problem stringlengths 10 5.15k | answer dict |
|---|---|
[asy]
draw((0,1)--(4,1)--(4,2)--(0,2)--cycle);
draw((2,0)--(3,0)--(3,3)--(2,3)--cycle);
draw((1,1)--(1,2));
label("1",(0.5,1.5));
label("2",(1.5,1.5));
label("32",(2.5,1.5));
label("16",(3.5,1.5));
label("8",(2.5,0.5));
label("6",(2.5,2.5));
[/asy]
The image above is a net of a unit cube. Let $n$ be a positive integer, and let $2n$ such cubes are placed to build a $1 \times 2 \times n$ cuboid which is placed on a floor. Let $S$ be the sum of all numbers on the block visible (not facing the floor). Find the minimum value of $n$ such that there exists such cuboid and its placement on the floor so $S > 2011$ . | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When drawing a histogram of the lifespans of 1000 people, if the class interval is uniformly 20, and the height of the vertical axis for the age range 60 to 80 years is 0.03, calculate the number of people aged 60 to 80. | {
"answer": "600",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define $F(x, y, z) = x \times y^z$. What positive value of $s$ is the solution to the equation $F(s, s, 2) = 1024$? | {
"answer": "8 \\cdot \\sqrt[3]{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the coefficients of all rational terms in the expansion of $$(2 \sqrt {x}- \frac {1}{x})^{6}$$ is \_\_\_\_\_\_ (answer with a number). | {
"answer": "365",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | {
"answer": "416",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many non-similar regular 720-pointed stars are there, given that a regular $n$-pointed star requires its vertices to not all align with vertices of a smaller regular polygon due to common divisors other than 1 between the step size and $n$? | {
"answer": "96",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? | {
"answer": "8\\pi - 16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $AHI$, which is equilateral, lines $\overline{BC}$, $\overline{DE}$, and $\overline{FG}$ are all parallel to $\overline{HI}$. The lengths satisfy $AB = BD = DF = FH$. However, the point $F$ on line segment $AH$ is such that $AF$ is half of $AH$. Determine the ratio of the area of trapezoid $FGIH$ to the area of triangle $AHI$. | {
"answer": "\\frac{3}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the coefficient of $x^{90}$ in the expansion of
\[(x - 1)(x^2 - 2)(x^3 - 3) \dotsm (x^{12} - 12)(x^{13} - 13).\] | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n$ that satisfy
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) < 0.\] | {
"answer": "47",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line $y = -2$ intersects the graph of $y = 5x^2 + 2x - 6$ at points $C$ and $D$. Find the distance between points $C$ and $D$, expressed in the form $\frac{\sqrt{p}}{q}$ where $p$ and $q$ are positive coprime integers. What is $p - q$? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, $a+b=11$. Choose one of the following two conditions as known, and find:<br/>$(Ⅰ)$ the value of $a$;<br/>$(Ⅱ)$ $\sin C$ and the area of $\triangle ABC$.<br/>Condition 1: $c=7$, $\cos A=-\frac{1}{7}$;<br/>Condition 2: $\cos A=\frac{1}{8}$, $\cos B=\frac{9}{16}$.<br/>Note: If both conditions 1 and 2 are answered separately, the first answer will be scored. | {
"answer": "\\frac{15\\sqrt{7}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive integer \(n\) for which there exists positive real numbers \(a\) and \(b\) such that
\[(a + 3bi)^n = (a - 3bi)^n,\]
and compute \(\frac{b}{a}\). | {
"answer": "\\frac{\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, a moving point $M(x,y)$ always satisfies the relation $2 \sqrt {(x-1)^{2}+y^{2}}=|x-4|$.
$(1)$ What is the trajectory of point $M$? Write its standard equation.
$(2)$ The distance from the origin $O$ to the line $l$: $y=kx+m$ is $1$. The line $l$ intersects the trajectory of $M$ at two distinct points $A$ and $B$. If $\overrightarrow{OA} \cdot \overrightarrow{OB}=-\frac{3}{2}$, find the area of triangle $AOB$. | {
"answer": "\\frac{3\\sqrt{7}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ points. If the expected value of the score $X$ obtained from drawing a ball from the bag is $\frac{5}{3}$. <br/>$(1)$ Find the value of the positive integer $a$; <br/>$(2)$ Draw $3$ balls from the bag at once, and find the probability that the sum of the scores obtained is $5$. | {
"answer": "\\frac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $θ$ is an angle in the second quadrant and $\tan(\begin{matrix}θ+ \frac{π}{4}\end{matrix}) = \frac{1}{2}$, find the value of $\sin(θ) + \cos(θ)$. | {
"answer": "-\\frac{\\sqrt{10}}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A line that passes through the two foci of the hyperbola $$\frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1 (a > 0, b > 0)$$ and is perpendicular to the x-axis intersects the hyperbola at four points, forming a square. Find the eccentricity of this hyperbola. | {
"answer": "\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find $c$ such that $\lfloor c \rfloor$ satisfies
\[3x^2 - 9x - 30 = 0\]
and $\{ c \} = c - \lfloor c \rfloor$ satisfies
\[4x^2 - 8x + 1 = 0.\] | {
"answer": "6 - \\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On the side \(AB\) of an equilateral triangle \(\mathrm{ABC}\), a right triangle \(\mathrm{AHB}\) is constructed (\(\mathrm{H}\) is the vertex of the right angle) such that \(\angle \mathrm{HBA}=60^{\circ}\). Let the point \(K\) lie on the ray \(\mathrm{BC}\) beyond the point \(\mathrm{C}\), and \(\angle \mathrm{CAK}=15^{\circ}\). Find the angle between the line \(\mathrm{HK}\) and the median of the triangle \(\mathrm{AHB}\) drawn from the vertex \(\mathrm{H}\). | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What percent of the positive integers less than or equal to $150$ have no remainders when divided by $6$? | {
"answer": "16.67\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $z$ be a complex number with $|z| = 2.$ Find the maximum value of
\[ |(z-2)^3(z+2)|. \] | {
"answer": "24\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a class of 50 students, each student must be paired with another student for a project. Due to prior groupings in other classes, 10 students already have assigned partners. If the pairing for the remaining students is done randomly, what is the probability that Alex is paired with her best friend, Jamie? Express your answer as a common fraction. | {
"answer": "\\frac{1}{29}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many of the first 512 smallest positive integers written in base 8 use the digit 5 or 6 (or both)? | {
"answer": "296",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the smallest number composed exclusively of ones that is divisible by 333...33 (where there are 100 threes in the number). | {
"answer": "300",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a straight road, there are an odd number of warehouses. The distance between adjacent warehouses is 1 kilometer, and each warehouse contains 8 tons of goods. A truck with a load capacity of 8 tons starts from the warehouse on the far right and needs to collect all the goods into the warehouse in the middle. It is known that after the truck has traveled 300 kilometers (the truck chose the optimal route), it successfully completed the task. There are warehouses on this straight road. | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A list of seven positive integers has a median of 4 and a mean of 10. What is the maximum possible value of the list's largest element? | {
"answer": "52",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A natural number greater than 1 is defined as nice if it is equal to the product of its distinct proper divisors. A number \( n \) is nice if:
1. \( n = pq \), where \( p \) and \( q \) are distinct prime numbers.
2. \( n = p^3 \), where \( p \) is a prime number.
3. \( n = p^2q \), where \( p \) and \( q \) are distinct prime numbers.
Determine the sum of the first ten nice numbers under these conditions. | {
"answer": "182",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A cube, all of whose surfaces are painted, is cut into $1000$ smaller cubes of the same size. Find the expected value $E(X)$, where $X$ denotes the number of painted faces of a small cube randomly selected. | {
"answer": "\\frac{3}{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let point O be a point inside triangle ABC with an area of 6, and it satisfies $$\overrightarrow {OA} + \overrightarrow {OB} + 2\overrightarrow {OC} = \overrightarrow {0}$$, then the area of triangle AOC is \_\_\_\_\_\_. | {
"answer": "\\frac {3}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The angle bisectors $\mathrm{AD}$ and $\mathrm{BE}$ of the triangle $\mathrm{ABC}$ intersect at point I. It turns out that the area of triangle $\mathrm{ABI}$ is equal to the area of quadrilateral $\mathrm{CDIE}$. Find the maximum possible value of angle $\mathrm{ACB}$. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Consider a large square of side length 60 units, subdivided into a grid with non-uniform rows and columns. The rows are divided into segments of 20, 20, and 20 units, and the columns are divided into segments of 15, 15, 15, and 15 units. A shaded region is created by connecting the midpoint of the leftmost vertical line to the midpoint of the top horizontal line, then to the midpoint of the rightmost vertical line, and finally to the midpoint of the bottom horizontal line. What is the ratio of the area of this shaded region to the area of the large square? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The volumes of a regular tetrahedron and a regular octahedron have equal edge lengths. Find the ratio of their volumes without calculating the volume of each polyhedron. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
$ABCD$ is a convex quadrilateral such that $AB=2$, $BC=3$, $CD=7$, and $AD=6$. It also has an incircle. Given that $\angle ABC$ is right, determine the radius of this incircle. | {
"answer": "\\frac{1+\\sqrt{13}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\left(\sqrt{625681 + 1000} - \sqrt{1000}\right)^2$. | {
"answer": "626681 - 2 \\cdot \\sqrt{626681} \\cdot 31.622776601683793 + 1000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the function $f(x)=x^{2}-m\cos x+m^{2}+3m-8$ has a unique zero, then the set of real numbers $m$ that satisfy this condition is \_\_\_\_\_\_. | {
"answer": "\\{2\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain high school encourages students to spontaneously organize various sports competitions. Two students, A and B, play table tennis matches in their spare time. The competition adopts a "best of seven games" format (i.e., the first player to win four games wins the match and the competition ends). Assuming that the probability of A winning each game is $\frac{1}{3}$. <br/>$(1)$ Find the probability that exactly $5$ games are played when the match ends; <br/>$(2)$ If A leads with a score of $3:1$, let $X$ represent the number of games needed to finish the match. Find the probability distribution and expectation of $X$. | {
"answer": "\\frac{19}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ has $\angle{A}=90^{\circ}$ , $AB=2$ , and $AC=4$ . Circle $\omega_1$ has center $C$ and radius $CA$ , while circle $\omega_2$ has center $B$ and radius $BA$ . The two circles intersect at $E$ , different from point $A$ . Point $M$ is on $\omega_2$ and in the interior of $ABC$ , such that $BM$ is parallel to $EC$ . Suppose $EM$ intersects $\omega_1$ at point $K$ and $AM$ intersects $\omega_1$ at point $Z$ . What is the area of quadrilateral $ZEBK$ ? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The radius of the Earth measures approximately $6378 \mathrm{~km}$ at the Equator. Suppose that a wire is adjusted exactly over the Equator.
Next, suppose that the length of the wire is increased by $1 \mathrm{~m}$, so that the wire and the Equator are concentric circles around the Earth. Can a standing man, an ant, or an elephant pass underneath this wire? | {
"answer": "0.159",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle ABC, let the lengths of the sides opposite to angles A, B, C be a, b, c, respectively. If a, b, c satisfy $a^2 + c^2 - b^2 = \sqrt{3}ac$,
(1) find angle B;
(2) if b = 2, c = $2\sqrt{3}$, find the area of triangle ABC. | {
"answer": "2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A ball with a diameter of 6 inches rolls along a complex track from start point A to endpoint B. The track comprises four semicircular arcs with radii $R_1 = 120$ inches, $R_2 = 50$ inches, $R_3 = 90$ inches, and $R_4 = 70$ inches respectively. The ball always stays in contact with the track and rolls without slipping. Calculate the distance traveled by the center of the ball from A to B.
A) $320\pi$ inches
B) $330\pi$ inches
C) $340\pi$ inches
D) $350\pi$ inches | {
"answer": "330\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $\binom{18}{8}=31824$, $\binom{18}{9}=48620$, and $\binom{18}{10}=43758$, calculate $\binom{20}{10}$. | {
"answer": "172822",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A sequence is defined recursively as follows: \( t_{1} = 1 \), and for \( n > 1 \):
- If \( n \) is even, \( t_{n} = 1 + t_{\frac{n}{2}} \).
- If \( n \) is odd, \( t_{n} = \frac{1}{t_{n-1}} \).
Given that \( t_{n} = \frac{19}{87} \), find the sum of the digits of \( n \).
(From the 38th American High School Mathematics Examination, 1987) | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system $xOy$, with the origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve $C$ is $\rho^{2}= \dfrac {16}{1+3\sin ^{2}\theta }$, and $P$ is a moving point on curve $C$, which intersects the positive half-axes of $x$ and $y$ at points $A$ and $B$ respectively.
$(1)$ Find the parametric equation of the trajectory of the midpoint $Q$ of segment $OP$;
$(2)$ If $M$ is a moving point on the trajectory of point $Q$ found in $(1)$, find the maximum value of the area of $\triangle MAB$. | {
"answer": "2 \\sqrt {2}+4",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets A = {-2, 1, 2} and B = {-1, 1, 3}, calculate the probability that a line represented by the equation ax - y + b = 0 will pass through the fourth quadrant. | {
"answer": "\\frac{5}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What are the first three digits to the right of the decimal point in the decimal representation of $\left(10^{1001}+1\right)^{3/2}$? | {
"answer": "743",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
1. Solve the trigonometric inequality: $\cos x \geq \frac{1}{2}$
2. In $\triangle ABC$, if $\sin A + \cos A = \frac{\sqrt{2}}{2}$, find the value of $\tan A$. | {
"answer": "-2 - \\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A shop specializing in small refrigerators sells an average of 50 refrigerators per month. The cost of each refrigerator is $1200. Since shipping is included in the sale, the shop needs to pay a $20 shipping fee for each refrigerator. In addition, the shop has to pay a "storefront fee" of $10,000 to the JD website each month, and $5000 for repairs. What is the minimum selling price for each small refrigerator in order for the shop to maintain a monthly sales profit margin of at least 20%? | {
"answer": "1824",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The slope angle of the tangent line to the curve $y= \sqrt {x}$ at $x= \frac {1}{4}$ is ______. | {
"answer": "\\frac {\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sequence ${a_n}$ satisfies the equation $a_{n+1} + (-1)^n a_n = 2n - 1$. Determine the sum of the first 80 terms of the sequence. | {
"answer": "3240",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \( ABC \), \( AB = 33 \), \( AC = 21 \), and \( BC = m \), where \( m \) is a positive integer. If point \( D \) can be found on \( AB \) and point \( E \) can be found on \( AC \) such that \( AD = DE = EC = n \), where \( n \) is a positive integer, what must the value of \( m \) be? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The distances from a certain point inside a regular hexagon to three of its consecutive vertices are 1, 1, and 2, respectively. What is the side length of this hexagon? | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An isosceles triangle $ABP$ with sides $AB = AP = 3$ inches and $BP = 4$ inches is placed inside a square $AXYZ$ with a side length of $8$ inches, such that $B$ is on side $AX$. The triangle is rotated clockwise about $B$, then $P$, and so on along the sides of the square until $P$ returns to its original position. Calculate the total path length in inches traversed by vertex $P$.
A) $\frac{24\pi}{3}$
B) $\frac{28\pi}{3}$
C) $\frac{32\pi}{3}$
D) $\frac{36\pi}{3}$ | {
"answer": "\\frac{32\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For what ratio of the bases of a trapezoid does there exist a line on which the six points of intersection with the diagonals, the lateral sides, and the extensions of the bases of the trapezoid form five equal segments? | {
"answer": "1:2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral $ABCD$, $\overrightarrow{AB}=(1,1)$, $\overrightarrow{DC}=(1,1)$, $\frac{\overrightarrow{BA}}{|\overrightarrow{BA}|}+\frac{\overrightarrow{BC}}{|\overrightarrow{BC}|}=\frac{\sqrt{3}\overrightarrow{BD}}{|\overrightarrow{BD}|}$, calculate the area of the quadrilateral. | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the terminal side of angle $α$ rotates counterclockwise by $\dfrac{π}{6}$ and intersects the unit circle at the point $\left( \dfrac{3 \sqrt{10}}{10}, \dfrac{\sqrt{10}}{10} \right)$, and $\tan (α+β)= \dfrac{2}{5}$.
$(1)$ Find the value of $\sin (2α+ \dfrac{π}{6})$,
$(2)$ Find the value of $\tan (2β- \dfrac{π}{3})$. | {
"answer": "\\dfrac{17}{144}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given two congruent squares $ABCD$ and $EFGH$, each with a side length of $12$, they overlap to form a rectangle $AEHD$ with dimensions $12$ by $20$. Calculate the percent of the area of rectangle $AEHD$ that is shaded. | {
"answer": "20\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If $f(x)=x^{2}+bx+c$, and $f(1)=0$, $f(3)=0$, find
(1) the value of $f(-1)$;
(2) the maximum and minimum values of $f(x)$ on the interval $[2,4]$. | {
"answer": "-1",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person named Jia and their four colleagues each own a car with license plates ending in 9, 0, 2, 1, and 5, respectively. To comply with the local traffic restriction rules from the 5th to the 9th day of a certain month (allowing cars with odd-ending numbers on odd days and even-ending numbers on even days), they agreed to carpool. Each day they can pick any car that meets the restriction, but Jia’s car can be used for one day at most. The number of different carpooling arrangements is __________. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have? | {
"answer": "1003",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Analogous to the exponentiation of rational numbers, we define the division operation of several identical rational numbers (all not equal to $0$) as "division exponentiation," denoted as $a^{ⓝ}$, read as "$a$ circle $n$ times." For example, $2\div 2\div 2$ is denoted as $2^{③}$, read as "$2$ circle $3$ times"; $\left(-3\right)\div \left(-3\right)\div \left(-3\right)\div \left(-3\right)$ is denoted as $\left(-3\right)^{④}$, read as "$-3$ circle $4$ times".<br/>$(1)$ Write down the results directly: $2^{③}=$______, $(-\frac{1}{2})^{④}=$______; <br/>$(2)$ Division exponentiation can also be converted into the form of powers, such as $2^{④}=2\div 2\div 2\div 2=2\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}=(\frac{1}{2})^{2}$. Try to directly write the following operation results in the form of powers: $\left(-3\right)^{④}=$______; ($\frac{1}{2})^{⑩}=$______; $a^{ⓝ}=$______; <br/>$(3)$ Calculate: $2^{2}\times (-\frac{1}{3})^{④}\div \left(-2\right)^{③}-\left(-3\right)^{②}$. | {
"answer": "-73",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For how many values of $n$ in the set $\{101, 102, 103, ..., 200\}$ is the tens digit of $n^2$ even? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the tetrahedron \( P-ABC \), \( \triangle ABC \) is an equilateral triangle with a side length of \( 2\sqrt{3} \), \( PB = PC = \sqrt{5} \), and the dihedral angle between \( P-BC \) and \( BC-A \) is \( 45^\circ \). Find the surface area of the circumscribed sphere around the tetrahedron \( P-ABC \). | {
"answer": "25\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a plane, points are colored in the following way:
1. Choose any positive integer \( m \), and let \( K_{1}, K_{2}, \cdots, K_{m} \) be circles with different non-zero radii such that \( K_{i} \subset K_{j} \) or \( K_{j} \subset K_{i} \) for \( i \neq j \).
2. Points chosen inside the circles are colored differently from the points outside the circles on the plane.
Given that there are 2019 points on the plane such that no three points are collinear, determine the maximum number of different colors possible that satisfy the given conditions. | {
"answer": "2019",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first term of a geometric sequence is 1024, and the 6th term is 125. What is the positive, real value for the 4th term? | {
"answer": "2000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The hypotenuse of a right triangle whose legs are consecutive even numbers is 50 units. What is the sum of the lengths of the two legs? | {
"answer": "70",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $\sqrt{3}\sin B + 2\cos^2\frac{B}{2} = 3$ and $\frac{\cos B}{b} + \frac{\cos C}{c} = \frac{\sin A \sin B}{6\sin C}$, find the area of the circumcircle of $\triangle ABC$. | {
"answer": "16\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $689\Box\Box\Box20312 \approx 69$ billion (rounded), find the number of ways to fill in the three-digit number. | {
"answer": "500",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The school club sells 200 tickets for a total of $2500. Some tickets are sold at full price, and the rest are sold for one-third the price of the full-price tickets. Determine the amount of money raised by the full-price tickets. | {
"answer": "1250",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the independent college admissions process, a high school has obtained 5 recommendation spots, with 2 for Tsinghua University, 2 for Peking University, and 1 for Fudan University. Both Peking University and Tsinghua University require the participation of male students. The school selects 3 male and 2 female students as candidates for recommendation. The total number of different recommendation methods is ( ). | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The number of trees in a park must be more than 80 and fewer than 150. The number of trees is 2 more than a multiple of 4, 3 more than a multiple of 5, and 4 more than a multiple of 6. How many trees are in the park? | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The value of $\tan {75}^{{o}}$ is $\dfrac{\sqrt{6}+\sqrt{2}}{4}$. | {
"answer": "2+\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs. | {
"answer": "1680",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$ , the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$ . If $P(3) = 89$ , what is the value of $P(10)$ ? | {
"answer": "859",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Define a $\it{great\ word}$ as a sequence of letters that consists only of the letters $D$, $E$, $F$, and $G$ --- some of these letters may not appear in the sequence --- and in which $D$ is never immediately followed by $E$, $E$ is never immediately followed by $F$, $F$ is never immediately followed by $G$, and $G$ is never immediately followed by $D$. How many six-letter great words are there? | {
"answer": "972",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In 1900, a reader asked the following question in 1930: He knew a person who died at an age that was $\frac{1}{29}$ of the year of his birth. How old was this person in 1900? | {
"answer": "44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, point $E$ is on $A_{1} D_{1}$, point $F$ is on $C D$, and $A_{1} E = 2 E D_{1}$, $D F = 2 F C$. Find the volume of the triangular prism $B-F E C_{1}$. | {
"answer": "\\frac{5}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the maximum possible value of \[\sin^2x_1+\sin^2x_2+\sin^2x_3+\cdots+\sin^2x_{2007},\] where, for $1\leq i\leq 2007$ , $x_i$ is a nonnegative real number, and \[x_1+x_2+x_3+\cdots+x_{2007}=\pi.\] Find the value of $a+b$ . | {
"answer": "2008",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x)=e^{x}(x^{3}-3x+3)-ae^{x}-x$, where $e$ is the base of the natural logarithm, find the minimum value of the real number $a$ such that the inequality $f(x)\leqslant 0$ has solutions in the interval $x\in\[-2,+\infty)$. | {
"answer": "1-\\frac{1}{e}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that acute angles $\alpha$ and $\beta$ satisfy $\sin\alpha=\frac{4}{5}$ and $\cos(\alpha+\beta)=-\frac{12}{13}$, determine the value of $\cos \beta$. | {
"answer": "-\\frac{16}{65}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the sum of the $x$-coordinates of the solutions to the system of equations $y=|x^2-8x+12|$ and $y=4-x$. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$ . | {
"answer": "$ \\frac {\\sqrt {10} }{2} $",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For all $m, n$ satisfying $1 \leqslant n \leqslant m \leqslant 5$, the number of distinct hyperbolas represented by the polar equation $\rho = \frac{1}{1 - c_{m}^{n} \cos \theta}$ is: | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + y < 3$? | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two numbers in the $4 \times 4$ grid can be swapped to create a Magic Square (in which all rows, all columns and both main diagonals add to the same total).
What is the sum of these two numbers?
A 12
B 15
C 22
D 26
E 28
\begin{tabular}{|c|c|c|c|}
\hline
9 & 6 & 3 & 16 \\
\hline
4 & 13 & 10 & 5 \\
\hline
14 & 1 & 8 & 11 \\
\hline
7 & 12 & 15 & 2 \\
\hline
\end{tabular} | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular coordinate system, the parametric equation of line $l$ is given by $\begin{cases}x=1+t\cos a \\ y=1+t\sin a\end{cases}$ ($t$ is the parameter). In the polar coordinate system, the equation of circle $C$ is $\rho =4\cos \theta$.
(1) Find the rectangular coordinate equation of circle $C$.
(2) If $P(1,1)$, and circle $C$ intersects with line $l$ at points $A$ and $B$, find the maximum or minimum value of $|PA|+|PB|$. | {
"answer": "2\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of ways to partition a set of $10$ elements, $S = \{1, 2, 3, . . . , 10\}$ into two parts; that is, the number of unordered pairs $\{P, Q\}$ such that $P \cup Q = S$ and $P \cap Q = \emptyset$ . | {
"answer": "511",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three people, A, B, and C, are playing a game: each person chooses a real number from the interval $[0,1]$. The person whose chosen number is between the numbers chosen by the other two wins. A randomly chooses a number from the interval $[0,1]$, B randomly chooses a number from the interval $\left[\frac{1}{2}, \frac{2}{3}\right]$. To maximize his probability of winning, what number should C choose? | {
"answer": "$\\frac{13}{24}$",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A regular 100-sided polygon is placed on a table, with the numbers $1, 2, \ldots, 100$ written at its vertices. These numbers are then rewritten in order of their distance from the front edge of the table. If two vertices are at an equal distance from the edge, the left number is listed first, followed by the right number. Form all possible sets of numbers corresponding to different positions of the 100-sided polygon. Calculate the sum of the numbers that occupy the 13th position from the left in these sets. | {
"answer": "10100",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A person forgot the last digit of a phone number and dialed randomly. Calculate the probability of connecting to the call in no more than 3 attempts. | {
"answer": "\\dfrac{3}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$ , $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$ .
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$ . In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$ , and Bob will tell Alice the value of $f(k)$ . Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
*Proposed by YaWNeeT* | {
"answer": "187",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Five numbers, $a_1$, $a_2$, $a_3$, $a_4$, are drawn randomly and without replacement from the set $\{1, 2, 3, \dots, 50\}$. Four other numbers, $b_1$, $b_2$, $b_3$, $b_4$, are then drawn randomly and without replacement from the remaining set of 46 numbers. Let $p$ be the probability that, after a suitable rotation, a brick of dimensions $a_1 \times a_2 \times a_3 \times a_4$ can be enclosed in a box of dimensions $b_1 \times b_2 \times b_3 \times b_4$, with the sides of the brick parallel to the sides of the box. Compute $p$ in lowest terms and determine the sum of the numerator and denominator. | {
"answer": "71",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, $c$, $d$ are the thousands, hundreds, tens, and units digits of a four-digit number, respectively, and the digits in lower positions are not less than those in higher positions. When $|a-b|+|b-c|+|c-d|+|d-a|$ reaches its maximum value, the minimum value of this four-digit number is ____. | {
"answer": "1119",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$? | {
"answer": "2592",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the rectangular table shown below, the number $1$ is written in the upper-left hand corner, and every number is the sum of the any numbers directly to its left and above. The table extends infinitely downwards and to the right.
\[
\begin{array}{cccccc}
1 & 1 & 1 & 1 & 1 & \cdots
1 & 2 & 3 & 4 & 5 & \cdots
1 & 3 & 6 & 10 & 15 & \cdots
1 & 4 & 10 & 20 & 35 & \cdots
1 & 5 & 15 & 35 & 70 & \cdots
\vdots & \vdots & \vdots & \vdots & \vdots & \ddots
\end{array}
\]
Wanda the Worm, who is on a diet after a feast two years ago, wants to eat $n$ numbers (not necessarily distinct in value) from the table such that the sum of the numbers is less than one million. However, she cannot eat two numbers in the same row or column (or both). What is the largest possible value of $n$ ?
*Proposed by Evan Chen* | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In quadrilateral \(ABCD\), we have \(AB=5\), \(BC=6\), \(CD=5\), \(DA=4\), and \(\angle ABC=90^\circ\). Let \(AC\) and \(BD\) meet at \(E\). Compute \(\frac{BE}{ED}\). | {
"answer": "\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many such five-digit Shenma numbers exist, where the middle digit is the smallest, the digits increase as they move away from the middle, and all the digits are different? | {
"answer": "1512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given functions $f(x)=-2x$ for $x<0$ and $g(x)=\frac{x}{\ln x}+x-2$. If $f(x_{1})=g(x_{2})$, find the minimum value of $x_{2}-2x_{1}$. | {
"answer": "4\\sqrt{e}-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum of
\[
\sqrt{x + 31} + \sqrt{17 - x} + \sqrt{x}
\]
for $0 \le x \le 17$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We define $N$ as the set of natural numbers $n<10^6$ with the following property:
There exists an integer exponent $k$ with $1\le k \le 43$ , such that $2012|n^k-1$ .
Find $|N|$ . | {
"answer": "1988",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given positive real numbers \( a, b, c, d \) that satisfy the equalities
\[ a^{2}+d^{2}-ad = b^{2}+c^{2}+bc \quad \text{and} \quad a^{2}+b^{2} = c^{2}+d^{2}, \]
find all possible values of the expression \( \frac{ab+cd}{ad+bc} \). | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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