problem
stringlengths 18
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0.92
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|---|---|---|
Point \( P \) is inside the right triangle \( \triangle ABC \) with \(\angle B = 90^\circ\), \( PA = 10 \), \( PB = 6 \), \(\angle APB = \angle BPC = \angle CPA\). Find \( PC \).
|
33
| 0.75 |
On the radius \( AO \) of a circle with center \( O \), a point \( M \) is selected. On one side of \( AO \) on the circle, points \( B \) and \( C \) are chosen such that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \) if the radius of the circle is 12 and \( \cos \alpha = \frac{1}{4} \).
|
6
| 0.25 |
Given a regular tetrahedron $P-ABC$ where all edges are of length $1$, and points $L$, $M$, and $N$ are the midpoints of edges $PA$, $PB$, and $PC$ respectively. Find the area of the cross-section of the circumscribed sphere of the tetrahedron made by the plane $LMN$.
|
\frac{\pi}{3}
| 0.25 |
Given two skew lines \( l \) and \( m \). On \( l \), there are three points \( A, B, \) and \( C \) such that \( AB = BC \). From points \( A, B,\) and \( C \), perpendicular lines \( AD, BE, \) and \( CF \) are dropped to \( m \) with feet \( D, E, \) and \( F \), respectively. It is known that \( AD = \sqrt{15}, BE = \frac{7}{2}, \) and \( CF = \sqrt{10} \). Find the distance between lines \( l \) and \( m \).
|
\sqrt{6}
| 0.375 |
Given an 8x8 chessboard, how many ways are there to place 6 rooks such that no two rooks are on the same row or column?
|
564480
| 0.5 |
Let \( n \) be a positive integer. When \( n > 100 \), what are the first two decimal places of the fractional part of \( \sqrt{n^2 + 3n + 1} \)?
|
49
| 0.625 |
For each natural number from 1 to 999, Damir subtracted the last digit from the first digit and wrote all the resulting 1000 differences on the board. For example, for the number 7, Damir wrote 0; for the number 105, he wrote (-4); for the number 61, he wrote 5.
What is the sum of all the numbers on the board?
|
495
| 0.75 |
Initially, the numbers 2 and 3 are written down. They are then multiplied, and their product, the number 6, is written down. This process continues by multiplying the two last single-digit numbers: \(3 \cdot 6 = 18\), then \(1 \cdot 8 = 8\), next \(8 \cdot 8 = 64\), and so on. As a result, we get the following sequence of digits:
\[2361886424 \ldots\]
Which digits cannot appear in this sequence? Which digit is in the thousandth place?
|
2
| 0.125 |
Find the largest negative root \( x_{0} \) of the equation \( \frac{\sin x}{1+\cos x} = 2 - \operatorname{ctg} x \). Write the answer as \( x_{0} \cdot \frac{3}{\pi} \).
|
-3.5
| 0.25 |
It is known that the remainder when a certain prime number is divided by 60 is a composite number. What is this composite number?
|
49
| 0.75 |
In a $4 \times 4$ grid, place candies according to the following requirements: (1) Each cell must contain candies; (2) In adjacent cells, the left cell has 1 fewer candy than the right cell and the upper cell has 2 fewer candies than the lower cell; (3) The bottom-right cell contains 20 candies. How many candies are there in total on the grid? (Adjacent cells are those that share a common edge)
|
248
| 0.25 |
Given that the equation \( xyz = 900 \) has all positive integer solutions \(\left(x_{i}, y_{i}, z_{i}\right)(1 \leqslant i \leqslant n)\), find the sum \(\sum_{k=1}^{n}\left(x_{k}+y_{k}+z_{k}\right)\).
|
22572
| 0.5 |
Let \(a_n\) be the sequence defined by \(a_1 = 3\) and \(a_{n+1} = 3^{k}\), where \(k = a_n\). Let \(b_n\) be the remainder when \(a_n\) is divided by 100. Which values \(b_n\) occur for infinitely many \(n\)?
|
87
| 0.625 |
Find the number of the form $7x36y5$ that is divisible by 1375.
|
713625
| 0.75 |
Let \( f(x) \) be an even function defined on \( \mathbf{R} \) with a period of 2, strictly decreasing on the interval \( [0, 1] \), and satisfying \( f(\pi)=1 \) and \( f(2 \pi)=2 \). Determine the solution set of the system of inequalities:
\[ \left\{\begin{array}{l}
1 \leq x \leq 2, \\
1 \leq f(x) \leq 2
\end{array}\right. \]
|
[\pi - 2, 8 - 2\pi]
| 0.125 |
Let \( A \) and \( B \) be the vertices of the major axis of ellipse \( \Gamma \). \( E \) and \( F \) are the foci of \( \Gamma \). Given that \( |AB| = 4 \) and \( |AF| = 2 + \sqrt{3} \). Point \( P \) lies on \( \Gamma \) and satisfies \( |PE| \cdot |PF| = 2 \). Find the area of \( \triangle PEF \).
|
1
| 0.875 |
A real number \( x \) is randomly chosen in the interval \(\left[-15 \frac{1}{2}, 15 \frac{1}{2}\right]\). Find the probability that the closest integer to \( x \) is odd.
|
\frac{16}{31}
| 0.875 |
Given that the real number \( m \) satisfies the following condition for the real coefficient quadratic equation \( a x^{2}+b x+c=0 \) with real roots,
$$(a-b)^{2}+(b-c)^{2}+(c-a)^{2} \geqslant m a^{2}$$
Then find the maximum value of \( m \).
|
\frac{9}{8}
| 0.875 |
At the World Meteorological Conference, each participant announced the average monthly temperature in their hometown in turn. Everyone else at that moment wrote down the product of the temperatures in their and the current speaker's city. A total of 78 positive and 54 negative numbers were recorded. What is the minimum number of times a positive temperature could have been announced?
|
3
| 0.375 |
On side \( AD \) of rectangle \( ABCD \), a point \( E \) is marked. On the segment \( EC \), there is a point \( M \) such that \( AB=BM \) and \( AE=EM \). Find the length of side \( BC \), knowing that \( ED=16 \) and \( CD=12 \).
|
20
| 0.875 |
Ten female tennis players arrived at a tournament, 4 of whom are from Russia. According to the rules, for the first round, the players are randomly paired. Find the probability that in the first round all the Russian players will play only with other Russian players.
|
\frac{1}{21}
| 0.875 |
On the sides $BC, CA, AB$ of an equilateral triangle $ABC$ with a side length of 7, points $A_1, B_1, C_1$ are chosen respectively. It is known that $AC_1 = BA_1 = CB_1 = 3$. Find the ratio of the area of triangle $ABC$ to the area of the triangle formed by the lines $AA_1, BB_1, CC_1$.
|
37
| 0.25 |
In trapezoid \(ABCD\) (\(AD\) is the longer base), the diagonal \(AC\) is perpendicular to side \(CD\) and bisects angle \(BAD\). It is known that \(\angle CDA = 60^\circ\), and the perimeter of the trapezoid is 2. Find \(AD\).
|
\frac{4}{5}
| 0.5 |
Ksyusha runs twice as fast as she walks (both speeds are constant).
On Tuesday, when she left home for school, she first walked, and then, when she realized she was late, she started running. The distance Ksyusha walked was twice the distance she ran. As a result, she reached the school from home in exactly 30 minutes.
On Wednesday, Ksyusha left home even later, so she had to run twice the distance she walked. How many minutes did it take her to get from home to school on Wednesday?
|
24
| 0.875 |
Given that a five-digit palindromic number equals the product of 45 and a four-digit palindromic number (i.e., $\overline{\mathrm{abcba}} = 45 \times \overline{\mathrm{deed}}$), what is the largest possible value of this five-digit palindromic number?
|
59895
| 0.375 |
Let \( n \) be a fixed integer with \( n \geq 2 \).
(1) Determine the smallest constant \( c \) such that the inequality
\[
\sum_{1 \leq i < j \leq n} x_{i} x_{j} \left(x_{i}^{2} + x_{j}^{2} \right) \leqslant c \left( \sum_{1 \leq i \leq n} x_{i} \right)^{4}
\]
holds for all non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{n} \).
(2) For this constant \( c \), determine the necessary and sufficient condition for the equality to hold.
|
\frac{1}{8}
| 0.75 |
If the inequality $0 \leq x^{2} + px + 5 \leq 1$ has exactly one real solution, determine the range of values for $p$.
|
p = \pm 4
| 0.5 |
\( x \sqrt{8} + \frac{1}{x \sqrt{8}} = \sqrt{8} \) has two real solutions \( x_1 \) and \( x_2 \). The decimal expansion of \( x_1 \) has the digit 6 in the 1994th place. What digit does \( x_2 \) have in the 1994th place?
|
3
| 0.375 |
Find the integer solutions of the equation
$$
x^{4}+y^{4}=3 x^{3} y
$$
|
(0,0)
| 0.75 |
Given that \( x, y, z \in \mathbb{R}_{+} \), and \( s = \sqrt{x+2} + \sqrt{y+5} + \sqrt{z+10} \), \( t = \sqrt{x+1} + \sqrt{y+1} + \sqrt{z+1} \), find the minimum value of \( s^{2} - t^{2} \).
|
36
| 0.25 |
Find all 4-digit numbers that are 7182 less than the number formed by reversing its digits.
|
1909
| 0.625 |
In the sum below, identical letters represent identical digits, and different letters represent different digits.
$$
\begin{array}{ccc}
& & X \\
+ & & X \\
& Y & Y \\
\hline Z & Z & Z
\end{array}
$$
What digit is represented by the letter $X$?
|
6
| 0.875 |
Given a function \( f(x) \) that satisfies the condition
\[ f(xy + 1) = f(x)f(y) - f(y) - x + 2, \]
what is \( f(2017) \) if it is known that \( f(0) = 1 \)?
|
2018
| 0.875 |
Consider five-dimensional Cartesian space
$$
\mathbb{R}^{5}=\left\{\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\right) \mid x_{i} \in \mathbb{R}\right\},
$$
and consider the hyperplanes with the following equations:
- \( x_{i} = x_{j} \) for every \( 1 \leq i < j \leq 5 \);
- \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = -1 \);
- \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 0 \);
- \( x_{1} + x_{2} + x_{3} + x_{4} + x_{5} = 1 \).
Into how many regions do these hyperplanes divide \( \mathbb{R}^{5} \)?
|
480
| 0.625 |
Given \( f(x) + g(x) = \sqrt{\frac{1 + \cos 2x}{1 - \sin x}} \) for \( x \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \), where \( f(x) \) is an odd function and \( g(x) \) is an even function, determine the value of \( [f(x)]^2 - [g(x)]^2 \).
|
-2 \cos x
| 0.875 |
From unit cubes, a large parallelepiped with sides greater than 4 was constructed. Two cubes are called adjacent if they touch each other by faces. Thus, one cube can have up to 6 neighbors. It is known that the number of cubes with exactly 6 neighbors is 836. Find the number of cubes with no more than four neighbors.
|
144
| 0.75 |
In a family, there are six children. Five of them are respectively 2, 6, 8, 12, and 14 years older than the youngest, and the age of each child is a prime number.
How old is the youngest?
|
5
| 0.875 |
If a class of 30 students is seated in the auditorium, at least two classmates will end up in the same row. If the same is done with a class of 26 students, at least three rows will be empty. How many rows are there in the auditorium?
|
29
| 0.75 |
Among all rectangles with a given perimeter \( P \), find the one with the maximum area. Calculate this area.
|
\frac{P^2}{16}
| 0.5 |
Given a sequence of natural numbers \( a_n \) whose terms satisfy the relation \( a_{n+1}=k \cdot \frac{a_n}{a_{n-1}} \) (for \( n \geq 2 \)). All terms of the sequence are integers. It is known that \( a_1=1 \) and \( a_{2018}=2020 \). Find the smallest natural \( k \) for which this is possible.
|
2020
| 0.5 |
Two linear functions \( f(x) \) and \( g(x) \) satisfy the properties that for all \( x \),
- \( f(x) + g(x) = 2 \)
- \( f(f(x)) = g(g(x)) \)
and \( f(0) = 2022 \). Compute \( f(1) \).
|
1
| 0.875 |
Compose the equation of a line passing through the point \( B(-1, 2) \) that is tangent to the parabola \( y = x^2 + 4x + 9 \).
|
y=6 x + 8
| 0.875 |
A teacher presents a function \( y = f(x) \). Four students, A, B, C, and D, each describe a property of this function:
- A: For \( x \in \mathbf{R} \), \( f(1 + x) = f(1 - x) \);
- B: The function is decreasing on \( (-\infty, 0] \);
- C: The function is increasing on \( (0, +\infty) \);
- D: \( f(0) \) is not the minimum value of the function.
If exactly three of these students are correct, write down such a function:
|
y = (x-1)^2
| 0.375 |
A drawer contains 7 red pencils and 4 blue pencils. Without looking into the drawer, a boy randomly draws the pencils. What is the minimum number of pencils he must draw to be sure to have pencils of both colors?
|
8
| 0.875 |
For each real number \( x \), the value of \( f(x) \) is the minimum of \( x^2 \), \( 6 - x \), and \( 2x + 15 \). What is the maximum value of \( f(x) \)?
|
9
| 0.875 |
Which number is greater: $\log _{135} 675$ or $\log _{45} 75$?
|
\log_{135} 675
| 0.5 |
Let \( n \) be a positive integer. Find the largest integer \( k \) such that it is possible to form \( k \) subsets from a set with \( n \) elements, where the intersection of any two subsets is non-empty.
|
2^{n-1}
| 0.75 |
Given that \( x \) and \( y \) are positive integers such that \( 56 \leq x + y \leq 59 \) and \( 0.9 < \frac{x}{y} < 0.91 \), find the value of \( y^2 - x^2 \).
|
177
| 0.75 |
A standard deck of cards has 52 cards after removing the Jokers. If 5 cards are drawn at random from this deck, what is the probability that at least two of them have the same number (or letter $J, Q, K, A$)? Calculate this probability to two decimal places.
|
0.49
| 0.625 |
Given \( f(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a \), \( b \), \( c \), and \( d \) are constants, if \( f(1) = 1 \), \( f(2) = 2 \), and \( f(3) = 3 \), find the value of \( f(0) + f(4) \).
|
28
| 0.75 |
Calculate the limit
$$
\lim _{x \rightarrow 0} \frac{2 x \sin x}{1-\cos x}
$$
|
4
| 0.875 |
At the vertices of a regular 2018-sided polygon, there are numbers: 2017 zeros and 1 one. In one move, it is allowed to add or subtract one from the numbers at the ends of any side of the polygon. Is it possible to make all the numbers divisible by 3?
|
\text{No}
| 0.875 |
Given a $54^{\circ}$ angle, divide it into three equal parts using a compass and straightedge.
|
18^\circ
| 0.125 |
The internal angle bisector of a triangle from vertex $A$ is parallel to the line $OM$, but not identical to it, where $O$ is the circumcenter and $M$ is the orthocenter. What is the angle at vertex $A$?
|
120^\circ
| 0.375 |
Philatelist Andrey decided to distribute all his stamps evenly into 2 envelopes, but it turned out that one stamp was left over. When he distributed them evenly into 3 envelopes, again one stamp was left over; when he distributed them evenly into 5 envelopes, 3 stamps were left over; finally, when he tried to distribute them evenly into 9 envelopes, 7 stamps were left over. How many stamps does Andrey have in total, given that he recently had to buy a second album for 150 stamps to store all his stamps, as one such album was not enough?
|
223
| 0.875 |
In the right triangle \(ABC\) with the right angle at \(A\), an altitude \(AH\) is drawn. The circle passing through points \(A\) and \(H\) intersects the legs \(AB\) and \(AC\) at points \(X\) and \(Y\) respectively. Find the length of segment \(AC\), given that \(AX = 5\), \(AY = 6\), and \(AB = 9\).
|
13.5
| 0.125 |
If the function \( f(x) \) is defined on \( \mathbb{R} \) and has a period (or the smallest positive period) of \( T \), then the function \( F(x) = f(\alpha x) \) has a period (or the smallest positive period) of \( \frac{T}{\alpha} \) for \( \alpha > 0 \).
|
\frac{T}{\alpha}
| 0.875 |
How many diagonals does a convex:
a) 10-sided polygon have?
b) k-sided polygon have (where $k > 3$)?
|
\frac{k(k-3)}{2}
| 0.875 |
How many integer solutions \( x, y \) does the equation \( 6x^2 + 2xy + y + x = 2019 \) have?
|
4
| 0.625 |
Masha lives in apartment No. 290, which is in the 4th entrance of a 17-story building. On which floor does Masha live? (The number of apartments is the same in all entrances of the building on all 17 floors; apartment numbers start from 1.)
|
7
| 0.75 |
Given a regular quadrangular prism \(ABCD-A_1B_1C_1D_1\) with the base \(ABCD\) being a unit square, if the dihedral angle \(A_1-BD-C_1\) is \(\frac{\pi}{3}\), find \(AA_1\).
|
\frac{\sqrt{6}}{2}
| 0.25 |
There is a special calculator. When a number is input, the calculator will multiply the number by 2, then reverse the digits of the result. Finally, it will add 2 and display the final result. If you input a two-digit number and the final displayed result is 27, what was the initial input?
|
26
| 0.875 |
A third of the sixth-grade students received C's on their math test. How many students received A's, if only one student received an F and $\frac{5}{13}$ of the sixth-grade students received B's?
|
10
| 0.75 |
A child gave Carlson 111 candies. They ate some of them right away, 45% of the remaining candies went to Carlson for lunch, and a third of the candies left after lunch were found by Freken Bok during cleaning. How many candies did she find?
|
11
| 0.75 |
Even-numbered throws belong to B. Bons. Therefore, Bons wins only if the total number of throws, including the last successful one, is even. The probability of rolling a six is $\frac{1}{6}$. The probability of the opposite event is $\frac{5}{6}$. Therefore, the probability that the total number of throws will be even is given by:
1) Initially, J. Silver did not roll a six, and then B. Bons rolled a six. The probability of this is $\frac{5}{6} \cdot \frac{1}{6}=\frac{5}{36}$.
2) Both Silver and Bons did not roll a six on their first tries. After this, the game essentially restarts, and B. Bons wins with probability $p$. The probability of this scenario is $\frac{5}{6} \cdot \frac{5}{6} \cdot p=\frac{25}{36} p$.
Therefore, $p=\frac{5}{36}+\frac{25}{36} p$, from which $p=\frac{5}{11}$.
|
\frac{5}{11}
| 0.875 |
The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out?
|
167
| 0.75 |
Let \( x_{i} \in \mathbf{R} \), \( x_{i} \geqslant 0 \) for \( i=1,2,3,4,5 \), and \( \sum_{i=1}^{5} x_{i} = 1 \). Find the minimum value of \( \max \left\{x_{1}+x_{2}, x_{2}+x_{3}, x_{3}+x_{4}, x_{4} + x_{5}\right\} \).
|
\frac{1}{3}
| 0.875 |
Given the sequence \( \left\{a_{1}, a_{2}, \cdots \right\} = \left\{ \frac{1}{1}, \frac{2}{1}, \frac{1}{2}, \frac{3}{1}, \frac{2}{2}, \frac{1}{3}, \frac{4}{1}, \frac{3}{2}, \frac{2}{3}, \frac{1}{4}, \cdots \right\} \), find the 1988th term \( a_{1988} \).
|
\frac{29}{35}
| 0.75 |
Let $(a, n) \in \mathbb{N}^{2}$ and $p$ a prime number such that $p \mid a^{n}$. Show that $p^{n} \mid a^{n}$.
|
p^n \mid a^n
| 0.75 |
There are 18 identical cars in a train. In some cars, exactly half of the seats are free, in others, exactly one-third of the seats are free, and in the remaining cars, all seats are occupied. At the same time, exactly one-ninth of all seats in the whole train are free. How many cars have all seats occupied?
|
13
| 0.75 |
Find the maximum value of the expression
$$
\begin{aligned}
& x_{1}+x_{2}+x_{3}+x_{4}-x_{1} x_{2}-x_{1} x_{3}-x_{1} x_{4}-x_{2} x_{3}-x_{2} x_{4}-x_{3} x_{4}+ \\
& +x_{1} x_{2} x_{3}+x_{1} x_{2} x_{4}+x_{1} x_{3} x_{4}+x_{2} x_{3} x_{4}-x_{1} x_{2} x_{3} x_{4}
\end{aligned}
$$
|
1
| 0.375 |
For the set $\{1,2, \cdots, n\}$ and each of its non-empty subsets, we define the "alternating sum" as follows: arrange the numbers in the set in ascending order, then alternately add and subtract each number starting from the largest one (for example, the alternating sum of $\{1,2,4,6,9\}$ is $9-6+4-2+1=6$, and the alternating sum of $\{5\}$ is just 5). For $n=7$, find the total sum of all these alternating sums.
|
448
| 0.125 |
Regular octagon \( CH I L D R E N \) has area 1. Determine the area of quadrilateral \( L I N E \).
|
\frac{1}{2}
| 0.5 |
Calculate the limit of the function:
$$\lim _{x \rightarrow-3} \frac{\left(x^{2}+2 x-3\right)^{2}}{x^{3}+4 x^{2}+3 x}$$
|
0
| 0.75 |
Given \(\theta = \arctan \frac{5}{12}\), find the principal value of the argument of the complex number \( z = \frac{\cos 2 \theta + \mathrm{i} \sin 2 \theta}{239 + \mathrm{i}} \).
|
\frac{\pi}{4}
| 0.75 |
Given that \(a b c=1\). Calculate the sum
\[
\frac{1}{1+a+ab}+\frac{1}{1+b+bc}+\frac{1}{1+c+ca}
\]
|
1
| 0.875 |
A cube is circumscribed around a sphere of radius 1. From one of the centers of the cube's faces, vectors are drawn to all other face centers and vertices. The dot products of each pair of these vectors are calculated, totaling 78. What is the sum of these dot products?
|
76
| 0.375 |
In figure 1, \( AB \) is parallel to \( DC \), \(\angle ACB\) is a right angle, \( AC = CB \), and \( AB = BD \). If \(\angle CBD = b^{\circ}\), find the value of \( b \).
|
15
| 0.625 |
Let the numbers \(x, y, u, v\) be distinct and satisfy the relationship \(\frac{x+u}{x+v}=\frac{y+v}{y+u}\). Find all possible values of the sum \(x+y+u+v\).
|
x+y+u+v=0
| 0.75 |
Given that \(ABCD\) is a square, points \(E\) and \(F\) lie on the side \(BC\) and \(CD\) respectively, such that \(BE = CF = \frac{1}{3} AB\). \(G\) is the intersection of \(BF\) and \(DE\). If
\[
\frac{\text{Area of } ABGD}{\text{Area of } ABCD} = \frac{m}{n}
\]
is in its lowest terms, find the value of \(m+n\).
|
23
| 0.875 |
For \( n \in \mathbf{Z}_{+}, n \geqslant 2 \), let
\[
S_{n}=\sum_{k=1}^{n} \frac{k}{1+k^{2}+k^{4}}, \quad T_{n}=\prod_{k=2}^{n} \frac{k^{3}-1}{k^{3}+1}
\]
Then, \( S_{n} T_{n} = \) .
|
\frac{1}{3}
| 0.75 |
Given that \(\mathrm{G}\) is the centroid of \(\triangle \mathrm{ABC}\), and the equation \(\sqrt{7 \mathrm{GA}} \sin A + 3 \overrightarrow{\mathrm{GB}} \sin B + 3 \sqrt{7 \mathrm{GC}} \sin C = \mathbf{0}\) holds, find \(\angle \mathrm{ABC}\).
|
60^\circ
| 0.625 |
Find the number of solutions in natural numbers to the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\).
|
110
| 0.625 |
A coin is tossed 10 times. Find the probability that heads will come up: a) between 4 and 6 times; b) at least once.
|
\frac{1023}{1024}
| 0.375 |
Given the point \( P(x, y) \) on the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{9}=1\), find the maximum value of \( 2x - y \).
|
5
| 0.625 |
On a highway, there are checkpoints D, A, C, and B arranged in sequence. A motorcyclist and a cyclist started simultaneously from A and B heading towards C and D, respectively. After meeting at point E, they exchanged vehicles and each continued to their destinations. As a result, the first person spent 6 hours traveling from A to C, and the second person spent 12 hours traveling from B to D. Determine the distance of path AB, given that the speed of anyone riding a motorcycle is 60 km/h, and the speed on a bicycle is 25 km/h. Additionally, the average speed of the first person on the path AC equals the average speed of the second person on the path BD.
|
340 \text{ km}
| 0.375 |
Given the function \( f(x) = a \sin x - \frac{1}{2} \cos 2x + a - \frac{3}{a} + \frac{1}{2} \), where \( a \in \mathbb{R} \) and \( a \neq 0 \):
(1) If \( f(x) \leq 0 \) for all \( x \in \mathbb{R} \), find the range of \( a \).
(2) If \( a \geq 2 \) and there exists an \( x \in \mathbb{R} \) such that \( f(x) \leq 0 \), find the range of \( a \).
|
[2, 3]
| 0.5 |
A person was asked how much he paid for a hundred apples and he answered the following:
- If a hundred apples cost 4 cents more, then for 1 dollar and 20 cents, he would get five apples less.
How much did 100 apples cost?
|
96 \text{ cents}
| 0.875 |
In a $7 \times 7$ grid, choose $k$ cells such that the centers of any 4 chosen cells do not form the vertices of a rectangle. Find the maximum value of $k$ that satisfies this condition.
|
21
| 0.25 |
Let \( x = \sqrt{1 + \frac{1}{1^{2}} + \frac{1}{2^{2}}} + \sqrt{1 + \frac{1}{2^{2}} + \frac{1}{3^{2}}} + \cdots + \sqrt{1 + \frac{1}{2012^{2}} + \frac{1}{2013^{2}}} \). Find the value of \( x - [x] \), where \( [x] \) denotes the greatest integer not exceeding \( x \).
|
\frac{2012}{2013}
| 0.875 |
Find the number of solutions in natural numbers for the equation \(\left\lfloor \frac{x}{10} \right\rfloor = \left\lfloor \frac{x}{11} \right\rfloor + 1\).
|
110
| 0.25 |
Let \( n \) be a fixed integer with \( n \geq 2 \). Determine the minimal constant \( c \) such that the inequality
\[ \sum_{1 \leq i < j \leq n} x_i x_j (x_i^2 + x_j^2) \leq c \left( \sum_{i=1}^{n} x_i \right)^4 \]
holds for all non-negative real numbers \( x_1, x_2, \ldots, x_n \). Additionally, determine the necessary and sufficient conditions for which equality holds.
|
\frac{1}{8}
| 0.875 |
You are standing at a pole and a snail is moving directly away from the pole at 1 cm/s. When the snail is 1 meter away, you start "Round 1". In Round \( n (n \geq 1) \), you move directly toward the snail at \( n+1 \) cm/s. When you reach the snail, you immediately turn around and move back to the starting pole at \( n+1 \) cm/s. When you reach the pole, you immediately turn around and Round \( n+1 \) begins. At the start of Round 100, how many meters away is the snail?
|
5050
| 0.75 |
In the right triangular prism $ABC - A_1B_1C_1$, $\angle ACB = 90^\circ$, $AC = 2BC$, and $A_1B \perp B_1C$. Find the sine of the angle between $B_1C$ and the lateral face $A_1ABB_1$.
|
\frac{\sqrt{10}}{5}
| 0.75 |
Given that \( 5^{\log 30} \times \left(\frac{1}{3}\right)^{\log 0.5} = d \), find the value of \( d \).
|
15
| 0.75 |
Calculate the improper integral of the second kind in the sense of the principal value:
\[ I = \mathrm{V} . \mathrm{p} . \int_{1 / e}^{e} \frac{d x}{x \ln x} \]
|
0
| 0.5 |
There is a sequence of 1999 numbers. The first number is equal to 1. It is known that every number, except the first and the last, is equal to the sum of its two neighbors.
Find the last number.
|
1
| 0.25 |
If \(a, b, c, d\) are positive real numbers such that \(\frac{5a + b}{5c + d} = \frac{6a + b}{6c + d}\) and \(\frac{7a + b}{7c + d} = 9\), calculate \(\frac{9a + b}{9c + d}\).
|
9
| 0.875 |
For which values of \( c \) are the numbers \(\sin \alpha\) and \(\cos \alpha\) roots of the quadratic equation \(10x^{2} - 7x - c = 0\) (where \(\alpha\) is a certain angle)?
|
2.55
| 0.375 |
A figure in the plane has exactly two axes of symmetry. Find the angle between these axes.
|
90^\circ
| 0.875 |
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