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|---|---|---|
In the arithmetic sequence \(\{a_{n}\}\), \(a_{20}=\frac{1}{a}, a_{201}=\frac{1}{b}, a_{2012}=\frac{1}{c}\). Find the value of \(1992 a c - 1811 b c - 181 a b\).
( Note: Given constants \(a\) and \(b\) satisfy \(a, b > 0, a \neq 1\), and points \(P(a, b)\) and \(Q(b, a)\) are on the curve \(y=\cos(x+c)\), where \(c\) is a constant. Find the value of \(\log _{a} b\).
|
0
| 0.5 |
What is the sum of all integers \( x \) such that \( |x+2| \leq 10 \)?
|
-42
| 0.75 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 1} \frac{\cos \left(\frac{\pi x}{2}\right)}{1-\sqrt{x}}$$
|
\pi
| 0.5 |
Find the number of digits in the decimal representation of the number \(2^{120}\), given that the decimal representation of the number \(2^{200}\) contains 61 digits.
|
37
| 0.75 |
Find the integer part of the expression
\[a=\sqrt{1981+\sqrt{1981+\sqrt{1981+\ldots+\sqrt{1981+\sqrt{1981}}}}},\]
if the number 1981 appears $n$ times ($n \geq 2$).
|
45
| 0.75 |
Consider the equation
$$
\sqrt{3 x^{2}-8 x+1}+\sqrt{9 x^{2}-24 x-8}=3.
$$
It is known that the largest root of the equation is $-k$ times the smallest root. Find $k$.
|
9
| 0.875 |
The Group of Twenty (G20) is an international economic cooperation forum with 20 member countries. These members come from Asia, Europe, Africa, Oceania, and America. The number of members from Asia is the highest, and the numbers from Africa and Oceania are equal and the least. The number of members from America, Europe, and Asia are consecutive natural numbers. How many members of the G20 are from Asia?
|
7
| 0.875 |
Into which curve is the unit circle $|z|=1$ mapped by the function $w=z^{2}$?
|
|w| = 1
| 0.75 |
There are 456 natives on an island, each of whom is either a knight who always tells the truth or a liar who always lies. All residents have different heights. Once, each native said, "All other residents are shorter than me!" What is the maximum number of natives who could have then said one minute later, "All other residents are taller than me?"
|
454
| 0.125 |
Calculate the area of the figure bounded by the graphs of the functions:
\[ x = \arccos y, \quad x = 0, \quad y = 0 \]
|
1
| 0.375 |
In a store, there are 9 headphones, 13 computer mice, and 5 keyboards for sale. Besides these, there are also 4 sets of "keyboard and mouse" and 5 sets of "headphones and mouse." How many ways can you buy three items: headphones, a keyboard, and a mouse? Answer: 646.
|
646
| 0.75 |
From the 16 integers ranging from 1 to 16, what is the minimum number of integers that must be chosen to ensure that among the chosen numbers, one of them is twice another?
|
12
| 0.125 |
ABCD is a quadrilateral with AB = CD and angle ABC > angle BCD. Show that AC > BD.
|
AC > BD
| 0.875 |
At a school, there are three clubs: mathematics, physics, and computer science. The principal observed that among the participants of the mathematics club, exactly $1 / 6$ also attend the physics club, and $1 / 8$ also attend the computer science club. Among the participants of the physics club, exactly $1 / 3$ also attend the mathematics club, and exactly $1 / 5$ attend the computer science club. Finally, among the participants of the computer science club, exactly $1 / 7$ attend the mathematics club. What fraction of the participants of the computer science club attend the physics club?
|
\frac{4}{35}
| 0.875 |
In a deck of 52 cards, each player makes one cut. A cut consists of taking the top $N$ cards and placing them at the bottom of the deck without changing their order.
- First, Andrey cut 28 cards,
- then Boris cut 31 cards,
- then Vanya cut 2 cards,
- then Gena cut an unknown number of cards,
- then Dima cut 21 cards.
The last cut restored the original order. How many cards did Gena cut?
|
22
| 0.5 |
Two natural numbers $x$ and $y$ are written on the board in increasing order ($x \leq y$). Petya writes $x^2$ (the square of the first number) on a piece of paper, then replaces the numbers on the board with $x$ and $y-x$, arranging them in ascending order. He repeats this operation with the new numbers on the board, and so on, until one of the numbers on the board becomes zero. What will be the sum of the numbers on Petya's paper at that moment?
|
xy
| 0.375 |
Inside a tetrahedron \( A B C D \) there is a point \( O \) such that the lines \( A O, B O, C O, D O \) intersect the faces \( B C D, A C D, A B D, A B C \) of the tetrahedron at points \( A_{1}, B_{1}, C_{1}, D_{1} \) respectively, and the ratios
\[
\frac{A O}{A_{1} O}, \frac{B O}{B_{1} O}, \frac{C O}{C_{1} O}, \frac{D O}{D_{1} O}
\]
are all equal to the same number. Find all the possible values that this number can take.
|
3
| 0.5 |
Find all the extrema of the function \( y = \sin^2(3x) \) on the interval \( (0, 0.6) \).
|
x = \frac{\pi}{6}
| 0.375 |
Carlson and Baby have several jars of jam, each weighing an integer number of pounds.
The total weight of all Carlson's jars of jam is 13 times the total weight of all Baby's jars. Carlson gave Baby the jar with the smallest weight (among those he had), after which the total weight of his jars turned out to be 8 times the total weight of Baby's jars.
What is the maximum possible number of jars Carlson could have initially had?
|
23
| 0.75 |
Given that \( a \) and \( b \) are real numbers and the sets \( A = \{a, a^{2}, ab\} \) and \( B = \{1, a, b\} \), if \( A = B \), find the value of \( a^{2004} + b^{2004} \).
|
1
| 0.5 |
This spring, three Hungarian women's handball teams reached the top eight in the EHF Cup. The teams were paired by drawing lots. All three Hungarian teams were paired with foreign opponents. What was the probability of this happening?
|
\frac{4}{7}
| 0.75 |
Let \( f(x) = \frac{a^x}{1 + a^x} \) where \( a > 0 \) and \( a \neq 1 \). Let \([m]\) denote the greatest integer less than or equal to the real number \( m \). Find the range of the function \( \left\lfloor f(x) - \frac{1}{2} \right\rfloor + \left\lfloor f(-x) - \frac{1}{2} \right\rfloor \).
|
\{-1, 0\}
| 0.75 |
There are three locations $A$, $O$, and $B$ on a road, with $O$ located between $A$ and $B$. The distance between $A$ and $O$ is 1620 meters. Two people, X and Y, respectively start at $A$ and $O$ towards $B$. After 12 minutes from the start, X and Y are equidistant from point $O$. They meet at point $B$ after 36 minutes from the start. What is the distance between points $O$ and $B$ in meters?
|
1620
| 0.875 |
The number of common terms (terms with the same value) in the arithmetic sequences $2, 5, 8, \cdots, 2015$ and $4, 9, 14, \cdots, 2014$ is $\qquad$ .
|
134
| 0.75 |
In 2016, the world record for completing a 5000m three-legged race was 19 minutes and 6 seconds. What was their approximate average speed in $\mathrm{km} / \mathrm{h}$?
A 10
B 12
C 15
D 18
E 25
|
C \text{ 15}
| 0.25 |
Find the smallest natural number \( N \) such that the number \( 99N \) consists only of threes.
|
3367
| 0.875 |
Find the minimum value of \(x^2 + 4y^2 - 2x\), where \(x\) and \(y\) are real numbers that satisfy \(2x + 8y = 3\).
|
-\frac{19}{20}
| 0.625 |
The price of a stadium entrance ticket is 400 rubles. After increasing the entrance fee, the number of spectators decreased by 20%, and the revenue increased by 5%. What is the price of the entrance ticket after the price increase?
|
525
| 0.875 |
In triangle \( ABC \), \( AB = 2 \), \( AC = 1 + \sqrt{5} \), and \( \angle CAB = 54^\circ \). Suppose \( D \) lies on the extension of \( AC \) through \( C \) such that \( CD = \sqrt{5} - 1 \). If \( M \) is the midpoint of \( BD \), determine the measure of \( \angle ACM \), in degrees.
|
63^\circ
| 0.375 |
In the following figure, the fractal geometry grows into a tree diagram according to a fractal pattern. Find the total number of solid circles in the first 20 rows.
As shown in the figure:
- The 1st row has 1 hollow circle and 0 solid circles.
- The 2nd row has 0 hollow circles and 1 solid circle.
- The 3rd row has 1 hollow circle and 1 solid circle.
Represent the number of hollow and solid circles in each row using “coordinates”. For example, the 1st row is represented as (1, 0), the 2nd row is represented as (0, 1), and so forth.
Determine the number of solid circles in the first 20 rows.
|
10945
| 0.5 |
A triangle \(ABC\) has an area of 944. Let \(D\) be the midpoint of \([AB]\), \(E\) the midpoint of \([BC]\), and \(F\) the midpoint of \([AE]\). What is the area of triangle \(DEF\)?
|
118
| 0.875 |
A square is inscribed in a circle of unit radius. A circle is then inscribed in this square, and an octagon is inscribed in this circle, followed by another circle inscribed in this octagon, and so on. Generally, in the $n$-th circle, a regular $2^{n+1}$-gon is inscribed, and within this, the $(n+1)$-th circle is inscribed. Let $R_{n}$ be the radius of the $n$-th circle. What is $\lim _{n \rightarrow \infty} R_{n}$ equal to?
|
\frac{2}{\pi}
| 0.875 |
In the figure, $\angle BAC = 70^\circ$ and $\angle FDE = x^\circ$. Find $x$.
A cuboid is $y$ cm wide, $6$ cm long, and $5$ cm high. Its surface area is $126$ cm². Find $y$.
If $\log_9(\log_2 k) = \frac{1}{2}$, find $k$.
If $a : b = 3 : 8$, $b : c = 5 : 6$ and $a : c = r : 16$, find $r$.
|
r = 5
| 0.875 |
There are 35 egg yolk mooncakes to be packed. There are two types of packaging: a large bag containing 9 mooncakes per bag and a small package containing 4 mooncakes per bag. If no mooncakes are to be left unpacked, how many packages were used in total?
|
5
| 0.875 |
Calculate the limit
$$
\lim _{x \rightarrow \pi} \frac{\cos 3 x-\cos x}{\operatorname{tg}^{2} 2 x}
$$
|
1
| 0.5 |
Given the quadratic polynomials \( f_{1}(x) = x^{2} - x + 2a \), \( f_{2}(x) = x^{2} + 2bx + 3 \), \( f_{3}(x) = 4x^{2} + (2b-3)x + 6a + 3 \), and \( f_{4}(x) = 4x^{2} + (6b-1)x + 9 + 2a \), let the differences of their roots be respectively \( A \), \( B \), \( C \), and \( D \). It is known that \( |A| \neq |B| \). Find the ratio \( \frac{C^{2} - D^{2}}{A^{2} - B^{2}} \). The values of \( A, B, C, D, a, \) and \( b \) are not given.
|
\frac{1}{2}
| 0.875 |
Calculate the arc length of the curve given by the equation
$$
y = 2 + \arcsin(\sqrt{x}) + \sqrt{x - x^2}, \quad \frac{1}{4} \leq x \leq 1
$$
|
1
| 0.75 |
On the side $AD$ of rectangle $ABCD$, a point $E$ is marked. On the segment $EC$, a point $M$ is found such that $AB = BM$ and $AE = EM$. Find the length of the side $BC$ if it is known that $ED = 16$ and $CD = 12$.
|
20
| 0.875 |
Given \( f(x) = 2^x m + x^2 + n x \), if
$$
\{x \mid f(x) = 0\} = \{x \mid f(f(x)) = 0\} \neq \varnothing,
$$
then the range of values for \( m + n \) is ______ .
|
[0,4)
| 0.625 |
A firecracker was thrown vertically upward with a speed of \(20 \text{ m/s}\). One second after the flight began, it exploded into two fragments of equal mass. The first fragment flew horizontally immediately after the explosion with a speed of 48 m/s. Find the magnitude of the speed of the second fragment (in m/s) immediately after the explosion. The acceleration due to gravity is \(10 \text{ m/s}^2\).
|
52
| 0.875 |
The scent of blooming lily of the valley bushes spreads within a radius of 20 meters around them. How many blooming lily of the valley bushes need to be planted along a straight 400-meter-long alley so that every point along the alley can smell the lily of the valley?
|
10
| 0.375 |
There were 28 cars in each of the two automobile columns. There were 11 "Zhiguli" cars in total in both columns, and the rest were "Moskvich" cars. How many "Moskvich" cars were in each automobile column if it is known that in the first column, for every "Zhiguli" car, there were half as many "Moskvich" cars as in the second column?
|
21 \text{, } 24
| 0.875 |
Given that \( x \) and \( y \) are natural numbers greater than 0 and \( x + y = 150 \). If \( x \) is a multiple of 3 and \( y \) is a multiple of 5, how many different pairs \( (x, y) \) are there?
|
9
| 0.5 |
Calculate:
1) \(x^{4} - 2x^{3} + 3x^{2} - 2x + 2\) given that \(x^{2} - x = 3\)
2) \(2x^{4} + 3x^{2}y^{2} + y^{4} + y^{2}\) given that \(x^{2} + y^{2} = 1\)
|
2
| 0.625 |
The minimum positive period of the function \( y = \sin^{2n} x - \cos^{2m-1} x \) where \( n, m \in \mathbf{N}^{-} \) is ____.
|
2\pi
| 0.875 |
Calculate
$$
\sqrt{1+2 \sqrt{1+3 \sqrt{1+\ldots+2017 \sqrt{1+2018 \cdot 2020}}}}
$$
|
3
| 0.125 |
Calculate: $99 \times \frac{5}{8} - 0.625 \times 68 + 6.25 \times 0.1 = \qquad$
|
20
| 0.75 |
In a right-angled triangle, the hypotenuse is \( c \). The centers of three circles with radius \( \frac{c}{5} \) are located at the vertices of the triangle. Find the radius of the fourth circle, which touches the three given circles and does not contain them within itself.
|
\frac{3c}{10}
| 0.875 |
Stephanie enjoys swimming. She goes for a swim on a particular date if, and only if, the day, month (where January is replaced by '01' through to December by '12') and year are all of the same parity (that is they are all odd, or all are even). On how many days will she go for a swim in the two-year period between January 1st of one year and December 31st of the following year inclusive?
|
183
| 0.25 |
How many multiples of 3 are there between 1 and 2015 whose units digit in the decimal representation is 2?
|
67
| 0.75 |
A number is the product of five 2's, three 3's, two 5's, and one 7. This number has many divisors, some of which are two-digit numbers. What is the largest two-digit divisor?
|
96
| 0.375 |
Given a right triangle \(ABC\) with a right angle at \(C\), a circle is drawn with diameter \(BC\) of length 26. A tangent \(AP\) from point \(A\) to this circle (distinct from \(AC\)) is drawn. The perpendicular \(PH\) dropped onto segment \(BC\) intersects segment \(AB\) at point \(Q\). Find the area of triangle \(BPQ\) given \(BH:CH = 4:9\).
|
24
| 0.75 |
Let point \( A(-2,0) \) and point \( B(2,0) \) be given, and let point \( P \) be on the unit circle. What is the maximum value of \( |PA| \cdot |PB| \)?
|
5
| 0.625 |
The rapid bus route has four stations. Arrange the distances between each pair of stations from smallest to largest as follows: \(2 \mathrm{~km}, 5 \mathrm{~km}, 7 \mathrm{~km}, ? \mathrm{~km}, 22 \mathrm{~km}, 24 \mathrm{~km}\). Determine the value of "?".
|
17
| 0.875 |
Find two natural numbers \( m \) and \( n \), given that \( m < n < 2m \) and \( mn = 2013 \).
|
m = 33, n = 61
| 0.5 |
Let \( S = \{1, 2, 3, 4, \ldots, 50\} \). A 3-element subset \(\{a, b, c\}\) of \(S\) is said to be good if \(a + b + c\) is divisible by 3. Determine the number of 3-element subsets of \(S\) which are good.
|
6544
| 0.625 |
Given the numerical sequence:
\[x_{0}=\frac{1}{n}, \quad x_{k}=\frac{1}{n-k}\left(x_{0}+x_{1}+\ldots+x_{k-1}\right), \quad k=1,2, \ldots, n-1.\]
Find \(S_{n}=x_{0}+x_{1}+\ldots+x_{n-1}\) for \(n=2021\).
|
1
| 0.625 |
Given two lines \( A_{1} x + B_{1} y = 1 \) and \( A_{2} x + B_{2} y = 1 \) intersect at the point \( P(-7,9) \), find the equation of the line passing through the two points \( P_{1}(A_{1}, B_{1}) \) and \( P_{2}(A_{2}, B_{2}) \).
|
-7x + 9y = 1
| 0.125 |
There are four cups with their mouth facing up. Each time, you may flip three cups. After \( n \) flips, all cups should have their mouths facing down. What is the minimum positive integer \( n \)?
Note: Flipping a cup is defined as changing it from mouth up (down) to mouth down (up).
|
4
| 0.875 |
Given that $A$ and $B$ are the left and right vertices of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, $F_{1}$ and $F_{2}$ are the left and right foci, and $C$ is a point on the hyperbola different from $A$ and $B$. If the lines $AC$ and $BC$ intersect the right directrix of the hyperbola at points $M$ and $N$, respectively, find $\overrightarrow{F_{1} M} \cdot \overrightarrow{F_{2} N}$.
|
-2 b^2
| 0.875 |
Let \( n \) be a positive integer greater than 3, such that \((n, 3) = 1\). Find the value of \(\prod_{k=1}^{m}\left(1+2 \cos \frac{2 a_{k} \pi}{n}\right)\), where \(a_{1}, a_{2}, \cdots, a_{m}\) are all positive integers less than or equal to \( n \) that are relatively prime to \( n \).
|
1
| 0.625 |
Smeshariki Kros, Yozhik, Nyusha, and Barash together ate 86 candies, with each of them eating at least 5 candies. It is known that:
- Nyusha ate more candies than any of the others;
- Kros and Yozhik together ate 53 candies.
How many candies did Nyusha eat?
|
28
| 0.75 |
For integers \(a, b, c, d\), let \(f(a, b, c, d)\) denote the number of ordered pairs of integers \((x, y) \in \{1,2,3,4,5\}^{2}\) such that \(ax + by\) and \(cx + dy\) are both divisible by 5. Find the sum of all possible values of \(f(a, b, c, d)\).
|
31
| 0.5 |
A triangle's two vertices, the center of its incircle, and its orthocenter lie on a circle. Calculate the angle at the triangle's third vertex!
|
60^\circ
| 0.5 |
Every city in a certain state is directly connected by air with at most three other cities in the state, but one can get from any city to any other city with at most one change of plane. What is the maximum possible number of cities?
|
10
| 0.5 |
Given two groups of numerical sequences, each containing 15 arithmetic progressions with 10 terms each. The first terms of the progressions in the first group are $1, 2, 3, \ldots, 15$, and their differences are respectively $2, 4, 6, \ldots, 30$. The second group of progressions has the same first terms $1, 2, 3, \ldots, 15$, but the differences are respectively $1, 3, 5, \ldots, 29$. Find the ratio of the sum of all elements of the first group to the sum of all elements of the second group.
|
\frac{160}{151}
| 0.875 |
Determine the functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that:
\[ f(f(x+1) + y - 1) = f(x) + y \]
|
f(x) = x
| 0.875 |
In the regular hexagon \( ABCDEF \) shown in the figure, point \( P \) is a point on \( AB \). It is known that the area of \( \triangle AFP = 8 \) and the area of \( \triangle CDP = 42 \). What is the area of \( \triangle EFP \)?
|
33
| 0.25 |
Let \(\lfloor x\rfloor\) denote the greatest integer which is less than or equal to \(x\). For example, \(\lfloor\pi\rfloor = 3\). \(S\) is the integer equal to the sum of the 100 terms shown:
$$
S = \lfloor\pi\rfloor + \left\lfloor\pi + \frac{1}{100}\right\rfloor + \left\lfloor\pi + \frac{2}{100}\right\rfloor + \left\lfloor\pi + \frac{3}{100}\right\rfloor + \cdots + \left\lfloor\pi + \frac{99}{100}\right\rfloor
$$
What is the value of \(S\)?
|
314
| 0.75 |
The four hydrogen atoms in the methane molecule $\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then $\sum_{1 \leq i < j \leq 4} \overrightarrow{C_{0} \vec{H}_{i}} \cdot \overrightarrow{C_{0} \vec{H}_{j}} = \quad$ .
|
-\frac{3}{4}
| 0.875 |
During training target practice, each soldier fired 10 times. One of them was successful and scored 90 points in total. How many times did he score 7 points if he scored 10 points 4 times and the results of the remaining shots were 7, 8, and 9 points? Note that there were no misses.
|
1
| 0.25 |
Once upon a time, a team of Knights and a team of Liars met in the park and decided to ride a circular carousel that can hold 40 people (the "Chain" carousel, where everyone sits one behind the other). When they took their seats, each person saw two others: one in front and one behind. Each person then said, "At least one of the people sitting in front of me or behind me belongs to my team." One spot turned out to be free, and they called one more Liar. This Liar said, "With me, we can arrange ourselves on the carousel so that this rule is met again." How many people were on the team of Knights? (A Knight always tells the truth, a Liar always lies).
|
26
| 0.25 |
In the rectangular coordinate system \( xOy \), let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (\( a > 0, b > 0 \)), respectively. Point \( P \) is a point on the right branch of the hyperbola, \( M \) is the midpoint of \( PF_{2} \), and \( OM \perp PF_{2} \). Given that \( 3PF_{1} = 4PF_{2} \), find the eccentricity of the hyperbola.
|
5
| 0.875 |
Solve the equation \( x-7 = \frac{4 \cdot |x-3|}{x-3} \). If the equation has multiple solutions, write down their sum.
|
11
| 0.875 |
An integer division is performed. If the dividend is increased by 65 and the divisor is increased by 5, then both the quotient and the remainder do not change. What is this quotient?
|
13
| 0.875 |
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 |
Compute the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\sqrt{1+\operatorname{tg} x}-\sqrt{1+\sin x}}{x^{3}}
$$
|
\frac{1}{4}
| 0.5 |
Solve the Cauchy problem for the heat conduction equation
$$
\begin{gathered}
u_{t}=u_{x x}, \quad x \in(-\infty,+\infty), \quad t \in(0,+\infty) \\
u(x, 0)=e^{-x} \quad x \in(-\infty,+\infty)
\end{gathered}
$$
|
u(x, t) = e^{t - x}
| 0.125 |
A circle touches the extensions of two sides $AB$ and $AD$ of a square $ABCD$ with a side length of $2-\sqrt{5-\sqrt{5}}$ cm. From the point $C$, two tangents are drawn to this circle. Find the radius of the circle if the angle between the tangents is $72^{\circ}$ and it is known that $\sin 36^{\circ}=\frac{\sqrt{5-\sqrt{5}}}{2 \sqrt{2}}$.
|
\sqrt{5 - \sqrt{5}}
| 0.375 |
There are 30 crickets and 30 grasshoppers in a cage. Each time the red-haired magician performs a trick, he transforms 4 grasshoppers into 1 cricket. Each time the green-haired magician performs a trick, he transforms 5 crickets into 2 grasshoppers. After the two magicians have performed a total of 18 tricks, there are only grasshoppers and no crickets left in the cage. How many grasshoppers are there at this point?
|
6
| 0.875 |
Twenty points are equally spaced around the circumference of a circle. Kevin draws all the possible chords that connect pairs of these points. How many of these chords are longer than the radius of the circle but shorter than its diameter?
|
120
| 0.625 |
Given \( x \in [0, \pi] \), find the range of values for the function
$$
f(x)=2 \sin 3x + 3 \sin x + 3 \sqrt{3} \cos x
$$
|
[-3\sqrt{3}, 8]
| 0.875 |
Let \( O \) be the vertex of a parabola, \( F \) be the focus, and \( PQ \) be a chord passing through \( F \). Given \( |OF| = a \) and \( |PQ| = b \), find the area of \(\triangle OPQ\).
|
a \sqrt{ab}
| 0.25 |
There are 7 boxes arranged in a row and numbered 1 through 7. You have a stack of 2015 cards, which you place one by one in the boxes. The first card is placed in box #1, the second in box #2, and so forth up to the seventh card which is placed in box #7. You then start working back in the other direction, placing the eighth card in box #6, the ninth in box #5, up to the thirteenth card being placed in box #1. The fourteenth card is then placed in box #2. This continues until every card is distributed. What box will the last card be placed in?
|
3
| 0.125 |
Knowing that the system
$$
\begin{aligned}
x+y+z & =3, \\
x^{3}+y^{3}+z^{3} & =15, \\
x^{4}+y^{4}+z^{4} & =35,
\end{aligned}
$$
has a real solution \( x, y, z \) for which \( x^{2}+y^{2}+z^{2}<10 \), find the value of \( x^{5}+y^{5}+z^{5} \) for that solution.
|
83
| 0.875 |
How many solutions does the equation
$$
\{x\}^{2}=\left\{x^{2}\right\}
$$
have in the interval $[1, 100]$? ($\{u\}$ denotes the fractional part of $u$, which is the difference between $u$ and the largest integer not greater than $u$.)
|
9901
| 0.625 |
Compute the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{1+2+\ldots+n}{n-n^{2}+3}
\]
|
-\frac{1}{2}
| 0.875 |
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 |
A rod is broken into three parts; two break points are chosen at random. What is the probability that a triangle can be formed from the three resulting parts?
|
\frac{1}{4}
| 0.625 |
Calculate the value of the expression
$$
\frac{\left(3^{4}+4\right) \cdot\left(7^{4}+4\right) \cdot\left(11^{4}+4\right) \cdot \ldots \cdot\left(2015^{4}+4\right) \cdot\left(2019^{4}+4\right)}{\left(1^{4}+4\right) \cdot\left(5^{4}+4\right) \cdot\left(9^{4}+4\right) \cdot \ldots \cdot\left(2013^{4}+4\right) \cdot\left(2017^{4}+4\right)}
$$
|
4080401
| 0.625 |
Calculate the arc length of the curve defined by the parametric equations:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x = 3.5(2 \cos t - \cos 2 t) \\
y = 3.5(2 \sin t - \sin 2 t)
\end{array}\right. \\
& 0 \leq t \leq \frac{\pi}{2}
\end{aligned}
$$
|
14(2 - \sqrt{2})
| 0.25 |
Find the largest natural number \( n \) that has the following property: for any odd prime number \( p \) less than \( n \), the difference \( n - p \) is also a prime number.
|
10
| 0.375 |
Find the angle between edge \(AB\) and face \(ACD\) in the trihedral angle \(ABCD\) with vertex \(A\) if the measures of the angles \(BAC\) are \(45^\circ\), \(CAD\) are \(90^\circ\), and \(BAD\) are \(60^\circ\).
|
30^\circ
| 0.75 |
Given a rectangle \(ABCD\). A circle intersects the side \(AB\) at the points \(K\) and \(L\) and the side \(CD\) at the points \(M\) and \(N\). Find the length of segment \(MN\) if \(AK=10\), \(KL=17\), and \(DN=7\).
|
23
| 0.125 |
The re-evaluation of the Council of Wise Men occurs as follows: the king lines them up in a single file and places a hat of either white, blue, or red color on each of their heads. All wise men can see the colors of the hats of everyone in front of them, but they cannot see their own hat or the hats of those behind them. Once per minute, each wise man must shout out one of the three colors (each wise man shouts out a color only once).
After this process is completed, the king will execute any wise man who shouts a color different from the color of his own hat.
The night before the re-evaluation, all one hundred members of the Council of Wise Men agreed and came up with a strategy to minimize the number of those executed. How many of them are guaranteed to avoid execution?
|
99
| 0.875 |
Let the natural number \( n \) be a three-digit number. The sum of all three-digit numbers formed by permuting its three non-zero digits minus \( n \) equals 1990. Find \( n \).
|
452
| 0.75 |
Find all three-digit numbers $\overline{\Pi В \Gamma}$, consisting of the distinct digits $П$, $В$, and $\Gamma$, for which the equality $\overline{\Pi В \Gamma}=(П+В+\Gamma) \times (П+В+\Gamma+1)$ holds.
|
156
| 0.375 |
If \( x, y, z \) are real numbers satisfying
\[
x + \frac{1}{y} = 2y + \frac{2}{z} = 3z + \frac{3}{x} = k \quad \text{and} \quad xyz = 3,
\]
then \( k = \) .
|
4
| 0.625 |
Dmitry is three times as old as Gregory was when Dmitry was as old as Gregory is now. When Gregory reaches the age that Dmitry is now, the sum of their ages will be 49 years. How old is Gregory?
|
14
| 0.875 |
Given \( a = \log_{4} e \), \( b = \log_{3} 4 \), and \( c = \log_{4} 5 \), determine the order of \( a \), \( b \), and \( c \).
|
a < c < b
| 0.875 |
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