problem stringlengths 10 5.15k | answer dict |
|---|---|
In the diagram below, $ABCD$ is a rectangle with side lengths $AB=3$ and $BC=11$, and $AECF$ is a rectangle with side lengths $AF=7$ and $FC=9,$ as shown. The area of the shaded region common to the interiors of both rectangles is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular parallelepiped shown, $AB = 3$, $BC = 1$, and $CG = 2$. Point $M$ is the midpoint of $\overline{FG}$. What is the volume of the rectangular pyramid with base $BCHE$ and apex $M$? | {
"answer": "\\frac{4}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The dimensions of a rectangle $R$ are $a$ and $b$, $a < b$. It is required to obtain a rectangle with dimensions $x$ and $y$, $x < a, y < a$, so that its perimeter is one-third that of $R$, and its area is one-third that of $R$. The number of such (different) rectangles is: | {
"answer": "0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $ABC$ the medians $AM$ and $CN$ to sides $BC$ and $AB$, respectively, intersect in point $O$. $P$ is the midpoint of side $AC$, and $MP$ intersects $CN$ in $Q$. If the area of triangle $OMQ$ is $n$, then the area of triangle $ABC$ is: | {
"answer": "24n",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twenty cubical blocks are arranged as shown. First, 10 are arranged in a triangular pattern; then a layer of 6, arranged in a triangular pattern, is centered on the 10; then a layer of 3, arranged in a triangular pattern, is centered on the 6; and finally one block is centered on top of the third layer. The blocks in the bottom layer are numbered 1 through 10 in some order. Each block in layers 2,3 and 4 is assigned the number which is the sum of numbers assigned to the three blocks on which it rests. Find the smallest possible number which could be assigned to the top block. | {
"answer": "114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If, in the expression $x^2 - 3$, $x$ increases or decreases by a positive amount of $a$, the expression changes by an amount: | {
"answer": "$\\pm 2ax + a^2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is | {
"answer": "2\\pi+4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What number is directly above $142$ in this array of numbers?
\[\begin{array}{cccccc}& & & 1 & &\\ & & 2 & 3 & 4 &\\ & 5 & 6 & 7 & 8 & 9\\ 10 & 11 & 12 &\cdots & &\\ \end{array}\] | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every integer $n\ge2$, let $\text{pow}(n)$ be the largest power of the largest prime that divides $n$. For example $\text{pow}(144)=\text{pow}(2^4\cdot3^2)=3^2$. What is the largest integer $m$ such that $2010^m$ divides
$\prod_{n=2}^{5300}\text{pow}(n)$? | {
"answer": "77",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $X$, $Y$ and $Z$ are different digits, then the largest possible $3-$digit sum for
$\begin{array}{ccc} X & X & X \ & Y & X \ + & & X \ \hline \end{array}$
has the form | {
"answer": "$YYZ$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? | {
"answer": "\\frac{3 + \\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? | {
"answer": "\\frac{223}{286}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The grading scale shown is used at Jones Junior High. The fifteen scores in Mr. Freeman's class were: \(\begin{tabular}[t]{lllllllll} 89, & 72, & 54, & 97, & 77, & 92, & 85, & 74, & 75, \\ 63, & 84, & 78, & 71, & 80, & 90. & & & \\ \end{tabular}\)
In Mr. Freeman's class, what percent of the students received a grade of C? | {
"answer": "33\\frac{1}{3}\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square of side length $1$ and a circle of radius $\frac{\sqrt{3}}{3}$ share the same center. What is the area inside the circle, but outside the square? | {
"answer": "\\frac{2\\pi}{9} - \\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
(6?3) + 4 - (2 - 1) = 5. To make this statement true, the question mark between the 6 and the 3 should be replaced by | {
"answer": "\\div",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each corner of a rectangular prism is cut off. Two (of the eight) cuts are shown. How many edges does the new figure have?
[asy] draw((0,0)--(3,0)--(3,3)--(0,3)--cycle); draw((3,0)--(5,2)--(5,5)--(2,5)--(0,3)); draw((3,3)--(5,5)); draw((2,0)--(3,1.8)--(4,1)--cycle,linewidth(1)); draw((2,3)--(4,4)--(3,2)--cycle,linewidth(1)); [/asy]
Assume that the planes cutting the prism do not intersect anywhere in or on the prism. | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What percent of the birds that were not swans were geese? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many positive factors does $23,232$ have? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five positive consecutive integers starting with $a$ have average $b$. What is the average of 5 consecutive integers that start with $b$? | {
"answer": "$a+4$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper whose length is $\sqrt{3}$ times the width has area $A$. The paper is divided into three equal sections along the opposite lengths, and then a dotted line is drawn from the first divider to the second divider on the opposite side as shown. The paper is then folded flat along this dotted line to create a new shape with area $B$. What is the ratio $\frac{B}{A}$? | {
"answer": "\\frac{4}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $i^2 = -1$, then the sum $\cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} + \cdots + i^{40}\cos{3645^\circ}$ equals | {
"answer": "\\frac{\\sqrt{2}}{2}(21 - 20i)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a regular tetrahedron and Let $E$ be a point inside the face $ABC.$ Denote by $s$ the sum of the distances from $E$ to the faces $DAB, DBC, DCA,$ and by $S$ the sum of the distances from $E$ to the edges $AB, BC, CA.$ Then $\frac{s}{S}$ equals | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Problems 8, 9 and 10 use the data found in the accompanying paragraph and figures
Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.
$\circ$ Art's cookies are trapezoids:
$\circ$ Roger's cookies are rectangles:
$\circ$ Paul's cookies are parallelograms:
$\circ$ Trisha's cookies are triangles:
Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Art's cookies sell for $60$ cents each. To earn the same amount from a single batch, how much should one of Roger's cookies cost in cents? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the shaded figure shown below? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex and Felicia each have cats as pets. Alex buys cat food in cylindrical cans that are $6$ cm in diameter and $12$ cm high. Felicia buys cat food in cylindrical cans that are $12$ cm in diameter and $6$ cm high. What is the ratio of the volume of one of Alex's cans to the volume of one of Felicia's cans? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What expression describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points? | {
"answer": "a>\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 $x$? | {
"answer": "$\\frac{3\\sqrt{7}-\\sqrt{3}}{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the radius of a circle is increased $100\%$, the area is increased: | {
"answer": "300\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The equation $x^{x^{x^{.^{.^.}}}}=2$ is satisfied when $x$ is equal to: | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins? | {
"answer": "48\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $n$ be the smallest nonprime integer greater than $1$ with no prime factor less than $10$. Then | {
"answer": "120 < n \\leq 130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The three row sums and the three column sums of the array \[
\left[\begin{matrix}4 & 9 & 2\\ 8 & 1 & 6\\ 3 & 5 & 7\end{matrix}\right]
\]
are the same. What is the least number of entries that must be altered to make all six sums different from one another? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x>y>0$ , then $\frac{x^y y^x}{y^y x^x}=$ | {
"answer": "{\\left(\\frac{x}{y}\\right)}^{y-x}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In their base $10$ representations, the integer $a$ consists of a sequence of $1985$ eights and the integer $b$ consists of a sequence of $1985$ fives. What is the sum of the digits of the base $10$ representation of $9ab$? | {
"answer": "17865",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\frac{x}{y}$? | {
"answer": "\\frac{37}{35}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The figure below shows line $\ell$ with a regular, infinite, recurring pattern of squares and line segments.
[asy] size(300); defaultpen(linewidth(0.8)); real r = 0.35; path P = (0,0)--(0,1)--(1,1)--(1,0), Q = (1,1)--(1+r,1+r); path Pp = (0,0)--(0,-1)--(1,-1)--(1,0), Qp = (-1,-1)--(-1-r,-1-r); for(int i=0;i <= 4;i=i+1) { draw(shift((4*i,0)) * P); draw(shift((4*i,0)) * Q); } for(int i=1;i <= 4;i=i+1) { draw(shift((4*i-2,0)) * Pp); draw(shift((4*i-1,0)) * Qp); } draw((-1,0)--(18.5,0),Arrows(TeXHead)); [/asy]
How many of the following four kinds of rigid motion transformations of the plane in which this figure is drawn, other than the identity transformation, will transform this figure into itself?
some rotation around a point of line $\ell$
some translation in the direction parallel to line $\ell$
the reflection across line $\ell$
some reflection across a line perpendicular to line $\ell$ | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $N$ be the greatest five-digit number whose digits have a product of $120$. What is the sum of the digits of $N$? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Azar and Carl play a game of tic-tac-toe. Azar places an \(X\) in one of the boxes in a \(3\)-by-\(3\) array of boxes, then Carl places an \(O\) in one of the remaining boxes. After that, Azar places an \(X\) in one of the remaining boxes, and so on until all boxes are filled or one of the players has of their symbols in a row—horizontal, vertical, or diagonal—whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third \(O\). How many ways can the board look after the game is over? | {
"answer": "148",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose that $p$ and $q$ are positive numbers for which $\log_{9}(p) = \log_{12}(q) = \log_{16}(p+q)$. What is the value of $\frac{q}{p}$? | {
"answer": "\\frac{1+\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that
$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1$
What is the maximum value of $P(a)?$ | {
"answer": "2 - \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fraction $\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}}$ is equal to | {
"answer": "\\frac43",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the numbers, $100,101,\cdots,999$ have three different digits in increasing order or in decreasing order? | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The degree measure of angle $A$ is | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \times 2 \times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\frac{m}{n}?$ | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle has vertices $(0,0)$, $(1,1)$, and $(6m,0)$. The line $y = mx$ divides the triangle into two triangles of equal area. What is the sum of all possible values of $m$? | {
"answer": "- \\frac {1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What percent of the area of rectangle $AQRD$ is shaded? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is: | {
"answer": "\\sqrt{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let side $AD$ of convex quadrilateral $ABCD$ be extended through $D$, and let side $BC$ be extended through $C$, to meet in point $E.$ Let $S$ be the degree-sum of angles $CDE$ and $DCE$, and let $S'$ represent the degree-sum of angles $BAD$ and $ABC.$ If $r=S/S'$, then: | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three young brother-sister pairs from different families need to take a trip in a van. These six children will occupy the second and third rows in the van, each of which has three seats. To avoid disruptions, siblings may not sit right next to each other in the same row, and no child may sit directly in front of his or her sibling. How many seating arrangements are possible for this trip? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. What is the area of the circle passing through $A$, $O$, and $C$? | {
"answer": "12\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three congruent isosceles triangles are constructed with their bases on the sides of an equilateral triangle of side length $1$. The sum of the areas of the three isosceles triangles is the same as the area of the equilateral triangle. What is the length of one of the two congruent sides of one of the isosceles triangles? | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
[asy]
dot((0,0)); dot((.5,.5)); dot((.5,0)); dot((.0,.5)); dot((1,0)); dot((1,.5)); dot((1.5,0)); dot((1.5,.5));
[/asy] | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A merchant bought some goods at a discount of $20\%$ of the list price. He wants to mark them at such a price that he can give a discount of $20\%$ of the marked price and still make a profit of $20\%$ of the selling price. The per cent of the list price at which he should mark them is: | {
"answer": "125",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The expression $(81)^{-2^{-2}}$ has the same value as: | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For every real number $x$, let $[x]$ be the greatest integer which is less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always | {
"answer": "-6[-W]",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? | {
"answer": "210",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $AB$ and $CD$ are perpendicular diameters of circle $Q$, $P$ in $\overline{AQ}$, and $\angle QPC = 60^\circ$, then the length of $PQ$ divided by the length of $AQ$ is | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A "stair-step" figure is made of alternating black and white squares in each row. Rows $1$ through $4$ are shown. All rows begin and end with a white square. The number of black squares in the $37\text{th}$ row is
[asy]
draw((0,0)--(7,0)--(7,1)--(0,1)--cycle);
draw((1,0)--(6,0)--(6,2)--(1,2)--cycle);
draw((2,0)--(5,0)--(5,3)--(2,3)--cycle);
draw((3,0)--(4,0)--(4,4)--(3,4)--cycle);
fill((1,0)--(2,0)--(2,1)--(1,1)--cycle,black);
fill((3,0)--(4,0)--(4,1)--(3,1)--cycle,black);
fill((5,0)--(6,0)--(6,1)--(5,1)--cycle,black);
fill((2,1)--(3,1)--(3,2)--(2,2)--cycle,black);
fill((4,1)--(5,1)--(5,2)--(4,2)--cycle,black);
fill((3,2)--(4,2)--(4,3)--(3,3)--cycle,black);
[/asy] | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square of perimeter 20 is inscribed in a square of perimeter 28. What is the greatest distance between a vertex of the inner square and a vertex of the outer square? | {
"answer": "\\sqrt{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\log 2 = .3010$ and $\log 3 = .4771$, the value of $x$ when $3^{x+3} = 135$ is approximately | {
"answer": "1.47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)--(1,2+sqrt(2))--(0,2+sqrt(2))--(-sqrt(2)/2,2+sqrt(2)/2)--(-sqrt(2)/2,1+sqrt(2)/2)--cycle; draw(p); draw(shift((1+sqrt(2)/2,-sqrt(2)/2-1))*p); draw(shift((0,-2-sqrt(2)))*p); draw(shift((-1-sqrt(2)/2,-sqrt(2)/2-1))*p);[/asy] | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE$? | {
"answer": "170",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A game board consists of $64$ squares that alternate in color between black and white. The figure below shows square $P$ in the bottom row and square $Q$ in the top row. A marker is placed at $P.$ A step consists of moving the marker onto one of the adjoining white squares in the row above. How many $7$-step paths are there from $P$ to $Q?$ (The figure shows a sample path.) | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let S be the statement "If the sum of the digits of the whole number $n$ is divisible by $6$, then $n$ is divisible by $6$." A value of $n$ which shows $S$ to be false is | {
"answer": "33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man buys a house for $10,000 and rents it. He puts $12\frac{1}{2}\%$ of each month's rent aside for repairs and upkeep; pays $325 a year taxes and realizes $5\frac{1}{2}\%$ on his investment. The monthly rent (in dollars) is: | {
"answer": "83.33",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x < a < 0$ means that $x$ and $a$ are numbers such that $x$ is less than $a$ and $a$ is less than zero, then: | {
"answer": "$x^2 > ax > a^2$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a geometric progression whose terms are positive, any term is equal to the sum of the next two following terms. then the common ratio is: | {
"answer": "\\frac{\\sqrt{5}-1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$? | {
"answer": "14",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The edges of a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ each have length one. Find the least possible distance between a pair of points $P$ and $Q$, where $P$ is on edge $AB$ and $Q$ is on edge $CD$. | {
"answer": "\\frac{\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: | {
"answer": "-12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A piece of string is cut in two at a point selected at random. The probability that the longer piece is at least x times as large as the shorter piece is | {
"answer": "\\frac{2}{x+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the volume of a cube whose surface area is twice that of a cube with volume 1? | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ace runs with constant speed and Flash runs $x$ times as fast, $x>1$. Flash gives Ace a head start of $y$ yards, and, at a given signal, they start off in the same direction. Then the number of yards Flash must run to catch Ace is: | {
"answer": "\\frac{xy}{x-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, then what fraction of the tiled floor is made of darker tiles? | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of $\angle GDA$ is | {
"answer": "120^{\\circ}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded? | {
"answer": "\\frac{7}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The quiz scores of a class with $k > 12$ students have a mean of $8$. The mean of a collection of $12$ of these quiz scores is $14$. What is the mean of the remaining quiz scores in terms of $k$? | {
"answer": "\\frac{8k-168}{k-12}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many rearrangements of $abcd$ are there in which no two adjacent letters are also adjacent letters in the alphabet? For example, no such rearrangements could include either $ab$ or $ba$. | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several students are seated at a large circular table. They pass around a bag containing $100$ pieces of candy. Each person receives the bag, takes one piece of candy and then passes the bag to the next person. If Chris takes the first and last piece of candy, then the number of students at the table could be | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ten chairs are evenly spaced around a round table and numbered clockwise from $1$ through $10$. Five married couples are to sit in the chairs with men and women alternating, and no one is to sit either next to or across from his/her spouse. How many seating arrangements are possible? | {
"answer": "480",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider this histogram of the scores for $81$ students taking a test:
The median is in the interval labeled | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $r_1$ and $r_2$ are the distinct real roots of $x^2+px+8=0$, then it must follow that: | {
"answer": "$|r_1+r_2|>4\\sqrt{2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square? | {
"answer": "29\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. How many of the $125$ sequences of three of these transformations (not necessarily distinct) will return $T$ to its original position? (For example, a $180^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by a reflection across the $y$-axis will return $T$ to its original position, but a $90^{\circ}$ rotation, followed by a reflection across the $x$-axis, followed by another reflection across the $x$-axis will not return $T$ to its original position.) | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\frac{1}{10} + \frac{2}{10} + \frac{3}{10} + \frac{4}{10} + \frac{5}{10} + \frac{6}{10} + \frac{7}{10} + \frac{8}{10} + \frac{9}{10} + \frac{55}{10}= | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to | {
"answer": "2x+1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$,
$AD = AB = 3$, and $DC = 6$. In addition, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. Find the area of $\triangle BEC$. | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pat Peano has plenty of 0's, 1's, 3's, 4's, 5's, 6's, 7's, 8's and 9's, but he has only twenty-two 2's. How far can he number the pages of his scrapbook with these digits? | {
"answer": "119",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the set of equations $z^x = y^{2x}$, $2^z = 2 \cdot 4^x$, $x + y + z = 16$, the integral roots in the order $x,y,z$ are: | {
"answer": "4,3,9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the margin made on an article costing $C$ dollars and selling for $S$ dollars is $M=\frac{1}{n}C$, then the margin is given by: | {
"answer": "\\frac{1}{n+1}S",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end? | {
"answer": "103",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$? | {
"answer": "11",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many polynomials of the form $x^5 + ax^4 + bx^3 + cx^2 + dx + 2020$, where $a$, $b$, $c$, and $d$ are real numbers, have the property that whenever $r$ is a root, so is $\frac{-1+i\sqrt{3}}{2} \cdot r$? (Note that $i=\sqrt{-1}$) | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall? | {
"answer": "353",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bar graph shows the number of hamburgers sold by a fast food chain each season. However, the bar indicating the number sold during the winter is covered by a smudge. If exactly $25\%$ of the chain's hamburgers are sold in the fall, how many million hamburgers are sold in the winter? | {
"answer": "2.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? | {
"answer": "65",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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