lengths 3, 5, 7, and 11. The sum of all the possible areas of the trapezoid can be written in the form of $r_1\sqrt{n_1}+r_2\sqrt{n_2}+r_3$, where $r_1$, $r_2$, and $r_3$ are rational numbers and $n_1$ and $n_2$ are positive integers not divisible by the square of any prime. What is the greatest integer less than or equal to $r_1+r_2+r_3+n_1+n_2$? | {
"answer": "63",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If 5 times a number is 2, then 100 times the reciprocal of the number is | {
"answer": "50",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The coordinates of $A, B$ and $C$ are $(5,5), (2,1)$ and $(0,k)$ respectively.
The value of $k$ that makes $\overline{AC}+\overline{BC}$ as small as possible is: | {
"answer": "2\\frac{1}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\log_{10}2=a$ and $\log_{10}3=b$, then $\log_{5}12=?$ | {
"answer": "\\frac{2a+b}{1-a}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$A$ and $B$ move uniformly along two straight paths intersecting at right angles in point $O$. When $A$ is at $O$, $B$ is $500$ yards short of $O$. In two minutes they are equidistant from $O$, and in $8$ minutes more they are again equidistant from $O$. Then the ratio of $A$'s speed to $B$'s speed is: | {
"answer": "5/6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? | {
"answer": "\\frac{n}{2n+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last year a bicycle cost $160 and a cycling helmet $40. This year the cost of the bicycle increased by $5\%$, and the cost of the helmet increased by $10\%$. The percent increase in the combined cost of the bicycle and the helmet is: | {
"answer": "6\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagram shows part of a scale of a measuring device. The arrow indicates an approximate reading of | {
"answer": "10.3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For distinct real numbers $x$ and $y$, let $M(x,y)$ be the larger of $x$ and $y$ and let $m(x,y)$ be the smaller of $x$ and $y$. If $a<b<c<d<e$, then
$M(M(a,m(b,c)),m(d,m(a,e)))=$ | {
"answer": "b",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 20^\circ$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then $\angle BDC =$ | {
"answer": "55^{\\circ}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{\text{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{\text{th}}$ pirate receive? | {
"answer": "1925",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a worker receives a $20$% cut in wages, he may regain his original pay exactly by obtaining a raise of: | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the unit circle shown in the figure, chords $PQ$ and $MN$ are parallel to the unit radius $OR$ of the circle with center at $O$. Chords $MP$, $PQ$, and $NR$ are each $s$ units long and chord $MN$ is $d$ units long.
Of the three equations
\begin{equation*} \label{eq:1} d-s=1, \qquad ds=1, \qquad d^2-s^2=\sqrt{5} \end{equation*}those which are necessarily true are | {
"answer": "I, II and III",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The statement $x^2 - x - 6 < 0$ is equivalent to the statement: | {
"answer": "-2 < x < 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set of values of $m$ for which $x^2+3xy+x+my-m$ has two factors, with integer coefficients, which are linear in $x$ and $y$, is precisely: | {
"answer": "0, 12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA=1$. The area of the "bat wings" (shaded area) is | {
"answer": "3 \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $\angle C = 90^\circ$ and $AB = 12$. Squares $ABXY$ and $CBWZ$ are constructed outside of the triangle. The points $X$, $Y$, $Z$, and $W$ lie on a circle. What is the perimeter of the triangle? | {
"answer": "12 + 12\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a set of four distinct lines in a plane, there are exactly $N$ distinct points that lie on two or more of the lines. What is the sum of all possible values of $N$? | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Distinct planes $p_1, p_2, \dots, p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $P = \bigcup_{j=1}^{k} p_j$. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Corners are sliced off a unit cube so that the six faces each become regular octagons. What is the total volume of the removed tetrahedra? | {
"answer": "\\frac{10-7\\sqrt{2}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $AB=13,BC=14$ and $AC=15$. Let $P$ be the point on $\overline{AC}$ such that $PC=10$. There are exactly two points $D$ and $E$ on line $BP$ such that quadrilaterals $ABCD$ and $ABCE$ are trapezoids. What is the distance $DE?$ | {
"answer": "12\\sqrt2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Segment $AB$ is both a diameter of a circle of radius $1$ and a side of an equilateral triangle $ABC$. The circle also intersects $AC$ and $BC$ at points $D$ and $E$, respectively. The length of $AE$ is | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? | {
"answer": "\\sqrt{95}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $i^2=-1$, then $(i-i^{-1})^{-1}=$ | {
"answer": "-\\frac{i}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}+\sqrt{5}} equals | {
"answer": "\\sqrt{2}+\\sqrt{3}-\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points $A=(6,13)$ and $B=(12,11)$ lie on circle $\omega$ in the plane. Suppose that the tangent lines to $\omega$ at $A$ and $B$ intersect at a point on the $x$-axis. What is the area of $\omega$? | {
"answer": "\\frac{85\\pi}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Raashan, Sylvia, and Ted play the following game. Each starts with $1. A bell rings every $15$ seconds, at which time each of the players who currently have money simultaneously chooses one of the other two players independently and at random and gives $1 to that player. What is the probability that after the bell has rung $2019$ times, each player will have $1? (For example, Raashan and Ted may each decide to give $1 to Sylvia, and Sylvia may decide to give her dollar to Ted, at which point Raashan will have $0, Sylvia will have $2, and Ted will have $1, and that is the end of the first round of play. In the second round Rashaan has no money to give, but Sylvia and Ted might choose each other to give their $1 to, and the holdings will be the same at the end of the second round.) | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Circles with centers $P, Q$ and $R$, having radii $1, 2$ and $3$, respectively, lie on the same side of line $l$ and are tangent to $l$ at $P', Q'$ and $R'$, respectively, with $Q'$ between $P'$ and $R'$. The circle with center $Q$ is externally tangent to each of the other two circles. What is the area of triangle $PQR$? | {
"answer": "0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In July 1861, $366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month? | {
"answer": "\\frac{366}{31 \\times 24}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that the roots of the polynomial $P(x)=x^3+ax^2+bx+c$ are $\cos \frac{2\pi}7,\cos \frac{4\pi}7,$ and $\cos \frac{6\pi}7$, where angles are in radians. What is $abc$? | {
"answer": "\\frac{1}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$. The area of the shaded "smile" $AEFBDA$, is | {
"answer": "$2\\pi-\\pi \\sqrt{2}-1$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jamal wants to save 30 files onto disks, each with 1.44 MB space. 3 of the files take up 0.8 MB, 12 of the files take up 0.7 MB, and the rest take up 0.4 MB. It is not possible to split a file onto 2 different disks. What is the smallest number of disks needed to store all 30 files? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Students from three middle schools worked on a summer project.
Seven students from Allen school worked for 3 days.
Four students from Balboa school worked for 5 days.
Five students from Carver school worked for 9 days.
The total amount paid for the students' work was 744. Assuming each student received the same amount for a day's work, how much did the students from Balboa school earn altogether? | {
"answer": "180.00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A contest began at noon one day and ended $1000$ minutes later. At what time did the contest end? | {
"answer": "4:40 a.m.",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then what is the value of $C$? | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The population of a small town is $480$. The graph indicates the number of females and males in the town, but the vertical scale-values are omitted. How many males live in the town? | {
"answer": "160",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0,n]$ for some positive integer $n$. The probability that no two of $x$, $y$, and $z$ are within 1 unit of each other is greater than $\frac {1}{2}$. What is the smallest possible value of $n$? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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