problem stringlengths 10 5.15k | answer dict |
|---|---|
Let $N$ be a positive multiple of $5$. One red ball and $N$ green balls are arranged in a line in random order. Let $P(N)$ be the probability that at least $\tfrac{3}{5}$ of the green balls are on the same side of the red ball. Observe that $P(5)=1$ and that $P(N)$ approaches $\tfrac{4}{5}$ as $N$ grows large. What is the sum of the digits of the least value of $N$ such that $P(N) < \tfrac{321}{400}$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The arithmetic mean of a set of $50$ numbers is $38$. If two numbers of the set, namely $45$ and $55$, are discarded, the arithmetic mean of the remaining set of numbers is: | {
"answer": "36.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of ordered pairs of integers $(m,n)$ for which $mn \ge 0$ and
$m^3 + n^3 + 99mn = 33^3$
is equal to | {
"answer": "35",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length 2 so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region ---- inside the hexagon but outside all of the semicircles? | {
"answer": "6\\sqrt{3} - 3\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \cdot CD = AB \cdot DA$. What is the largest possible value of $BD$? | {
"answer": "\\sqrt{\\dfrac{425}{2}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$(1+x^2)(1-x^3)$ equals | {
"answer": "1+x^2-x^3-x^5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Side $AB$ of triangle $ABC$ has length 8 inches. Line $DEF$ is drawn parallel to $AB$ so that $D$ is on segment $AC$, and $E$ is on segment $BC$. Line $AE$ extended bisects angle $FEC$. If $DE$ has length $5$ inches, then the length of $CE$, in inches, is: | {
"answer": "\\frac{40}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The interior of a quadrilateral is bounded by the graphs of $(x+ay)^2 = 4a^2$ and $(ax-y)^2 = a^2$, where $a$ is a positive real number. What is the area of this region in terms of $a$, valid for all $a > 0$? | {
"answer": "\\frac{8a^2}{a^2+1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 16-step path is to go from $(-4,-4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. How many such paths stay outside or on the boundary of the square $-2 < x < 2$, $-2 < y < 2$ at each step? | {
"answer": "1698",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is: | {
"answer": "77\\% of 250.00",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are four more girls than boys in Ms. Raub's class of $28$ students. What is the ratio of number of girls to the number of boys in her class? | {
"answer": "4 : 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from A to B? | {
"answer": "238\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of solutions to \{1,~2\} \subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}, where $X$ is a subset of \{1,~2,~3,~4,~5\} is | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two candles of the same length are made of different materials so that one burns out completely at a uniform rate in $3$ hours and the other in $4$ hours. At what time P.M. should the candles be lighted so that, at 4 P.M., one stub is twice the length of the other? | {
"answer": "1:36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The units digit of $3^{1001} 7^{1002} 13^{1003}$ is | {
"answer": "3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many non-congruent right triangles with positive integer leg lengths have areas that are numerically equal to $3$ times their perimeters? | {
"answer": "7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The increasing sequence $3, 15, 24, 48, \ldots$ consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? | {
"answer": "935",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants: | {
"answer": "I and II",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum $2\frac{1}{7} + 3\frac{1}{2} + 5\frac{1}{19}$ is between | {
"answer": "$10\\frac{1}{2} \\text{ and } 11$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ is the cube of a positive integer and $d$ is the number of positive integers that are divisors of $x$, then $d$ could be | {
"answer": "202",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is \(\sum^{100}_{i=1} \sum^{100}_{j=1} (i+j) \)? | {
"answer": "1{,}010{,}000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.) | {
"answer": "1260",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$\log p+\log q=\log(p+q)$ only if: | {
"answer": "p=\\frac{q}{q-1}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the ratio of the legs of a right triangle is $1: 2$, then the ratio of the corresponding segments of the hypotenuse made by a perpendicular upon it from the vertex is: | {
"answer": "1: 4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\frac{x}{y}=\frac{3}{4}$, then the incorrect expression in the following is: | {
"answer": "$\\frac{1}{4}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A convex polyhedron $Q$ has vertices $V_1,V_2,\ldots,V_n$, and $100$ edges. The polyhedron is cut by planes $P_1,P_2,\ldots,P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. How many edges does $R$ have? | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What 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"
} |
If $f(2x)=\frac{2}{2+x}$ for all $x>0$, then $2f(x)=$ | {
"answer": "\\frac{8}{4+x}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$. | {
"answer": "\\frac{1}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$.
If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ, \measuredangle ABC = 100^\circ, \measuredangle ACB = 20^\circ$ and
$\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals | {
"answer": "\\frac{\\sqrt{3}}{8}",
"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": "\\sqrt{6}-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider the $12$-sided polygon $ABCDEFGHIJKL$, as shown. Each of its sides has length $4$, and each two consecutive sides form a right angle. Suppose that $\overline{AG}$ and $\overline{CH}$ meet at $M$. What is the area of quadrilateral $ABCM$? | {
"answer": "88/5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum to infinity of the terms of an infinite geometric progression is $6$. The sum of the first two terms is $4\frac{1}{2}$. The first term of the progression is: | {
"answer": "9 or 3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several sets of prime numbers, such as $\{7,83,421,659\}$ use each of the nine nonzero digits exactly once. What is the smallest possible sum such a set of primes could have? | {
"answer": "207",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the largest number of solid $2 \times 2 \times 1$ blocks that can fit in a $3 \times 2 \times 3$ box? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 < a - b \le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$? | {
"answer": "855",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In how many ways can $345$ be written as the sum of an increasing sequence of two or more consecutive positive integers? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A closed box with a square base is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet of paper, as shown in the figure on the left. The four corners of the wrapping paper are to be folded up over the sides and brought together to meet at the center of the top of the box, point $A$ in the figure on the right. The box has base length $w$ and height $h$. What is the area of the sheet of wrapping paper?
[asy] size(270pt); defaultpen(fontsize(10pt)); filldraw(((3,3)--(-3,3)--(-3,-3)--(3,-3)--cycle),lightgrey); dot((-3,3)); label("$A$",(-3,3),NW); draw((1,3)--(-3,-1),dashed+linewidth(.5)); draw((-1,3)--(3,-1),dashed+linewidth(.5)); draw((-1,-3)--(3,1),dashed+linewidth(.5)); draw((1,-3)--(-3,1),dashed+linewidth(.5)); draw((0,2)--(2,0)--(0,-2)--(-2,0)--cycle,linewidth(.5)); draw((0,3)--(0,-3),linetype("2.5 2.5")+linewidth(.5)); draw((3,0)--(-3,0),linetype("2.5 2.5")+linewidth(.5)); label('$w$',(-1,-1),SW); label('$w$',(1,-1),SE); draw((4.5,0)--(6.5,2)--(8.5,0)--(6.5,-2)--cycle); draw((4.5,0)--(8.5,0)); draw((6.5,2)--(6.5,-2)); label("$A$",(6.5,0),NW); dot((6.5,0)); [/asy] | {
"answer": "2(w+h)^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The letters $\text{A}$, $\text{J}$, $\text{H}$, $\text{S}$, $\text{M}$, $\text{E}$ and the digits $1$, $9$, $8$, $9$ are "cycled" separately as follows and put together in a numbered list:
\[\begin{tabular}[t]{lccc} & & AJHSME & 1989 \ & & & \ 1. & & JHSMEA & 9891 \ 2. & & HSMEAJ & 8919 \ 3. & & SMEAJH & 9198 \ & & ........ & \end{tabular}\]
What is the number of the line on which $\text{AJHSME 1989}$ will appear for the first time? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, \dots, k_n$ for which
\[k_1^2 + k_2^2 + \dots + k_n^2 = 2002?\] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$ | {
"answer": "\\frac{\\sqrt{5}}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Horses $X, Y$ and $Z$ are entered in a three-horse race in which ties are not possible. The odds against $X$ winning are $3:1$ and the odds against $Y$ winning are $2:3$, what are the odds against $Z$ winning? (By "odds against $H$ winning are $p:q$" we mean the probability of $H$ winning the race is $\frac{q}{p+q}$.) | {
"answer": "17:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the figure below, $3$ of the $6$ disks are to be painted blue, $2$ are to be painted red, and $1$ is to be painted green. Two paintings that can be obtained from one another by a rotation or a reflection of the entire figure are considered the same. How many different paintings are possible?
[asy]
size(100);
pair A, B, C, D, E, F;
A = (0,0);
B = (1,0);
C = (2,0);
D = rotate(60, A)*B;
E = B + D;
F = rotate(60, A)*C;
draw(Circle(A, 0.5));
draw(Circle(B, 0.5));
draw(Circle(C, 0.5));
draw(Circle(D, 0.5));
draw(Circle(E, 0.5));
draw(Circle(F, 0.5));
[/asy] | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vertex $E$ of equilateral $\triangle ABE$ is in the interior of square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and line segment $AE$. If length $AB$ is $\sqrt{1+\sqrt{3}}$ then the area of $\triangle ABF$ is | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an $h$-meter race, Sunny is exactly $d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. How many meters ahead is Sunny when Sunny finishes the second race? | {
"answer": "\\frac {d^2}{h}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Sara makes a staircase out of toothpicks as shown:
[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } } [/asy]
This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Chandra pays an on-line service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was $12.48, but in January her bill was $17.54 because she used twice as much connect time as in December. What is the fixed monthly fee? | {
"answer": "6.24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Travis has to babysit the terrible Thompson triplets. Knowing that they love big numbers, Travis devises a counting game for them. First Tadd will say the number $1$, then Todd must say the next two numbers ($2$ and $3$), then Tucker must say the next three numbers ($4$, $5$, $6$), then Tadd must say the next four numbers ($7$, $8$, $9$, $10$), and the process continues to rotate through the three children in order, each saying one more number than the previous child did, until the number $10,000$ is reached. What is the $2019$th number said by Tadd? | {
"answer": "5979",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tony works $2$ hours a day and is paid $\$0.50$ per hour for each full year of his age. During a six month period Tony worked $50$ days and earned $\$630$. How old was Tony at the end of the six month period? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $\log_8{3}=p$ and $\log_3{5}=q$, then, in terms of $p$ and $q$, $\log_{10}{5}$ equals | {
"answer": "\\frac{3pq}{1+3pq}",
"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"
} |
In $\triangle ABC$, $\angle C = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ 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"
} |
A recipe that makes $5$ servings of hot chocolate requires $2$ squares of chocolate, $\frac{1}{4}$ cup sugar, $1$ cup water and $4$ cups milk. Jordan has $5$ squares of chocolate, $2$ cups of sugar, lots of water, and $7$ cups of milk. If he maintains the same ratio of ingredients, what is the greatest number of servings of hot chocolate he can make? | {
"answer": "8 \\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $P$ is the product of $n$ quantities in Geometric Progression, $S$ their sum, and $S'$ the sum of their reciprocals, then $P$ in terms of $S, S'$, and $n$ is | {
"answer": "$(S/S')^{\\frac{1}{2}n}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fly trapped inside a cubical box with side length $1$ meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. What is the maximum possible length, in meters, of its path? | {
"answer": "$4\\sqrt{2}+4\\sqrt{3}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the $whatsis$ is $so$ when the $whosis$ is $is$ and the $so$ and $so$ is $is \cdot so$, what is the $whosis \cdot whatsis$ when the $whosis$ is $so$, the $so$ and $so$ is $so \cdot so$ and the $is$ is two ($whatsis, whosis, is$ and $so$ are variables taking positive values)? | {
"answer": "$so \\text{ and } so$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Call a positive integer an uphill integer if every digit is strictly greater than the previous digit. For example, $1357$, $89$, and $5$ are all uphill integers, but $32$, $1240$, and $466$ are not. How many uphill integers are divisible by $15$? | {
"answer": "6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run 100 meters. They next meet after Sally has run 150 meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters? | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Bernardo randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks 3 distinct numbers from the set $\{1,2,3,4,5,6,7,8\}$ and also arranges them in descending order to form a 3-digit number. What is the probability that Bernardo's number is larger than Silvia's number? | {
"answer": "\\frac{37}{56}",
"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"
} |
The number of significant digits in the measurement of the side of a square whose computed area is $1.1025$ square inches to the nearest ten-thousandth of a square inch is: | {
"answer": "5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle? | {
"answer": "\\frac{10}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Regular polygons with $5, 6, 7,$ and $8$ sides are inscribed in the same circle. No two of the polygons share a vertex, and no three of their sides intersect at a common point. At how many points inside the circle do two of their sides intersect? | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to
[asy] unitsize(3mm); defaultpen(linewidth(0.8pt)); path p1=(0,0)--(3,0)--(3,3)--(0,3)--(0,0); path p2=(0,1)--(1,1)--(1,0); path p3=(2,0)--(2,1)--(3,1); path p4=(3,2)--(2,2)--(2,3); path p5=(1,3)--(1,2)--(0,2); path p6=(1,1)--(2,2); path p7=(2,1)--(1,2); path[] p=p1^^p2^^p3^^p4^^p5^^p6^^p7; for(int i=0; i<3; ++i) { for(int j=0; j<3; ++j) { draw(shift(3*i,3*j)*p); } } [/asy] | {
"answer": "56",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When a bucket is two-thirds full of water, the bucket and water weigh $a$ kilograms. When the bucket is one-half full of water the total weight is $b$ kilograms. In terms of $a$ and $b$, what is the total weight in kilograms when the bucket is full of water? | {
"answer": "3a - 2b",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}\left(a+b\sqrt{2}+\pi\right)$, where $a$ and $b$ are positive integers. What is $a+b$? | {
"answer": "68",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The set of $x$-values satisfying the inequality $2 \leq |x-1| \leq 5$ is: | {
"answer": "-4\\leq x\\leq-1\\text{ or }3\\leq x\\leq 6",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A plane flew straight against a wind between two towns in 84 minutes and returned with that wind in 9 minutes less than it would take in still air. The number of minutes (2 answers) for the return trip was | {
"answer": "63 or 12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The roots of the equation $2\sqrt{x} + 2x^{-\frac{1}{2}} = 5$ can be found by solving: | {
"answer": "4x^2-17x+4 = 0",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sides of a right triangle are $a$ and $b$ and the hypotenuse is $c$. A perpendicular from the vertex divides $c$ into segments $r$ and $s$, adjacent respectively to $a$ and $b$. If $a : b = 1 : 3$, then the ratio of $r$ to $s$ is: | {
"answer": "1 : 9",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \hdots \cdot 9!$? | {
"answer": "672",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs? | {
"answer": "2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $70^2$ and $73^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of the $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{6}L_j$, and for every circle $C$ denote by $r(C)$ its radius. What is
\[\sum_{C\in S} \frac{1}{\sqrt{r(C)}}?\]
[asy] import olympiad; size(350); defaultpen(linewidth(0.7)); // define a bunch of arrays and starting points pair[] coord = new pair[65]; int[] trav = {32,16,8,4,2,1}; coord[0] = (0,73^2); coord[64] = (2*73*70,70^2); // draw the big circles and the bottom line path arc1 = arc(coord[0],coord[0].y,260,360); path arc2 = arc(coord[64],coord[64].y,175,280); fill((coord[0].x-910,coord[0].y)--arc1--cycle,gray(0.75)); fill((coord[64].x+870,coord[64].y+425)--arc2--cycle,gray(0.75)); draw(arc1^^arc2); draw((-930,0)--(70^2+73^2+850,0)); // We now apply the findCenter function 63 times to get // the location of the centers of all 63 constructed circles. // The complicated array setup ensures that all the circles // will be taken in the right order for(int i = 0;i<=5;i=i+1) { int skip = trav[i]; for(int k=skip;k<=64 - skip; k = k + 2*skip) { pair cent1 = coord[k-skip], cent2 = coord[k+skip]; real r1 = cent1.y, r2 = cent2.y, rn=r1*r2/((sqrt(r1)+sqrt(r2))^2); real shiftx = cent1.x + sqrt(4*r1*rn); coord[k] = (shiftx,rn); } // Draw the remaining 63 circles } for(int i=1;i<=63;i=i+1) { filldraw(circle(coord[i],coord[i].y),gray(0.75)); }[/asy] | {
"answer": "\\frac{143}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $(1+x+x^2)^n=a_0 + a_1x+a_2x^2+ \cdots + a_{2n}x^{2n}$ be an identity in $x$. If we let $s=a_0+a_2+a_4+\cdots +a_{2n}$, then $s$ equals: | {
"answer": "\\frac{3^n+1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles
(not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is: | {
"answer": "\\frac{y^3}{x^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Several students are competing in a series of three races. A student earns $5$ points for winning a race, $3$ points for finishing second and $1$ point for finishing third. There are no ties. What is the smallest number of points that a student must earn in the three races to be guaranteed of earning more points than any other student? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to the center to form triangle $B$. [See figure for $n=5, m=7$.] What is the ratio of the area of triangle $A$ to the area of triangle $B$? | {
"answer": "\\frac{m}{n}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Rectangle $ABCD$, pictured below, shares $50\%$ of its area with square $EFGH$. Square $EFGH$ shares $20\%$ of its area with rectangle $ABCD$. What is $\frac{AB}{AD}$? | {
"answer": "10",
"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"
} |
The triangular plot of ACD lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land ACD? | {
"answer": "4.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are two positive numbers that may be inserted between $3$ and $9$ such that the first three are in geometric progression while the last three are in arithmetic progression. The sum of those two positive numbers is | {
"answer": "11\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many odd positive $3$-digit integers are divisible by $3$ but do not contain the digit $3$? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $P$ units be the increase in circumference of a circle resulting from an increase in $\pi$ units in the diameter. Then $P$ equals: | {
"answer": "\\pi^2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Jane can walk any distance in half the time it takes Hector to walk the same distance. They set off in opposite directions around the outside of the 18-block area as shown. When they meet for the first time, they will be closest to | {
"answer": "D",
"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 area of the ring between two concentric circles is $12\frac{1}{2}\pi$ square inches. The length of a chord of the larger circle tangent to the smaller circle, in inches, is: | {
"answer": "5\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the remainder when $2^{202} + 202$ is divided by $2^{101} + 2^{51} + 1$? | {
"answer": "201",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible by the square of any prime number. What is $p+q+r+s?$ | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $a$ and $b$ are two unequal positive numbers, then: | {
"answer": "\\frac {a + b}{2} > \\sqrt {ab} > \\frac {2ab}{a + b}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular piece of paper 6 inches wide is folded as in the diagram so that one corner touches the opposite side. The length in inches of the crease L in terms of angle $\theta$ is | {
"answer": "$3\\sec ^2\\theta\\csc\\theta$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The bar graph shows the results of a survey on color preferences. What percent preferred blue? | {
"answer": "24\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The table below gives the percent of students in each grade at Annville and Cleona elementary schools:
\[\begin{tabular}{rccccccc}&\textbf{\underline{K}}&\textbf{\underline{1}}&\textbf{\underline{2}}&\textbf{\underline{3}}&\textbf{\underline{4}}&\textbf{\underline{5}}&\textbf{\underline{6}}\\ \textbf{Annville:}& 16\% & 15\% & 15\% & 14\% & 13\% & 16\% & 11\%\\ \textbf{Cleona:}& 12\% & 15\% & 14\% & 13\% & 15\% & 14\% & 17\%\end{tabular}\]
Annville has 100 students and Cleona has 200 students. In the two schools combined, what percent of the students are in grade 6? | {
"answer": "15\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $y=(x-a)^2+(x-b)^2, a, b$ constants. For what value of $x$ is $y$ a minimum? | {
"answer": "\\frac{a+b}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose $a$, $b$, $c$ are positive integers such that $a+b+c=23$ and $\gcd(a,b)+\gcd(b,c)+\gcd(c,a)=9.$ What is the sum of all possible distinct values of $a^2+b^2+c^2$? | {
"answer": "438",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $S=(x-1)^4+4(x-1)^3+6(x-1)^2+4(x-1)+1$. Then $S$ equals: | {
"answer": "x^4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For a certain complex number $c$, the polynomial
\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\]has exactly 4 distinct roots. What is $|c|$? | {
"answer": "\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)
How many different routes can Paula take? | {
"answer": "4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many different patterns can be made by shading exactly two of the nine squares? Patterns that can be matched by flips and/or turns are not considered different. For example, the patterns shown below are not considered different. | {
"answer": "8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. What is the value of $ab$? | {
"answer": "\\frac{1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the markings on the number line are equally spaced, what is the number $\text{y}$? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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