Dataset Viewer
problem
stringclasses 60
values | answer
stringclasses 50
values | reward
float64 0.06
0.44
|
|---|---|---|
Suppose that $a$ and $b$ are nonzero integers such that two of the roots of
\[x^3 + ax^2 + bx + 9a\]coincide, and all three roots are integers. Find $|ab|.$
|
1344
| 0.4375 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.
|
69
| 0.3125 |
Alice and Bob are playing a game where Alice declares, "My number is 36." Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in Alice's. What is the smallest possible number Bob can choose?
|
72
| 0.1875 |
Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Find $N$.
|
772
| 0.125 |
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator?
|
11
| 0.1875 |
Determine the largest multiple of 36 that consists of all even and distinct digits.
|
8640
| 0.375 |
Let \( S = \{1, 2, \cdots, 2005\} \). If any \( n \) pairwise coprime numbers in \( S \) always include at least one prime number, find the minimum value of \( n \).
|
16
| 0.125 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.
|
-332
| 0.375 |
There are 100 people in a room with ages $1,2, \ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?
|
43
| 0.0625 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}
|
135
| 0.0625 |
It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number).
|
336
| 0.125 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
|
6
| 0.375 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
If \( x = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^{6}}} \), then the value of \([x]\) is
|
1998
| 0.4375 |
It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number).
|
336
| 0.125 |
Given a triangle $ABC$ with angles $\angle A = 60^{\circ}, \angle B = 75^{\circ}, \angle C = 45^{\circ}$ , let $H$ be its orthocentre, and $O$ be its circumcenter. Let $F$ be the midpoint of side $AB$ , and $Q$ be the foot of the perpendicular from $B$ onto $AC$ . Denote by $X$ the intersection point of the lines $FH$ and $QO$ . Suppose the ratio of the length of $FX$ and the circumradius of the triangle is given by $\dfrac{a + b \sqrt{c}}{d}$ , then find the value of $1000a + 100b + 10c + d$ .
|
1132
| 0.125 |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[16^{x^2 + y} + 16^{x + y^2} = 1.\]
|
1
| 0.1875 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.
|
-332
| 0.375 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week.
|
36
| 0.125 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.
|
69
| 0.3125 |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is
|
13
| 0.0625 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
|
6
| 0.375 |
The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)?
|
30
| 0.3125 |
Find the smallest positive integer $n$ for which $$1!2!\cdots(n-1)!>n!^{2}$$
|
8
| 0.375 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week.
|
36
| 0.125 |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.
|
24
| 0.4375 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
|
256
| 0.1875 |
Suppose that $a$ and $b$ are nonzero integers such that two of the roots of
\[x^3 + ax^2 + bx + 9a\]coincide, and all three roots are integers. Find $|ab|.$
|
1344
| 0.4375 |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$.
|
-1
| 0.4375 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.
|
-332
| 0.375 |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));
[/asy]
[asy]
draw((18,0)--(0,0)--(0,18));
label("6",(3,-1));
label("7",(6,-1));
label("8",(9,-1));
label("9",(12,-1));
label("10",(15,-1));
fill((3,.5)..(3.5,1)..(3,1.5)..(2.5,1)..cycle);
fill((3,2)..(3.5,2.5)..(3,3)..(2.5,2.5)..cycle);
fill((3,3.5)..(3.5,4)..(3,4.5)..(2.5,4)..cycle);
fill((3,5)..(3.5,5.5)..(3,6)..(2.5,5.5)..cycle);
fill((3,6.5)..(3.5,7)..(3,7.5)..(2.5,7)..cycle);
fill((3,8)..(3.5,8.5)..(3,9)..(2.5,8.5)..cycle);
fill((6,.5)..(6.5,1)..(6,1.5)..(5.5,1)..cycle);
fill((6,2)..(6.5,2.5)..(6,3)..(5.5,2.5)..cycle);
fill((6,3.5)..(6.5,4)..(6,4.5)..(5.5,4)..cycle);
fill((6,5)..(6.5,5.5)..(6,6)..(5.5,5.5)..cycle);
fill((9,.5)..(9.5,1)..(9,1.5)..(8.5,1)..cycle);
fill((9,2)..(9.5,2.5)..(9,3)..(8.5,2.5)..cycle);
fill((9,3.5)..(9.5,4)..(9,4.5)..(8.5,4)..cycle);
fill((15,.5)..(15.5,1)..(15,1.5)..(14.5,1)..cycle);
label("Number of Home Runs",(9,-3));
picture perpLabel;
label(perpLabel,"Number of Top Hitters");
add(rotate(90)*perpLabel,(-1,9));
[/asy]
|
7
| 0.0625 |
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.
|
76
| 0.4375 |
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator?
|
11
| 0.1875 |
(In the coordinate system and parametric equations optional question) In the polar coordinate system, it is known that the line $l: p(\sin\theta - \cos\theta) = a$ divides the region enclosed by the curve $C: p = 2\cos\theta$ into two parts with equal area. Find the value of the constant $a$.
|
-1
| 0.4375 |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$?
|
15
| 0.125 |
Given $a= \int_{ 0 }^{ \pi }(\sin x-1+2\cos ^{2} \frac {x}{2})dx$, find the constant term in the expansion of $(a \sqrt {x}- \frac {1}{ \sqrt {x}})^{6}\cdot(x^{2}+2)$.
|
-332
| 0.375 |
The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)?
|
30
| 0.3125 |
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing by 3. How can you obtain the number 11 from the number 1 using this calculator?
|
11
| 0.1875 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 0.3125 |
In triangle $ABC,\,$ angle $C$ is a right angle and the altitude from $C\,$ meets $\overline{AB}\,$ at $D.\,$ The lengths of the sides of $\triangle ABC\,$ are integers, $BD=29^3,\,$ and $\cos B=m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. Find $m+n.\,$
|
450
| 0.3125 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}
|
135
| 0.0625 |
Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Find $N$.
|
772
| 0.125 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
Sean is a biologist, and is looking at a string of length 66 composed of the letters $A, T, C, G$. A substring of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has 10 substrings: $A, G, T, C, AG, GT, TC, AGT, GTC, AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?
|
2100
| 0.4375 |
A pyramid \( S A B C D \) has a trapezoid \( A B C D \) as its base, with bases \( B C \) and \( A D \). Points \( P_1, P_2, P_3 \) lie on side \( B C \) such that \( B P_1 < B P_2 < B P_3 < B C \). Points \( Q_1, Q_2, Q_3 \) lie on side \( A D \) such that \( A Q_1 < A Q_2 < A Q_3 < A D \). Let \( R_1, R_2, R_3, \) and \( R_4 \) be the intersection points of \( B Q_1 \) with \( A P_1 \); \( P_2 Q_1 \) with \( P_1 Q_2 \); \( P_3 Q_2 \) with \( P_2 Q_3 \); and \( C Q_3 \) with \( P_3 D \) respectively. It is known that the sum of the volumes of the pyramids \( S R_1 P_1 R_2 Q_1 \) and \( S R_3 P_3 R_4 Q_3 \) equals 78. Find the minimum value of
\[ V_{S A B R_1}^2 + V_{S R_2 P_2 R_3 Q_2}^2 + V_{S C D R_4}^2 \]
and give the closest integer to this value.
|
2028
| 0.1875 |
In triangle $ABC,\,$ angle $C$ is a right angle and the altitude from $C\,$ meets $\overline{AB}\,$ at $D.\,$ The lengths of the sides of $\triangle ABC\,$ are integers, $BD=29^3,\,$ and $\cos B=m/n\,$, where $m\,$ and $n\,$ are relatively prime positive integers. Find $m+n.\,$
|
450
| 0.3125 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
Given the sets
$$
\begin{array}{l}
M=\{x, x y, \lg (x y)\} \\
N=\{0,|x|, y\},
\end{array}
$$
and $M=N$, determine the value of:
$$\left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right).$$
|
-2
| 0.1875 |
It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.
|
69
| 0.3125 |
An ellipse is defined parametrically by
\[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right).\]Then the equation of the ellipse can be written in the form
\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\gcd(|A|,|B|,|C|,|D|,|E|,|F|) = 1.$ Find $|A| + |B| + |C| + |D| + |E| + |F|.$
|
1381
| 0.375 |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$.
|
2889
| 0.125 |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$?
|
15
| 0.125 |
Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.
|
147
| 0.0625 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
The stem-and-leaf plot shows the number of minutes and seconds of one ride on each of the 21 top-rated water slides in the world. In the stem-and-leaf plot, $1 \ 45$ represents 1 minute, 45 seconds, which is equivalent to 105 seconds. What is the median of this data set? Express your answer in seconds.
\begin{tabular}{c|cccccc}
0&15&30&45&55&&\\
1&00&20&35&45&55&\\
2&10&15&30&45&50&55\\
3&05&10&15&&&\\
\end{tabular}
|
135
| 0.0625 |
If \( x = 1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{10^{6}}} \), then the value of \([x]\) is
|
1998
| 0.4375 |
It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number).
|
336
| 0.125 |
How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?
|
616
| 0.1875 |
Sean is a biologist, and is looking at a string of length 66 composed of the letters $A, T, C, G$. A substring of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has 10 substrings: $A, G, T, C, AG, GT, TC, AGT, GTC, AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?
|
2100
| 0.4375 |
The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
|
12
| 0.0625 |
Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive?
|
12
| 0.3125 |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.
|
24
| 0.4375 |
Given that the positive integers \( a, b, c \) satisfy \( 2017 \geqslant 10a \geqslant 100b \geqslant 1000c \), find the number of possible triples \( (a, b, c) \).
|
574
| 0.3125 |
There are 100 people in a room with ages $1,2, \ldots, 100$. A pair of people is called cute if each of them is at least seven years older than half the age of the other person in the pair. At most how many pairwise disjoint cute pairs can be formed in this room?
|
43
| 0.0625 |
The circular region of the sign now has an area of 50 square inches. To decorate the edge with a ribbon, Vanessa plans to purchase 5 inches more than the circle’s circumference. How many inches of ribbon should she buy if she estimates \(\pi = \frac{22}{7}\)?
|
30
| 0.3125 |
An ellipse is defined parametrically by
\[(x,y) = \left( \frac{2 (\sin t - 1)}{2 - \cos t}, \frac{3 (\cos t - 5)}{2 - \cos t} \right).\]Then the equation of the ellipse can be written in the form
\[Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,\]where $A,$ $B,$ $C,$ $D,$ $E,$ and $F$ are integers, and $\gcd(|A|,|B|,|C|,|D|,|E|,|F|) = 1.$ Find $|A| + |B| + |C| + |D| + |E| + |F|.$
|
1381
| 0.375 |
In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is
|
60
| 0.375 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week.
|
36
| 0.125 |
Acme Corporation has released a new version of its vowel soup where each vowel (A, E, I, O, U) appears six times, and additionally, each bowl contains one wildcard character that can represent any vowel. How many six-letter "words" can be formed from a bowl of this new Acme Enhanced Vowel Soup?
|
46656
| 0.125 |
Find the number of ordered pairs $(x,y)$ of real numbers such that
\[16^{x^2 + y} + 16^{x + y^2} = 1.\]
|
1
| 0.1875 |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));
[/asy]
[asy]
draw((18,0)--(0,0)--(0,18));
label("6",(3,-1));
label("7",(6,-1));
label("8",(9,-1));
label("9",(12,-1));
label("10",(15,-1));
fill((3,.5)..(3.5,1)..(3,1.5)..(2.5,1)..cycle);
fill((3,2)..(3.5,2.5)..(3,3)..(2.5,2.5)..cycle);
fill((3,3.5)..(3.5,4)..(3,4.5)..(2.5,4)..cycle);
fill((3,5)..(3.5,5.5)..(3,6)..(2.5,5.5)..cycle);
fill((3,6.5)..(3.5,7)..(3,7.5)..(2.5,7)..cycle);
fill((3,8)..(3.5,8.5)..(3,9)..(2.5,8.5)..cycle);
fill((6,.5)..(6.5,1)..(6,1.5)..(5.5,1)..cycle);
fill((6,2)..(6.5,2.5)..(6,3)..(5.5,2.5)..cycle);
fill((6,3.5)..(6.5,4)..(6,4.5)..(5.5,4)..cycle);
fill((6,5)..(6.5,5.5)..(6,6)..(5.5,5.5)..cycle);
fill((9,.5)..(9.5,1)..(9,1.5)..(8.5,1)..cycle);
fill((9,2)..(9.5,2.5)..(9,3)..(8.5,2.5)..cycle);
fill((9,3.5)..(9.5,4)..(9,4.5)..(8.5,4)..cycle);
fill((15,.5)..(15.5,1)..(15,1.5)..(14.5,1)..cycle);
label("Number of Home Runs",(9,-3));
picture perpLabel;
label(perpLabel,"Number of Top Hitters");
add(rotate(90)*perpLabel,(-1,9));
[/asy]
|
7
| 0.0625 |
Let $Q$ be the product of the first $150$ positive odd integers. Find the largest integer $k'$ such that $Q$ is divisible by $3^{k'}$.
|
76
| 0.4375 |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));
[/asy]
[asy]
draw((18,0)--(0,0)--(0,18));
label("6",(3,-1));
label("7",(6,-1));
label("8",(9,-1));
label("9",(12,-1));
label("10",(15,-1));
fill((3,.5)..(3.5,1)..(3,1.5)..(2.5,1)..cycle);
fill((3,2)..(3.5,2.5)..(3,3)..(2.5,2.5)..cycle);
fill((3,3.5)..(3.5,4)..(3,4.5)..(2.5,4)..cycle);
fill((3,5)..(3.5,5.5)..(3,6)..(2.5,5.5)..cycle);
fill((3,6.5)..(3.5,7)..(3,7.5)..(2.5,7)..cycle);
fill((3,8)..(3.5,8.5)..(3,9)..(2.5,8.5)..cycle);
fill((6,.5)..(6.5,1)..(6,1.5)..(5.5,1)..cycle);
fill((6,2)..(6.5,2.5)..(6,3)..(5.5,2.5)..cycle);
fill((6,3.5)..(6.5,4)..(6,4.5)..(5.5,4)..cycle);
fill((6,5)..(6.5,5.5)..(6,6)..(5.5,5.5)..cycle);
fill((9,.5)..(9.5,1)..(9,1.5)..(8.5,1)..cycle);
fill((9,2)..(9.5,2.5)..(9,3)..(8.5,2.5)..cycle);
fill((9,3.5)..(9.5,4)..(9,4.5)..(8.5,4)..cycle);
fill((15,.5)..(15.5,1)..(15,1.5)..(14.5,1)..cycle);
label("Number of Home Runs",(9,-3));
picture perpLabel;
label(perpLabel,"Number of Top Hitters");
add(rotate(90)*perpLabel,(-1,9));
[/asy]
|
7
| 0.0625 |
A mason has bricks with dimensions $2\times5\times8$ and other bricks with dimensions $2\times3\times7$ . She also has a box with dimensions $10\times11\times14$ . The bricks and the box are all rectangular parallelepipeds. The mason wants to pack bricks into the box filling its entire volume and with no bricks sticking out.
Find all possible values of the total number of bricks that she can pack.
|
24
| 0.4375 |
Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines?
|
6
| 0.375 |
We have created a convex polyhedron using pentagons and hexagons where three faces meet at each vertex. Each pentagon shares its edges with 5 hexagons, and each hexagon shares its edges with 3 pentagons. How many faces does the polyhedron have?
|
32
| 0.375 |
The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).
Find the angle \( \angle AMB \).
|
72
| 0.3125 |
For any real number $x$, the symbol $\lfloor x \rfloor$ represents the integer part of $x$, that is, $\lfloor x \rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\lfloor \log_{2}1 \rfloor + \lfloor \log_{2}2 \rfloor + \lfloor \log_{2}3 \rfloor + \lfloor \log_{2}4 \rfloor + \ldots + \lfloor \log_{2}1024 \rfloor$.
|
8204
| 0.4375 |
A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement?
|
11
| 0.375 |
The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).
Find the angle \( \angle AMB \).
|
72
| 0.3125 |
Record the outcome of hitting or missing for 6 consecutive shots in order.
① How many possible outcomes are there?
② How many outcomes are there where exactly 3 shots hit the target?
③ How many outcomes are there where 3 shots hit the target, and exactly two of those hits are consecutive?
|
12
| 0.3125 |
In the diagram, \( PQ \) is perpendicular to \( QR \), \( QR \) is perpendicular to \( RS \), and \( RS \) is perpendicular to \( ST \). If \( PQ=4 \), \( QR=8 \), \( RS=8 \), and \( ST=3 \), then the distance from \( P \) to \( T \) is
|
13
| 0.0625 |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$.
|
2889
| 0.125 |
Let $ABCDEF$ be an equiangular hexagon such that $AB=6, BC=8, CD=10$, and $DE=12$. Denote by $d$ the diameter of the largest circle that fits inside the hexagon. Find $d^2$.
|
147
| 0.0625 |
The graph below shows the number of home runs in April for the top hitters in the league. What is the mean (average) number of home runs hit by these players?
[asy]
draw((0,0)--(0,7)--(24,7)--(24,0)--cycle);
label("KEY:",(3,5));
fill((3,2.5)..(3.5,2)..(3,1.5)..(2.5,2)..cycle);
label("- one(1) baseball player",(14,2));
[/asy]
[asy]
draw((18,0)--(0,0)--(0,18));
label("6",(3,-1));
label("7",(6,-1));
label("8",(9,-1));
label("9",(12,-1));
label("10",(15,-1));
fill((3,.5)..(3.5,1)..(3,1.5)..(2.5,1)..cycle);
fill((3,2)..(3.5,2.5)..(3,3)..(2.5,2.5)..cycle);
fill((3,3.5)..(3.5,4)..(3,4.5)..(2.5,4)..cycle);
fill((3,5)..(3.5,5.5)..(3,6)..(2.5,5.5)..cycle);
fill((3,6.5)..(3.5,7)..(3,7.5)..(2.5,7)..cycle);
fill((3,8)..(3.5,8.5)..(3,9)..(2.5,8.5)..cycle);
fill((6,.5)..(6.5,1)..(6,1.5)..(5.5,1)..cycle);
fill((6,2)..(6.5,2.5)..(6,3)..(5.5,2.5)..cycle);
fill((6,3.5)..(6.5,4)..(6,4.5)..(5.5,4)..cycle);
fill((6,5)..(6.5,5.5)..(6,6)..(5.5,5.5)..cycle);
fill((9,.5)..(9.5,1)..(9,1.5)..(8.5,1)..cycle);
fill((9,2)..(9.5,2.5)..(9,3)..(8.5,2.5)..cycle);
fill((9,3.5)..(9.5,4)..(9,4.5)..(8.5,4)..cycle);
fill((15,.5)..(15.5,1)..(15,1.5)..(14.5,1)..cycle);
label("Number of Home Runs",(9,-3));
picture perpLabel;
label(perpLabel,"Number of Top Hitters");
add(rotate(90)*perpLabel,(-1,9));
[/asy]
|
7
| 0.0625 |
Convert the quadratic equation $3x=x^{2}-2$ into general form and determine the coefficients of the quadratic term, linear term, and constant term.
|
-2
| 0.3125 |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
|
256
| 0.1875 |
In the diagram, \(ABCD\) is a rectangle with \(AD = 13\), \(DE = 5\), and \(EA = 12\). The area of \(ABCD\) is
|
60
| 0.375 |
How many positive integers, not exceeding 200, are multiples of 3 or 5 but not 6?
|
60
| 0.375 |
Given the set $A=\{x|x=a_0+a_1\times3+a_2\times3^2+a_3\times3^3\}$, where $a_k\in\{0,1,2\}$ ($k=0,1,2,3$), and $a_3\neq0$, calculate the sum of all elements in set $A$.
|
2889
| 0.125 |
It is known that the optimal amount of a certain material to be added is between 100g and 1100g. If the 0.618 method is used to arrange the experiment and the first and second trials are at points $x_1$ and $x_2$ ($x_1 > x_2$), then when $x_2$ is considered the better point, the third trial point $x_3$ should be __g (answer with a number).
|
336
| 0.125 |
The circle is divided by points \(A\), \(B\), \(C\), and \(D\) such that \(AB: BC: CD: DA = 3: 2: 13: 7\). Chords \(AD\) and \(BC\) are extended to intersect at point \(M\).
Find the angle \( \angle AMB \).
|
72
| 0.3125 |
For any positive integer \( k \), let \( f_{1}(k) \) be the square of the sum of the digits of \( k \) when written in decimal notation. For \( n > 1 \), let \( f_{n}(k) = f_{1}\left(f_{n-1}(k)\right) \). What is \( f_{1992}\left(2^{1991}\right) \)?
|
256
| 0.1875 |
Let $N$ be the number of ordered pairs of nonempty sets $\mathcal{A}$ and $\mathcal{B}$ that have the following properties:
$\mathcal{A} \cup \mathcal{B} = \{1,2,3,4,5,6,7,8,9,10,11,12\}$,
$\mathcal{A} \cap \mathcal{B} = \emptyset$,
The number of elements of $\mathcal{A}$ is not an element of $\mathcal{A}$,
The number of elements of $\mathcal{B}$ is not an element of $\mathcal{B}$.
Find $N$.
|
772
| 0.125 |
A certain high school has three mathematics teachers. For the convenience of the students, they arrange for a math teacher to be on duty every day from Monday to Friday, and two teachers are scheduled to be on duty on Monday. If each teacher is on duty for two days per week, there are ________ possible duty arrangements for the week.
|
36
| 0.125 |
Determine the largest multiple of 36 that consists of all even and distinct digits.
|
8640
| 0.375 |
Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2).
|
-1
| 0.125 |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy = 1$ and both branches of the hyperbola $xy = -1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.)
|
4
| 0.0625 |
A pyramid \( S A B C D \) has a trapezoid \( A B C D \) as its base, with bases \( B C \) and \( A D \). Points \( P_1, P_2, P_3 \) lie on side \( B C \) such that \( B P_1 < B P_2 < B P_3 < B C \). Points \( Q_1, Q_2, Q_3 \) lie on side \( A D \) such that \( A Q_1 < A Q_2 < A Q_3 < A D \). Let \( R_1, R_2, R_3, \) and \( R_4 \) be the intersection points of \( B Q_1 \) with \( A P_1 \); \( P_2 Q_1 \) with \( P_1 Q_2 \); \( P_3 Q_2 \) with \( P_2 Q_3 \); and \( C Q_3 \) with \( P_3 D \) respectively. It is known that the sum of the volumes of the pyramids \( S R_1 P_1 R_2 Q_1 \) and \( S R_3 P_3 R_4 Q_3 \) equals 78. Find the minimum value of
\[ V_{S A B R_1}^2 + V_{S R_2 P_2 R_3 Q_2}^2 + V_{S C D R_4}^2 \]
and give the closest integer to this value.
|
2028
| 0.1875 |
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