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|---|---|---|
The function \( f \) satisfies
\[
f(x) + f(2x + y) + 5xy = f(3x - y) + 2x^2 + 1
\]
for all real numbers \( x \) and \( y \). Determine the value of \( f(10) \).
|
-49
| 0.75 |
Let the parabola $y = ax^2 + bx + c$ pass through the points $A(-1, -3)$, $B(4, 2)$, and $C(0, 2)$. Let $P$ be a moving point on the axis of symmetry of the parabola. If $P A + P C$ reaches its minimum value at the point $P$ with coordinates $(m, n)$, find $n$.
|
0
| 0.75 |
In how many ways can the number \( n \) be represented as the sum of several terms, each equal to 1 or 2? (Representations that differ in the order of terms are considered different.)
|
F_{n+1}
| 0.75 |
Let \( p(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( a, b, c, d \) are constants, and given that \( p(1) = 1993 \), \( p(2) = 3986 \), and \( p(3) = 5979 \). Calculate \( \frac{1}{4} [p(11) + p(-7)] \).
|
5233
| 0.75 |
Calculate the arc lengths of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\
y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t
\end{array}\right. \\
& \frac{\pi}{2} \leq t \leq \frac{2 \pi}{3}
\end{aligned}
$$
|
\sqrt{2} -1
| 0.625 |
A merchant has two types of tea: Ceylon tea at 10 rubles per pound and Indian tea at 6 rubles per pound. To increase profit, the merchant decides to mix the two types but sell the mixture at the same price of 10 rubles per pound. In what proportion should he mix them to make an additional profit of 3 rubles per pound?
|
1:3
| 0.75 |
Find the smallest prime number $p$ such that $n^{2}+n+11$ is divisible by $p$ for some integer $n$.
|
11
| 0.625 |
Anya calls a date beautiful if all 6 digits of its record are different. For example, 19.04.23 is a beautiful date, while 19.02.23 and 01.06.23 are not. How many beautiful dates are there in the year 2023?
|
30
| 0.25 |
Construct on the plane the set of points whose coordinates satisfy the inequality \( |3x + 4| + |4y - 3| \leq 12 \). Indicate the area of the resulting figure.
|
24
| 0.625 |
The diagonals of a trapezoid are 6 and 8, and the midline is 5. Find the area of the trapezoid.
|
24
| 0.625 |
Given the points \( B_1 \) and \( B_2 \) are the lower and upper vertices of the ellipse \( C: \frac{x^2}{8} + \frac{y^2}{4} = 1 \), respectively. The line \( l \) passing through the point \( A(0, -2\sqrt{2}) \) intersects the ellipse \( C \) at points \( P \) and \( Q \) (different from \( B_1 \) and \( B_2 \)).
1. If \( \tan \angle B_1 P B_2 = 2 \tan \angle B_1 Q B_2 \), find the equation of the line \( l \).
2. Let \( R \) be the intersection point of the lines \( B_1 P \) and \( B_2 Q \). Find the value of \( \overrightarrow{A R} \cdot \overrightarrow{B_1 B_2} \).
|
4\sqrt{2}
| 0.125 |
There were 15 buns on a plate. Karlsson took three times as many buns as Little Boy, and Little Boy's dog Bimbo took three times fewer buns than Little Boy. How many buns are left on the plate? Explain your answer.
|
2
| 0.75 |
In triangle \( A B C \) with side \( A C = 8 \), a bisector \( B L \) is drawn. It is known that the areas of triangles \( A B L \) and \( B L C \) are in the ratio \( 3: 1 \). Find the bisector \( B L \), for which the height dropped from vertex \( B \) to the base \( A C \) will be the greatest.
|
3\sqrt{2}
| 0.75 |
Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\).
|
\frac{\sqrt{5}}{5}
| 0.75 |
What is the remainder when \(2^{99}\) is divided by 7?
|
1
| 0.875 |
A square with a side length of 100 was cut into two equal rectangles. These rectangles were then placed next to each other as shown in the picture. Find the perimeter of the resulting figure.
|
500
| 0.625 |
Find all prime numbers \( p \) not exceeding 1000 such that \( 2p + 1 \) is a perfect power (i.e., there exist natural numbers \( m \) and \( n \geq 2 \) such that \( 2p + 1 = m^n \)).
|
13
| 0.25 |
Calculate the surface area of the part of the paraboloid of revolution \( 3y = x^2 + z^2 \) that is located in the first octant and bounded by the plane \( y = 6 \).
|
\frac{39 \pi}{4}
| 0.5 |
A caterpillar is climbing a 20-meter pole. During the day, it climbs 5 meters, and during the night, it slides down 4 meters. How long will it take for the caterpillar to reach the top of the pole?
|
16 \text{ days}
| 0.875 |
For what values of the parameter \( a \) does the equation \( x^{3} - 15x^{2} + ax - 64 = 0 \) have three distinct real roots that form a geometric progression?
|
60
| 0.625 |
Given a triangle \(ABC\) with a perimeter of 1. A circle is inscribed in angle \(BAC\), lying outside the triangle \(ABC\) and touching side \(BC\) (and the extensions of sides \(AB\) and \(AC\)). Find the length of segment \(AM\), where \(M\) is the point where the circle touches line \(AC\).
|
1/2
| 0.875 |
Given \( 0 \leqslant a_{k} \leqslant 1 \) for \( k = 1, 2, \cdots, 2002 \), and let \( a_{2003} = a_{1} \) and \( a_{2004} = a_{2} \), find the maximum value of \( \sum_{k=1}^{20002} \left( a_{k} - a_{k+1} a_{k+2} \right) \).
|
1001
| 0.875 |
In triangle \( ABC \), \( AB = 14 \), \( BC = 6 \), and \( CA = 9 \). Point \( D \) lies on line \( BC \) such that \( BD:DC = 1:9 \). The circles inscribed in triangles \( ADC \) and \( ADB \) touch side \( AD \) at points \( E \) and \( F \) respectively. Find the length of segment \( EF \).
|
4.9
| 0.625 |
Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square).
|
\frac{1}{33}
| 0.25 |
Find the value of the expression \(\cos ^{4} \frac{5 \pi}{24}+\cos ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{19 \pi}{24}+\sin ^{4} \frac{13 \pi}{24}\).
|
\frac{3}{2}
| 0.875 |
How many 10-digit positive integers consisting only of the digits 0 and 1 are divisible by 11?
|
126
| 0.25 |
Given real numbers \( a, b, \) and \( c \) that satisfy
\[ f(x) = a \cos x + b \cos 2x + c \cos 3x \geq -1 \]
for any real number \( x \). What is the maximum value of \( a + b + c \)?
|
3
| 0.625 |
Let \( A = \{1, 2, \cdots, 10\} \). If the equation \( x^2 - bx - c = 0 \) satisfies \( b, c \in A \) and the equation has at least one root \( a \in A \), then the equation is called a "beautiful equation". Find the number of "beautiful equations".
|
12
| 0.25 |
Inside a convex $n$-gon there are 100 points positioned in such a way that no three of these $n+100$ points are collinear. The polygon is divided into triangles, each having vertices among any 3 of the $n+100$ points. For what maximum value of $n$ can no more than 300 triangles be formed?
|
102
| 0.875 |
\(ABC\) is a triangle with \(AB = 15\), \(BC = 14\), and \(CA = 13\). The altitude from \(A\) to \(BC\) is extended to meet the circumcircle of \(ABC\) at \(D\). Find \(AD\).
|
\frac{63}{4}
| 0.875 |
In a compartment, any \( m (m \geqslant 3) \) passengers have a unique common friend (if A is a friend of B, then B is also a friend of A, and no one is their own friend). How many friends does the person with the most friends have in this compartment?
|
m
| 0.25 |
Given $x, y \in \mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15}.
$$
|
112
| 0.125 |
$a, b$ and $c$ are the lengths of the opposite sides $\angle A, $\angle B$ and $\angle C$ of the $\triangle ABC$ respectively. If $\angle C = 60^{\circ}$ and $\frac{a}{b+c} + \frac{b}{a+c} = P$, find the value of $P$.
|
1
| 0.875 |
Given a fixed triangle \( \triangle ABC \) and a point \( P \), find the maximum value of
\[
\frac{AB^{2} + BC^{2} + CA^{2}}{PA^{2} + PB^{2} + PC^{2}}
\]
|
3
| 0.5 |
How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing?
|
128
| 0.375 |
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take?
|
10000
| 0.375 |
There is a table of size $8 \times 8$ representing a chessboard. In each step, you are allowed to swap any two columns or any two rows. Is it possible, in several steps, to make the upper half of the table white and the lower half black?
|
\text{No}
| 0.75 |
A sequence of positive integers \(a_{1}, a_{2}, \ldots\) is such that for each \(m\) and \(n\) the following holds: if \(m\) is a divisor of \(n\) and \(m < n\), then \(a_{m}\) is a divisor of \(a_{n}\) and \(a_{m} < a_{n}\). Find the least possible value of \(a_{2000}\).
|
128
| 0.125 |
Given two sets of points \(A = \left\{(x, y) \mid (x-3)^{2}+(y-4)^{2} \leqslant \left(\frac{5}{2}\right)^{2}\right\}\) and \(B = \left\{(x, y) \mid (x-4)^{2}+(y-5)^{2} > \left(\frac{5}{2}\right)^{2}\right\}\), the number of lattice points (i.e., points with integer coordinates) in the set \(A \cap B\) is ...
|
7
| 0.625 |
Let \( x \in \mathbb{R} \). The function \( f(x)=|2x-1| + |3x-2| + |4x-3| + |5x-4| \). What is the minimum value of the function?
|
1
| 0.75 |
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be an odd function, i.e., a function that satisfies \( -f(x) = f(-x) \) for all \( x \in \mathbb{R} \). Suppose that \( f(x+5) = f(x) \) for all \( x \in \mathbb{R} \) and that \( f(1/3) = 1 \). Determine the value of the sum:
\[
f(16/3) + f(29/3) + f(12) + f(-7)
\]
|
0
| 0.875 |
Let \( a^{\frac{2}{3}} + b^{\frac{2}{3}} = 17 \frac{1}{2} \), \( x = a + 3a^{\frac{1}{3}} b^{\frac{2}{3}} \) and \( y = b + 3a^{\frac{2}{3}} b^{\frac{1}{3}} \). If \( P = (x+y)^{\frac{2}{3}} + (x-y)^{\frac{2}{3}} \), find the value of \( P \).
|
35
| 0.875 |
Find all non-empty finite sets \( S \) of positive integers such that if \( m, n \in S \), then \( \frac{m+n}{\gcd(m,n)} \in S \).
|
\{2\}
| 0.75 |
In the vertices of a unit square, perpendiculars are erected to its plane. On them, on one side of the plane of the square, points are taken at distances of 3, 4, 6, and 5 from this plane (in order of traversal). Find the volume of the polyhedron whose vertices are the specified points and the vertices of the square.
|
4.5
| 0.25 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \),
\[ f(f(x)+2y) = 6x + f(f(y)-x). \]
|
f(x) = 2x + c
| 0.25 |
How many times longer is the staircase to the fourth floor of a building compared to the staircase to the second floor of the same building?
|
3
| 0.125 |
Let \(\pi\) be a randomly chosen permutation of the numbers from 1 through 2012. Find the probability that \(\pi(\pi(2012))=2012\).
|
\frac{1}{1006}
| 0.75 |
The sum of all natural numbers not exceeding 200 that leave a remainder of 7 when divided by 11 and a remainder of 5 when divided by 7 is $\qquad$
|
351
| 0.875 |
Let \( x \) and \( y \) be positive real numbers, and let \( \theta \neq \frac{n \pi}{2} \) (where \( n \) is an integer). If \( \frac{\sin \theta}{x}=\frac{\cos \theta}{y} \), and \( \frac{\cos ^{4} \theta}{x^{4}}+\frac{\sin ^{4} \theta}{y^{4}}=\frac{97 \sin 2 \theta}{x^{3} y+y^{3} x} \), then find the value of \( \frac{y}{x}+\frac{x}{y} \).
|
4
| 0.75 |
Calculate the following:
$$\frac{1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \ldots + 2013 \cdot 2014}{(1 + 2 + 3 + \ldots + 2014) \cdot \frac{1}{5}}$$
If necessary, round the answer to two decimal places.
|
6710
| 0.5 |
The side \( AD \) of the rectangle \( ABCD \) is equal to 2. On the extension of the side \( AD \) beyond point \( A \), a point \( E \) is taken such that \( EA = 1 \), and \( \angle BEC = 30^\circ \). Find \( BE \).
|
2
| 0.875 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 4} \frac{\sqrt{1+2 x}-3}{\sqrt{x}-2}
\]
|
\frac{4}{3}
| 0.875 |
As shown in the figure, first place 5 pieces (4 black and 1 white) on a circle. Then, place a white piece between two pieces of the same color, and place a black piece between two pieces of different colors, while removing the original 5 pieces. Continuously perform these operations. What is the maximum number of white pieces that can be on the circle among the 5 pieces?
|
3
| 0.25 |
A dandelion blooms in the morning, remains yellow for three days, turns white on the fourth morning, and sheds its seeds by the evening of the fifth day. On Monday afternoon, there were 20 yellow and 14 white dandelions in the meadow. On Wednesday, there were 15 yellow and 11 white dandelions. How many white dandelions will be in the meadow on Saturday?
|
6
| 0.125 |
To enter a park, a group of two men, four women, and two children paid 226 reais, while a group of three men, three women, and one child paid 207 reais.
a) How much would a group of 8 men, 10 women, and 4 children pay to enter the park?
b) If the ticket prices are all natural numbers, how many possible prices are there for the tickets?
|
21
| 0.25 |
A math professor asks 100 math students to guess an integer between 0 and 100, inclusive, such that their guess is 'two-thirds of the average of all the responses.' Each student who guesses the highest integer that is not higher than two-thirds of the average of all responses will receive a prize. Assuming that it is common knowledge that all students will write down the best response, and there is no communication between students, what single integer should every student write down?
|
0
| 0.25 |
In the expression \((x + y + z)^{2018} + (x - y - z)^{2018}\), the brackets were expanded, and like terms were combined. How many monomials \(x^{a} y^{b} z^{c}\) have a nonzero coefficient?
|
1020100
| 0.625 |
Mr. Rychlý and Mr. Louda started the same hiking route at the same time. Mr. Rychlý walked from a mountain hut down to the town, and Mr. Louda walked from the town up to the mountain hut. They passed each other at 10 AM. Mr. Rychlý hurried and reached his destination at 12 PM, while Mr. Louda moved slowly and reached the hut at 6 PM. At what time did they start their hike, given that each traveled at a constant speed?
|
6 \text{ AM}
| 0.875 |
Find all positive integers \( k \) such that the polynomial \( x^{2k+1} + x + 1 \) is divisible by \( x^k + x + 1 \). For each \( k \) that satisfies this condition, find the positive integers \( n \) such that \( x^n + x + 1 \) is divisible by \( x^k + x + 1 \).
(British Mathematical Olympiad, 1991)
|
2
| 0.75 |
There is a set of integers: two 2's, three 3's, four 4's, and five 5's (14 numbers in total). These numbers are placed in the circles of a star as shown in the diagram. Could it be such that the sums of the numbers along each segment (each with 4 circles) are equal to each other?
|
\text{No}
| 0.375 |
Find the limit as \( x \) approaches \( \frac{\pi}{2} \) of \((x - \frac{\pi}{2}) \cdot \tan(x)\).
|
-1
| 0.75 |
If \( f(x) = \sum_{k=0}^{4034} a_k x^k \) is the expansion of \( \left(x^2 + x + 2\right)^{2017} \), calculate \( \sum_{k=0}^{1344} \left(2 a_{3k} - a_{3k+1} - a_{3k+2}\right) \).
|
2
| 0.625 |
60 explorers need to cross a river using a rubber boat that can carry 6 people (one trip across the river and back counts as two crossings). Each crossing takes 3 minutes. How many minutes will it take for all explorers to reach the other side of the river?
|
69
| 0.5 |
The Wolf and Ivan Tsarevich are 20 versts away from a source of living water, and the Wolf is taking Ivan Tsarevich there at a speed of 3 versts per hour. To revive Ivan Tsarevich, one liter of water is needed, which flows from the source at a rate of half a liter per hour. At the source, there is a Raven with unlimited carrying capacity; it must gather the water, after which it will fly towards the Wolf and Ivan Tsarevich at a speed of 6 versts per hour, spilling a quarter liter of water every hour. After how many hours will it be possible to revive Ivan Tsarevich?
|
4 \text{ hours}
| 0.375 |
Given that \( a, b, c \) are positive integers, and the parabola \( y = ax^2 + bx + c \) intersects the x-axis at two distinct points \( A \) and \( B \). If the distances from \( A \) and \( B \) to the origin are both less than 1, find the minimum value of \( a + b + c \).
|
11
| 0.625 |
The numbers \(2^{n}\) and \(5^{n}\) (in decimal notation) are written one after the other. How many digits does the resulting number have?
|
n+1
| 0.375 |
Calculate the limit
$$
\lim _{x \rightarrow 0} \frac{1+\sin 2x - \cos 2x}{1 - \sin 2x - \cos 2x}
$$
|
-1
| 0.875 |
There are 37 people lined up in a row, and they are counting off one by one. The first person says 1, and each subsequent person says the number obtained by adding 3 to the previous person’s number. At one point, someone makes a mistake and subtracts 3 from the previous person's number instead. The sum of all the numbers reported by the 37 people is 2011. Which person made the mistake?
|
34
| 0.5 |
Find all sets of positive integers \((x, y, z)\) such that \(y\) is a prime number, neither \(y\) nor 3 divides \(z\), and \(x^3 - y^3 = z^2\).
|
(8,7,13)
| 0.75 |
During the vacation, for eight classes of a school, each with an equal number of students, a museum excursion was organized, while for the remaining students, who turned out to be $15\%$ more, a field trip to the puppet theater was arranged. How many students are in the school, given that there are no more than 520 of them, and more than 230 students participated in the excursion?
|
516
| 0.875 |
Find the strictly increasing functions \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that \( f(2)=2 \) and for all \( n, m \geq 1 \), we have
\[ f(n m) = f(n) f(m). \]
|
f(n) = n
| 0.75 |
A regular hexagon \( A B C D E K \) is inscribed in a circle of radius \( 3 + 2\sqrt{3} \). Find the radius of the circle inscribed in the triangle \( B C D \).
|
\frac{3}{2}
| 0.875 |
On the board, 34 ones are written. Each minute, Karlson erases two random numbers and writes their sum on the board, and then eats a number of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 34 minutes?
|
561
| 0.375 |
Brian has a 20-sided die with faces numbered from 1 to 20, and George has three 6-sided dice with faces numbered from 1 to 6. Brian and George simultaneously roll all their dice. What is the probability that the number on Brian's die is larger than the sum of the numbers on George's dice?
|
\frac{19}{40}
| 0.625 |
Let \( F_1 \) and \( F_2 \) be the foci of the ellipse \( \frac{x^{2}}{9}+\frac{y^{2}}{4}=1 \), and \( P \) be a point on the ellipse such that \( \left|P F_1\right|:\left|P F_2\right| = 2:1 \). Find the area of the triangle \( \triangle P F_1 F_2 \).
|
4
| 0.875 |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-100)^{2}
$$
If the result is a non-integer, round it to the nearest whole number.
|
44200
| 0.375 |
Anya calls a date beautiful if all 6 digits in its recording are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not.
a) How many beautiful dates will there be in April 2023?
b) How many beautiful dates will there be in the entire year of 2023?
|
30
| 0.25 |
Let $\triangle ABC$ have three internal angles $\angle A, \angle B, \angle C$, and the side lengths opposite to these angles are $a, b, c$ respectively. Given that $a < b < c$, and
\[
\begin{cases}
\frac{b}{a} = \frac{\left|b^{2} + c^{2} - a^{2}\right|}{bc} \\
\frac{c}{b} = \frac{\left|c^{2} + a^{2} - b^{2}\right|}{ca} \\
\frac{a}{c} = \frac{\left|a^{2} + b^{2} - c^{2}\right|}{ab}
\end{cases}
\]
find the ratio of the radian measures of the angles $\angle A, \angle B, \angle C$.
|
1:2:4
| 0.875 |
The famous skater Tony Hawk rides a skateboard (segment \( AB \)) on a ramp, which is a semicircle with diameter \( PQ \). Point \( M \) is the midpoint of the skateboard, and \( C \) is the foot of the perpendicular dropped from point \( A \) to the diameter \( PQ \). What values can the angle \( \angle ACM \) take, given that the angular measure of arc \( AB \) is \( 24^\circ \)?
|
12^\circ
| 0.875 |
Four points \( A, O, B, O' \) are aligned in this order on a line. Let \( C \) be the circle centered at \( O \) with radius 2015, and \( C' \) be the circle centered at \( O' \) with radius 2016. Suppose that \( A \) and \( B \) are the intersection points of two common tangents to the two circles. Calculate \( AB \) given that \( AB \) is an integer \( < 10^7 \) and that \( AO \) and \( AO' \) are integers.
|
8124480
| 0.125 |
At the namesake festival, 45 Alexanders, 122 Borises, 27 Vasily, and several Gennady attended. At the beginning of the festival, all of them lined up so that no two people with the same name stood next to each other. What is the minimum number of Gennadys that could have attended the festival?
|
49
| 0.5 |
Find \(\log _{\sqrt{3}} \sqrt[6]{a}\), given that \(\log _{a} 27 = b\).
|
\frac{1}{b}
| 0.875 |
Given that \( m \) and \( n \) are both positive integers and satisfy \( 24m = n^4 \), find the minimum value of \( m \).
|
54
| 0.75 |
If a natural number \(N\) can be expressed as the sum of 3 consecutive natural numbers, the sum of 11 consecutive natural numbers, and the sum of 12 consecutive natural numbers, then the smallest value of \(N\) is \(\qquad\). (Note: The smallest natural number is 0.)
|
66
| 0.5 |
A porter needs to transport 200 steamed buns from the kitchen to the construction site (he is currently in the kitchen). He can carry 40 buns each time. However, because he is very greedy, he will eat 1 bun each time he travels from the kitchen to the construction site or from the construction site to the kitchen. What is the maximum number of buns he can transport to the construction site?
|
191
| 0.125 |
Bag $A$ contains 2 ten-yuan banknotes and 3 one-yuan banknotes, and bag $B$ contains 4 five-yuan banknotes and 3 one-yuan banknotes. Now, if two banknotes are randomly drawn from each of the bags, what is the probability that the sum of the remaining banknotes in bag $A$ is greater than the sum of the remaining banknotes in bag $B$?
|
\frac{9}{35}
| 0.25 |
At a temperature of $13^{\circ}$, a 41% sugar solution in water becomes saturated. How many grams of sugar will not dissolve in water (and will precipitate) if (at a temperature of $15^{\circ}$) you thoroughly mix a glass of water (220 g) with 280 g of sugar?
|
127
| 0.125 |
For the "Skillful Hands" club, Boris needs to cut several identical pieces of wire (the length of each piece is an integer number of centimeters). Initially, Boris took a piece of wire 10 meters long and was able to cut only 14 required pieces from it. Then, Boris took a piece that was 50 centimeters longer, but it was also only enough for 14 pieces. What length did Boris need to cut the pieces? Express your answer in centimeters.
|
71
| 0.625 |
Find the remainder when \( 2^{1999} + 1 \) is divided by 17.
|
10
| 0.5 |
The largest divisor of a natural number \( N \), smaller than \( N \), was added to \( N \), producing a power of ten. Find all such \( N \).
|
75
| 0.5 |
Let \( x_{1}, x_{2}, \ldots, x_{100} \) be natural numbers greater than 1 (not necessarily distinct). In a \( 100 \times 100 \) table, numbers are placed as follows: at the intersection of the \( i \)-th row and the \( k \)-th column, the number \( \log _{x_{k}} \frac{x_{i}}{4} \) is written. Find the smallest possible value of the sum of all the numbers in the table.
|
-10000
| 0.875 |
There are 4 points \( A, B, C, D \) in space, satisfying \( AB = BC = CD \). Given that \( \angle ABC = \angle BCD = \angle CDA = 36^\circ \), what is the angle between line \( AC \) and line \( BD \)?
|
90^\circ
| 0.5 |
In $\triangle ACD$, $\angle A=45^{\circ}$, $CD=7$, $B$ is a point on $AD$, $CB=5$, $BD=3$, find $AC$.
|
\frac{5\sqrt{6}}{2}
| 0.5 |
Calculate the value of the expression \(\arccos \frac{\sqrt{6}+1}{2 \sqrt{3}} - \arccos \sqrt{\frac{2}{3}}\). Express the result in the form \(\frac{a \pi}{b}\), where \(a\) and \(b\) are integers that are coprime, and indicate the value of \(|a-b|\).
|
7
| 0.375 |
Evaluate the following limits:
1. $\lim_{x \rightarrow 0} \frac{1- \sqrt{x+1}}{x}$
2. $\lim_{x \rightarrow 0} \frac{\operatorname{tg} x}{1-\sqrt{1+\operatorname{tg} x}}$
3. $\lim_{x \rightarrow 1} \frac{2- \sqrt{x}}{3- \sqrt{2x+1}}$
4. $\lim_{x \rightarrow 1} \frac{1- \sqrt{x}}{1- \sqrt[3]{x}}$
|
\frac{3}{2}
| 0.5 |
Given a parallelogram \(ABCD\) with \(\angle B = 60^\circ\). Point \(O\) is the center of the circumcircle of triangle \(ABC\). Line \(BO\) intersects the bisector of the exterior angle \(\angle D\) at point \(E\). Find the ratio \(\frac{BO}{OE}\).
|
\frac{1}{2}
| 0.25 |
An edge of an inclined parallelepiped is equal to $l$. It is bounded by two adjacent faces, whose areas are $m^{2}$ and $n^{2}$, and the planes of these faces form an angle of $30^{\circ}$. Calculate the volume of the parallelepiped.
|
\frac{m^2 n^2}{2 l}
| 0.875 |
Every third student in the sixth grade is a member of the math club, every fourth student is a member of the history club, and every sixth student is a member of the chemistry club. The rest of the students are members of the literature club. How many people are in the chemistry club if the number of members in the math club exceeds the number of members in the literature club by three?
|
6
| 0.875 |
The common difference of the arithmetic sequence $\left\{a_{n}\right\}$ is $d=2$, and the sum of the first 30 terms is $a_{1}+a_{2}+\cdots+a_{30}=100$. Find the sum of every third term from $a_{3}$ to $a_{30}$: $a_{3}+a_{6}+\cdots+a_{30}$.
|
\frac{160}{3}
| 0.75 |
Given a positive integer \( n \), and a set \( M = \{1, 2, \cdots, 2n\} \), find the smallest positive integer \( k \) such that for any \( k \)-element subset of \( M \), it must contain 4 different elements whose sum equals \( 4n + 1 \).
|
n + 3
| 0.125 |
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