problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Let \( f:[0,1) \rightarrow \mathbb{R} \) be a function that satisfies the following condition: if
\[ x = \sum_{n=1}^{\infty} \frac{a_n}{10^n} = .a_1 a_2 a_3 \ldots \]
is the decimal expansion of \( x \) and there does not exist a positive integer \( k \) such that \( a_n = 9 \) for all \( n \geq k \), then
\[ f(x) = \sum_{n=1}^{\infty} \frac{a_n}{10^{2n}} . \]
Determine \( f'\left(\frac{1}{3}\right) \).
|
0
| 0.375 |
Solve the following equations:
a) $\log _{1 / 5} \frac{2+x}{10}=\log _{1 / 5} \frac{2}{x+1}$;
b) $\log _{3}\left(x^{2}-4 x+3\right)=\log _{3}(3 x+21)$;
c) $\log _{1 / 10} \frac{2 x^{2}-54}{x+3}=\log _{1 / 10}(x-4)$;
d) $\log _{(5+x) / 3} 3=\log _{-1 /(x+1)} 3$.
|
-4
| 0.75 |
Calculate: $3752 \div(39 \times 2)+5030 \div(39 \times 10)=$
|
61
| 0.625 |
Find all odd integers \( n \geq 1 \) such that \( n \) divides \( 3^{n} + 1 \).
|
n = 1
| 0.75 |
A bag of rice takes Liu Bei 5 days to finish eating alone, and Guan Yu 3 days to finish eating alone. A bag of wheat takes Guan Yu 5 days to finish eating alone, and Zhang Fei 4 days to finish eating alone. Liu Bei's daily food consumption is $\qquad$% less than Zhang Fei's daily food consumption.
|
52
| 0.25 |
Let
$$
A = \sqrt{5} + \sqrt{22+2 \sqrt{5}}, \quad B = \sqrt{11+2 \sqrt{29}} + \sqrt{16-2 \sqrt{29} + 2 \sqrt{55-10 \sqrt{29}}}
$$
What is the relative magnitude of the numbers $A$ and $B$?
|
A = B
| 0.875 |
Four boys and three girls went to the forest to collect mushrooms. Each of them found several mushrooms, and in total they collected 70 mushrooms. No two girls gathered the same number of mushrooms, and any three boys together brought at least 43 mushrooms. The number of mushrooms collected by any two children differed by no more than 5 times. Masha collected the most mushrooms among the girls. How many mushrooms did she gather?
|
5
| 0.25 |
On a straight line, there are \(2n\) points, and the distance between any two adjacent points is 1. A person starts from the 1st point and jumps to other points, jumping \(2n\) times and returning to the 1st point. During these \(2n\) jumps, the person must visit all \(2n\) points. How should they jump to maximize the total distance jumped?
|
2n^2
| 0.5 |
Let \( a, b, c, d \) be strictly positive real numbers satisfying \( (a+c)(b+d)=ac+bd \). Determine the smallest value that
\[
\frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a}
\]
can take.
|
8
| 0.625 |
Find the number of 7-tuples \(\left(n_{1}, \ldots, n_{7}\right)\) of integers such that
\[
\sum_{i=1}^{7} n_{i}^{6} = 96957
\]
|
2688
| 0.5 |
In the quadrilateral \(ABCD\), it is known that \(AB = BD\) and \(\angle ABD = \angle DBC\), with \(\angle BCD = 90^\circ\). A point \(E\) on segment \(BC\) is such that \(AD = DE\). What is the length of segment \(BD\), given that \(BE = 7\) and \(EC = 5\)?
|
17
| 0.25 |
Two cars, Car A and Car B, travel towards each other from cities A and B, which are 330 kilometers apart. Car A starts from city A first. After some time, Car B starts from city B. The speed of Car A is $\frac{5}{6}$ of the speed of Car B. When the two cars meet, Car A has traveled 30 kilometers more than Car B. Determine how many kilometers Car A had traveled before Car B started.
|
55
| 0.75 |
In triangle \( \triangle ABC \), what is the maximum value of \( \sin \frac{A}{2} \sin \frac{B}{2} \sin \frac{C}{2} \)?
|
\frac{1}{8}
| 0.75 |
Among the positive integers not greater than 2017, how many are divisible by 12 but not divisible by 20?
|
135
| 0.625 |
The diagonal \(AC\) of a convex quadrilateral \(ABCD\) is the diameter of the circumcircle. Find the ratio of the areas of triangles \(ABC\) and \(ACD\) if it is known that the diagonal \(BD\) divides \(AC\) in the ratio 2:1 (counting from point \(A\)), and \(\angle BAC = 30^{\circ}\).
|
\frac{7}{8}
| 0.25 |
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
Points:
$M_{1}(-3, -5, 6)$
$M_{2}(2, 1, -4)$
$M_{3}(0, -3, -1)$
$M_{0}(3, 6, 68)$
|
\sqrt{573}
| 0.75 |
A chocolate bar has a size of $4 \times 10$ tiles. In one move, it is allowed to break one of the existing pieces into two along a straight break line. What is the minimum number of moves required to break the entire chocolate bar into pieces of size one tile each?
|
39
| 0.875 |
Given a triangle with area \( A \) and perimeter \( p \), let \( S \) be the set of all points that are a distance 5 or less from a point of the triangle. Find the area of \( S \).
|
A + 5p + 25\pi
| 0.125 |
If the real numbers $\alpha, \beta, \gamma$ form a geometric sequence with a common ratio of 2, and $\sin \alpha, \sin \beta, \sin \gamma$ form a geometric sequence, find the value of $\cos \alpha$.
|
-\frac{1}{2}
| 0.875 |
In triangle \( ABC \), \( AC = 3 AB \). Let \( AD \) bisect angle \( A \) with \( D \) lying on \( BC \), and let \( E \) be the foot of the perpendicular from \( C \) to \( AD \). Find \( \frac{[ABD]}{[CDE]} \). (Here, \([XYZ]\) denotes the area of triangle \( XYZ \)).
|
1/3
| 0.375 |
\[\frac{\tan 96^{\circ} - \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)}{1 + \tan 96^{\circ} \tan 12^{\circ} \left( 1 + \frac{1}{\sin 6^{\circ}} \right)} =\]
|
\frac{\sqrt{3}}{3}
| 0.25 |
Petya showed Vasya 37 identical-looking cards laid out in a row. He said that the hidden sides of the cards contain all the numbers from 1 to 37 (each exactly once) in such a way that the number on any card starting from the second one is a divisor of the sum of the numbers written on all the preceding cards. Then Petya revealed to Vasya that the number 37 is written on the first card and the number 1 is written on the second card. Vasya said that he then knows what number is written on the third card. What number is written on the third card?
|
2
| 0.375 |
The expression \( x + \frac{1}{x} \) has a maximum for \( x < 0 \) and a minimum for \( x > 0 \). Find the area of the rectangle whose sides are parallel to the axes and two of whose vertices are the maximum and minimum values of \( x + \frac{1}{x} \).
|
8
| 0.875 |
If $x+y-2$ is a factor of the polynomial $x^{2}+a x y+b y^{2}-5 x + y + 6$ with respect to $x$ and $y$, then what is the value of $a-b$?
|
1
| 0.875 |
Inside an angle of $30^{\circ}$ with vertex $A$, a point $K$ is chosen such that its distances to the sides of the angle are 1 and 2. Through point $K$, all possible lines intersecting the sides of the angle are drawn. Find the minimum area of a triangle that is cut off by one of these lines from the angle.
|
8
| 0.375 |
Given that \( F_{1} \) and \( F_{2} \) are the left and right foci of the hyperbola \( C: \frac{x^{2}}{4}-\frac{y^{2}}{12}=1 \), and point \( P \) lies on the hyperbola \( C \). Let \( G \) and \( I \) be the centroid and incenter of \( \triangle F_{1} P F_{2} \) respectively. If \( G I \) is parallel to the \( x \)-axis, find the circumradius \( R \) of \( \triangle F_{1} P F_{2} \).
|
5
| 0.5 |
Let \( x = \cos \theta \). Express \( \cos 3\theta \) in terms of \( x \).
|
4x^3 - 3x
| 0.875 |
1) It is known that the sequence \( u_{1}=\sqrt{2}, u_{2}=\sqrt{2+\sqrt{2}}, u_{3}=\sqrt{2+\sqrt{2+\sqrt{2}}}, \ldots, u_{n}=\sqrt{2+u_{n-1}} \) converges. Find its limit.
2) Find the limit of the sequence \( u_{n}=\frac{2^{n}}{n!} \).
|
0
| 0.875 |
Place 48 chess pieces into 9 boxes, with each box containing at least 1 piece and every box containing a different number of pieces. What is the maximum number of pieces that can be placed in the box with the most pieces?
|
12
| 0.5 |
Find the odd prime number \( p \) that satisfies the following condition: There exists a permutation \( b_1, b_2, \cdots, b_{p-1} \) of \( 1, 2, \cdots, p-1 \) such that \( 1^{b_1}, 2^{b_2}, \cdots, (p-1)^{b_{p-1}} \) forms a reduced residue system modulo \( p \).
|
3
| 0.875 |
We have a two-pan balance and want to be able to weigh any object with an integer mass between 1 and 40 kg. What is the minimum number of weights with integer masses needed to achieve this:
1. if the weights can only be placed on one pan of the balance?
2. if the weights can be placed on both pans of the balance?
|
4 \text{ weights}
| 0.625 |
Find the derivative of the scalar field \( u = x z^{2} + 2 y z \) at the point \( M_{0}(1,0,2) \) along the curve described by
\[
\left\{
\begin{array}{l}
x = 1 + \cos t \\
y = \sin t - 1 \\
z = 2
\end{array}
\right.
\]
|
-4
| 0.875 |
1. In what base is 16324 the square of 125?
2. In what base does \( 4 \cdot 13 = 100 \)?
|
b=6
| 0.5 |
Given \( x, y, z \in \mathbf{R} \) such that \( x^2 + y^2 + xy = 1 \), \( y^2 + z^2 + yz = 2 \), \( x^2 + z^2 + xz = 3 \), find \( x + y + z \).
|
\sqrt{3 + \sqrt{6}}
| 0.625 |
Given an odd function \( f(x) \) defined on \( \mathbf{R} \) whose graph is symmetric about the line \( x=2 \), and when \( 0 < x \leq 2 \), \( f(x) = x + 1 \). Find the value of \( f(-100) + f(-101) \).
|
2
| 0.5 |
An engineer arrives at the train station every day at 8 AM. At exactly 8 AM, a car arrives at the station and takes the engineer to the factory. One day, the engineer arrived at the station at 7 AM and started walking towards the car. Upon meeting the car, he got in and arrived at the factory 20 minutes earlier than usual. How long did the engineer walk? The speeds of the car and the engineer are constant.
|
50
| 0.375 |
In a football tournament, each team is supposed to play one match against each of the other teams. However, during the tournament, half of the teams were disqualified and did not participate further. As a result, a total of 77 matches were played, and the disqualified teams managed to play all their matches against each other, with each disqualified team having played the same number of matches. How many teams were there at the beginning of the tournament?
|
14
| 0.75 |
An infinite sequence of positive real numbers is defined by \(a_{0}=1\) and \(a_{n+2}=6 a_{n}-a_{n+1}\) for \(n=0,1,2, \cdots\). Find the possible value(s) of \(a_{2007}\).
|
2^{2007}
| 0.5 |
As shown in the diagram, through point \( D \) draw \( D O \) bisecting \( \angle A D C \). Through point \( B \) draw \( B O \) bisecting \( \angle A B C \), with \( B O \) and \( D O \) intersecting at point \( O \). Given that \( \angle A = 35^\circ \) and \( \angle O = 42^\circ \), find \( \angle C \).
|
49^\circ
| 0.75 |
Let \( F \) be a finite nonempty set of integers and let \( n \) be a positive integer. Suppose that:
- Any \( x \in F \) may be written as \( x = y + z \) for some \( y, z \in F \);
- If \( 1 \leq k \leq n \) and \( x_{1}, \ldots, x_{k} \in F \), then \( x_{1}+ \cdots + x_{k} \neq 0 \).
Show that \( F \) has at least \( 2n+2 \) distinct elements.
|
2n + 2
| 0.625 |
Given \(x, y, z \in (0, 2]\), find the maximum value of the expression
\[
A = \frac{\left(x^{3} - 6\right) \sqrt[3]{x + 6} + \left(y^{3} - 6\right) \sqrt[3]{y + 6} + \left(z^{3} - 6\right) \sqrt[3]{z + 6}}{x^{2} + y^{2} + z^{2}}
\]
|
1
| 0.875 |
Given the real numbers \( x \) and \( y \) satisfy the equation \( 2x^2 + 3xy + 2y^2 = 1 \), find the minimum value of \( x + y + xy \).
|
-\frac{9}{8}
| 0.875 |
Define the sequence \(\left\{a_n\right\}\) as follows:
\[
\begin{aligned}
& a_1 = 1, \\
& a_2 = 3, \\
& a_3 = 5, \\
& a_n = a_{n-1} - a_{n-2} + a_{n-3} \quad \text{for} \; n = 4, 5, \ldots
\end{aligned}
\]
Determine the sum of the first 2015 terms of this sequence, \( S_{2015} \).
|
6045
| 0.125 |
Find the number of positive integers less than 10000 that contain the digit 1.
|
3439
| 0.25 |
Suppose \( f \) is a function satisfying \( f\left(x+x^{-1}\right)=x^{6}+x^{-6} \), for all \( x \neq 0 \). Determine \( f(3) \).
|
322
| 0.5 |
The medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular to each other. Find the area of the square with side $AB$, given that $BC = 28$ and $AC = 44$.
|
544
| 0.375 |
An isosceles trapezoid is circumscribed around a circle. The lateral side of the trapezoid is equal to $a$, and the segment connecting the points of tangency of the lateral sides with the circle is equal to $b$. Find the diameter of the circle.
|
D = \sqrt{ab}
| 0.125 |
In how many ways can all ten digits be arranged in a row such that the digits $2, 5, 8$ are placed next to each other?
|
241920
| 0.875 |
In a $3 \times 4$ grid, you need to place 4 crosses so that there is at least one cross in each row and each column. How many ways are there to do this?
|
36
| 0.125 |
There were 7 boxes in a warehouse. Some of them were filled with 7 more boxes, and this was done several times. In the end, there were 10 non-empty boxes. How many boxes are there in total?
|
77
| 0.125 |
As shown in the figure, a large square is divided into six smaller squares. If two smaller squares share more than one common point, they are called adjacent. Fill the numbers $1, 2, 3, 4, 5, 6$ into the smaller squares in the right figure, with each smaller square containing one number such that the difference between the numbers in any two adjacent smaller squares is never 3. How many different ways can this be done?
|
96
| 0.375 |
Reimu and Sanae play a game using 4 fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?
|
\frac{5}{16}
| 0.25 |
Determine the value of \(\lambda\) for which the equation
\[
\lambda x^{2}+4xy+y^{2}-4x-2y-3=0
\]
represents a pair of straight lines.
|
4
| 0.75 |
Given \( w=\sqrt{2p-q}+\sqrt{3q-2p}+\sqrt{6-2q} \), where \( p \) and \( q \) are real numbers that make \( w \) meaningful, determine the maximum value of \( w \).
|
3\sqrt{2}
| 0.625 |
The vulgar fraction \(\frac{1}{221}\) is represented as a repeating decimal. Find the length of the repeating period. (For example, the length of the repeating period for the fraction \(\frac{25687}{99900}=0.25712712712\ldots=0.25(712)\) is 3.)
|
48
| 0.5 |
In triangle \( \triangle ABC \), the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that \( \angle ABC = 120^\circ \), the bisector of \( \angle ABC \) intersects \( AC \) at point \( D \) and \( BD = 1 \). Find the minimum value of \( 4a + c \).
|
9
| 0.75 |
A quadrilateral $ABCD$ is such that $AC = DB$. Let $M$ be any point in the plane. Show that $MA < MB + MC + MD$.
|
MA < MB + MC + MD
| 0.875 |
Points \(E\) and \(F\) are on sides \(AD\) and \(BC\), respectively, of square \(ABCD\). Given that \(BE = EF = FD = 30\), determine the area of the square.
|
810
| 0.5 |
The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{1}=1\), and for each \(n \in \mathbf{N}^{*}\), \(a_{n}\) and \(a_{n+1}\) are the two roots of the equation \(x^{2}+3nx+b_{n}=0\). Find \(\sum_{k=1}^{20} b_{k}\).
|
6385
| 0.375 |
The numbers \(a\) and \(b\) are such that the polynomial \(x^{4} + x^{3} - x^{2} + ax + b\) is the square of some other polynomial. Find \(b\).
|
\frac{25}{64}
| 0.75 |
The team members' numbers are uniquely selected from the positive integers 1 to 100. If the number of any team member is neither the sum of the numbers of any two other team members nor twice the number of another team member, what is the maximum number of members in this sports team?
|
50
| 0.875 |
Among the applicants who passed the entrance exams to a technical university, the grades of "excellent" were received as follows: 48 people in mathematics, 37 people in physics, 42 people in literature, 75 people in mathematics or physics, 76 people in mathematics or literature, 66 people in physics or literature, and 4 people in all three subjects. How many applicants received only one "excellent" grade? Exactly two "excellent" grades? At least one "excellent" grade?
|
94
| 0.25 |
Given that the radius of the inscribed circle of triangle \( \triangle ABC \) is 2 and \(\tan A = -\frac{4}{3}\), find the minimum value of the area of triangle \( \triangle ABC \).
|
18 + 8 \sqrt{5}
| 0.75 |
Distribute 12 different objects among 3 people so that each person receives 4 objects. In how many ways is this possible?
|
34650
| 0.875 |
Two cars start simultaneously towards each other from cities $A$ and $Б$, which are 220 km apart. Their speeds are 60 km/h and 80 km/h. At what distance from the point $C$, located halfway between $A$ and $Б$, will the cars meet? Provide the answer in kilometers, rounding to the nearest hundredth if necessary.
|
15.71
| 0.875 |
If you get 58 out of 84 questions correct on a test, what is your accuracy percentage?
|
69.05\%
| 0.25 |
Calculate: $5 \times 13 \times 31 \times 73 \times 137$
|
20152015
| 0.5 |
For an acute triangle \(ABC\), find the values of \(\alpha\) such that \(A\) is equidistant from the circumcenter \(O\) and the orthocenter \(H\).
|
60^\circ
| 0.875 |
How many digits are in the number \(2^{1000}\)?
|
302
| 0.875 |
Points \( E \) and \( F \) lie on sides \( AB \) and \( BC \) respectively of rhombus \( ABCD \), such that \( AE = 5BE \) and \( BF = 5CF \). It is known that triangle \( DEF \) is equilateral. Find the angle \( BAD \).
|
60^\circ
| 0.875 |
A circle is tangent to both branches of the hyperbola \( x^{2} - 20y^{2} = 24 \) as well as the \( x \)-axis. Compute the area of this circle.
|
504 \pi
| 0.75 |
Does there exist a positive integer \( m \) such that the equation \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{abc} = \frac{m}{a+b+c}\) has infinitely many solutions in positive integers \( (a, b, c) \)?
|
12
| 0.625 |
Let \( x_{i} \in \{\sqrt{2}-1, \sqrt{2}+1\} \), where \( i = 1, 2, 3, \dots, 2012 \). Define
\[ S = x_{1} x_{2} + x_{3} x_{4} + x_{5} x_{6} + \cdots + x_{2000} x_{2010} + x_{2011} x_{2012}. \]
How many different positive integer values can \( S \) attain?
|
504
| 0.75 |
Let \( d = \overline{xyz} \) be a three-digit number that cannot be divisible by 10. If the sum of \( \overline{xyz} \) and \( \overline{zyx} \) is divisible by \( c \), find the largest possible value of this integer \( d \).
|
979
| 0.125 |
Find the sum of squares of all distinct complex numbers \(x\) satisfying the equation
$$
0 = 4x^{10} - 7x^{9} + 5x^{8} - 8x^{7} + 12x^{6} - 12x^{5} + 12x^{4} - 8x^{3} + 5x^{2} - 7x + 4.
$$
|
-\frac{7}{16}
| 0.25 |
Find the largest natural number in which all digits are different and each pair of adjacent digits differs by 6 or 7.
|
60718293
| 0.25 |
Given an $n$-degree polynomial $P(x)$ such that $P(j)=2^{j-1}$ for $j=1, 2, \ldots, n, n+1$, what is the value of $P(n+2)$?
|
2^{n+1} - 1
| 0.75 |
Let \(a\), \(b\), \(c\) be nonzero real numbers such that \(a+b+c=0\) and \(a^3 + b^3 + c^3 = a^5 + b^5 + c^5\). Find the value of \(a^2 + b^2 + c^2\).
|
\frac{6}{5}
| 0.875 |
Find a positive integer whose last digit is 2, and if this digit is moved from the end to the beginning, the new number is twice the original number.
|
105263157894736842
| 0.375 |
What is the smallest natural number $n$ such that the decimal representation of $n$! ends with ten zeros?
|
45
| 0.75 |
Let \( x \) and \( y \) be real numbers such that \( x^2 + y^2 = 2 \) and \( |x| \neq |y| \). Find the minimum value of \( \frac{1}{(x+y)^2} + \frac{1}{(x-y)^2} \).
|
1
| 0.875 |
A contest has six problems worth seven points each. On any given problem, a contestant can score either 0, 1, or 7 points. How many possible total scores can a contestant achieve over all six problems?
|
28
| 0.375 |
Calculate
$$
\frac{6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1 - 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}{4 \cdot 3 \cdot 2 \cdot 1}
$$
|
25
| 0.875 |
A root of unity is a complex number that is a solution to \( z^{n}=1 \) for some positive integer \( n \). Determine the number of roots of unity that are also roots of \( z^{2}+a z+b=0 \) for some integers \( a \) and \( b \).
|
8
| 0.75 |
Let it be known that for a random variable \(\xi\), its expectation \(\mathrm{E} \xi\) is negative, and \(\mathrm{E} e^{\theta \xi} = 1\) for some \(\theta \neq 0\). Show that \(\theta > 0\).
|
\theta > 0
| 0.75 |
At what speed does the distance traveled at uniform motion ($s = v t$) equal the same number as the time?
|
1
| 0.75 |
Given a numerical sequence:
\[ x_{0}=\frac{1}{n} \]
\[ x_{k}=\frac{1}{n-k}\left(x_{0}+x_{1}+\ldots+x_{k-1}\right) \quad \text{for}\ k=1,2,\ldots,n-1 \]
Find \( S_{n} = x_{0} + x_{1} + \ldots + x_{n-1} \) when \( n = 2022 \).
|
1
| 0.75 |
The general solution of the equation \(\cos \frac{x}{4}=\cos x\) is \((\quad)\). In the interval \((0, 24 \pi)\), there are ( ) distinct solutions.
|
20
| 0.625 |
Let \(ABC\) be a triangle with \(CA = CB = 5\) and \(AB = 8\). A circle \(\omega\) is drawn such that the interior of triangle \(ABC\) is completely contained in the interior of \(\omega\). Find the smallest possible area of \(\omega\).
|
16\pi
| 0.125 |
In triangle \( \triangle ABC \), \( M \) is the midpoint of side \( BC \), and \( N \) is the midpoint of line segment \( BM \). Given that \( \angle A = \frac{\pi}{3} \) and the area of \( \triangle ABC \) is \( \sqrt{3} \), find the minimum value of \( \overrightarrow{AM} \cdot \overrightarrow{AN} \).
|
\sqrt{3} + 1
| 0.625 |
A ball with a radius of 1 is placed inside a cube with an edge length of 4. The ball can move freely inside the cube, and the cube can also be flipped in any direction. What is the area of the inner surface of the cube that the ball cannot touch?
|
72
| 0.25 |
Given the real number sets \( A_n = \{ x \mid n < x^n < n+1, n \in \mathbf{N} \} \), if the intersection \( A_1 \cap A_2 \cap \cdots \cap A_n \neq \emptyset \), determine the maximum value of \( n \).
|
4
| 0.75 |
In 1968, a high school graduate returned from a written university exam and shared at home that he could not solve the following problem:
Several identical books and identical albums were purchased. 10 rubles and 56 kopecks were paid for the books, and 56 kopecks for the albums. The number of books bought was 6 more than the number of albums. How many books were bought if the price of a book is more than a ruble higher than the price of an album?
How did his younger brother, a fifth-grader, solve the problem and conclude that 8 books were bought?
If you find solving this problem challenging, try to understand and remember the conditions clearly. Seek new approaches to solving the problem. Try to "guess" the answer or approximate it. How many solutions does the problem have? Write down the solution and verify your answer.
|
8
| 0.375 |
Find the lateral surface area of a regular triangular pyramid if the plane angle at its apex is $90^{\circ}$ and the area of the base is $S$.
|
S \sqrt{3}
| 0.125 |
Is it possible? Can three people cover a distance of 60 km in 3 hours if they have a two-seater motorcycle at their disposal? The speed of the motorcycle is 50 km/h, and the speed of a pedestrian is 5 km/h.
|
Yes
| 0.25 |
Let \( D \) be a point inside the acute triangle \( \triangle ABC \), such that \( \angle ADB = \angle ACB + 90^\circ \) and \( AC \cdot BD = AD \cdot BC \). Calculate the ratio \( \frac{AB \cdot CD}{AC \cdot BD} \).
|
\sqrt{2}
| 0.75 |
In the triangle \(ABC\), angle \(C\) is obtuse, \(D\) is the intersection point of the line \(DB\) perpendicular to \(AB\) and the line \(DC\) perpendicular to \(AC\). The altitude of triangle \(ADC\) drawn from vertex \(C\) intersects side \(AB\) at point \(M\). Given that \(AM = a\) and \(MB = b\), find \(AC\).
|
\sqrt{a(a+b)}
| 0.375 |
The teacher wrote the number 1818 on the board. Vasya noticed that if you insert a multiplication sign between the hundreds and tens places, the value of the resulting expression is a perfect square (18 × 18 = 324 = 18²). What is the next four-digit number after 1818 that has the same property?
|
1832
| 0.375 |
Given that the sequence \(\left\{a_{n}\right\}\) is a geometric sequence with all positive terms, and \(a_{50}\) and \(a_{51}\) are two different solutions to the equation \(100 \lg^{2} x = \lg (100 x)\). Find the value of \(a_{1} a_{2} \cdots a_{100}\).
|
\sqrt{10}
| 0.75 |
Each of the 33 warriors either always lies or always tells the truth. It is known that each warrior has exactly one favorite weapon: a sword, a spear, an axe, or a bow.
One day, Uncle Chernomor asked each warrior four questions:
- Is your favorite weapon a sword?
- Is your favorite weapon a spear?
- Is your favorite weapon an axe?
- Is your favorite weapon a bow?
13 warriors answered "yes" to the first question, 15 warriors answered "yes" to the second question, 20 warriors answered "yes" to the third question, and 27 warriors answered "yes" to the fourth question.
How many of the warriors always tell the truth?
|
12
| 0.75 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.