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stringlengths 18
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0.92
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|---|---|---|
Two cars covered the same distance. The speed of the first car was constant and three times less than the initial speed of the second car. The second car traveled the first half of the journey without changing speed, then its speed was suddenly halved, then traveled with constant speed for another quarter of the journey, and halved its speed again for the next eighth part of the journey, and so on. After the eighth decrease in speed, the car did not change its speed until the end of the trip. By how many times did the second car take more time to complete the entire journey than the first car?
|
\frac{5}{3}
| 0.875 |
An integer \( n \) is chosen uniformly at random from the set \( \{1, 2, 3, \ldots, 2023!\} \). Compute the probability that
$$
\operatorname{gcd}(n^n + 50, n + 1) = 1
$$
|
\frac{265}{357}
| 0.625 |
A group consisting of 6 young men and 6 young women was randomly paired up. Find the probability that at least one pair consists of two young women. Round your answer to two decimal places.
|
0.93
| 0.625 |
A three-digit number is 56 times greater than its last digit. By how many times is it greater than its first digit? Justify your answer.
|
112
| 0.5 |
How many paths are there on $\mathbb{Z}^{2}$ starting from $(0,0)$, making steps of $+(1,0)$ or $+(0,1)$, and ending at $(m, n)$, where $m, n \geq 0$?
|
\binom{m+n}{m}
| 0.75 |
An archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by at most one bridge. It is known that no more than 5 bridges lead from each island, and among any 7 islands, there are always at least two connected by a bridge. What is the maximum possible value of $N$?
|
36
| 0.25 |
Given that \( A_{n} \) and \( B_{n} \) are the sums of the first \( n \) terms of the arithmetic sequences \( \{a_{n}\} \) and \( \{b_{n}\} \) respectively, and that \(\frac{A_{n}}{B_{n}}=\frac{5n-3}{n+9} \), find \(\frac{a_{8}}{b_{8}}\).
|
3
| 0.875 |
It is required to decipher what numbers are encoded, given that the same digits are replaced by the same letters:
$$
\text{KIS} + \text{KSI} = \text{ISK}.
$$
|
495 + 459 = 954
| 0.375 |
a) The lengths of the edges \( A B, A C, A D \), and \( B C \) of an orthocentric tetrahedron are respectively 5, 7, 8, and 6 cm. Find the lengths of the remaining two edges.
b) Is the tetrahedron \( A B C D \) orthocentric if \( A B = 8 \) cm, \( B C = 12 \) cm, and \( D C = 6 \) cm?
|
\text{No}
| 0.25 |
Find the number of eight-digit integers comprising the eight digits from 1 to 8 such that \( (i+1) \) does not immediately follow \( i \) for all \( i \) that runs from 1 to 7.
|
16687
| 0.25 |
Given \( n \), how many ways can we write \( n \) as a sum of one or more positive integers \( a_1 \leq a_2 \leq \cdots \leq a_k \) with \( a_k - a_1 = 0 \) or \( 1 \)?
|
n
| 0.375 |
In the infinite sequence \(\left\{ a_{n} \right\}\), \(a_{1} = 0\) and \(a_{n} = \frac{a_{n-1} + 4}{a_{n-1} - 2}\) for \(n \geq 2\). If \(\lim a_{n} = A\), then \(A =\)?
|
-1
| 0.625 |
Let \( D \) be the set of divisors of 100. Let \( Z \) be the set of integers between 1 and 100, inclusive. Mark chooses an element \( d \) of \( D \) and an element \( z \) of \( Z \) uniformly at random. What is the probability that \( d \) divides \( z \)?
|
\frac{217}{900}
| 0.75 |
There are 29 students in a class: some are honor students who always tell the truth, and some are troublemakers who always lie.
All the students in this class sat at a round table.
- Several students said: "There is exactly one troublemaker next to me."
- All other students said: "There are exactly two troublemakers next to me."
What is the minimum number of troublemakers that can be in the class?
|
10
| 0.75 |
In which number systems is there a number written with two identical digits whose square is written with four identical digits?
|
7
| 0.25 |
The numbers \( p \) and \( q \) are chosen such that the parabola \( y = p x - x^2 \) intersects the hyperbola \( x y = q \) at three distinct points \( A, B, \) and \( C \). The sum of the squares of the sides of triangle \( ABC \) is 324, and the intersection point of its medians is at a distance of 2 from the origin. Find the product \( pq \).
|
42
| 0.25 |
In how many different ways can the digits $0,1,2,3,4,5,6$ form a seven-digit number that is divisible by 4? (The number cannot start with 0.)
|
1248
| 0.125 |
Let $[x]$ denote the largest integer not greater than the real number $x$. Define $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2016}}{8}\right]$. Find the remainder when $A$ is divided by 50.
|
42
| 0.5 |
In triangle $K L M$, the lengths of the sides are $8$, $3\sqrt{17}$, and $13$. Find the area of the figure consisting of those points $P$ inside the triangle $K L M$ for which the condition $P K^{2} + P L^{2} + P M^{2} \leq 145$ is satisfied.
|
\frac{49\pi}{9}
| 0.875 |
Compute the sum of all integers \(1 \leq a \leq 10\) with the following property: there exist integers \(p\) and \(q\) such that \(p, q, p^{2}+a\) and \(q^{2}+a\) are all distinct prime numbers.
|
20
| 0.5 |
Let \( S = \{1, 2, 3, 4, \ldots, 16\} \). Each of the following subsets of \( S \):
\[ \{6\},\{1, 2, 3\}, \{5, 7, 9, 10, 11, 12\}, \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \]
has the property that the sum of all its elements is a multiple of 3. Find the total number of non-empty subsets \( A \) of \( S \) such that the sum of all elements in \( A \) is a multiple of 3.
|
21855
| 0.375 |
Which integers from 1 to 80000 (inclusive) are more numerous and by how many: those containing only even digits or those containing only odd digits?
|
780
| 0.25 |
A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?
|
115
| 0.75 |
The smallest of three consecutive natural numbers is a multiple of 9, the middle number is a multiple of 8, and the largest number is a multiple of 7. What is the smallest possible sum of these three numbers?
|
1488
| 0.75 |
Find the point $M^{\prime}$ that is symmetric to point $M$ relative to the line.
$$
\begin{aligned}
& M(-1, 0, -1) \\
& \frac{x}{-1}=\frac{y-1.5}{0}=\frac{z-2}{1}
\end{aligned}
$$
|
(3, 3, 3)
| 0.75 |
Among the numbers $1^{2}, 2^{2}, 3^{2}, \cdots, 95^{2}$, how many of these 95 numbers have an odd tens digit?
|
19
| 0.5 |
Let \( r \) be a positive integer. Show that if a graph \( G \) has no cycles of length at most \( 2r \), then it has at most \(|V|^{2016}\) cycles of length exactly \( 2016r \), where \(|V|\) denotes the number of vertices in the graph \( G \).
|
|V|^{2016}
| 0.5 |
A kilogram of beef with bones costs 78 rubles, a kilogram of boneless beef costs 90 rubles, and a kilogram of bones costs 15 rubles. How many grams of bones are in a kilogram of beef?
|
160 \text{ grams}
| 0.75 |
On Namek, each hour has 100 minutes. On the planet's clocks, the hour hand completes one full circle in 20 hours and the minute hand completes one full circle in 100 minutes. How many hours have passed on Namek from 0:00 until the next time the hour hand and the minute hand overlap?
|
\frac{20}{19}
| 0.625 |
A permutation of \(\{1, \ldots, n\}\) is chosen at random. How many fixed points does it have on average?
|
1
| 0.625 |
For all real $x$ and $y$, the equality $f\left(x^{2}+y\right)=f(x)+f\left(y^{2}\right)$ holds. Find $f(-1)$.
|
0
| 0.875 |
Find $\frac{a^{8}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}$, given that $\frac{a}{3}-\frac{3}{a}=4$.
|
72
| 0.5 |
In the Cartesian coordinate system, there is an ellipse with foci at $(9,20)$ and $(49,55)$, and it is tangent to the $x$-axis. What is the length of the major axis of the ellipse?
|
85
| 0.75 |
Given \( a, b > 0 \), determine the solutions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) such that \( f(f(x)) + a f(x) = b(a + b) x \).
|
f(x) = bx
| 0.875 |
Given a right triangle \( ABC \) with a right angle at vertex \( C \) and leg lengths in the ratio of \( 1:3 \). Points \( K \) and \( L \) are the centers of squares that share one side with leg \( AC \) and \( BC \) respectively, and these squares do not overlap with triangle \( ABC \). Point \( M \) is the midpoint of the hypotenuse \( AB \).
a) Justify that point \( C \) lies on the line segment \( KL \).
b) Calculate the ratio of the areas of triangles \( ABC \) and \( KLM \).
(J. Švrček)
|
\frac{3}{4}
| 0.25 |
The probability that exactly two out of three red, yellow, and blue balls randomly placed into five different boxes $A, B, C, D, E$ end up in the same box is $\qquad$ .
|
\frac{12}{25}
| 0.875 |
During the university entrance exams, each applicant is assigned a cover code consisting of five digits. The exams were organized by a careful but superstitious professor who decided to exclude from all possible codes (i.e., 00000 to 99999) those that contained the number 13, that is, the digit 3 immediately following the digit 1. How many codes did the professor have to exclude?
|
3970
| 0.125 |
Find the smallest integer \( n > 1 \) such that \(\frac{1^2 + 2^2 + 3^2 + \ldots + n^2}{n}\) is a square.
|
337
| 0.875 |
Шоссе Долгое пересекается с улицей Узкой и с улицей Тихой. На обоих перекрёстках стоят светофоры. Первый светофор $x$ секунд разрешает движение по шоссе, а полминуты - по ул. Узкой. Второй светофор две минуты разрешает движение по шоссе, а $x$ секунд - по ул. Тихой. Светофоры работают независимо друг от друга. При каком значении $x$ вероятность проехать по Долгому шоссе оба перекрестка, не останавливаясь на светофорах, будет наибольшей? Чему равна эта наибольшая вероятность?
|
\frac{4}{9}
| 0.75 |
A square with an integer side length was cut into 2020 smaller squares. It is known that the areas of 2019 of these squares are 1, while the area of the 2020th square is not equal to 1. Find all possible values that the area of the 2020th square can take. Provide the smallest of these possible area values in the answer.
|
112225
| 0.25 |
How many four-digit palindromes \( a b b a \) have the property that the two-digit integer \( a b \) and the two-digit integer \( b a \) are both prime numbers? (For example, 2332 does not have this property, since 23 is prime but 32 is not.)
|
9
| 0.125 |
Given \((a + b i)^2 = 3 + 4i\), where \(a, b \in \mathbf{R}\), and \(i\) is the imaginary unit, find the value of \(a^2 + b^2\).
|
5
| 0.875 |
A trapezoid \(AEFG\) (\(EF \parallel AG\)) is positioned inside a square \(ABCD\) with a side length of 14, such that points \(E\), \(F\), and \(G\) lie on sides \(AB\), \(BC\), and \(CD\) respectively. The diagonals \(AF\) and \(EG\) are perpendicular, and \(EG = 10\sqrt{2}\). Find the perimeter of the trapezoid.
|
45
| 0.5 |
The number of proper subsets of the set
\[
\left\{x \left\lvert\,-1 \leqslant \log _{\frac{1}{x}} 10<-\frac{1}{2}\right., x \in \mathbf{N}\right\}
\]
is ____.
|
2^{90} - 1
| 0.625 |
Let $D$ be a point inside triangle $\triangle ABC$ such that $AB = DC$, $\angle DCA = 24^\circ$, $\angle DAC = 31^\circ$, and $\angle ABC = 55^\circ$. Find $\angle DAB$.
|
63^\circ
| 0.125 |
Let \( M \) be the centroid of \( \triangle ABC \), and \( AM = 3 \), \( BM = 4 \), \( CM = 5 \). Find the area of \( \triangle ABC \).
|
18
| 0.375 |
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^{2} y \).
|
\frac{10}{13}
| 0.875 |
Inside a grid rectangle with a perimeter of 50 cells, a rectangular hole with a perimeter of 32 cells is cut out along the cell borders (the hole does not contain boundary cells). If you cut this figure along all the horizontal grid lines, you get 20 strips that are 1 cell wide. How many strips will you get if you cut it along all the vertical grid lines instead? (A $1 \times 1$ square is also considered a strip!)
|
21
| 0.125 |
There are $N$ natural numbers written on a board, where $N \geq 5$. It is known that the sum of all the numbers is 80, and the sum of any five of them is no more than 19. What is the smallest possible value of $N$?
|
26
| 0.5 |
Quadrilateral $F$ under the mapping $f:(x, y) \rightarrow (x-1, y+2)$ results in quadrilateral $F'$. The area of quadrilateral $F$ is 6. What is the area of quadrilateral $F'$?
|
6
| 0.875 |
Let \( R \) be the radius of the sphere circumscribed around a right square pyramid, \( r \) be the radius of the sphere inscribed in the pyramid, and \( d \) be the distance between the centers of these spheres. Show that
\[
d^{2} + (R + r)^{2} = 2 R^{2}
\]
What can be inferred about the maximum value of the ratio \( r / R \)? (Cf. F. 2096.)
|
\sqrt{2} - 1
| 0.5 |
Suppose \(\alpha, \beta \in(0, \pi / 2)\). If \(\tan \beta = \frac{\cot \alpha - 1}{\cot \alpha + 1}\), find \(\alpha + \beta\).
|
\frac{\pi}{4}
| 0.875 |
The city has the shape of a \(5 \times 5\) square. What is the minimum length a route can have if it needs to go through each street of this city and return to the original location? (Each street can be traversed any number of times.)
|
68
| 0.625 |
Calculate the integral \(I=\oint_{L} \frac{y}{x^{2}+y^{2}} dx - \frac{x}{x^{2}+y^{2}} dy\), where \(L\) is the circle:
a) \(x^{2} + y^{2} = 1\), b) \((x-1)^{2} + y^{2} = 1\), c) \((x-1)^{2} + (y-1)^{2} = 1\).
|
0
| 0.125 |
Calculate the flux of the vector field \( \mathbf{a} = y \mathbf{i} + z \mathbf{j} + x \mathbf{k} \) through the closed surface bounded by the cylinder \( x^{2} + y^{2} = R^{2} \) and the planes \( z = x \) and \( z = 0 \) (for \( z \geq 0 \)).
|
0
| 0.875 |
90 students arrived at the camp. It is known that among any 10 students, there are necessarily two friends. A group of students is said to form a chain of friendships if the children in the group can be numbered from 1 to \( k \) such that all students can be divided into no more than 9 groups, each of which forms a chain of friendships. (A group of one student also forms a chain of friendships.)
|
9
| 0.5 |
The sequence $\left\{a_{n}\right\}$ is defined by the conditions $a_{1}=0$ and $a_{n+1}=\sum_{k=1}^{n}\left(a_{k}+1\right)$ for $n \geqslant 1$. Find an explicit formula for this sequence.
|
2^{n-1} - 1
| 0.875 |
Kolya and Zhenya agreed to meet in the subway during the first hour of the afternoon. Kolya arrives at the meeting place between noon and 1 PM, waits for 10 minutes, and leaves. Zhenya does the same.
a) What is the probability that they will meet?
b) How does the probability of meeting change if Zhenya decides to arrive earlier than 12:30 PM, while Kolya still arrives between noon and 1 PM?
c) How does the probability of meeting change if Zhenya decides to arrive at any random time between 12:00 PM and 12:50 PM, while Kolya still arrives between 12:00 PM and 1:00 PM?
|
\frac{19}{60}
| 0.125 |
The numbers \(a\) and \(b\) are such that the polynomial \(x^{4} - x^{3} + x^{2} + a x + b\) is the square of some other polynomial. Find \(b\).
|
\frac{9}{64}
| 0.875 |
When $x^{2}$ was added to the quadratic polynomial $f(x)$, its minimum value increased by 1. When $x^{2}$ was subtracted from it, its minimum value decreased by 3. How will the minimum value of $f(x)$ change if $2x^{2}$ is added to it?
|
\frac{3}{2}
| 0.375 |
Consider an alphabet of 2 letters. A word is any finite combination of letters. We will call a word unpronounceable if it contains more than two identical letters in a row. How many unpronounceable words of 7 letters are there?
|
86
| 0.5 |
Find all values of the parameter \( a \) for which the equation \( a x^{2}+\sin^{2} x=a^{2}-a \) has a unique solution.
|
a = 1
| 0.875 |
Let $a$ and $b$ be two real numbers. Show that $a^{2} + b^{2} \geq 2ab$.
|
a^2 + b^2 \geq 2ab
| 0.375 |
What is the smallest three-digit number \( K \) which can be written as \( K = a^b + b^a \), where both \( a \) and \( b \) are one-digit positive integers?
|
100
| 0.5 |
Given the following table filled with signs:
$$
\begin{aligned}
& ++-+ \\
& --++ \\
& ++++ \\
& +-+-
\end{aligned}
$$
A move consists of choosing a row or a column and changing the signs within it. Is it possible to achieve a table filled with '+' signs?
|
\text{No}
| 0.25 |
Find the smallest positive integer \( n \) such that \( 107n \) has the same last two digits as \( n \).
|
50
| 0.875 |
Let \( x > 0 \). Show that
\[ x + \frac{1}{x} \geq 2 \]
For which values of \( x \) does equality hold?
|
x = 1
| 0.75 |
If \(\cos ^{4} \theta + \sin ^{4} \theta + (\cos \theta \cdot \sin \theta)^{4} + \frac{1}{\cos ^{4} \theta + \sin ^{4} \theta} = \frac{41}{16}\), find the value of \(\sin ^{2} \theta\).
|
\frac{1}{2}
| 0.875 |
Maxim came up with a new method for dividing numbers by a two-digit number \( N \). To divide an arbitrary number \( A \) by the number \( N \), you need to do the following steps:
1) Divide \( A \) by the sum of the digits of \( N \);
2) Divide \( A \) by the product of the digits of \( N \);
3) Subtract the second result from the first.
For which numbers \( N \) will Maxim’s method give the correct result? (20 points)
|
24
| 0.75 |
Given \( n = p \cdot q \cdot r \cdot s \), where \( p, q, r, s \) are distinct primes such that:
1. \( s = p + r \)
2. \( p(p + q + r + s) = r(s - q) \)
3. \( qs = 1 + qr + s \)
Find \( n \).
|
2002
| 0.375 |
For the four-digit number $\overline{a b c d}$, if $a > b$, $b < c$, and $c > d$, it is classified as a $P$ type number. If $a < b$, $b > c$, and $c < d$, it is classified as a $Q$ type number. What is the difference between the total number of $P$ type numbers and $Q$ type numbers?
|
285
| 0.25 |
For a soccer match, three types of tickets (A, B, and C) were sold, totaling 400 tickets. The prices for type A, type B, and type C tickets are 50 yuan, 40 yuan, and 30 yuan respectively. The total revenue from ticket sales is 15,500 yuan. Additionally, the number of type B and type C tickets sold is the same. How many tickets of each type were sold?
|
150
| 0.375 |
For what value of \( p \) is the ratio of the roots of the equation \( x^{2} + p x - 16 = 0 \) equal to \( -4 \)?
|
\pm 6
| 0.125 |
Find the value of the function \( f(x) \) at the point \( x_{0} = 1000 \), given that \( f(0) = 1 \) and for any \( x \), the equality \( f(x+2) = f(x) + 4x + 2 \) holds.
|
999001
| 0.75 |
The center of a cube is reflected off all its faces, and the 6 points obtained, together with the vertices of the cube, determine 14 vertices of a polyhedron, all 12 faces of which are congruent rhombuses. (This polyhedron is called a rhombic dodecahedron—a shape naturally occurring in garnet single crystals.) Calculate the surface area and volume of the constructed polyhedron if the edge of the original cube has a length of \( a \).
|
2a^3
| 0.25 |
From one point, a perpendicular and two oblique lines are drawn to a given straight line.
Find the length of the perpendicular if the oblique lines are 41 and 50, and their projections onto the given straight line are in the ratio \(3: 10\).
|
40
| 0.875 |
Given that \( \tan \left(\frac{\pi}{4}+\alpha\right)=2 \), find the value of \( \frac{1}{2 \sin \alpha \cos \alpha+\cos ^{2} \alpha} \).
|
\frac{2}{3}
| 0.875 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 1}\left(\frac{1}{x}\right)^{\frac{\ln (x+1)}{\ln (2-x)}}$$
|
2
| 0.875 |
Let \( p \) be an integer such that both roots of the equation
\[ 5x^2 - 5px + (66p - 1) = 0 \]
are positive integers. Find the value of \( p \).
|
76
| 0.75 |
Find the smallest six-digit number that is divisible by 11, where the sum of the first and fourth digits is equal to the sum of the second and fifth digits, and equal to the sum of the third and sixth digits.
|
100122
| 0.75 |
Indicate the integer closest to the number: \(\sqrt{2012-\sqrt{2013 \cdot 2011}}+\sqrt{2010-\sqrt{2011 \cdot 2009}}+\ldots+\sqrt{2-\sqrt{3 \cdot 1}}\).
|
31
| 0.625 |
We place 9 integers in the cells of a $3 \times 3$ grid in such a way that the sum of the numbers in any row or column is always odd. What values can the number of even cells in such a configuration take?
|
0, 4, 6
| 0.125 |
The sides of a regular 18-sided polygon inscribed in a unit circle have length \(a\). Show that
$$
a^{3}=3a-1
$$
|
a^3 = 3a - 1
| 0.875 |
Beginner millionaire Bill buys a bouquet of 7 roses for $20. Then, he can sell a bouquet of 5 roses for $20 per bouquet. How many bouquets does he need to buy to "earn" a difference of $1000?
|
125
| 0.625 |
Let's call a natural number a "snail" if its representation consists of the representations of three consecutive natural numbers, concatenated in some order: for example, 312 or 121413. "Snail" numbers can sometimes be squares of natural numbers: for example, $324=18^{2}$ or $576=24^{2}$. Find a four-digit "snail" number that is the square of some natural number.
|
1089
| 0.625 |
Find the number of 11-digit positive integers such that the digits from left to right are non-decreasing. (For example, 12345678999, 55555555555, 23345557889.)
|
75582
| 0.75 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(\frac{x^{2}+2 x-3}{x^{2}+4 x-5}\right)^{\frac{1}{2-x}}
$$
|
\frac{2}{3}
| 0.75 |
In the Cartesian coordinate plane, circles \( C_{1} \) and \( C_{2} \) intersect at points \( P \) and \( Q \). The coordinates of point \( P \) are \( (3, 2) \), and the product of the radii of the two circles is \( \frac{13}{2} \). If both circles are tangent to the line \( l: y = kx \) and the x-axis, find the equation of the line \( l \).
|
y = 2 \sqrt{2} x
| 0.5 |
Given a triangle \( \triangle ABC \) with \( BC=a \), \( AC=b \), and \( AB=c \), if \( \angle A + \angle C = 2 \angle B \), find the largest positive integer \( n \) such that \( a^n + c^n \leq 2b^n \) holds for any such triangle.
|
4
| 0.75 |
The midsegment of a trapezoid divides it into two quadrilaterals. The difference in the perimeters of these two quadrilaterals is 24, and the ratio of their areas is $\frac{20}{17}$. Given that the height of the trapezoid is 2, what is the area of this trapezoid?
|
148
| 0.375 |
Solve the equation \(2 x \log x + x - 1 = 0\) in the set of real numbers.
|
x = 1
| 0.625 |
In how many ways can the cells of a \(4 \times 4\) table be filled in with the digits \(1,2, \ldots, 9\) so that each of the 4-digit numbers formed by the columns is divisible by each of the 4-digit numbers formed by the rows?
|
9
| 0.625 |
How many 7-digit positive integers are made up of the digits 0 and 1 only, and are divisible by 6?
|
11
| 0.875 |
The total \( T \) is obtained as the sum of the integers from 2006 to 2036 inclusive. What is the sum of all the prime factors of \( T \)?
|
121
| 0.875 |
In the cube \(ABCDE_{1}B_{1}C_{1}D_{1}\) as shown in Figure 7-10, find the degree of the dihedral angle \(A-BD_{1}-A_{1}\).
|
60^\circ
| 0.25 |
The area of an isosceles trapezoid is 100, and its diagonals are mutually perpendicular. Find the height of this trapezoid.
|
10
| 0.75 |
The base of the quadrangular pyramid \( M A B C D \) is a parallelogram \( A B C D \). Given that \( \overline{D K} = \overline{K M} \) and \(\overline{B P} = 0.25 \overline{B M}\), the point \( X \) is the intersection of the line \( M C \) and the plane \( A K P \). Find the ratio \( M X: X C \).
|
M X : X C = 3 : 4
| 0.375 |
In the diagram, \( A B C D E F \) is a regular hexagon with side length 2. Points \( E \) and \( F \) are on the \( x \)-axis and points \( A, B, C, \) and \( D \) lie on a parabola. What is the distance between the two \( x \)-intercepts of the parabola?
|
2\sqrt{7}
| 0.125 |
Show that \( 3 < \pi < 4 \) without using approximate values of \(\pi\).
|
3 < \pi < 4
| 0.875 |
Given that both \( p \) and \( q \) are prime numbers, and that \( 7p + q \) and \( 2q + 11 \) are also prime numbers, find the value of \( p^q + q^p \).
|
17
| 0.875 |
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