Unnamed: 0
int64 0
40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
12,500 |
There are 111 balls in a box, each being red, green, blue, or white. It is known that if 100 balls are drawn, it ensures getting balls of all four colors. Find the smallest integer $N$ such that if $N$ balls are drawn, it can ensure getting balls of at least three different colors.
|
88
| 0 |
12,501 |
For the one-variable quadratic equation $x^{2}+3x+m=0$ with two real roots for $x$, determine the range of values for $m$.
|
\frac{9}{4}
| 0 |
12,502 |
Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $39x + 3y = 2070$. Find the number of such distinct triangles whose area is a positive integer that is also a multiple of three.
|
676
| 0 |
12,503 |
Given square $PQRS$ with side length $12$ feet, a circle is drawn through vertices $P$ and $S$, and tangent to side $QR$. If the point of tangency divides $QR$ into segments of $3$ feet and $9$ feet, calculate the radius of the circle.
|
\sqrt{(6 - 3\sqrt{2})^2 + 9^2}
| 0 |
12,504 |
Simplify $\cos 20^\circ - \cos 40^\circ.$
|
\frac{\sqrt{5} - 1}{4}
| 0 |
12,505 |
Given that \( a_{k} \) is the number of integer terms in \( \log_{2} k, \log_{3} k, \cdots, \log_{2018} k \). Calculate \( \sum_{k=1}^{2018} a_{k} \).
|
4102
| 71.09375 |
12,506 |
Let \( p, q, r, \) and \( s \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 2500, \\
pr &=& qs &=& 1200.
\end{array}
\]
If \( T = p + q + r + s \), compute the value of \( \lfloor T \rfloor \).
|
120
| 0 |
12,507 |
Simplify $(2^8+4^5)(2^3-(-2)^3)^{10}$.
|
1342177280
| 0 |
12,508 |
Let $(x_1,y_1),$ $(x_2,y_2),$ $\dots,$ $(x_n,y_n)$ be the solutions to
\begin{align*}
|x - 5| &= |y - 12|, \\
|x - 12| &= 3|y - 5|.
\end{align*}
Find $x_1 + y_1 + x_2 + y_2 + \dots + x_n + y_n.$
|
70
| 0.78125 |
12,509 |
How many ways can you arrange 15 dominoes (after removing all dominoes with five or six pips) in a single line according to the usual rules of the game, considering arrangements from left to right and right to left as different? As always, the dominoes must be placed such that matching pips (e.g., 1 to 1, 6 to 6, etc.) are adjacent.
|
126760
| 0 |
12,510 |
Given Angela has 4 files that are each 1.2 MB, 8 files that are 0.9 MB each, and 10 files that are 0.6 MB each, calculate the minimum number of disks Angela will need to store all her files.
|
11
| 1.5625 |
12,511 |
A large number \( y \) is defined by \( 2^33^54^45^76^57^38^69^{10} \). Determine the smallest positive integer that, when multiplied with \( y \), results in a product that is a perfect square.
|
70
| 0 |
12,512 |
Given the integers a and b, where a consists of a sequence of 1986 nines and b consists of a sequence of 1986 fours, calculate the sum of the digits of the base 10 representation of 9ab.
|
15880
| 0 |
12,513 |
Vasya wrote down 11 natural numbers in one notebook. Petya wrote down the greatest common divisors of each pair of numbers recorded in Vasya's notebook in another notebook. It turned out that every number written in one of the two notebooks is also in the other notebook. What is the maximum number of distinct numbers that could have been written in Vasya's notebook?
|
10
| 1.5625 |
12,514 |
In triangle \(ABC\), a circle \(\omega\) with center \(O\) passes through \(B\) and \(C\) and intersects segments \(\overline{AB}\) and \(\overline{AC}\) again at \(B'\) and \(C'\), respectively. Suppose that the circles with diameters \(BB'\) and \(CC'\) are externally tangent to each other at \(T\). If \(AB = 18\), \(AC = 36\), and \(AT = 12\), compute \(AO\).
|
65/3
| 0 |
12,515 |
Simplify the expression, then evaluate: $$(1- \frac {a}{a+1})\div \frac {1}{1-a^{2}}$$ where $a=-2$.
|
\frac {1}{3}
| 0 |
12,516 |
The equation
\[(x - \sqrt[3]{17})(x - \sqrt[3]{67})(x - \sqrt[3]{97}) = \frac{1}{2}\]
has three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$
|
181.5
| 6.25 |
12,517 |
Let $ABCD$ be a rectangle such that $\overline{AB}=\overline{CD}=30$, $\overline{BC}=\overline{DA}=50$ and point $E$ lies on line $AB$, 20 units from $A$. Find the area of triangle $BEC$.
|
1000
| 0 |
12,518 |
Let \( z \) be a complex number that satisfies
\[ |z - 2| + |z - 7i| = 10. \]
Find the minimum value of \( |z| \).
|
1.4
| 0 |
12,519 |
Given the coordinates of points $A(3, 0)$, $B(0, -3)$, and $C(\cos\alpha, \sin\alpha)$, where $\alpha \in \left(\frac{\pi}{2}, \frac{3\pi}{2}\right)$. If $\overrightarrow{OC}$ is parallel to $\overrightarrow{AB}$ and $O$ is the origin, find the value of $\alpha$.
|
\frac{3\pi}{4}
| 0 |
12,520 |
Ben received a bill for $\$600$. If a 2% late charge is applied for each 30-day period past the due date, and he pays 90 days after the due date, what is his total bill?
|
636.53
| 0 |
12,521 |
Rationalize the denominator of $\frac{\sqrt[3]{27} + \sqrt[3]{2}}{\sqrt[3]{3} + \sqrt[3]{2}}$ and express your answer in simplest form.
|
7 - \sqrt[3]{54} + \sqrt[3]{6}
| 0 |
12,522 |
Evaluate the polynomial \[ x^3 - 3x^2 - 12x + 9, \] where \(x\) is the positive number such that \(x^2 - 3x - 12 = 0\) and \(x \neq -2\).
|
-23
| 0 |
12,523 |
What is the value of $x + y$ if the sequence $3, ~9, ~15, \ldots, ~x, ~y, ~39$ is an arithmetic sequence?
|
60
| 20.3125 |
12,524 |
Let $\mathrm{C}$ be a circle in the $\mathrm{xy}$-plane with a radius of 1 and its center at $O(0,0,0)$. Consider a point $\mathrm{P}(3,4,8)$ in space. If a sphere is completely contained within the cone with $\mathrm{C}$ as its base and $\mathrm{P}$ as its apex, find the maximum volume of this sphere.
|
\frac{4}{3}\pi(3-\sqrt{5})^3
| 0 |
12,525 |
The shaded design shown in the diagram is made by drawing eight circular arcs, all with the same radius. The centers of four arcs are the vertices of the square; the centers of the four touching arcs are the midpoints of the sides of the square. The diagonals of the square have length 1. Calculate the total length of the border of the shaded design.
|
\frac{5}{2} \pi
| 0 |
12,526 |
A function \( f: \{a, b, c, d\} \rightarrow \{1, 2, 3\} \) is given. If \( 10 < f(a) \cdot f(b) \) and \( f(c) \cdot f(d) < 20 \), how many such mappings exist?
|
25
| 0.78125 |
12,527 |
Given that 6 rational numbers are placed sequentially on a given circle, now select any 3 adjacent numbers $a$, $b$, $c$ in a clockwise direction, satisfying $a = |b - c|$. It is also known that the total sum of all the placed numbers is 1. What are the values of these 6 numbers?
|
\frac{1}{4}
| 0 |
12,528 |
Given a prism \(ABC-A'B'C'\) with a base that is an equilateral triangle with side length 2, the lateral edge \(AA'\) forms a 45-degree angle with the edges \(AB\) and \(AC\) of the base. Point \(A'\) is equidistant from the planes \(ABC\) and \(BB'C'C\). Find \(A'A = \_\_\_\_\_ \).
|
\sqrt{6}
| 17.1875 |
12,529 |
An equilateral triangle with side length 10 cm is inscribed in a circle. The triangle's side is the diameter of the circle. In two opposite sectors of the circle, there are small shaded regions. If the sum of the areas of the two small shaded regions is in the form \(a\pi - b\sqrt{c}\), calculate \(a+b+c\).
|
28
| 3.125 |
12,530 |
The intersection of two squares with perimeter $8$ is a rectangle with diagonal length $1$ . Given that the distance between the centers of the two squares is $2$ , the perimeter of the rectangle can be expressed as $P$ . Find $10P$ .
|
25
| 7.03125 |
12,531 |
Let \( x, y, z \) be nonnegative real numbers. Define:
\[
A = \sqrt{x + 3} + \sqrt{y + 6} + \sqrt{z + 12},
\]
\[
B = \sqrt{x + 2} + \sqrt{y + 2} + \sqrt{z + 2}.
\]
Find the minimum value of \( A^2 - B^2 \).
|
36
| 3.90625 |
12,532 |
What is the smallest four-digit number that is divisible by $45$?
|
1008
| 0 |
12,533 |
In a parking lot, there are seven parking spaces numbered from 1 to 7. Now, two different trucks and two different buses are to be parked at the same time, with each parking space accommodating at most one vehicle. If vehicles of the same type are not parked in adjacent spaces, there are a total of ▲ different parking arrangements.
|
840
| 2.34375 |
12,534 |
A glass is filled to the brim with salty water. Fresh ice with mass \( m = 502 \) floats on the surface. What volume \( \Delta V \) of water will spill out of the glass by the time the ice melts? Neglect surface tension. The density of fresh ice is \( \rho_{n} = 0.92 \, \text{g/cm}^3 \), the density of salty ice is \( \rho_{c} = 0.952 \, \text{g/cm}^3 \), and the density of fresh water is \( \rho_{ns} = 12 \, \text{g/cm}^3 \). Neglect the change in total volume when mixing the two liquids.
|
2.63
| 0 |
12,535 |
What integer \( n \) satisfies \( 0 \le n < 23 \) and
$$
54126 \equiv n \pmod{23}~?
$$
|
13
| 3.90625 |
12,536 |
Ms. Carr expands her reading list to 12 books and asks each student to choose any 6 books. Harold and Betty each randomly select 6 books from this list. Calculate the probability that there are exactly 3 books that they both select.
|
\frac{405}{2223}
| 0 |
12,537 |
Given vectors $\overrightarrow{a}=(2\cos x,1)$, $\overrightarrow{b}=(\sqrt{3}\sin x+\cos x,-1)$, and the function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.
1. Find the maximum and minimum values of $f(x)$ in the interval $[0,\frac{\pi}{4}]$.
2. If $f(x_{0})=\frac{6}{5}$, $x_{0}\in[\frac{\pi}{4},\frac{\pi}{2}]$, find the value of $\cos 2x_{0}$.
3. If the function $y=f(\omega x)$ is monotonically increasing in the interval $(\frac{\pi}{3},\frac{2\pi}{3})$, find the range of positive values for $\omega$.
|
\frac{1}{4}
| 0.78125 |
12,538 |
Consider a rectangle with dimensions 6 units by 7 units. A triangle is formed with its vertices on the sides of the rectangle. Vertex $A$ is on the left side, 3 units from the bottom. Vertex $B$ is on the bottom side, 5 units from the left. Vertex $C$ is on the top side, 2 units from the right. Calculate the area of triangle $ABC$.
|
17.5
| 0.78125 |
12,539 |
Two boards, one five inches wide and the other eight inches wide, are nailed together to form an X. The angle at which they cross is 45 degrees. If this structure is painted and the boards are separated, what is the area of the unpainted region on the five-inch board? (Neglect the holes caused by the nails.)
|
40 \sqrt{2}
| 0.78125 |
12,540 |
Given the equation about $x$, $2x^{2}-( \sqrt {3}+1)x+m=0$, its two roots are $\sin θ$ and $\cos θ$, where $θ∈(0,π)$. Find:
$(1)$ the value of $m$;
$(2)$ the value of $\frac {\tan θ\sin θ}{\tan θ-1}+ \frac {\cos θ}{1-\tan θ}$;
$(3)$ the two roots of the equation and the value of $θ$ at this time.
|
\frac {1}{2}
| 0.78125 |
12,541 |
Let \( C_1 \) and \( C_2 \) be circles defined by
\[
(x-12)^2 + y^2 = 25
\]
and
\[
(x+18)^2 + y^2 = 64,
\]
respectively. What is the length of the shortest line segment \( \overline{RS} \) that is tangent to \( C_1 \) at \( R \) and to \( C_2 \) at \( S \)?
|
30
| 0 |
12,542 |
Find $\overrightarrow{a}+2\overrightarrow{b}$, where $\overrightarrow{a}=(2,0)$ and $|\overrightarrow{b}|=1$, and then calculate the magnitude of this vector.
|
2\sqrt{3}
| 1.5625 |
12,543 |
Calculate $\sin 9^\circ \sin 45^\circ \sin 69^\circ \sin 81^\circ.$
|
\frac{0.6293 \sqrt{2}}{4}
| 0 |
12,544 |
How many six-digit numbers exist that have three even and three odd digits?
|
281250
| 0.78125 |
12,545 |
In the "Nine Chapters on the Mathematical Art," a tetrahedron with all four faces being right-angled triangles is referred to as a "turtle's knee." Given that in the turtle's knee $M-ABC$, $MA \perp$ plane $ABC$, and $MA=AB=BC=2$, the sum of the surface areas of the circumscribed sphere and the inscribed sphere of the turtle's knee is ______.
|
24\pi-8\sqrt{2}\pi
| 0.78125 |
12,546 |
On a straight segment of a one-way, single-lane highway, cars travel at the same speed and follow a safety rule where the distance from the back of one car to the front of the next is equal to the car’s speed divided by 10 kilometers per hour, rounded up to the nearest whole number (e.g., a car traveling at 52 kilometers per hour maintains a 6 car length distance to the car in front). Assume each car is 5 meters long, and the cars can travel at any speed. A sensor by the side of the road counts the number of cars that pass in one hour. Let $N$ be the maximum whole number of cars that can pass the sensor in one hour. Find $N$ divided by 20.
|
92
| 0.78125 |
12,547 |
Compute the number of ordered pairs of integers $(x,y)$ with $1\le x<y\le 150$ such that $i^x+i^y$ is a real number.
|
3515
| 0 |
12,548 |
(1) Given that $\log_2{2} = a$, express $\log_8{20} - 2\log_2{20}$ in terms of $a$.
(2) Evaluate the expression: $(\ln{4})^0 + (\frac{9}{4})^{-0.5} + \sqrt{(1 - \sqrt{3})^2} - 2^{\log_4{3}}$.
|
\frac{9}{2} - 2\sqrt{3}
| 0 |
12,549 |
The Student council has 24 members: 12 boys and 12 girls. A 5-person committee is selected at random. What is the probability that the committee includes at least one boy and at least one girl?
|
\frac{455}{472}
| 0 |
12,550 |
In triangle \(ABC\), point \(N\) lies on side \(AB\) such that \(AN = 3NB\); the median \(AM\) intersects \(CN\) at point \(O\). Find \(AB\) if \(AM = CN = 7\) cm and \(\angle NOM = 60^\circ\).
|
4\sqrt{7}
| 0 |
12,551 |
Find the smallest positive integer that is both an integer power of 13 and is not a palindrome.
|
169
| 39.0625 |
12,552 |
The sides of rectangle $ABCD$ have lengths $12$ (height) and $15$ (width). An equilateral triangle is drawn so that no point of the triangle lies outside $ABCD$. Find the maximum possible area of such a triangle.
|
369 \sqrt{3} - 540
| 0 |
12,553 |
Compute the sum:
\[
\sin^2 3^\circ + \sin^2 9^\circ + \sin^2 15^\circ + \dots + \sin^2 177^\circ.
\]
|
10
| 0 |
12,554 |
Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g.
|
2020
| 2.34375 |
12,555 |
Can we find \( N \) such that all \( m \times n \) rectangles with \( m, n > N \) can be tiled with \( 4 \times 6 \) and \( 5 \times 7 \) rectangles?
|
840
| 3.125 |
12,556 |
An ellipse has a major axis of length 12 and a minor axis of 10. Using one focus as a center, an external circle is tangent to the ellipse. Find the radius of the circle.
|
\sqrt{11}
| 1.5625 |
12,557 |
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take?
|
10000
| 3.90625 |
12,558 |
Let $\alpha$ be a nonreal root of $x^4 = 1.$ Compute
\[(1 - \alpha + \alpha^2 - \alpha^3)^4 + (1 + \alpha - \alpha^2 + \alpha^3)^4.\]
|
32
| 2.34375 |
12,559 |
How many integers between $123$ and $789$ have at least two identical digits, when written in base $10?$
|
180
| 0 |
12,560 |
7.61 log₂ 3 + 2 log₄ x = x^(log₉ 16 / log₃ x).
|
16/3
| 0 |
12,561 |
Given $f(x)=1-2x^{2}$ and $g(x)=x^{2}-2x$, let $F(x) = \begin{cases} f(x), & \text{if } f(x) \geq g(x) \\ g(x), & \text{if } f(x) < g(x) \end{cases}$. Determine the maximum value of $F(x)$.
|
\frac{7}{9}
| 28.90625 |
12,562 |
Regular hexagon $ABCDEF$ has its center at $G$. Each of the vertices and the center are to be associated with one of the digits $1$ through $7$, with each digit used once, in such a way that the sums of the numbers on the lines $AGC$, $BGD$, and $CGE$ are all equal. In how many ways can this be done?
|
144
| 50 |
12,563 |
Let \( a, b, c, d \) be positive integers such that \( \gcd(a, b) = 24 \), \( \gcd(b, c) = 36 \), \( \gcd(c, d) = 54 \), and \( 70 < \gcd(d, a) < 100 \). Which of the following numbers is a factor of \( a \)?
|
13
| 5.46875 |
12,564 |
The cubic polynomial $q(x)$ satisfies $q(1) = 5,$ $q(6) = 20,$ $q(14) = 12,$ and $q(19) = 30.$ Find
\[q(0) + q(1) + q(2) + \dots + q(20).\]
|
357
| 0.78125 |
12,565 |
Given that the first term of a geometric sequence $\{a\_n\}$ is $\frac{3}{2}$, and the sum of the first $n$ terms is $S\_n$, where $n \in \mathbb{N}^*$. Also, $-2S\_2$, $S\_3$, and $4S\_4$ form an arithmetic sequence.
1. Find the general term formula for the sequence $\{a\_n\}$.
2. For a sequence $\{A\_n\}$, if there exists an interval $M$ such that $A\_i \in M$ for all $i = 1, 2, 3, ...$, then $M$ is called the "range interval" of sequence $\{A\_n\}$. Let $b\_n = S\_n + \frac{1}{S\_n}$, find the minimum length of the "range interval" of sequence $\{b\_n\}$.
|
\frac{1}{6}
| 10.9375 |
12,566 |
Let \( y = \cos \frac{2 \pi}{9} + i \sin \frac{2 \pi}{9} \). Compute the value of
\[
(3y + y^3)(3y^3 + y^9)(3y^6 + y^{18})(3y^2 + y^6)(3y^5 + y^{15})(3y^7 + y^{21}).
\]
|
112
| 0 |
12,567 |
A vessel with a capacity of 100 liters is filled with a brine solution containing 10 kg of dissolved salt. Every minute, 3 liters of water flows into it, and the same amount of the resulting mixture is pumped into another vessel of the same capacity, initially filled with water, from which the excess liquid overflows. At what point in time will the amount of salt in both vessels be equal?
|
333.33
| 0 |
12,568 |
Compute \[\sum_{n=1}^{500} \frac{1}{n^2 + 2n}.\]
|
\frac{1499}{2008}
| 0 |
12,569 |
Given a right triangle \( \triangle ABC \) with legs \( AC=12 \), \( BC=5 \). Point \( D \) is a moving point on the hypotenuse \( AB \). The triangle is folded along \( CD \) to form a right dihedral angle \( A-CD-B \). When the length of \( AB \) is minimized, let the plane angle of the dihedral \( B-AC-D \) be \( \alpha \). Find \( \tan^{10} \alpha \).
|
32
| 0.78125 |
12,570 |
A polynomial $p(x)$ leaves a remainder of $2$ when divided by $x - 3,$ a remainder of 1 when divided by $x - 4,$ and a remainder of 5 when divided by $x + 4.$ Let $r(x)$ be the remainder when $p(x)$ is divided by $(x - 3)(x - 4)(x + 4).$ Find $r(5).$
|
\frac{1}{13}
| 0 |
12,571 |
How many distinct sequences of five letters can be made from the letters in COMPUTER if each letter can be used only once, each sequence must begin with M, end with R, and the third letter must be a vowel (A, E, I, O, U)?
|
36
| 2.34375 |
12,572 |
Let \( O \) be the origin, \( A_1, A_2, A_3, \ldots \) be points on the curve \( y = \sqrt{x} \) and \( B_1, B_2, B_3, \ldots \) be points on the positive \( x \)-axis such that the triangles \( O B_1 A_1, B_1 B_2 A_2, B_2 B_3 A_3, \ldots \) are all equilateral, with side lengths \( l_1, l_2, l_3, \ldots \) respectively. Find the value of \( l_1 + l_2 + l_3 + \cdots + l_{2005} \).
|
4022030/3
| 0 |
12,573 |
Let $\{b_k\}$ be a sequence of integers such that $b_1=2$ and $b_{m+n}=b_m+b_n+mn^2,$ for all positive integers $m$ and $n.$ Find $b_{12}$.
|
98
| 0 |
12,574 |
A convex pentagon $P=ABCDE$ is inscribed in a circle of radius $1$ . Find the maximum area of $P$ subject to the condition that the chords $AC$ and $BD$ are perpendicular.
|
1 + \frac{3\sqrt{3}}{4}
| 0 |
12,575 |
The "Hua Luogeng" Golden Cup Junior Math Invitational Contest was first held in 1986, the second in 1988, and the third in 1991, and has subsequently been held every 2 years. The sum of the digits of the year of the first "Hua Cup" is: \( A_1 = 1 + 9 + 8 + 6 = 24 \). The sum of the digits of the years of the first two "Hua Cup" contests is: \( A_2 = 1 + 9 + 8 + 6 + 1 + 9 + 8 + 8 = 50 \). Find the sum of the digits of the years of the first 50 "Hua Cup" contests, \( A_{50} \).
|
629
| 86.71875 |
12,576 |
Given the sequence defined by $O = \begin{cases} 3N + 2, & \text{if } N \text{ is odd} \\ \frac{N}{2}, & \text{if } N \text{ is even} \end{cases}$, for a given integer $N$, find the sum of all integers that, after being inputted repeatedly for 7 more times, ultimately result in 4.
|
1016
| 0 |
12,577 |
Given $b-a=-6$ and $ab=7$, find the value of $a^2b-ab^2$.
|
-42
| 1.5625 |
12,578 |
How many of the natural numbers from 1 to 700, inclusive, contain the digit 5 at least once?
|
214
| 0 |
12,579 |
A regular triangular prism has a triangle $ABC$ with side $a$ as its base. Points $A_{1}, B_{1}$, and $C_{1}$ are taken on the lateral edges and are located at distances of $a / 2, a, 3a / 2$ from the base plane, respectively. Find the angle between the planes $ABC$ and $A_{1}B_{1}C_{1}$.
|
\frac{\pi}{4}
| 22.65625 |
12,580 |
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \]
|
7.9
| 0 |
12,581 |
When the set of natural numbers is listed in ascending order, what is the smallest prime number that occurs after a sequence of seven consecutive positive integers, all of which are nonprime?
|
53
| 1.5625 |
12,582 |
In a vertical vessel with straight walls closed by a piston, there is water. Its height is $h=2$ mm. There is no air in the vessel. To what height must the piston be raised for all the water to evaporate? The density of water is $\rho=1000$ kg / $\mathrm{m}^{3}$, the molar mass of water vapor is $M=0.018$ kg/mol, the pressure of saturated water vapor at a temperature of $T=50{ }^{\circ} \mathrm{C}$ is $p=12300$ Pa. The temperature of water and vapor is maintained constant.
|
24.258
| 0 |
12,583 |
Given that the number of rabbits in a farm increases such that the difference between the populations in year $n+2$ and year $n$ is directly proportional to the population in year $n+1$, and the populations in the years $2001$, $2002$, and $2004$ were $50$, $80$, and $170$, respectively, determine the population in $2003$.
|
120
| 54.6875 |
12,584 |
Let $q(x) = 2x^6 - 3x^4 + Dx^2 + 6$ be a polynomial. When $q(x)$ is divided by $x - 2$, the remainder is 14. Find the remainder when $q(x)$ is divided by $x + 2$.
|
158
| 0 |
12,585 |
Given \( x_{0} > 0 \), \( x_{0} \neq \sqrt{3} \), a point \( Q\left( x_{0}, 0 \right) \), and a point \( P(0, 4) \), the line \( PQ \) intersects the hyperbola \( x^{2} - \frac{y^{2}}{3} = 1 \) at points \( A \) and \( B \). If \( \overrightarrow{PQ} = t \overrightarrow{QA} = (2-t) \overrightarrow{QB} \), then \( x_{0} = \) _______.
|
\frac{\sqrt{2}}{2}
| 0 |
12,586 |
In the country of Anchuria, a day can either be sunny, with sunshine all day, or rainy, with rain all day. If today's weather is different from yesterday's, the Anchurians say that the weather has changed. Scientists have established that January 1st is always sunny, and each subsequent day in January will be sunny only if the weather changed exactly one year ago on that day. In 2015, January in Anchuria featured a variety of sunny and rainy days. In which year will the weather in January first change in exactly the same pattern as it did in January 2015?
|
2047
| 0 |
12,587 |
Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is an Ace and the second card is a $\spadesuit$?
|
\dfrac{1}{52}
| 60.9375 |
12,588 |
Amy rolls six fair 8-sided dice, each numbered from 1 to 8. What is the probability that exactly three of the dice show a prime number and at least one die shows an 8?
|
\frac{2899900}{16777216}
| 0 |
12,589 |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal?
|
18
| 9.375 |
12,590 |
A room has a floor with dimensions \(7 \times 8\) square meters, and the ceiling height is 4 meters. A fly named Masha is sitting in one corner of the ceiling, while a spider named Petya is in the opposite corner of the ceiling. Masha decides to travel to visit Petya by the shortest route that includes touching the floor. Find the length of the path she travels.
|
\sqrt{265}
| 0 |
12,591 |
Sharik and Matroskin ski on a circular track, half of which is an uphill slope and the other half is a downhill slope. Their speeds are identical on the uphill slope and are four times less than their speeds on the downhill slope. The minimum distance Sharik falls behind Matroskin is 4 km, and the maximum distance is 13 km. Find the length of the track.
|
24
| 4.6875 |
12,592 |
A regular triangular prism \( A B C A_{1} B_{1} C_{1} \) is inscribed in a sphere, where \( A B C \) is the base and \( A A_{1}, B B_{1}, C C_{1} \) are the lateral edges. The segment \( C D \) is the diameter of this sphere, and point \( K \) is the midpoint of the edge \( A A_{1} \). Find the volume of the prism if \( C K = 2 \sqrt{3} \) and \( D K = 2 \sqrt{2} \).
|
9\sqrt{2}
| 0.78125 |
12,593 |
How many ways are there to put 6 balls in 4 boxes if the balls are not distinguishable but the boxes are?
|
84
| 100 |
12,594 |
The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \ldots$), and that the sum of the house numbers from the beginning of the street to his friend's house matches the sum of the house numbers from his friend's house to the end of the street. It is also known that on the side of the street where his friend's house is located, there are more than 50 but fewer than 500 houses.
What is the house number where the storyteller's friend lives?
|
204
| 78.90625 |
12,595 |
A gymnastics team consists of 48 members. To form a square formation, they need to add at least ____ people or remove at least ____ people.
|
12
| 14.0625 |
12,596 |
Let $S$ be the set of all positive integer divisors of $129,600$. Calculate the number of numbers that are the product of two distinct elements of $S$.
|
488
| 0 |
12,597 |
Let \( m \) be an integer greater than 1, and let's define a sequence \( \{a_{n}\} \) as follows:
\[
\begin{array}{l}
a_{0}=m, \\
a_{1}=\varphi(m), \\
a_{2}=\varphi^{(2)}(m)=\varphi(\varphi(m)), \\
\vdots \\
a_{n}=\varphi^{(n)}(m)=\varphi\left(\varphi^{(n-1)}(m)\right),
\end{array}
\]
where \( \varphi(m) \) is the Euler's totient function.
If for any non-negative integer \( k \), \( a_{k+1} \) always divides \( a_{k} \), find the greatest positive integer \( m \) not exceeding 2016.
|
1944
| 28.90625 |
12,598 |
What is the largest positive integer that is not the sum of a positive integral multiple of $36$ and a positive composite integer that is not a multiple of $4$?
|
147
| 0 |
12,599 |
Let $P(x) = (x-1)(x-4)(x-5)$. Determine how many polynomials $Q(x)$ there exist such that there exists a polynomial $R(x)$ of degree 3 with $P(Q(x)) = P(x) \cdot R(x)$, and the coefficient of $x$ in $Q(x)$ is 6.
|
22
| 0 |
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