problem stringlengths 10 5.15k | answer dict |
|---|---|
How many $7$-digit palindromes (numbers that read the same backward as forward) can be formed using the digits $2$, $2$, $3$, $3$, $5$, $5$, $5$? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. What is the probability that after they are painted, the cubes can be rotated to be identical in appearance? | {
"answer": "\\frac{147}{1024}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x<0$, then $|x-\sqrt{(x-1)^2}|$ equals | {
"answer": "1-2x",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Of the following sets, the one that includes all values of $x$ which will satisfy $2x - 3 > 7 - x$ is: | {
"answer": "$x >\\frac{10}{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An athlete's target heart rate, in beats per minute, is $80\%$ of the theoretical maximum heart rate. The maximum heart rate is found by subtracting the athlete's age, in years, from $220$. To the nearest whole number, what is the target heart rate of an athlete who is $26$ years old? | {
"answer": "134",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a certain school, there are $3$ times as many boys as girls and $9$ times as many girls as teachers. Using the letters $b, g, t$ to represent the number of boys, girls, and teachers, respectively, then the total number of boys, girls, and teachers can be represented by the expression | {
"answer": "\\frac{37b}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The marked price of a book was 30% less than the suggested retail price. Alice purchased the book for half the marked price at a Fiftieth Anniversary sale. What percent of the suggested retail price did Alice pay? | {
"answer": "35\\%",
"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"
} |
Square $ABCD$ has area $36,$ and $\overline{AB}$ is parallel to the x-axis. Vertices $A,$ $B$, and $C$ are on the graphs of $y = \log_{a}x,$ $y = 2\log_{a}x,$ and $y = 3\log_{a}x,$ respectively. What is $a?$ | {
"answer": "\\sqrt[6]{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is ? | {
"answer": "-13.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}{2}$. The length of the hypotenuse is | {
"answer": "\\frac{3\\sqrt{5}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A truck travels $\frac{b}{6}$ feet every $t$ seconds. There are $3$ feet in a yard. How many yards does the truck travel in $3$ minutes? | {
"answer": "\\frac{10b}{t}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five unit squares are arranged in the coordinate plane as shown, with the lower left corner at the origin. The slanted line, extending from $(c,0)$ to $(3,3)$, divides the entire region into two regions of equal area. What is $c$? | {
"answer": "\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Michael walks at the rate of $5$ feet per second on a long straight path. Trash pails are located every $200$ feet along the path. A garbage truck traveling at $10$ feet per second in the same direction as Michael stops for $30$ seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How many times will Michael and the truck meet? | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Aaron the ant walks on the coordinate plane according to the following rules. He starts at the origin $p_0=(0,0)$ facing to the east and walks one unit, arriving at $p_1=(1,0)$. For $n=1,2,3,\dots$, right after arriving at the point $p_n$, if Aaron can turn $90^\circ$ left and walk one unit to an unvisited point $p_{n+1}$, he does that. Otherwise, he walks one unit straight ahead to reach $p_{n+1}$. Thus the sequence of points continues $p_2=(1,1), p_3=(0,1), p_4=(-1,1), p_5=(-1,0)$, and so on in a counterclockwise spiral pattern. What is $p_{2015}$? | {
"answer": "(13,-22)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The medians of a right triangle which are drawn from the vertices of the acute angles are $5$ and $\sqrt{40}$. The value of the hypotenuse is: | {
"answer": "2\\sqrt{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A particular $12$-hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a $1$, it mistakenly displays a $9$. For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Convex quadrilateral $ABCD$ has $AB = 18$, $\angle A = 60^\circ$, and $\overline{AB} \parallel \overline{CD}$. In some order, the lengths of the four sides form an arithmetic progression, and side $\overline{AB}$ is a side of maximum length. The length of another side is $a$. What is the sum of all possible values of $a$? | {
"answer": "66",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between | {
"answer": "7 and 8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When the sum of the first ten terms of an arithmetic progression is four times the sum of the first five terms, the ratio of the first term to the common difference is: | {
"answer": "1: 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$, and $C$ are sets for which $n(A)+n(B)+n(C)=n(A\cup B\cup C)$ and $|A|=|B|=100$, then what is the minimum possible value of $|A\cap B\cap C|$? | {
"answer": "97",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A paper triangle with sides of lengths $3,4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease? | {
"answer": "$\\frac{15}{8}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Aunt Anna is $42$ years old. Caitlin is $5$ years younger than Brianna, and Brianna is half as old as Aunt Anna. How old is Caitlin? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$.
If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, then: | {
"answer": "$a>9, b>6$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At Typico High School, $60\%$ of the students like dancing, and the rest dislike it. Of those who like dancing, $80\%$ say that they like it, and the rest say that they dislike it. Of those who dislike dancing, $90\%$ say that they dislike it, and the rest say that they like it. What fraction of students who say they dislike dancing actually like it? | {
"answer": "25\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numerator of a fraction is $6x + 1$, then denominator is $7 - 4x$, and $x$ can have any value between $-2$ and $2$, both included. The values of $x$ for which the numerator is greater than the denominator are: | {
"answer": "\\frac{3}{5} < x \\le 2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When simplified, $(x^{-1}+y^{-1})^{-1}$ is equal to: | {
"answer": "\\frac{xy}{x+y}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the $20$ balls is tossed independently and at random into one of the $5$ bins. Let $p$ be the probability that some bin ends up with $3$ balls, another with $5$ balls, and the other three with $4$ balls each. Let $q$ be the probability that every bin ends up with $4$ balls. What is $\frac{p}{q}$? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which one of the following is not equivalent to $0.000000375$? | {
"answer": "$\\frac{3}{8} \\times 10^{-7}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each side of the large square in the figure is trisected (divided into three equal parts). The corners of an inscribed square are at these trisection points, as shown. The ratio of the area of the inscribed square to the area of the large square is
[asy]
draw((0,0)--(3,0)--(3,3)--(0,3)--cycle);
draw((1,0)--(1,0.2));
draw((2,0)--(2,0.2));
draw((3,1)--(2.8,1));
draw((3,2)--(2.8,2));
draw((1,3)--(1,2.8));
draw((2,3)--(2,2.8));
draw((0,1)--(0.2,1));
draw((0,2)--(0.2,2));
draw((2,0)--(3,2)--(1,3)--(0,1)--cycle);
[/asy] | {
"answer": "\\frac{5}{9}",
"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"
} |
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? | {
"answer": "991",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Frieda the frog begins a sequence of hops on a $3 \times 3$ grid of squares, moving one square on each hop and choosing at random the direction of each hop-up, down, left, or right. She does not hop diagonally. When the direction of a hop would take Frieda off the grid, she "wraps around" and jumps to the opposite edge. For example if Frieda begins in the center square and makes two hops "up", the first hop would place her in the top row middle square, and the second hop would cause Frieda to jump to the opposite edge, landing in the bottom row middle square. Suppose Frieda starts from the center square, makes at most four hops at random, and stops hopping if she lands on a corner square. What is the probability that she reaches a corner square on one of the four hops? | {
"answer": "\\frac{13}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center? | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24$? | {
"answer": "\\frac{60}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month? | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $m>0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$, then $m=$ | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\sqrt2 \cdot K.$ | {
"answer": "270",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with area $A_1$ is contained in the interior of a larger circle with area $A_1+A_2$. If the radius of the larger circle is $3$, and if $A_1 , A_2, A_1 + A_2$ is an arithmetic progression, then the radius of the smaller circle is | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of polygon $ABCDEF$, in square units, is | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two medians of a triangle with unequal sides are $3$ inches and $6$ inches. Its area is $3 \sqrt{15}$ square inches. The length of the third median in inches, is: | {
"answer": "3\\sqrt{6}",
"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"
} |
A sample consisting of five observations has an arithmetic mean of $10$ and a median of $12$. The smallest value that the range (largest observation minus smallest) can assume for such a sample is | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABC$ be an equilateral triangle. Extend side $\overline{AB}$ beyond $B$ to a point $B'$ so that $BB'=3 \cdot AB$. Similarly, extend side $\overline{BC}$ beyond $C$ to a point $C'$ so that $CC'=3 \cdot BC$, and extend side $\overline{CA}$ beyond $A$ to a point $A'$ so that $AA'=3 \cdot CA$. What is the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$? | {
"answer": "16:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A bug travels from A to B along the segments in the hexagonal lattice pictured below. The segments marked with an arrow can be traveled only in the direction of the arrow, and the bug never travels the same segment more than once. How many different paths are there? | {
"answer": "2400",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sides $AB$, $BC$, $CD$ and $DA$ of convex quadrilateral $ABCD$ are extended past $B$, $C$, $D$ and $A$ to points $B'$, $C'$, $D'$ and $A'$, respectively. Also, $AB = BB' = 6$, $BC = CC' = 7$, $CD = DD' = 8$ and $DA = AA' = 9$. The area of $ABCD$ is $10$. The area of $A'B'C'D'$ is | {
"answer": "114",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? | {
"answer": "\\frac{4}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ | {
"answer": "23",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles right triangle with legs of length $8$ is partitioned into $16$ congruent triangles as shown. The shaded area is | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using only the paths and the directions shown, how many different routes are there from $\text{M}$ to $\text{N}$? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$. | {
"answer": "\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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 that 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 $s$? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which triplet of numbers has a sum NOT equal to 1? | {
"answer": "1.1 + (-2.1) + 1.0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular hexagon has side length 6. Congruent arcs with radius 3 are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown What is the area of the shaded region? | {
"answer": "54\\sqrt{3}-18\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An $n$-digit positive integer is cute if its $n$ digits are an arrangement of the set $\{1,2,...,n\}$ and its first $k$ digits form an integer that is divisible by $k$, for $k = 1,2,...,n$. For example, $321$ is a cute $3$-digit integer because $1$ divides $3$, $2$ divides $32$, and $3$ divides $321$. How many cute $6$-digit integers are there? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fourth power of $\sqrt{1+\sqrt{1+\sqrt{1}}}$ is | {
"answer": "3+2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What expression is never a prime number when $p$ is a prime number? | {
"answer": "$p^2+26$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Eight spheres of radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? | {
"answer": "1+\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle of radius $r$ is concentric with and outside a regular hexagon of side length $2$. The probability that three entire sides of hexagon are visible from a randomly chosen point on the circle is $1/2$. What is $r$? | {
"answer": "$3\\sqrt{2}+\\sqrt{6}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\frac{x^2-bx}{ax-c}=\frac{m-1}{m+1}$ has roots which are numerically equal but of opposite signs, the value of $m$ must be: | {
"answer": "\\frac{a-b}{a+b}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many sets of two or more consecutive positive integers have a sum of $15$? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a circle with center $O$ and radius $r$, chord $AB$ is drawn with length equal to $r$ (units). From $O$, a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. In terms of $r$ the area of triangle $MDA$, in appropriate square units, is: | {
"answer": "\\frac{r^2\\sqrt{3}}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The Fibonacci sequence $1,1,2,3,5,8,13,21,\ldots$ starts with two 1s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci sequence? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\angle A = 60^\circ$, $\angle E = 40^\circ$ and $\angle C = 30^\circ$, then $\angle BDC =$ | {
"answer": "50^\\circ",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is | {
"answer": "320",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a trapezoid with $AB \parallel CD$, $AB=11$, $BC=5$, $CD=19$, and $DA=7$. Bisectors of $\angle A$ and $\angle D$ meet at $P$, and bisectors of $\angle B$ and $\angle C$ meet at $Q$. What is the area of hexagon $ABQCDP$? | {
"answer": "$30\\sqrt{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $y=x+\frac{1}{x}$, then $x^4+x^3-4x^2+x+1=0$ becomes: | {
"answer": "$x^2(y^2+y-6)=0$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "4:1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A telephone number has the form \text{ABC-DEF-GHIJ}, where each letter represents a different digit. The digits in each part of the number are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find $A$. | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vertex $E$ of equilateral $\triangle{ABE}$ is in the interior of unit square $ABCD$. Let $R$ be the region consisting of all points inside $ABCD$ and outside $\triangle{ABE}$ whose distance from $AD$ is between $\frac{1}{3}$ and $\frac{2}{3}$. What is the area of $R$? | {
"answer": "\\frac{3-\\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a parallelogram with $\angle{ABC}=120^\circ$, $AB=16$ and $BC=10$. Extend $\overline{CD}$ through $D$ to $E$ so that $DE=4$. If $\overline{BE}$ intersects $\overline{AD}$ at $F$, then $FD$ is closest to | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers. How many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $10\%$? | {
"answer": "1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is a right triangle with $\angle ACB$ as its right angle, $m\angle ABC = 60^\circ$ , and $AB = 10$. Let $P$ be randomly chosen inside $ABC$ , and extend $\overline{BP}$ to meet $\overline{AC}$ at $D$. What is the probability that $BD > 5\sqrt2$? | {
"answer": "\\frac{3-\\sqrt3}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The tower function of twos is defined recursively as follows: $T(1) = 2$ and $T(n + 1) = 2^{T(n)}$ for $n\ge1$. Let $A = (T(2009))^{T(2009)}$ and $B = (T(2009))^A$. What is the largest integer $k$ for which $\underbrace{\log_2\log_2\log_2\ldots\log_2B}_{k\text{ times}}$ is defined? | {
"answer": "2010",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window? | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Recall that the conjugate of the complex number $w = a + bi$, where $a$ and $b$ are real numbers and $i = \sqrt{-1}$, is the complex number $\overline{w} = a - bi$. For any complex number $z$, let $f(z) = 4i\overline{z}$. The polynomial $P(z) = z^4 + 4z^3 + 3z^2 + 2z + 1$ has four complex roots: $z_1$, $z_2$, $z_3$, and $z_4$. Let $Q(z) = z^4 + Az^3 + Bz^2 + Cz + D$ be the polynomial whose roots are $f(z_1)$, $f(z_2)$, $f(z_3)$, and $f(z_4)$, where the coefficients $A,$ $B,$ $C,$ and $D$ are complex numbers. What is $B + D?$ | {
"answer": "208",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $F=\log\dfrac{1+x}{1-x}$. Find a new function $G$ by replacing each $x$ in $F$ by $\dfrac{3x+x^3}{1+3x^2}$, and simplify.
The simplified expression $G$ is equal to: | {
"answer": "3F",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic mean (ordinary average) of the fifty-two successive positive integers beginning at 2 is: | {
"answer": "27\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of the $5$ sides and the $5$ diagonals of a regular pentagon are randomly and independently colored red or blue with equal probability. What is the probability that there will be a triangle whose vertices are among the vertices of the pentagon such that all of its sides have the same color? | {
"answer": "\\frac{253}{256}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two candles of the same height are lighted at the same time. The first is consumed in $4$ hours and the second in $3$ hours.
Assuming that each candle burns at a constant rate, in how many hours after being lighted was the first candle twice the height of the second? | {
"answer": "2\\frac{2}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. The number of checkerboard squares which are completely covered by the disc is | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$? | {
"answer": "130",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If a number eight times as large as $x$ is increased by two, then one fourth of the result equals | {
"answer": "2x + \\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can have? | {
"answer": "9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat? | {
"answer": "27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $P(x)$ denotes a polynomial of degree $n$ such that $P(k)=\frac{k}{k+1}$ for $k=0,1,2,\ldots,n$, determine $P(n+1)$. | {
"answer": "$\\frac{n+1}{n+2}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a narrow alley of width $w$ a ladder of length $a$ is placed with its foot at point $P$ between the walls.
Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground.
Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ is equal to | {
"answer": "$h$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Farmer Pythagoras has a field in the shape of a right triangle. The right triangle's legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted? | {
"answer": "\\frac{145}{147}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The area of the region bounded by the graph of \[x^2+y^2 = 3|x-y| + 3|x+y|\] is $m+n\pi$, where $m$ and $n$ are integers. What is $m + n$? | {
"answer": "54",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
100 \times 19.98 \times 1.998 \times 1000= | {
"answer": "(1998)^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular dodecagon ($12$ sides) is inscribed in a circle with radius $r$ inches. The area of the dodecagon, in square inches, is: | {
"answer": "3r^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 1991 the population of a town was a perfect square. Ten years later, after an increase of 150 people, the population was 9 more than a perfect square. Now, in 2011, with an increase of another 150 people, the population is once again a perfect square. What is the percent growth of the town's population during this twenty-year period? | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "\\frac{5}{8}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? | {
"answer": "45",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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? | {
"answer": "0.4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chloe and Zoe are both students in Ms. Demeanor's math class. Last night, they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only $80\%$ of the problems she solved alone, but overall $88\%$ of her answers were correct. Zoe had correct answers to $90\%$ of the problems she solved alone. What was Zoe's overall percentage of correct answers? | {
"answer": "93",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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