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|---|---|---|
In triangle \(ABC\), if \(\cos^{2} A + \cos^{2} B + \cos^{2} C = \sin^{2} B\), then \(\tan A \tan C =\)
|
3
| 0.75 |
What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers?
|
77
| 0.5 |
As shown in the diagram, the abacus has three sections, each with 10 beads. By dividing the beads in each section into top and bottom parts, two three-digit numbers are formed. The requirement is that the three-digit number in the top part must have distinct digits and be a multiple of the three-digit number in the bottom part. Determine the three-digit number in the top part.
|
925
| 0.5 |
The sum of four numbers in a geometric progression is -40, and the sum of their squares is 3280. Find this progression.
|
2, -6, 18, -54
| 0.75 |
The number of positive integers \( k \) such that there exists a pair of positive integers \( (a, b) \) satisfying
$$
\frac{k}{2015}(a+b)=\text{lcm}(a, b)
$$
where \(\text{lcm}(a, b)\) denotes the least common multiple of \(a\) and \(b\), is ______.
|
1007
| 0.125 |
Let a three-digit number \( n = \overline{abc} \), where \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle as the lengths of its sides. How many such three-digit numbers \( n \) are there?
|
165
| 0.25 |
Let \( x_{1} \) and \( x_{2} \) be the largest roots of the polynomials \( f(x) = 1 - x - 4x^{2} + x^{4} \) and \( g(x) = 16 - 8x - 16x^{2} + x^{4} \) respectively. Find \( \frac{x_{2}}{x_{1}} \).
|
2
| 0.875 |
A shopkeeper wants to sell his existing stock of a popular item at one and a half times its current price. To achieve this, he plans to implement a phased price increase. He starts with a certain price increase, and when two-thirds of the stock has been sold, he raises the price again by the same percentage. This new price remains in effect until the entire stock is sold. What percentage price increases are needed to achieve the desired goal?
|
34.5\%
| 0.375 |
For a positive integer \( n \), let the sum of its digits be denoted as \( s(n) \), and the product of its digits be denoted as \( p(n) \). If \( s(n) + p(n) = n \) holds true, then \( n \) is called a "magic number." Find the sum of all magic numbers.
|
531
| 0.75 |
A circle is tangent to the extensions of two sides \(AB\) and \(AD\) of the square \(ABCD\), and the points of tangency cut off segments of length 4 cm from vertex \(A\). From point \(C\), two tangents are drawn to this circle. Find the side length of the square if the angle between the tangents is \(60^\circ\).
|
4(\sqrt{2}-1)
| 0.25 |
Knowing that segment \( CD \) has a length of 6 and its midpoint is \( M \), two triangles \( \triangle ACD \) and \( \triangle BCD \) are constructed on the same side with \( CD \) as a common side and both having a perimeter of 16, such that \( \angle AMB = 90^{\circ} \). Find the minimum area of \( \triangle AMB \).
|
\frac{400}{41}
| 0.375 |
Given that for any \( x \in [0,1] \), the inequality
\[ f(x) = k \left( x^2 - x + 1 \right) - x^4 (1 - x)^4 \geqslant 0, \]
holds, find the minimum value of \( k \).
|
\frac{1}{192}
| 0.75 |
In the diagram, there is a map consisting of 8 cities connected by 12 roads. How many ways are there to close 5 roads simultaneously for repairs such that it is still possible to travel from any city to any other city?
|
384
| 0.75 |
In an isosceles trapezoid, the bases are 40 and 24, and its diagonals are mutually perpendicular. Find the area of the trapezoid.
|
1024
| 0.625 |
Compute the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\cos (1+x)}{\left(2+\sin \left(\frac{1}{x}\right)\right) \ln (1+x)+2}
$$
|
\frac{\cos 1}{2}
| 0.75 |
There are four individuals: Jia, Yi, Bing, and Ding. It is known that the average age of Jia, Yi, and Bing is 1 year more than the average age of all four people. The average age of Jia and Yi is 1 year more than the average age of Jia, Yi, and Bing. Jia is 4 years older than Yi, and Ding is 17 years old. How old is Jia?
|
24
| 0.875 |
Let \( x \) be a non-zero real number such that \( x + \frac{1}{x} = \sqrt{2019} \). What is the value of \( x^{2} + \frac{1}{x^{2}} \)?
|
2017
| 0.875 |
Aunt Martha bought some nuts. She gave Tommy one nut and a quarter of the remaining nuts, Bessie received one nut and a quarter of the remaining nuts, Bob also got one nut and a quarter of the remaining nuts, and finally, Jessie received one nut and a quarter of the remaining nuts. It turned out that the boys received 100 more nuts than the girls.
How many nuts did Aunt Martha keep for herself?
|
321
| 0.375 |
What is the smallest number of triangular pyramids (tetrahedrons) into which a cube can be divided?
|
5
| 0.75 |
Egor borrowed 28 rubles from Nikita and then repaid them in four payments. It turned out that Egor always returned a whole number of rubles, and each payment amount always increased and was exactly divisible by the previous one. What amount did Egor pay back the last time?
|
18
| 0.75 |
Determine the number of integers \( n \) such that \( 1 \leq n \leq 10^{10} \), and for all \( k = 1, 2, \ldots, 10 \), the integer \( n \) is divisible by \( k \).
|
3968253
| 0.125 |
If Xiao Zhang's daily sleep time is uniformly distributed between 6 hours and 9 hours, what is the probability that Xiao Zhang's average sleep time over two consecutive days is at least 7 hours?
|
\frac{7}{9}
| 0.25 |
How many ordered pairs $(A, B)$ are there where $A$ and $B$ are subsets of a fixed $n$-element set and $A \subseteq B$?
|
3^n
| 0.875 |
Compute the limit of the numerical sequence:
$$\lim _{n \rightarrow \infty} \frac{2-5+4-7+\ldots+2 n-(2 n+3)}{n+3}$$
|
-3
| 0.875 |
Some of the 20 metal cubes, which are identical in size and appearance, are made of aluminum, and the rest are made of duralumin (which is heavier). How can you determine the number of duralumin cubes using 11 weighings on a balance scale without weights?
|
11
| 0.5 |
Let \( f(m) \) be the product of the digits of the positive integer \( m \). Find the positive integer solutions to the equation \( f(m) = m^2 - 10m - 36 \).
|
13
| 0.5 |
The apex of a regular pyramid with a square base $ABCD$ of unit side length is $E$. Point $P$ lies on the base edge $AB$ and point $Q$ lies on the lateral edge $EC$ such that $PQ$ is perpendicular to both $AB$ and $EC$. Additionally, we know that $AP : PB = 6 : 1$. What are the lengths of the lateral edges?
|
\sqrt{2}
| 0.75 |
What is the condition for two medians of a triangle to be perpendicular to each other?
|
a^2 + b^2 = 5c^2
| 0.125 |
Let \( S = \{1, 2, \ldots, 2013\} \). Find the number of ordered triples \((A, B, C)\) of subsets of \(S\) such that \(A \subseteq B\) and \(A \cup B \cup C = S\).
|
5^{2013}
| 0.625 |
50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hair between any two students with brown hair. What is the minimum number of students with red hair that can be sitting at this table?
|
17
| 0.5 |
In triangle \(ABC\), the side \(AC\) is longer than \(AB\). The line \(l\) is the bisector of the external angle at \(C\). A line passing through the midpoint of \(AB\) and parallel to \(l\) intersects \(AC\) at point \(E\). Find \(CE\) if \(AC = 7\) and \(CB = 4\). (The external angle of a triangle is the angle adjacent to the internal angle at a given vertex.)
|
\frac{11}{2}
| 0.5 |
What is the highest power of 3 that divides the number whose decimal representation consists of $3^n$ ones?
|
3^n
| 0.5 |
The numbers \( a, b, c, d \) belong to the interval \([-13.5, 13.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \).
|
756
| 0.625 |
Given the sequence \(\left\{a_{n}\right\}\), satisfy the equation \(\frac{a_{n+1}+a_{n}-1}{a_{n+1}-a_{n}+1}=n\) for \(n \in \mathbf{N}^{*}\), and \(a_{2}=6\).
|
n(2n-1)
| 0.125 |
We have \( n \geq 3 \) points in the plane. We suppose that the area of any triangle formed by 3 of these points does not exceed 1.
Show that all these points can be placed inside a triangle with an area of at most 4.
|
4
| 0.5 |
Let \( ABC \) be a triangle with \(\angle A = 60^\circ\). Line \(\ell\) intersects segments \( AB \) and \( AC \) and splits triangle \( ABC \) into an equilateral triangle and a quadrilateral. Let \( X \) and \( Y \) be on \(\ell\) such that lines \( BX \) and \( CY \) are perpendicular to \(\ell\). Given that \( AB = 20 \) and \( AC = 22 \), compute \( XY \).
|
21
| 0.5 |
After adding the coefficients \( p \) and \( q \) with the roots of the quadratic polynomial \( x^2 + px + q \), the result is 2, and after multiplying them, the result is 12. Find all such quadratic polynomials.
|
x^2 + 3x + 2
| 0.25 |
One vertex of an equilateral triangle lies on a circle, and the other two vertices divide a certain chord into three equal parts.
At what angle is the chord seen from the center of the circle?
|
120^\circ
| 0.875 |
Find the smallest positive integer $n$ such that the polynomial $(x+1)^{n}-1$ is divisible by $x^{2}+1$ modulo 3.
|
8
| 0.375 |
Find the largest constant \( K \) such that for all positive real numbers \( a, b \), and \( c \), the following inequality holds
$$
\sqrt{\frac{ab}{c}} + \sqrt{\frac{bc}{a}} + \sqrt{\frac{ac}{b}} \geqslant K \sqrt{a+b+c}
$$
|
\sqrt{3}
| 0.875 |
Let \(ABC\) be a triangle such that \(AB=7\), and let the angle bisector of \(\angle BAC\) intersect line \(BC\) at \(D\). If there exist points \(E\) and \(F\) on sides \(AC\) and \(BC\), respectively, such that lines \(AD\) and \(EF\) are parallel and divide triangle \(ABC\) into three parts of equal area, determine the number of possible integral values for \(BC\).
|
13
| 0.25 |
Let \( n = 1990 \), then
$$
\begin{aligned}
& \frac{1}{2^{n}}\left(1-3 \binom{n}{2}+3^{2} \binom{n}{4}-3^{3} \binom{n}{6}+\cdots+3^{994} \binom{n}{1988}-3^{995} \binom{n}{1990}\right) \\
= &
\end{aligned}
$$
|
-\frac{1}{2}
| 0.75 |
1. Find \(\lim_{x \to \infty} \frac{x^2}{e^x}\).
2. Find \(\lim_{k \to 0} \frac{1}{k} \int_{0}^{k} (1 + \sin{2x})^{1/x} \, dx\).
|
e^2
| 0.875 |
Given that the complex number \( z \) satisfies \( |z+\sqrt{3}i| + |z-\sqrt{3}i| = 4 \), find the minimum value of \( |z - \mathrm{i}| \).
|
\frac{\sqrt{6}}{3}
| 0.75 |
Find the number of triples of natural numbers \((a, b, c)\) that satisfy the system of equations:
\[
\left\{
\begin{array}{l}
\gcd(a, b, c) = 6 \\
\operatorname{lcm}(a, b, c) = 2^{15} \cdot 3^{16}
\end{array}
\right.
\]
|
7560
| 0.375 |
The numbers from 1 to 8 are arranged at the vertices of a cube in such a way that the sum of the numbers at any three vertices on the same face is at least 10. What is the minimum possible sum of the numbers on the vertices of one face?
|
16
| 0.625 |
Draw a convex pentagon \(ABCDE\) where \(AB = CD\), \(BC = ED\), \(AC = BD = EC\), and the diagonal \(BE\) intersects diagonal \(AC\) at a \(90^\circ\) angle, while \(AD\) intersects \(AC\) at a \(75^\circ\) angle.
|
ABCDE
| 0.5 |
Point \( A \) lies on the line \( y = \frac{5}{12} x - 11 \), and point \( B \) lies on the parabola \( y = x^{2} \). What is the minimum length of segment \( AB \)?
|
\frac{6311}{624}
| 0.375 |
Among all integer solutions of the equation \(20x + 19y = 2019\), find the one for which the value \(|x - y|\) is minimized. Write down the product \(xy\) as the answer.
|
2623
| 0.375 |
Camp Koeller offers exactly three water activities: canoeing, swimming, and fishing. None of the campers is able to do all three of the activities. In total, 15 of the campers go canoeing, 22 go swimming, 12 go fishing, and 9 do not take part in any of these activities. Determine the smallest possible number of campers at Camp Koeller.
|
34
| 0.5 |
Suppose \( x_{1}, x_{2}, \ldots, x_{49} \) are real numbers such that
\[ x_{1}^{2} + 2 x_{2}^{2} + \cdots + 49 x_{49}^{2} = 1. \]
Find the maximum value of \( x_{1} + 2 x_{2} + \cdots + 49 x_{49} \).
|
35
| 0.625 |
For non-negative real numbers $x_{0}$, $x_{1}, \cdots, x_{n}$ such that their sum is 1 and $x_{n+1}=x_{1}$, find the maximum value of
$$
S=\sqrt{x_{0}+\sum_{i=1}^{n} \frac{\left(x_{i}-x_{i+1}\right)^{2}}{2n}} + \sum_{i=1}^{n} \sqrt{x_{i}} \quad (n \geqslant 3).
$$
|
\sqrt{n+1}
| 0.875 |
$$
\frac{2(a+(a+1)+(a+2)+\ldots+2a)}{a^{2}+3a+2}+\frac{6\left(a^{1/2}+b^{1/2}\right)}{(a-b)^{0.6}(a+2)}:\left(\left(a^{1/2}-b^{1/2}\right)(a-b)^{-2/5}\right)^{-1}
$$
|
3
| 0.625 |
In triangle \(ABC\), angle \(C\) is \(75^\circ\) and angle \(B\) is \(60^\circ\). The vertex \(M\) of an isosceles right triangle \(BCM\) with hypotenuse \(BC\) is located inside triangle \(ABC\). Find angle \(MAC\).
|
30^\circ
| 0.5 |
The sum of the ages of five people \( A, B, C, D, \) and \( E \) is 256 years. The age difference between any two people is not less than 2 years and not more than 10 years. What is the minimum age of the youngest person among them?
|
46
| 0.375 |
Name three whole numbers whose product equals their sum.
Find 12.5% of the number 44.
|
5.5
| 0.875 |
There are 16 students who form a $4 \times 4$ square matrix. In an examination, their scores are all different. After the scores are published, each student compares their score with the scores of their adjacent classmates (adjacent refers to those directly in front, behind, left, or right; for example, a student sitting in a corner has only 2 adjacent classmates). A student considers themselves "happy" if at most one classmate has a higher score than them. What is the maximum number of students who will consider themselves "happy"?
|
12
| 0.125 |
For all real numbers \( x \), let
\[ f(x) = \frac{1}{\sqrt[2011]{1 - x^{2011}}}. \]
Evaluate \( (f(f(\ldots(f(2011)) \ldots)))^{2011} \), where \( f \) is applied 2010 times.
|
2011^{2011}
| 0.75 |
Two pedestrians started walking at dawn. Each was walking at a constant speed. One walked from point $A$ to point $B$, and the other from point $B$ to point $A$. They met at noon and continued walking without stopping. One arrived at $B$ at 4 PM, and the other arrived at $A$ at 9 PM. At what hour was dawn on that day?
|
6
| 0.75 |
1. What is the units digit of \(3^{1789}\)?
2. What is the units digit of \(1777^{1777^{1777}}\)?
|
7
| 0.875 |
The diagonals of a quadrilateral are equal, and the lengths of its midlines are p and q. Find the area of the quadrilateral.
|
pq
| 0.25 |
There are four cups, each with the mouth facing up. Each time, three cups are flipped, and cups that have been flipped before are allowed to be flipped again. After $n$ flips, all the cups have the mouth facing down. What is the smallest value of the positive integer $n$?
Note: Flipping a cup means turning it from mouth up to mouth down or from mouth down to mouth up.
|
4
| 0.5 |
Find a six-digit number which, when multiplied by 2, 3, 4, 5, and 6, results in six-digit numbers that use the same digits in a different order.
|
142857
| 0.625 |
There are 30 volumes of an encyclopedia arranged in some order on a bookshelf. In one operation, it is allowed to swap any two adjacent volumes. What is the minimum number of operations required to guarantee arranging all the volumes in the correct order (from the first to the thirtieth from left to right) regardless of the initial arrangement?
|
435
| 0.875 |
Given a circle, two points \( P \) and \( Q \) on this circle, and a line. Find a point \( M \) on the circle such that the lines \( MP \) and \( MQ \) intercept a segment \( AB \) of a given length on the given line.
|
M
| 0.375 |
Given a four-digit number that satisfies the following conditions: (1) If the units digit and the hundreds digit are swapped, and the tens digit and the thousands digit are swapped, the number increases by 5940; (2) When divided by 9, the remainder is 8. Find the smallest odd number that meets these conditions.
|
1979
| 0.125 |
In triangle \( \triangle ABC \), \( BC = a, AC = b, AB = c \), and \( D \) is the midpoint of \( AC \). If \( a^2 + b^2 + c^2 = ab + bc + ca \), then \( \angle CBD = \quad \).
|
30^\circ
| 0.875 |
Solve the following equation in the set of integer pairs:
$$
(x+2)^{4}-x^{4}=y^{3} \text {. }
$$
|
(-1, 0)
| 0.625 |
In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________.
|
24
| 0.125 |
How many integers at a minimum must be selected from the set $\{1,2, \ldots, 20\}$ to ensure that this selection includes two integers \( a \) and \( b \) such that \( a - b = 2 \)?
|
11
| 0.875 |
The sum of the distances from a point inside an equilateral triangle of perimeter length \( p \) to the sides of the triangle is \( s \). Show that \( s \sqrt{12} = p \).
|
s \sqrt{12} = p
| 0.875 |
As shown in the figure, $A$ and $B$ are endpoints of a diameter of a circular track. Three micro-robots, Alpha, Beta, and Gamma, start simultaneously on the circular track and move uniformly in a circular motion. Alpha and Beta start from point $A$, and Gamma starts from point $B$. Beta moves clockwise, while Alpha and Gamma move counterclockwise. After 12 seconds, Alpha reaches point $B$. Nine seconds later, Alpha catches up with Gamma for the first time and also coincides with Beta for the first time. When Gamma reaches point $A$ for the first time, how many seconds will it take for Beta to reach point $B$ for the first time?
|
56
| 0.75 |
What values can the expression \((x-y)(y-z)(z-x)\) take, if it is known that \(\sqrt{x-y+z}=\sqrt{x}-\sqrt{y}+\sqrt{z}\)?
|
0
| 0.875 |
Students from grades 9A, 9B, and 9C gathered for a ceremony. Mary Ivanovna counted 27 students from grade 9A, 29 students from grade 9B, and 30 students from grade 9C. Ilia Grigorievich decided to count the total number of students present from all three grades and ended up with 96 students.
It turned out that Mary Ivanovna's count for each grade could be off by no more than 2 students. Additionally, Ilia Grigorievich's total count could be off by no more than 4 students.
How many students from grade 9A were present at the ceremony?
|
29
| 0.625 |
Find the two-digit number whose digits are distinct and whose square is equal to the cube of the sum of its digits.
|
27
| 0.625 |
A tree is 24 meters tall. A snail at the bottom of the tree wants to climb to the top. During the day, it climbs 6 meters up, and at night, it slides down 4 meters. After how many days can the snail reach the top of the tree?
|
10
| 0.875 |
Find the largest natural number \( n \) for which the system of inequalities
\[ 1 < x < 2, \]
\[ 2 < x^2 < 3, \]
\[ \vdots \]
\[ n < x^n < n+1 \]
has a solution.
|
n = 4
| 0.75 |
Find all values of \( a \) for which the system
$$
\left\{\begin{array}{l}
2^{b x}+(a+1) b y^{2}=a^{2} \\
(a-1) x^{3}+y^{3}=1
\end{array}\right.
$$
has at least one solution for any value of \( b \) (\(a, b, x, y \in \mathbf{R}\)).
|
-1
| 0.875 |
The base of a quadrangular pyramid is a rhombus \(ABCD\) in which \(\angle BAD = 60^\circ\). It is known that \(SD = SB\) and \(SA = SC = AB\). Point \(E\) is taken on edge \(DC\) such that the area of triangle \(BSE\) is the smallest among all sections of the pyramid containing segment \(BS\) and intersecting segment \(DC\). Find the ratio \(DE : EC\).
|
2:5
| 0.25 |
The length of the diagonal of a rectangular parallelepiped is 3. What is the maximum possible surface area of such a parallelepiped?
|
18
| 0.75 |
Several children purchase two types of items priced at 3 yuan and 5 yuan each. Each child buys at least one item, but the total amount spent by each child must not exceed 15 yuan. Xiaomin stated that among the children, there are definitely at least three children who buy the same quantity of each type of item. What is the minimum number of children?
|
25
| 0.875 |
Let \( A B C D \) be a convex quadrilateral inscribed in a circle \( \Gamma \). The line parallel to \( (B C) \) passing through \( D \) intersects \( (C A) \) at \( P \), \( (A B) \) at \( Q \), and intersects the circle \( \Gamma \) again at \( R \). The line parallel to \( (A B) \) passing through \( D \) intersects \( (C A) \) at \( S \), \( (B C) \) at \( T \), and intersects the circle \( \Gamma \) again at \( U \).
Show that if \( P Q = Q R \), then \( S T = T U \).
|
TS = TU
| 0.875 |
Given the plane vectors $a, b,$ and $c$ such that:
$$
\begin{array}{l}
|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=2, \boldsymbol{a}+\boldsymbol{b}+\boldsymbol{c}=\mathbf{0}.
\end{array}
$$
If $0 \leqslant x \leqslant \frac{1}{2} \leqslant y \leqslant 1$, then find the minimum value of
$$
|x(\boldsymbol{a}-\boldsymbol{c})+y(\boldsymbol{b}-\boldsymbol{c})+\boldsymbol{c}|.
$$
|
\frac{1}{2}
| 0.25 |
As shown in the right image, a frog stands at position 1. On the first jump, it jumps 1 step to position 2; on the second jump, it jumps 2 steps to position 4; on the third jump, it jumps 3 steps to position 1; ...; on the nth jump, it jumps n steps. After the frog has jumped 20 times in a clockwise direction, it will be at position ( ).
|
1
| 0.25 |
Calculate the sum
$$
S=\frac{2014}{2 \cdot 5}+\frac{2014}{5 \cdot 8}+\frac{2014}{8 \cdot 11}+\ldots+\frac{2014}{2012 \cdot 2015}
$$
In the answer, indicate the remainder when the even number closest to the obtained value of $S$ is divided by 5.
|
1
| 0.75 |
An apple, pear, orange, and banana were placed in four boxes (one fruit in each box). Labels were made on the boxes:
1. An orange is here.
2. A pear is here.
3. If a banana is in the first box, then either an apple or a pear is here.
4. An apple is here.
It is known that none of the labels correspond to reality.
Determine which fruit is in which box. In the answer, record the box numbers sequentially, without spaces, where the apple, pear, orange, and banana are located, respectively (the answer should be a 4-digit number).
|
2431
| 0.875 |
Every day, from Monday to Friday, an old man went to the blue sea and cast his net into the sea. Each day, the number of fish caught in the net was no greater than the number caught on the previous day. In total, over the five days, the old man caught exactly 100 fish. What is the minimum total number of fish he could have caught over three specific days: Monday, Wednesday, and Friday?
|
50
| 0.25 |
What is the smallest area a right triangle can have if its hypotenuse contains the point \(M(1, 0)\) and its legs lie on the lines \(y = -2\) and \(x = 0\)?
|
S_{min} = 4
| 0.75 |
From 1999 to 2010, over 12 years, the price increase was $150\%$ (i.e., in 1999, an item costing 100 yuan would cost 150 yuan more in 2010 to purchase the same item). If the wages of a company's frontline employees have not changed in these 12 years, calculate the percentage decrease in their wages in terms of purchasing power.
|
60 \%
| 0.875 |
There are 16 people in Nastya's room, each of whom either is friends with or hostile to every other person. Upon entering the room, each person writes down the number of friends who are already there, and upon leaving, writes down the number of enemies still remaining in the room. What can the sum of all the recorded numbers be after everyone has first entered and then left the room?
|
120
| 0.375 |
The diagram shows a right-angled triangle \( ACD \) with a point \( B \) on the side \( AC \). The sides of triangle \( ABD \) have lengths 3, 7, and 8. What is the area of triangle \( BCD \)?
|
2\sqrt{3}
| 0.875 |
Given a natural number \( n \), find the number of distinct triplets of natural numbers whose sum is \( 6n \).
|
3n^2
| 0.125 |
Calculate $\Delta f\left(P_{0}\right)$ and $d f\left(P_{0}\right)$ for the function $f(x, y) = x^{2} y$ at the point $P_{0} = (5, 4)$ with $\Delta x = 0.1$ and $\Delta y = -0.2$.
|
-1.162
| 0.25 |
A subset \( H \) of the set of numbers \(\{1, 2, \ldots, 100\}\) has the property that if an element is in \( H \), then ten times that element is not in \( H \). What is the maximum number of elements that \( H \) can have?
|
91
| 0.125 |
Let \( a \) and \( b \) be positive numbers. Find the minimum value of the fraction \(\frac{(a+b)(a+2)(b+2)}{16ab}\). Justify your answer.
|
1
| 0.75 |
Find the sum of all factors \( d \) of \( N = 19^{88} - 1 \) that are of the form \( d = 2^a \cdot 3^b \) (where \( a \) and \( b \) are natural numbers).
|
744
| 0.375 |
Given set \( S \) such that \( |S| = 10 \). Let \( A_1, A_2, \cdots, A_k \) be non-empty subsets of \( S \), and the intersection of any two subsets contains at most two elements. Find the maximum value of \( k \).
|
175
| 0.375 |
Point \( F \) is the midpoint of side \( BC \) of square \( ABCD \). A perpendicular \( AE \) is drawn to segment \( DF \). Find the angle \( CEF \).
|
45^{\circ}
| 0.875 |
In a convex quadrilateral, the lengths of the diagonals are 2 cm and 4 cm. Find the area of the quadrilateral, given that the lengths of the segments connecting the midpoints of the opposite sides are equal.
|
4 \text{ cm}^2
| 0.75 |
Points \( P \) and \( Q \) are located on the sides \( AB \) and \( AC \) of triangle \( ABC \) such that \( AP:PB = 1:4 \) and \( AQ:QC = 3:1 \). Point \( M \) is chosen randomly on side \( BC \). Find the probability that the area of triangle \( ABC \) exceeds the area of triangle \( PQM \) by no more than two times. Find the mathematical expectation of the random variable - the ratio of the areas of triangles \( PQM \) and \( ABC \).
|
\frac{13}{40}
| 0.625 |
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