problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
Petrov and Vasechkin were solving the same arithmetic problem. A certain number had to be divided by 2, multiplied by 7, and subtracted by 1001. Petrov performed all the operations correctly, but Vasechkin mixed up everything: he divided by 8, squared it, and also subtracted 1001. It is known that Petrov ended up with a prime number. What number did Vasechkin get?
|
295
| 0.875 |
The master and his apprentice were tasked with producing a batch of identical parts. After the master worked for 7 hours and the apprentice for 4 hours, it was found that they had completed $5 / 9$ of the total work. After working together for an additional 4 hours, they determined that $1 / 18$ of the work remained. How long would it take for the apprentice to complete the entire work if working alone?
|
24 \text{ hours}
| 0.875 |
A poplar tree, a willow tree, a locust tree, a birch tree, and a phoenix tree are planted in a row, with a distance of 1 meter between any two adjacent trees. The distance between the poplar tree and the willow tree is equal to the distance between the poplar tree and the locust tree. The distance between the birch tree and the poplar tree is equal to the distance between the birch tree and the locust tree. What is the distance between the phoenix tree and the birch tree? $\qquad$ meters.
|
2
| 0.75 |
Vasya throws three dice (each die has numbers from 1 to 6 on its faces) and sums up the rolled numbers. Furthermore, if all three rolled numbers are different, he can roll all three dice again and add the rolled numbers to the already accumulated sum. This process continues until at least two of the three rolled numbers are the same. What is the expected value of Vasya's total result?
|
23.625
| 0.25 |
On weekdays, a scatterbrained scientist takes the Moscow metro's circular line from the "Taganskaya" station to the "Kievskaya" station, and back in the evening. He boards the first arriving train. Trains run at approximately equal intervals in both directions. The northern route (via "Belorusskaya") takes 17 minutes between "Kievskaya" and "Taganskaya" or vice-versa, and the southern route (via "Paveletskaya") takes 11 minutes.
The scientist has calculated that:
- A counterclockwise train arrives at "Kievskaya" on average 1 minute and 15 seconds after a clockwise train. The same is true for "Taganskaya";
- The average journey from home to work takes 1 minute less than the return journey.
Find the expected interval between trains going in the same direction.
|
3 \text{ minutes}
| 0.125 |
For positive integer \(n\), let \(D(n)\) be the eventual value obtained when the digits of \(n\) (in base 10) are added up recursively until a one-digit number is obtained. For example \(D(4)=4\), \(D(2012)=D(5)=5\) and \(D(1997)=D(26)=D(8)=8\). If \(x\) denotes the 2012th Fibonacci number (i.e. the 2012th term of the sequence \(1, 1, 2, 3, 5, 8, 13, \ldots)\), find \(D(x)\).
|
6
| 0.875 |
2015 people are sitting around a round table, each of whom is either a knight or a liar. Knights always tell the truth, liars always lie. They were each given a card, with a unique number written on it. After looking at the cards of their neighbors, each person at the table said: "My number is greater than that of each of my two neighbors." After this, $k$ of the people said: "My number is less than that of each of my two neighbors." What is the maximum possible value of $k?
|
2013
| 0.125 |
Solve the following equation:
$$
1+\log x=\log (1+x)
$$
|
\frac{1}{9}
| 0.625 |
We place a coin in one of three identical boxes and then put these three boxes into two drawers such that each drawer contains at least one box. What is the probability that if someone randomly selects a drawer and then takes a box out from that drawer, they will find the coin in that box,
a) if the box containing the coin is alone in one drawer?
b) if the box containing the coin is in a drawer with one other box?
c) if the 3 boxes are randomly placed into the two drawers without knowing which box contains the coin?
|
\frac{1}{3}
| 0.875 |
The base side length of the regular triangular prism \( ABC-A_1B_1C_1 \) is \( a \), and the side edge length is \( \sqrt{2}a \). Find the angle between \( AC_1 \) and the lateral face \( ABB_1A_1 \).
|
30^\circ
| 0.875 |
Given the ellipse \(\frac{x^{2}}{6}+\frac{y^{2}}{3}=1\) and a point \(P\left(1, \frac{1}{2}\right)\) inside the ellipse, draw a line through \(P\) that does not pass through the origin and intersects the ellipse at points \(A\) and \(B\). Find the maximum area of triangle \(OAB\), where \(O\) is the origin.
|
\frac{3\sqrt{6}}{4}
| 0.375 |
Let the ellipse with foci \( F_1(-1,0) \) and \( F_2(1,0) \) have an eccentricity \( e \). A parabola with vertex at \( F_1 \) and focus at \( F_2 \) intersects the ellipse at a point \( P \). If \( \frac{|P F_1|}{|P F_2|} = e \), what is the value of \( e \)?
|
\frac{\sqrt{3}}{3}
| 0.5 |
Calculate the integral
$$
\int_{L} \frac{\sin z}{z\left(z-\frac{\pi}{2}\right)} d z
$$
where \( L \) is the rectangle bounded by the lines \( x=2 \), \( x=-1 \), \( y=2 \), and \( y=-1 \).
|
4i
| 0.875 |
A square is drawn on a plane with its sides parallel to the horizontal and vertical axes. Inside the square, several line segments parallel to its sides are drawn such that no two segments lie on the same line or intersect at an interior point of both segments. These segments divide the square into rectangles in such a manner that any vertical line that intersects the square and does not contain any of the dividing segments intersects exactly $k$ of these rectangles, and any horizontal line that intersects the square and does not contain any of the dividing segments intersects exactly $\ell$ of these rectangles. What could be the total number of rectangles formed by this division?
|
k \ell
| 0.25 |
Is it possible to arrange the digits $1, 2, \ldots, 8$ in the cells of a) the letter "Ш"; b) the strips (see figure), such that for any cutting of the figure into two parts, the sum of all digits in one part is divisible by the sum of all digits in the other? (Cuts can only be made along cell borders. Each cell must contain one digit, and each digit can only be used once.)
a)
b)
|
\text{No}
| 0.375 |
In the Cartesian coordinate plane \(xOy\), points \(A\) and \(B\) are on the parabola \(y^2 = 4x\), satisfying \(\overrightarrow{OA} \cdot \overrightarrow{OB} = -4\). \(F\) is the focus of the parabola. Find \(S_{\triangle OFA} \cdot S_{\triangle OFB} = \) ________.
|
2
| 0.75 |
Integers \(a, b, c, d,\) and \(e\) satisfy the following three properties:
(i) \(2 \leq a<b<c<d<e<100\)
(ii) \(\operatorname{gcd}(a, e)=1\)
(iii) \(a, b, c, d, e\) form a geometric sequence.
What is the value of \(c\)?
|
36
| 0.75 |
A sphere with a radius of \(\sqrt{3}\) has a cylindrical hole drilled through it; the axis of the cylinder passes through the center of the sphere, and the diameter of the base of the cylinder is equal to the radius of the sphere. Find the volume of the remaining part of the sphere.
|
\frac{9 \pi}{2}
| 0.375 |
Polina custom makes jewelry for a jewelry store. Each piece of jewelry consists of a chain, a stone, and a pendant. The chains can be silver, gold, or iron. Polina has stones - cubic zirconia, emerald, quartz - and pendants in the shape of a star, sun, and moon. Polina is happy only when three pieces of jewelry are laid out in a row from left to right on the showcase according to the following rules:
- There must be a piece of jewelry with a sun pendant on an iron chain.
- Next to the jewelry with the sun pendant there must be gold and silver jewelry.
- The three pieces of jewelry in the row must have different stones, pendants, and chains.
How many ways are there to make Polina happy?
|
24
| 0.5 |
Find the number of pairs of integers \((x, y)\) that satisfy the equation \(x^{2} + 7xy + 6y^{2} = 15^{50}\).
|
4998
| 0.25 |
What is the smallest positive integer $n$ such that
$$
\sqrt{n}-\sqrt{n-1}<0.01 ?
$$
|
2501
| 0.625 |
A point \( M \) is chosen on the diameter \( AB \). Points \( C \) and \( D \), lying on the circumference on one side of \( AB \), are chosen such that \(\angle AMC=\angle BMD=30^{\circ}\). Find the diameter of the circle given that \( CD=12 \).
|
8\sqrt{3}
| 0.375 |
Let \( S \) be a set of sequences of length 15 formed by using the letters \( a \) and \( b \) such that every pair of sequences in \( S \) differ in at least 3 places. What is the maximum number of sequences in \( S \)?
|
2048
| 0.75 |
Find all the ways to assign an integer to each vertex of a 100-gon such that among any three consecutive numbers written down, one of the numbers is the sum of the other two.
|
0
| 0.75 |
The edge length of cube \( ABCD-A_1B_1C_1D_1 \) is 1. Find the distance between the line \( A_1C_1 \) and the line \( BD_1 \).
|
\frac{\sqrt{6}}{6}
| 0.75 |
Find all prime numbers \( p \) such that both \( p + 28 \) and \( p + 56 \) are also prime.
|
3
| 0.125 |
There is a bag of grapes with more than one hundred. If each student in the choir group is given 3 grapes, there will be 10 grapes left; if each student in the art group is given 5 grapes, there will be 12 grapes left; if each student in the math group is given 7 grapes, there will be 18 grapes left. How many grapes are there in the bag at most?
|
172
| 0.375 |
Let \( S \) denote the set of all triples \( (i, j, k) \) of positive integers where \( i + j + k = 17 \). Compute
$$
\sum_{(i, j, k) \in S} i j k
$$
|
11628
| 0.625 |
Point \( C \) divides the chord \( AB \) of a circle with radius 6 into segments \( AC = 4 \) and \( CB = 5 \). Find the minimum distance from point \( C \) to the points on the circle.
|
2
| 0.625 |
Select some numbers from \(1, 2, 3, \cdots, 9, 10\) such that each number from \(1, 2, 3, \cdots, 19, 20\) is equal to the sum of one or two of the selected numbers (they can be the same). Find the minimum number of selected numbers needed.
|
6
| 0.5 |
At most, how many interior angles greater than $180^\circ$ can a 2006-sided polygon have?
|
2003
| 0.625 |
Find the integer part of the number \(a + \frac{9}{b}\), where \(a\) and \(b\) are the integer and fractional parts, respectively, of the number \(\sqrt{76 - 42 \sqrt{3}}\).
|
12
| 0.75 |
The length of the escalator is 200 steps. When Petya walks down the escalator, he counts 50 steps. How many steps will he count if he runs twice as fast?
|
80 \text{ steps}
| 0.75 |
In an isosceles triangle \( \triangle ABC \), the base \( AB \) is the diameter of the circle, which intersects the legs \( AC \) and \( CB \) at points \( D \) and \( E \) respectively. Find the perimeter of the triangle \( ABC \), given \( AD = 2 \) and \( AE = \frac{8}{3} \).
|
\frac{80}{9}
| 0.25 |
There are fewer than 30 students in a class. The probability that a randomly selected girl is an honor student is $\frac{3}{13}$, and the probability that a randomly selected boy is an honor student is $\frac{4}{11}$. How many honor students are there in the class?
|
7
| 0.875 |
Find the smallest natural number \( n \) for which \( n^{2} + 20n + 19 \) is divisible by 2019.
|
2000
| 0.25 |
There is a checkers championship in Anchuria that consists of several rounds. The days and cities where the rounds are held are determined by a draw. According to the rules of the championship, no two rounds can take place in the same city, and no two rounds can take place on the same day. A lottery is held among the fans: the main prize is awarded to the person who correctly guesses, before the championship begins, in which cities and on which days all the rounds will take place. If no one guesses correctly, the main prize will be retained by the organizing committee. There are eight cities in Anchuria, and the championship is allocated a total of eight days. How many rounds should the championship consist of to maximize the probability that the main prize will be retained by the organizing committee?
|
6 \text{ rounds}
| 0.375 |
The two diagonals of a quadrilateral are perpendicular. The lengths of the diagonals are 14 and 30. What is the area of the quadrilateral?
|
210
| 0.875 |
Integer \( n \) such that the polynomial \( f(x) = 3x^3 - nx - n - 2 \) can be factored into a product of two non-constant polynomials with integer coefficients. Find the sum of all possible values of \( n \).
|
192
| 0.625 |
If real numbers \(x\) and \(y\) satisfy \(x^2 + y^2 = 20\), what is the maximum value of \(xy + 8x + y\)?
|
42
| 0.5 |
100 people participated in a quick calculation test consisting of 10 questions. The number of people who answered each question correctly is given in the table below:
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Problem Number & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
Number of Correct Answers & 93 & 90 & 86 & 91 & 80 & 83 & 72 & 75 & 78 & 59 \\
\hline
\end{tabular}
Criteria: To pass, one must answer at least 6 questions correctly. Based on the table, calculate the minimum number of people who passed.
|
62
| 0.625 |
Find the number of real values of \( a \) such that for each \( a \), the cubic equation \( x^{3} = ax + a + 1 \) has an even root \( x \) with \( |x| < 1000 \).
|
999
| 0.75 |
Several people were seated around a round table such that the distances between neighboring people were equal. One of them was given a card with the number 1, and the rest were given cards with numbers 2, 3, and so on, in a clockwise direction.
The person with the card numbered 31 noticed that the distance from him to the person with the card numbered 7 is the same as the distance to the person with the card numbered 14. How many people are seated at the table in total?
|
41
| 0.5 |
Given the universal set \( U = \{1, 2, 3, 4, 5\} \) and the set \( I = \{X \mid X \subseteq U\} \), two different elements \( A \) and \( B \) are randomly selected from set \(I\). What is the probability that the intersection \( A \cap B \) has exactly 3 elements?
|
\frac{5}{62}
| 0.125 |
Consider a $4 \times 4$ square grid with 25 points. Determine the number of distinct lines passing through at least 3 of these points.
|
32
| 0.25 |
Given a polynomial \( P(x) \) with integer coefficients. It is known that \( P(1) = 2013 \), \( P(2013) = 1 \), and \( P(k) = k \), where \( k \) is some integer. Find \( k \).
|
1007
| 0.625 |
Given the curve \( x y^{2} = 1 (y > 0) \), the slope of the tangent line at a point \((x_0, y_0)\) on the curve is \( k = -\frac{1}{2 \sqrt{x_0^{3}}} \). A line parallel to the \( y \)-axis through the point \( P_1 (1, 0) \) intersects the curve at \( Q_1 \). The tangent line to the curve at \( Q_1 \) intersects the \( x \)-axis at \( P_2 \). A line parallel to the \( y \)-axis through \( P_2 \) intersects the curve at \( Q_2 \). This process is repeated, generating a sequence of points \( P_1, P_2, \cdots \) and \( Q_1, Q_2, \cdots \). Let \( l_n = \left| P_n Q_n \right| \). Find the value of \( l_{2003} \).
|
\frac{1}{3^{1001}}
| 0.5 |
How many integer solutions does the equation \((2x + y)^{2} = 2017 + x^{2}\) have for \(x\) and \(y\)?
|
4
| 0.5 |
Find the positive integer solution of the equation \( x^{2y} + (x+1)^{2y} = (x+2)^{2y} \).
|
(3, 1)
| 0.75 |
There are some pieces in a box, less than 50 in total. Xiaoming and Xiaoliang take turns taking pieces from the box. If Xiaoming takes 2 pieces, Xiaoliang takes 2 pieces, Xiaoming takes 2 pieces, and Xiaoliang takes 2 pieces, in this manner, Xiaoming will have 2 more pieces than Xiaoliang in the end. If Xiaoming takes 3 pieces, Xiaoliang takes 3 pieces, Xiaoming takes 3 pieces, and Xiaoliang takes 3 pieces, in this manner, both will end up taking the same number of pieces. How many pieces are there at most in the box?
|
42
| 0.5 |
Given the complex number \( z \) satisfying
$$
\left|\frac{z^{2}+1}{z+\mathrm{i}}\right|+\left|\frac{z^{2}+4 \mathrm{i}-3}{z-\mathrm{i}+2}\right|=4,
$$
find the minimum value of \( |z - 1| \).
|
\sqrt{2}
| 0.375 |
Several consecutive natural numbers are written on the board. Exactly 52% of them are even. How many even numbers are written on the board?
|
13
| 0.5 |
In a tetrahedron \( ABCD \), \( AB = AC = AD = 5 \), \( BC = 3 \), \( CD = 4 \), \( DB = 5 \). Find the volume of this tetrahedron.
|
5\sqrt{3}
| 0.75 |
Given that \( b\left[\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{1999 \times 2001}\right]=2 \times\left[\frac{1^{2}}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{1000^{2}}{1999 \times 2001}\right] \), find the value of \( b \).
|
1001
| 0.5 |
Find the smallest value \( x \) such that, given any point inside an equilateral triangle of side 1, we can always choose two points on the sides of the triangle, collinear with the given point and a distance \( x \) apart.
|
\frac{2}{3}
| 0.125 |
What is the maximum number of sides of a convex polygon that can be divided into right triangles with acute angles measuring 30 and 60 degrees?
|
12
| 0.5 |
Consider the collection of all 5-digit numbers whose sum of the digits is 43. One of these numbers is chosen at random. What is the probability that it is a multiple of 11?
|
\frac{1}{5}
| 0.5 |
Seven distinct balls are to be placed into four labeled boxes. It is required that boxes 1 and 2 must contain an even number of balls, and box 3 must contain an odd number of balls. Find the number of ways to do this.
|
2080
| 0.25 |
A quadrilateral has three sides with lengths \(a=4 \sqrt{3}\), \(b=9\), and \(c=\sqrt{3}\). The angle between sides \(a\) and \(b\) is \(30^{\circ}\), and the angle between sides \(b\) and \(c\) is \(90^{\circ}\). What is the angle between the diagonals of the quadrilateral?
|
60^\circ
| 0.875 |
Given that the modulus of the complex number \( z \) is 1, find the minimum value of \( |z-4|^{2} + |z+3i|^{2} \).
|
17
| 0.375 |
A rectangular painting is placed in a rectangular frame made of strips of the same width. The area of the painting is equal to the area of the frame. What is the ratio of the length to the width of the painting in the frame if, without the frame, the ratio of the length of the painting to its width is $2:3$?
|
\frac{3}{4}
| 0.875 |
Find all polynomials \( P \in \mathbb{R}[X] \) such that \( P(X^2 + 1) = P(X)^2 + 1 \) with \( P(0) = 0 \).
Bonus (difficult): Remove the condition \( P(0) = 0 \).
|
P(X) = X
| 0.375 |
There are 11 children sitting in a circle playing a game. They are numbered clockwise from 1 to 11. The game starts with child number 1, and each child has to say a two-digit number. The number they say cannot have a digit sum of 6 or 9, and no child can repeat a number that has already been said. The game continues until someone cannot say a new number, and the person who cannot say a new number loses the game. Who will be the last person in the game?
|
10
| 0.25 |
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\).
|
\frac{1}{4}
| 0.375 |
If both \( 7n+1 \) and \( 8n+1 \) can be represented as the sum of three distinct positive integers in a geometric progression, what is the smallest value of the positive integer \( n \)?
|
6
| 0.625 |
We colored every side of unit squares using one of the colors red, yellow, blue, and green in such a way that each square has all four sides in different colors. The squares colored in this way can be assembled along their sides with the same color.
Can a $99 \times 99$ square and a $100 \times 100$ square be assembled from such colored squares such that the outer sides of the assembled square are single-colored and each side has a different color?
|
\text{Yes}
| 0.125 |
Determine the order of operations in the expression
$$
1891-(1600: a+8040: a) \times c
$$
and calculate its value when \( a = 40 \) and \( c = 4 \). Show how the expression can be modified without changing its numerical value.
|
927
| 0.75 |
In how many ways can the numbers \(1, 2, 3, 4, 5, 6\) be arranged in a row so that for any three consecutive numbers \(a, b, c\), the expression \(ac - b^2\) is divisible by 7? Answer: 12.
|
12
| 0.875 |
Find the distance from the point \( M_{0} \) to the plane passing through the three points \( M_{1}, M_{2}, M_{3} \).
\( M_{1}(1, 3, 0) \)
\( M_{2}(4, -1, 2) \)
\( M_{3}(3, 0, 1) \)
\( M_{0}(4, 3, 0) \)
|
\sqrt{6}
| 0.75 |
Edge \( AB \) of tetrahedron \( ABCD \) is the diagonal of the base of a quadrilateral pyramid. Edge \( CD \) is parallel to the other diagonal of this base, and its ends lie on the lateral edges of the pyramid. Find the smallest possible volume of the pyramid if the volume of the tetrahedron is \( V \).
|
4V
| 0.25 |
a) Find all the divisors of the number 572 based on its prime factorization.
b) How many divisors does the number \(572 a^{3} b c\) have if:
I. \(a, b\), and \(c\) are prime numbers greater than 20 and different from each other?
II. \(a = 31\), \(b = 32\), and \(c = 33\)?
|
384
| 0.5 |
On the circle, points $A, B, C, D, E, F, G$ are arranged clockwise as shown in the diagram. It is known that $AE$ is the diameter of the circle. It is also known that $\angle ABF = 81^{\circ}$ and $\angle EDG = 76^{\circ}$. How many degrees is the angle $FCG$?
|
67^\circ
| 0.375 |
Given the sequence $\left\{a_{n}\right\}$ which satisfies $a_{1}=0$ and $a_{n+1}=a_{n}+4 \sqrt{a_{n}+1}+4$ for $n \geq 1$, find $a_{n}$.
|
a_n = 4n^2 - 4n
| 0.375 |
Given \(\sin x \sin 2x \sin 3x + \cos x \cos 2x \cos 3x = 1\), find the value of \(x\).
|
x = k\pi
| 0.125 |
There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what ratio should these alloys be combined to obtain a new alloy in which zinc is twice as much as copper?
|
1 : 2
| 0.875 |
From the vertex of the obtuse angle \( A \) of triangle \( ABC \), the height \( AD \) is dropped. A circle with center \( D \) and radius \( DA \) is drawn, which intersects the sides \( AB \) and \( AC \) again at points \( M \) and \( N \) respectively. Find \( AC \), given that \( AB = c \), \( AM = m \), and \( AN = n \).
|
\frac{mc}{n}
| 0.5 |
Calculate the area of the figures bounded by the curves given in polar coordinates.
$$
r=2 \sin 4 \phi
$$
|
2\pi
| 0.5 |
Money placed in the 1st month will remain on deposit for 6 months and yield a nominal income of $8700\left((1+0.06 / 12)^{\wedge}-1\right)=264.28$ rubles.
Funds placed in the 2nd month will be on deposit for 5 months and yield a nominal income of $8700\left((1+0.06 / 12)^{\wedge} 5-1\right)=219.69$ rubles.
Similarly, for the following months:
$8700\left((1+0.06 / 12)^{\wedge 4-1}\right)=175.31$ rubles.
$8700\left((1+0.06 / 12)^{\wedge} 3-1\right)=131.15$ rubles.
$8700\left((1+0.06 / 12)^{\wedge} 2-1\right)=87.22$ rubles.
$8700((1+0.06 / 12)-1)=43.5$ rubles.
Therefore, the nominal income for 6 months will be:
$264.28+219.69+175.31+131.15+87.22+43.5=921.15$ rubles.
|
921.15
| 0.75 |
The sequence of consecutive positive integers starting from 1 was written on the board up to a certain number. Then, one number was erased. The arithmetic mean of the remaining numbers is $\frac{602}{17}$. What was the erased number?
|
7
| 0.875 |
Karel and Vojta discovered that the kitchen clock at the cottage runs 1.5 minutes fast every hour, and the bedroom clock runs 0.5 minutes slow every hour. At exactly noon, they synchronized both clocks to the correct time. Both the kitchen and bedroom clocks have a standard twelve-hour dial. Determine when the clocks will next show the correct time without further adjustments:
1. The kitchen clock will again show the correct time.
2. The bedroom clock will again show the correct time.
3. Both clocks will again show the same (possibly incorrect) time.
(M. Volfová)
|
15 \text{ days}
| 0.5 |
a) Determine the sum of the multiplicative contents of all subsets of \(\{1,2,3,4\}\).
b) Determine the sum of the multiplicative contents of all subsets of
\[
\left\{1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{2016}\right\}
\]
|
2017
| 0.75 |
Two cars, A and B, start from points A and B respectively and travel towards each other at the same time. They meet at point C after 6 hours. If car A maintains its speed and car B increases its speed by 5 km/h, they will meet 12 km away from point C. If car B maintains its speed and car A increases its speed by 5 km/h, they will meet 16 km away from point C. What was the original speed of car A?
|
30
| 0.875 |
\((\cos 8 \alpha \cdot \tan 4 \alpha - \sin 8 \alpha)(\cos 8 \alpha \cdot \cot 4 \alpha + \sin 8 \alpha)\).
|
-1
| 0.875 |
The largest angle of a right trapezoid is $120^{\circ}$, and the longer leg is 12. Find the difference between the bases of the trapezoid.
|
6
| 0.375 |
A novice economist-cryptographer received a cryptogram from the ruler, which contained the next secret decree about introducing a per unit tax in a certain market. The cryptogram specified the amount of tax revenue that needed to be collected and emphasized that it would be impossible to collect a higher amount of tax revenue in this market. Unfortunately, the economist-cryptographer deciphered the cryptogram incorrectly — the digits in the tax revenue amount were determined in the wrong order. Based on these erroneous data, a decision was made to introduce a per unit tax on the consumer at a rate of 30 monetary units per unit of goods. The market supply is given by $Qs=6P-312$, and the market demand is linear. Additionally, it is known that when the price changes by one unit, the change in demand is 1.5 times less than the change in supply. After the tax was introduced, the consumer price rose to 118 monetary units.
1) Restore the market demand function.
2) Determine the amount of tax revenue collected at the chosen rate.
3) Determine the rate of the per unit tax that would achieve the ruler's decree.
4) What is the amount of tax revenue the ruler indicated to collect?
|
T_{\max} = 8640
| 0.75 |
The Mad Hatter's clock gains 15 minutes per hour, while the March Hare's clock loses 10 minutes per hour. One day, they set their clocks according to the Dormouse's clock, which is stopped and always shows 12:00, and agreed to meet at 5 o'clock in the evening for their traditional five-o'clock tea. How long will the Mad Hatter wait for the March Hare if both arrive exactly at 17:00 according to their own clocks?
|
2 \text{ hours}
| 0.125 |
A chessboard has 13 rows and 17 columns. Each small square contains a number, starting from the top-left corner. The first row is filled sequentially with $1, 2, \cdots, 17$; the second row with $18, 19, \cdots, 34$, and so on, until the last row. Now, the numbers are rewritten starting from the top-left corner. The first column is filled sequentially from top to bottom with $1, 2, \cdots, 13$; the second column with $14, 15, \cdots, 26$, and so on, until the last column.
Some of the small squares will have the same number in both the original and the rewritten grid. Find the sum of all such numbers.
|
555
| 0.375 |
On the board is written the number 98. Every minute the number is erased and replaced with the product of its digits increased by 15. What number will be on the board in an hour?
|
23
| 0.5 |
Given 100 numbers. Each number is increased by 2. The sum of the squares of the numbers remains unchanged. Each resulting number is then increased by 2 again. How has the sum of the squares changed now?
|
800
| 0.75 |
Find the value of the expression $\frac{p}{q}+\frac{q}{p}$, where $p$ and $q$ are the largest and smallest roots of the equation $x^{3}+6x^{2}+6x=-1$ respectively.
|
23
| 0.75 |
There are $n$ people, and it is known that any two of them can make at most one phone call to each other. For any $n-2$ people among them, the total number of phone calls is equal and is $3^k$ (where $k$ is a positive integer). Find all possible values of $n$.
|
5
| 0.875 |
Given the vectors \(\boldsymbol{a} = (x, 1)\), \(\boldsymbol{b} = (2, y)\), and \(\boldsymbol{c} = (1,1)\), and knowing that \(\boldsymbol{a} - \boldsymbol{b}\) is collinear with \(\boldsymbol{c}\). Find the minimum value of \( |\boldsymbol{a}| + 2|\boldsymbol{b}| \).
|
3\sqrt{5}
| 0.875 |
\(\frac{\sqrt{2}(x-a)}{2 x-a} - \left(\left(\frac{\sqrt{x}}{\sqrt{2 x}+\sqrt{a}}\right)^2+\left(\frac{\sqrt{2 x}+\sqrt{a}}{2 \sqrt{a}}\right)^{-1}\right)^{1/2}\)
Given: \(a = 0.32, x = 0.08\)
|
1
| 0.75 |
In the complex plane, the complex number \( z_{1} \) moves along the line segment connecting \( 1+2i \) and \( 1-2i \), and the complex number \( z_{2} \) moves along the circumference of a circle centered at the origin with radius 1. What is the area of the shape formed by the trajectory of \( z_{1} + z_{2} \)?
|
8+\pi
| 0.625 |
A square is divided into 2016 triangles, with no vertex of any triangle lying on the sides or inside any other triangle. The sides of the square are sides of some of the triangles in the division. How many total points, which are the vertices of the triangles, are located inside the square?
|
1007
| 0.5 |
Find the ratio \(\frac{b^{2}}{a c}\) given that one of the roots of the equation \(a x^{2} + b x + c = 0\) is 4 times the other root \((a \neq 0, c \neq 0)\).
|
\frac{25}{4}
| 0.875 |
Suppose I have a closed cardboard box in the shape of a cube. By cutting 7 out of its 12 edges with a razor blade (it must be exactly 7 edges), I can unfold the box onto a plane, and the unfolding can take various forms. For instance, if I cut along the edges shown by the bold lines in the diagram and along the invisible edge indicated by a dashed line, I will get unfolding $A$. By cutting the box differently, one can obtain unfolding $B$ or $C$. It is easy to see that unfolding $D$ is simply a flipped version of unfolding $C$, so we consider these two unfoldings to be identical.
How many distinct unfoldings can be obtained in this way?
|
11
| 0.375 |
Primes like $2, 3, 5, 7$ are natural numbers greater than 1 that can only be divided by 1 and themselves. We split 2015 into the sum of 100 prime numbers, requiring that the largest of these prime numbers be as small as possible. What is this largest prime number?
|
23
| 0.125 |
Divide each three-digit (natural) number by the sum of its digits in your mind. What is the smallest resulting quotient?
|
\frac{199}{19}
| 0.625 |
a) In the cube \( ABCD A_{1} B_{1} C_{1} D_{1} \), a common perpendicular \( MN \) is drawn to the lines \( A_{1} B \) and \( B_{1} C \) (point \( M \) lies on the line \( A_{1} B \) ). Find the ratio \( A_{1} M : M B \).
b) Given a cube \( ABCD A_{1} B_{1} C_{1} D_{1} \). Points \( M \) and \( N \) are taken on the segments \( A A_{1} \) and \( B C_{1} \) respectively, such that the lines \( MN \) and \( B_{1} D \) intersect. Find the difference between the ratios \( B C_{1} : B N \) and \( A M : A A_{1} \).
|
1
| 0.375 |
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