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|---|---|---|
If \( x \in \left(0, \frac{\pi}{2}\right) \), then the maximum value of the function \( f(x) = 2 \cos^3 x + 3 \cos^2 x - 6 \cos x - 2 \cos 3x \) is .
|
\frac{1}{9}
| 0.875 |
Concatenate the natural number $N$ to the right of each natural number. If the resulting new number is always divisible by $N$, then $N$ is called a magic number. How many magic numbers are there among the natural numbers less than 130?
|
9
| 0.75 |
In a right triangle \(ABC\) (with \(\angle C = 90^\circ\)), height \(CD\) is drawn. The radii of the circles inscribed in triangles \(ACD\) and \(BCD\) are 0.6 cm and 0.8 cm, respectively. Find the radius of the circle inscribed in triangle \(ABA\).
|
1
| 0.125 |
Given that each carbon pen costs 1 yuan 8 jiao, each notebook costs 3 yuan 5 jiao, and each pencil case costs 4 yuan 2 jiao, and Jing Jing spent exactly 20 yuan to buy these three kinds of stationery, find out how many notebooks she bought.
|
4
| 0.25 |
In a right triangle \(ABC\) with a right angle at \(B\) and \(\angle A = 30^\circ\), a height \(BD\) is drawn. Then, in triangle \(BDC\), a median \(DE\) is drawn, and in triangle \(DEC\), an angle bisector \(EF\) is drawn. Find the ratio \( \frac{FC}{AC} \).
|
\frac{1}{8}
| 0.75 |
Given that \( \sin \alpha - \cos \alpha = \frac{1}{3} \), find the value of \( \sin 3\alpha + \cos 3\alpha \).
|
-\frac{25}{27}
| 0.875 |
Given that \( a \) and \( b \) are positive integers, and \( b - a = 2013 \). If the equation \( x^2 - ax + b = 0 \) has a positive integer solution for \( x \), then find the minimum value of \( a \).
|
93
| 0.375 |
The birth date of Albert Einstein is 14 March 1879. If we denote Monday by 1, Tuesday by 2, Wednesday by 3, Thursday by 4, Friday by 5, Saturday by 6, and Sunday by 7, which day of the week was Albert Einstein born? Give your answer as an integer from 1 to 7.
|
5
| 0.125 |
Given a polygon drawn on graph paper with a perimeter of 2014 units, and whose sides follow the grid lines, what is the maximum possible area of this polygon?
|
253512
| 0.25 |
A dog and a cat both grabbed a sausage from different ends at the same time. If the dog bites off his piece and runs away, the cat will get 300 grams more than the dog. If the cat bites off his piece and runs away, the dog will get 500 grams more than the cat. How much sausage will be left if both bite off their pieces and run away?
|
400
| 0.5 |
In quadrilateral \(ABCD\), the diagonals intersect at point \(O\). It is known that \(S_{ABO} = S_{CDO} = \frac{3}{2}\), \(BC = 3\sqrt{2}\), and \(\cos \angle ADC = \frac{3}{\sqrt{10}}\). Find the minimum area of such a quadrilateral.
|
6
| 0.875 |
In the triangle \( \triangle ABC \), if \( \frac{\overrightarrow{AB} \cdot \overrightarrow{BC}}{3} = \frac{\overrightarrow{BC} \cdot \overrightarrow{CA}}{2} = \frac{\overrightarrow{CA} \cdot \overrightarrow{AB}}{1} \), find \( \tan A \).
|
\sqrt{11}
| 0.875 |
On the island of Misfortune with a population of 96 people, the government decided to implement five reforms. Each reform is disliked by exactly half of the citizens. A citizen protests if they are dissatisfied with more than half of all the reforms. What is the maximum number of people the government can expect at the protest?
|
80
| 0.875 |
A $10 \times 10 \times 10$-inch wooden cube is painted red on the outside and then cut into its constituent 1-inch cubes. How many of these small cubes have at least one red face?
|
488
| 0.75 |
Find all polynomials \( P \) such that the identity
\[ P(x+1) = P(x) + 2x + 1 \]
is satisfied.
|
P(x) = x^2 + c
| 0.625 |
The Russian Chess Championship is held in a single round. How many games are played if 18 chess players participate?
|
153 \text{ games}
| 0.75 |
In a certain country, the airline network is such that any city is connected by flight routes with no more than three other cities, and from any city to any other city, you can travel with no more than one transfer.
What is the maximum number of cities that can be in this country?
|
10
| 0.375 |
What is the remainder on dividing \(1234^{567} + 89^{1011}\) by 12?
|
9
| 0.75 |
Write 2004 numbers on the blackboard: $1, 2, \cdots, 2004$. In each step, erase some numbers and write their sum modulo 167. After several steps, among the two numbers remaining on the blackboard, one of them is 999. What is the other number?
|
3
| 0.5 |
In triangle \( ABC \), the medians \( AD \) and \( BE \) meet at the centroid \( G \). Determine the ratio of the area of quadrilateral \( CDGE \) to the area of triangle \( ABC \).
|
\frac{1}{3}
| 0.625 |
The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$. Find the integer part of the following sum: $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{200}}$.
|
1
| 0.5 |
Given real numbers \( a, b, c, d \) that satisfy \( 5a + 6b - 7c + 4d = 1 \), what is the minimum value of \( 3a^2 + 2b^2 + 5c^2 + d^2 \)?
|
\frac{15}{782}
| 0.5 |
Let \( p \) and \( q \) be two prime numbers such that \( q \) divides \( 3^p - 2^p \). Show that \( p \) divides \( q - 1 \).
|
p \mid q - 1
| 0.5 |
Find the last two digits of \( 7 \times 19 \times 31 \times \cdots \times 1999 \). (Here \( 7, 19, 31, \ldots, 1999 \) form an arithmetic sequence of common difference 12.)
|
75
| 0.75 |
Calculate: \( 8.0 \dot{\dot{1}} + 7.1 \dot{2} + 6.2 \dot{3} + 5.3 \dot{4} + 4.4 \dot{5} + 3.5 \dot{6} + 2.6 \dot{7} + 1.7 \dot{8} = \)
|
39.2
| 0.375 |
Show that the numbers \( p \) and \( 8p^2 + 1 \) can be prime simultaneously only if \( p = 3 \).
|
3
| 0.75 |
Francesca chooses an integer from the list \(-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6\) and then a second integer that is larger than the first. How many such pairs of integers can she choose so that the sum of the pair is 3?
|
5
| 0.75 |
Three candles can burn for 30, 40, and 50 minutes, respectively (but are not ignited simultaneously). It is known that the three candles are burning simultaneously for 10 minutes, and only one candle is burning for 20 minutes. How long are exactly two candles burning simultaneously?
|
35
| 0.75 |
Let \( B \) be a point on the circle centered at \( O \) with diameter \( AC \) and let \( D \) and \( E \) be the circumcenters of the triangles \( OAB \) and \( OBC \) respectively. Given that \( \sin \angle BOC = \frac{4}{5} \) and \( AC = 24 \), find the area of the triangle \( BDE \).
|
45
| 0.875 |
Find the smallest natural number $n$ such that the natural number $n^2 + 14n + 13$ is divisible by 68.
|
21
| 0.5 |
a) \((2+3i)^{2}\)
b) \((3-5i)^{2}\)
c) \((5+3i)^{3}\)
|
-10 + 198i
| 0.75 |
If the product of a 4-digit number abSd and 9 is equal to another 4-digit number dSba, find the value of \( S \).
|
8
| 0.625 |
Five distinct digits from 1 to 9 are given. Arnaldo forms the largest possible number using three of these 5 digits. Then, Bernaldo writes the smallest possible number using three of these 5 digits. What is the units digit of the difference between Arnaldo's number and Bernaldo's number?
|
0
| 0.75 |
Katya and Pasha ask each other four tricky questions and answer them randomly without thinking. The probability that Katya lies in response to any question from Pasha is $\frac{1}{3}$ and does not depend on the question number. Pasha answers Katya’s questions truthfully with a probability of $\frac{3}{5}$, regardless of the question order. After the dialogue, it was revealed that Pasha gave two more truthful answers than Katya. What is the probability that this could have happened?
|
\frac{48}{625}
| 0.625 |
Xiao Zhang drives a car from the foot of the mountain at point $A$, reaches the top of the mountain, then descends to the foot of the mountain at point $B$, and finally returns to point $A$ along the same route. The car's speed uphill is 30 kilometers per hour, and its speed downhill is 40 kilometers per hour. When Xiao Zhang returns to point $A$, he finds that the return trip took 10 minutes less than the outbound trip, and the car's odometer increased by 240 kilometers. How many hours did Xiao Zhang spend on this round trip?
|
7
| 0.625 |
Given a sequence that starts with one, in which each succeeding term is equal to double the sum of all previous terms. Find the smallest number such that the element at this position is divisible by \(3^{2017}\).
|
2019
| 0.875 |
Let \( a, b, c \geqslant 1 \), and the positive real numbers \( x, y, z \) satisfy
\[
\begin{cases}
a^x + b^y + c^z = 4, \\
x a^x + y b^y + z c^z = 6, \\
x^2 a^x + y^2 b^y + z^2 c^z = 9.
\end{cases}
\]
Then the maximum possible value of \( c \) is \_\_\_\_\_
|
\sqrt[3]{4}
| 0.125 |
Eight numbers \( a_{1}, a_{2}, a_{3}, a_{4} \) and \( b_{1}, b_{2}, b_{3}, b_{4} \) satisfy the equations:
\[
\left\{
\begin{array}{l}
a_{1} b_{1} + a_{2} b_{3} = 1 \\
a_{1} b_{2} + a_{2} b_{4} = 0 \\
a_{3} b_{1} + a_{4} b_{3} = 0 \\
a_{3} b_{2} + a_{4} b_{4} = 1
\end{array}
\right.
\]
Given that \( a_{2} b_{3} = 7 \), find \( a_{4} b_{4} \).
|
-6
| 0.875 |
Compute the number of tuples \(\left(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right)\) of (not necessarily positive) integers such that \(a_{i} \leq i\) for all \(0 \leq i \leq 5\) and
\[ a_{0} + a_{1} + \cdots + a_{5} = 6. \]
|
2002
| 0.75 |
Anya is arranging stones on the sand. First, she placed one stone, then added stones to form a pentagon, then made a larger outer pentagon with the stones, then another outer pentagon, and so on, as shown in the picture. The number of stones she had arranged in the first four pictures is 1, 5, 12, and 22. If she continues making such pictures, how many stones will be in the 10th picture?
|
145
| 0.875 |
Given a set $S$ of $n$ points in the plane such that no three points from $S$ are collinear, show that the number of triangles of area 1 whose vertices are in $S$ is at most:
$$
\frac{2 n(n-1)}{3}
$$
|
\frac{2 n(n-1)}{3}
| 0.375 |
On the leg \(AC\) of the right triangle \(ABC\), a circle is constructed with \(AC\) as its diameter, intersecting the hypotenuse \(AB\) at point \(K\).
Find \(CK\) if \(AC = 2\) and \(\angle A = 30^\circ\).
|
CK = 1
| 0.875 |
If \( a < b < c < d \) are distinct positive integers such that \( a+b+c+d \) is a square, what is the minimum value of \( c+d \)?
|
11
| 0.75 |
When the boy Clive approached his grandfather's wall cuckoo clock, it was 12:05.
Clive started spinning the minute hand with his finger until the hour hand returned to its original position. How many "cuckoos" did the grandfather count in the neighboring room during this time?
|
78
| 0.25 |
Dasha poured 9 grams of food into the aquarium for the fish. In the first minute, they ate half of the food, in the second minute - a third of the remaining food, in the third minute - a quarter of the remaining food, and so on, in the ninth minute - a tenth of the remaining food. How many grams of food are left in the aquarium?
|
0.9
| 0.625 |
In a competition, there were 50 shooters. The first shooter scored 60 points; the second - 80; the third - the arithmetic mean of the points of the first two; the fourth - the arithmetic mean of the points of the first three. Each subsequent shooter scored the arithmetic mean of the points of all the previous ones. How many points did the 42nd shooter score? And the 50th?
|
70 \text{ points}
| 0.75 |
My 24-hour digital clock displays hours and minutes only. For how many different times does the display contain at least one occurrence of the digit 5 in a 24-hour period?
|
450
| 0.5 |
What is the largest positive integer $n$ for which
$$
\sin ^{n} x+\cos ^{n} x \geq \frac{1}{n}
$$
holds for any real number $x$?
|
8
| 0.875 |
If \( a > 0 \) and \( b > 0 \), and \( \arcsin a + \arcsin b = \frac{\pi}{2} \), with \( m = \log_2 \frac{1}{a} + \log_2 \frac{1}{b} \), what is the range of values for \( m \)?
|
[1, +\infty)
| 0.25 |
Find the numerical value of the monomial \(0.007 a^{7} b^{9}\) if \(a = -5\) and \(b = 2\).
|
-280000
| 0.5 |
Provide an example of a non-zero polynomial with integer coefficients that has the number $\cos 18^{\circ}$ as one of its roots.
|
16x^4 - 20x^2 + 5
| 0.625 |
For which smallest natural number \( k \) is the expression \( 2018 \cdot 2019 \cdot 2020 \cdot 2021 + k \) a square of a natural number?
|
1
| 0.875 |
From letter cards, the word "КАРАКАТИЦА" can be formed. How many different words (not necessarily meaningful) can be formed from these cards where the letters "Р" and "Ц" are adjacent?
|
15120
| 0.5 |
A truck left the settlement of Mirny at a speed of 40 km/h. At the same time, a car left the town of Tikhiy in the same direction as the truck. In the first hour of the journey, the car covered 50 km, and in each subsequent hour, it covered 5 km more than in the previous hour. After how many hours will the car catch up with the truck if the distance between the settlement and the town is 175 km?
|
7
| 0.75 |
In a trapezoid, where the diagonals intersect at a right angle, it is known that the midline is 6.5 and one of the diagonals is 12. Find the length of the other diagonal.
|
5
| 0.875 |
For two sets \( A \) and \( B \), define
\[
A \Delta B = \{x \mid x \in A \text{ and } x \notin B\} \cup \{x \mid x \in B \text{ and } x \notin A\}.
\]
Let \( n > 1 \) be an integer, and let \( A_{1}, A_{2}, \ldots, A_{2n} \) be all the subsets of
\[
S = \{1, 2, \cdots, n\}.
\]
Define \( M \) to be a \( 2^n \times 2^n \) table. For any \( i, j \in \{1, 2, \cdots, 2^n\} \), place the sum of all elements in the set \( A_i \Delta A_j \) (with the convention that the sum of elements in an empty set is 0) in the \(i\)-th row and \(j\)-th column of the table \( M \).
Find the sum of all the numbers in the table \( M \).
|
2^{2n-2} n (n + 1)
| 0.125 |
Given $\boldsymbol{a}=\left(\lambda+2, \lambda^{2}-\cos^2 \alpha\right)$ and $\boldsymbol{b}=\left(m, \frac{m}{2}+\sin \alpha\right)$, where $\lambda$, $m$, and $\alpha$ are real numbers, if $\boldsymbol{a}=2\boldsymbol{b}$, find the range of $\frac{\lambda}{m}$.
|
[-6, 1]
| 0.875 |
Let \( f(n) = 5n^{13} + 13n^{5} + 9an \). Find the smallest positive integer \( a \) such that \( f(n) \) is divisible by 65 for every integer \( n \).
|
63
| 0.25 |
Determine \( q \) such that one root of the second equation is twice as large as one root of the first equation.
\[
x^{2} - 5x + q = 0 \quad \text{and} \quad x^{2} - 7x + 2q = 0
\]
|
6
| 0.625 |
In a "level passing game" rule, in the \( n \)-th level, a die needs to be rolled \( n \) times. If the sum of the points appearing in these \( n \) rolls exceeds \( 2^n \), then the level is considered passed. Questions:
(I) What is the maximum number of levels a person can pass in this game?
(II) What is the probability of passing the first three levels consecutively?
(Note: A die is a uniform cube with points \( 1, 2, 3, 4, 5, 6 \) on each face, and the number facing up after the die comes to rest is the point that appears.)
(2004 Annual High School Math Competition)
|
\frac{100}{243}
| 0.375 |
Two cyclists were riding on a highway, each with their own constant speed. It turned out that the faster one travels 6 km in 5 minutes less and in 20 minutes travels 4 km more than the slower one. Find the product of the cyclists' speeds, expressed in kilometers per hour.
|
864
| 0.875 |
According to experts' forecasts, apartment prices in Moscow will decrease by 20% in rubles and by 40% in euros over the next year. In Sochi, apartment prices will decrease by 10% in rubles over the next year. By what percentage will apartment prices decrease in Sochi in euros? It is assumed that the euro-to-ruble exchange rate (i.e., the cost of one euro in rubles) is the same in both Moscow and Sochi, but it may change over time. Justify the answer.
|
32.5\%
| 0.5 |
How many natural numbers with up to six digits contain the digit 1?
|
468559
| 0.5 |
If a two-digit number is divided by the sum of its digits, the quotient is 3 and the remainder is 7. If you then take the sum of the squares of its digits and subtract the product of the same digits, you get the original number. Find this number.
|
37
| 0.875 |
Formulate the equation of a plane that is parallel to the $Oz$ axis and passes through the points $(1, 0, 1)$ and $(-2, 1, 3)$.
|
x + 3y - 1 = 0
| 0.5 |
A confectionery factory received 5 spools of ribbon, each 60 meters long, for packaging cakes. How many cuts are needed to obtain pieces of ribbon, each 1 meter 50 centimeters long?
|
195
| 0.75 |
Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 boxes, and it is known that only one of them contains the treasure, while the rest trigger a stone slab to fall on the person trying to open them. The boxes are numbered from left to right. The first, fourth, and fifth boxes are made of cedar, while the second and third boxes are made of sandalwood. The inscriptions on the boxes read as follows:
- First box: "The treasure is either in me or in the fourth box."
- Second box: "The treasure is in the box to my left."
- Third box: "The treasure is either in me or in the rightmost box."
- Fourth box: "There is no treasure in the boxes to the left of me."
- Fifth box: "The writings on all other boxes are lies."
The last guardian of the temple, before dying, revealed to Indiana a secret: The same number of false statements are written on the cedar boxes as on the sandalwood boxes. In which box is the treasure?
|
2
| 0.875 |
One school library has 24,850 books, and another has 55,300 books. When these books were arranged evenly on shelves, 154 books were left in the first library, and 175 books were left in the second library. How many books were placed on each shelf?
|
441
| 0.75 |
The lines containing the lateral sides of a trapezoid intersect at a right angle. The longer lateral side of the trapezoid is 8, and the difference between the bases is 10. Find the shorter lateral side.
|
6
| 0.75 |
Using the relationship between the domain and range of inverse functions, find the range of the function \( y = \frac{3x^2 - 2}{x^2 + 1} \) for \( x \geqslant 0 \).
|
[-2, 3)
| 0.875 |
There are 8 identical balls in a box, consisting of three balls numbered 1, three balls numbered 2, and two balls numbered 3. A ball is randomly drawn from the box, returned, and then another ball is randomly drawn. The product of the numbers on the balls drawn first and second is denoted by $\xi$. Find the expected value $E(\xi)$.
|
\frac{225}{64}
| 0.625 |
One mole of an ideal monatomic gas is first heated isobarically. In this process, it performs work of 30 J. Then it is heated isothermally, receiving the same amount of heat as in the first case. What work (in Joules) does the gas perform in the second case?
|
75
| 0.875 |
Given $\triangle ABC$ and $\triangle A'B'C'$ with side lengths $a, b, c$ and $a', b', c'$ respectively, and $\angle B = \angle B'$, $\angle A + \angle A' = 180^\circ$, then $a a' = b b' + c c'$.
|
a a' = b b' + c c'
| 0.875 |
Consecutive odd numbers are grouped as follows: $1 ;(3,5) ;(7,9,11) ;(13, 15, 17, 19) ; \ldots$. Find the sum of the numbers in the $n$-th group.
|
n^3
| 0.875 |
Calculate: $3 \times 995 + 4 \times 996 + 5 \times 997 + 6 \times 998 + 7 \times 999 - 4985 \times 3=$
|
9980
| 0.5 |
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 \( z^2 + az + b = 0 \) for some integers \( a \) and \( b \).
|
8
| 0.875 |
A table of numbers consists of several rows. Starting from the second row, each number in a row is the sum of the two numbers directly above it. The last row contains only one number. The first row is the first 100 positive integers arranged in ascending order. What is the number in the last row? (You may express the answer using exponents).
|
101 \cdot 2^{98}
| 0.125 |
Calculate the limit of the function:
\[
\lim_{x \rightarrow 0} \frac{e^{5x} - e^{3x}}{\sin 2x - \sin x}
\]
|
2
| 0.875 |
Let \( z \) be a complex number such that \( |z| = 1 \) and \( |z - 1.45| = 1.05 \). Compute the real part of \( z \).
|
\frac{20}{29}
| 0.625 |
Optimus Prime in the shape of a robot travels from point $A$ to point $B$ and arrives on time. If he starts by transforming into a car, his speed increases by $\frac{1}{4}$ and he arrives at point $B$ 1 hour earlier. If he travels the first 150 kilometers as a robot and then transforms into a car, increasing his speed by $\frac{1}{5}$, he arrives 40 minutes earlier. Determine the distance between points $A$ and $B$ in kilometers.
|
750 \text{ km}
| 0.875 |
Find the general solution of the differential equation
$$
d r - r d \varphi = 0
$$
|
r = C e^{\varphi}
| 0.5 |
There are 9 cards, each with a digit from 1 to 9 written on it (one digit per card). Three cards are randomly chosen and placed in a sequence. What is the probability that the resulting three-digit number is divisible by 3?
|
\frac{5}{14}
| 0.5 |
In triangle \( ABC \), point \( M \) is taken on side \( AC \) such that \( AM : MC = 1 : 3 \). Find the area of triangle \( ABM \) if the area of triangle \( ABC \) is 1.
|
\frac{1}{4}
| 0.625 |
As shown in the figure, \( P_{1} \) is a semicircular paperboard with a radius of 1. By cutting out a semicircle with a radius of \( \frac{1}{2} \) from the lower left end of \( P_{1} \), we obtain the figure \( P_{2} \). Then, successively cut out smaller semicircles (each having a diameter equal to the radius of the previously cut semicircle) to obtain figures \( P_{3}, P_{4}, \cdots, P_{n}, \cdots \). Let \( S_{n} \) be the area of the paperboard \( P_{n} \). Find \( \lim _{n \rightarrow \infty} S_{n} \).
|
\frac{\pi}{3}
| 0.875 |
In a singing contest, a Rooster, a Crow, and a Cuckoo were contestants. Each jury member voted for one of the three contestants. The Woodpecker tallied that there were 59 judges, and that the sum of votes for the Rooster and the Crow was 15, the sum of votes for the Crow and the Cuckoo was 18, and the sum of votes for the Cuckoo and the Rooster was 20. The Woodpecker does not count well, but each of the four numbers mentioned is off by no more than 13. How many judges voted for the Crow?
|
13
| 0.5 |
A and B agreed to meet at a restaurant for a meal. Due to the restaurant being very busy, A arrived first and took a waiting number and waited for B. After a while, B also arrived but did not see A, so he took another waiting number. While waiting, B saw A, and they took out their waiting numbers and discovered that the digits of these two numbers are reversed two-digit numbers, and the sum of the digits of both numbers (for example, the sum of the digits of 23 is $2+3=5$) is 8, and B's number is 18 greater than A's. What is A's number?
|
35
| 0.75 |
Anna Alexandrovna's age is 60 years, 60 months, 60 weeks, 60 days, and 60 hours. How many full years old is Anna Alexandrovna?
|
66
| 0.875 |
Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13?
|
10
| 0.625 |
Find the curve that passes through the point \( M_{0}(1,4) \) and has the property that the segment of any of its tangents, enclosed between the coordinate axes, is bisected at the point of tangency.
|
xy = 4
| 0.25 |
Suppose that \( x_{1} \) and \( x_{2} \) are the two roots of the equation \( (x-2)^{2} = 3(x+5) \). What is the value of the expression \( x_{1} x_{2} + x_{1}^{2} + x_{2}^{2} \) ?
|
60
| 0.875 |
One mole of an ideal monoatomic gas is first heated isobarically. In this process, it does 20 J of work. Then, it is heated isothermally, receiving the same amount of heat as in the first case. How much work (in joules) does the gas do in the second case?
|
50 \text{ J}
| 0.875 |
The cells of a $9 \times 9$ board are painted in black and white in a checkerboard pattern. How many ways are there to place 9 rooks on cells of the same color on the board such that no two rooks attack each other? (A rook attacks any cell that is in the same row or column as it.)
|
2880
| 0.375 |
Find two natural numbers \( m \) and \( n \), given that \( m < n < 2m \) and \( mn = 2013 \).
|
m = 33, n = 61
| 0.5 |
Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13?
|
10
| 0.625 |
A school admits art-talented students based on their entrance exam scores. It enrolls $\frac{2}{5}$ of the candidates, whose average score is 15 points higher than the admission cutoff score. The average score of the candidates who were not admitted is 20 points lower than the cutoff score. If the average score of all candidates is 90 points, determine the admission cutoff score.
|
96
| 0.875 |
On the island, \( \frac{2}{3} \) of all men are married, and \( \frac{3}{5} \) of all women are married. What fraction of the island's population is married?
|
\frac{12}{19}
| 0.875 |
A gardener needs to plant 10 trees over the course of three days. In how many ways can he distribute the work over the days if he plants at least one tree each day?
|
36
| 0.875 |
If \( A \) is the area of a square inscribed in a circle of diameter 10, find \( A \).
If \( a+\frac{1}{a}=2 \), and \( S=a^{3}+\frac{1}{a^{3}} \), find \( S \).
An \( n \)-sided convex polygon has 14 diagonals. Find \( n \).
If \( d \) is the distance between the 2 points \( (2,3) \) and \( (-1,7) \), find \( d \).
|
5
| 0.875 |
The decimal representation of a 2015-digit natural number \( N \) contains the digits 5, 6, 7, and no other digits. Find the remainder when the number \( N \) is divided by 9, given that the number of fives in the representation is 15 more than the number of sevens.
|
6
| 0.875 |
On a blackboard, a stranger writes the values of \( s_{7}(n)^2 \) for \( n = 0, 1, \ldots, 7^{20} - 1 \), where \( s_{7}(n) \) denotes the sum of the digits of \( n \) in base 7. Compute the average value of all the numbers on the board.
|
3680
| 0.875 |
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