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Two friends agree to meet at a specific place between 12:00 PM and 12:30 PM. The first one to arrive waits for the other for 20 minutes before leaving. Find the probability that the friends will meet, assuming each chooses their arrival time randomly (between 12:00 PM and 12:30 PM) and independently.
|
\frac{8}{9}
| 0.875 |
A reservoir has 400 tons of water. At midnight each day, the inlet and outlet gates are opened simultaneously. The amount of water $w$ (tons) flowing out through the outlet gate is a function of time $t$ (hours): $w = 120 \sqrt{6t} \text{ for } (0 \leq t \leq 24)$.
(1) In order to ensure that the reservoir still contains 400 tons of water by midnight the next day, how many tons of water need to be added to the reservoir per hour (assuming the rate of inflow is constant each hour)?
(2) Under the conditions of part (1), at what time of day will the amount of water in the reservoir be at its minimum, and how many tons of water will be in the reservoir at that time?
|
40
| 0.125 |
Eight people sit around a circular table, each of whom is either a knight or a liar. When asked about their neighbors, each of them answered: "My neighbors are a liar and a knight." How many liars are among them? How would the answer change if there were nine people at the table?
|
3
| 0.125 |
Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beautiful wings. If the areas of the two triangles that she cut off are 4 square centimeters and 9 square centimeters respectively, then the area of the wings that Sasa made is $\qquad$ square centimeters.
|
12
| 0.75 |
If the function \( f(x) = x^3 - 6x^2 + 9x \) has a maximum value in the interval \( \left(3-a^2, a\right) \), then the range of the real number \( a \) is \(\qquad\)
|
(\sqrt{2}, 4]
| 0.125 |
Let \( P \) and \( Q \) be points on line \( l \) with \( PQ = 12 \). Two circles, \( \omega \) and \( \Omega \), are both tangent to \( l \) at \( P \) and are externally tangent to each other. A line through \( Q \) intersects \( \omega \) at \( A \) and \( B \), with \( A \) closer to \( Q \) than \( B \), such that \( AB = 10 \). Similarly, another line through \( Q \) intersects \( \Omega \) at \( C \) and \( D \), with \( C \) closer to \( Q \) than \( D \), such that \( CD = 7 \). Find the ratio \( AD / BC \).
|
\frac{8}{9}
| 0.125 |
How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \]
|
12
| 0.875 |
Find all pairs of natural numbers \( x \) and \( y \) such that the ratio \(\frac{x y^{3}}{x+y}\) is a prime number.
|
(14, 2)
| 0.625 |
Given the complex number \( z_{1}=\mathrm{i}(1-\mathrm{i})^{3} \), when the complex number \( z \) satisfies \( |z|=1 \), find the maximum value of \( \left|z-z_{1}\right| \).
|
1 + 2\sqrt{2}
| 0.125 |
For which values of the parameter \(a\) does the equation \(x^{3} + 16x^{2} + ax + 64 = 0\) have three distinct real roots that form a geometric progression?
|
64
| 0.375 |
In a notebook, there is a grid rectangle of size $3 \times 7$. Igor's robot was asked to trace all the lines with a marker, and it took him 26 minutes (the robot draws lines at a constant speed). How many minutes will it take him to trace all the lines in a $5 \times 5$ grid square with the marker?
|
30 \text{ minutes}
| 0.875 |
For what smallest natural number \( k \) does the quadratic trinomial
\[ y = kx^2 - px + q \]
with natural coefficients \( p \) and \( q \) have two distinct positive roots less than 1?
|
5
| 0.75 |
In figure 1, \( \triangle ABC \) and \( \triangle EBC \) are two right-angled triangles with \( \angle BAC = \angle BEC = 90^\circ \). Given that \( AB = AC \) and \( EDB \) is the angle bisector of \( \angle ABC \), find the value of \( \frac{BD}{CE} \).
|
2
| 0.125 |
The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).
|
984
| 0.5 |
A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?
|
P_{2007}
| 0.375 |
In the coordinate plane, draw the following lines \( y = k \), \( y = \sqrt{3} x + 2k \), and \( y = -\sqrt{3} x + 2k \), where \( k = 0, \pm 1, \pm 2, \ldots, \pm 10 \). These 63 lines divide the plane into several equilateral triangles. Find the number of equilateral triangles with a side length of \( \frac{2}{\sqrt{3}} \).
|
660
| 0.125 |
Given a regular triangular pyramid $P-ABC$ with the side length of the base equilateral triangle being 1, the circumsphere center $O$ satisfies $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{0}$. Find the volume of this regular triangular pyramid.
|
\frac{1}{12}
| 0.25 |
Find the square numbers such that when divided by 11, the quotient is a prime number and the remainder is 4.
|
n^2 = 81
| 0.125 |
In triangle $ABC$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is given that $AC - AB = 5$ cm. Find the length of the angle bisector of angle $B$.
|
5 \text{ cm}
| 0.875 |
Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left focus \( F \), and a line \( l \) passing through \((1,1)\) that intersects the ellipse at points \( A \) and \( B \). When the perimeter of triangle \( \triangle FAB \) is at its maximum, what is the area of triangle \( \triangle FAB \)?
|
3
| 0.75 |
In the drawing, there is a grid consisting of 25 small equilateral triangles.
How many rhombuses can be formed from two adjacent small triangles?
|
30
| 0.125 |
Given that the graph of the function \( y=\frac{\left|x^{2}-1\right|}{x-1} \) intersects with the graph of the function \( y=kx-2 \) at two points. Determine the range of the real number \( k \).
|
(0,1)\cup(1,4)
| 0.375 |
By calculating the limit $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$, find the derivatives of the following functions:
1) $y = 3x^2 - 4x$
2) $y = \frac{1}{x}$
3) $y = \sqrt{x}$
4) $y = \cos 3x$
|
f'(x) = -3\sin (3x)
| 0.125 |
Let the random variables $\xi$ and $\eta$ denote the lifetimes of the blue and red light bulbs, respectively. The lifetime of the flashlight is equal to the minimum of these two values. Clearly, $\min (\xi, \eta) \leq \xi$. Let's move to the expected values: $\operatorname{E} \min (\xi, \eta) \leq \mathrm{E} \xi=2$. Therefore, the expected lifetime of the flashlight is no more than 2 years.
|
2
| 0.875 |
In the coordinate plane, a point whose x-coordinate and y-coordinate are both integers is called a lattice point. For any natural number \( n \), let us connect the origin \( O \) with the point \( A_{n}(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the segment \( OA_{n} \) excluding the endpoints. Find the value of \( f(1) + f(2) + \cdots + f(1990) \).
|
1326
| 0.75 |
There are 90 players in a league. Each of the 90 players plays exactly one match against each of the other 89 players. Each match ends with a win for one player and a loss for the other player, or with a tie for both players. Each player earns 1 point for a win, 0 points for a loss, and 0.5 points for a tie. After all matches have been played, the points earned by each player are added up. What is the greatest possible number of players whose total score can be at least 54 points?
|
71
| 0.5 |
Calculate the volumes of solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.
$$
y = (x-1)^{2}, x=0, x=2, y=0
$$
|
\frac{4\pi}{3}
| 0.625 |
Seventeen people dine every Saturday night around a round table. How many times is it possible for them to dine if each person wants to have two new neighbors each time? What is the result for eighteen people?
|
8
| 0.75 |
Let \( f(x) = x^2 + ax + b \) have two real roots in the interval \([0,1]\). Then, find the range of values of \( a^2 - 2b \).
|
[0, 2]
| 0.375 |
Given a cyclic quadrilateral \(A B C D\) with side lengths \(AB = 1\), \(BC = 3\), \(CD = DA = 2\), find the area of quadrilateral \(A B C D\).
|
2\sqrt{3}
| 0.5 |
Let \( x > 0 \), plane vectors \( \overrightarrow{A B} = (0,1), \overrightarrow{B C} = (1,2) \), and \( \overrightarrow{C D} = \left(\log_{3} x, \log_{9} x\right) \). If \( \overrightarrow{A C} \cdot \overrightarrow{B D} = 2 \), then find the value of \( x \).
|
\frac{1}{9}
| 0.875 |
Numbers \( p \) and \( q \) are such that the parabolas \( y = -2x^2 \) and \( y = x^2 + px + q \) intersect at two points, enclosing a certain region.
Find the equation of the vertical line that divides the area of this region in half.
|
x = -\frac{p}{6}
| 0.75 |
Arrange the numbers
$$
\begin{gathered}
x=(a+b)(c+d) \\
y=(a+c)(b+d) \\
z=(a+d)(b+c)
\end{gathered}
$$
in ascending order, given that \(a < b < c < d\).
|
x < y < z
| 0.875 |
Kiril Konstantinovich's age is 48 years, 48 months, 48 weeks, 48 days, and 48 hours. How many full years old is Kiril Konstantinovich?
|
53
| 0.75 |
Given that the real numbers \(a\), \(b\), and \(c\) satisfy \(\left|a x^{2} + b x + c\right|\) with a maximum value of 1 on the interval \(x \in [-1,1]\), what is the maximum possible value of \(\left|c x^{2} + b x + a\right|\) on the interval \(x \in [-1,1]\)?
|
2
| 0.875 |
Given the set \( A = \{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\} \), calculate the sums of the two-element subsets of \( A \) to form the set \( B = \{4, 5, 6, 7, 8, 9, 10, 12, 13, 14\} \). Determine the set \( A \).
|
\{1, 3, 4, 5, 9\}
| 0.75 |
Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the plane.
\[ M(2, -1, 1) \]
\[ x - y + 2z - 2 = 0 \]
|
(1, 0, -1)
| 0.75 |
Does there exist a function \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(f(n)) = n + 1987 \) for every natural number \( n \)?
|
\text{No}
| 0.125 |
Using 125 small cube blocks to form a larger cube, some of the small cube blocks on the surface of the larger cube are painted. The painting on the opposite pairs of faces - top and bottom, left and right, front and back - corresponds to each other. Among these 125 small cube blocks, there are $\qquad$ blocks that have none of their faces painted.
|
27
| 0.875 |
There are some stamps with denominations of 0.5 yuan, 0.8 yuan, and 1.2 yuan, with a total value of 60 yuan. The number of 0.8 yuan stamps is four times the number of 0.5 yuan stamps. How many 1.2 yuan stamps are there?
|
13
| 0.625 |
Diligent Masha wrote out all natural numbers from 372 to 506 inclusive. Then she calculated two sums: first the sum of all odd numbers in this range, and then the sum of all even numbers. After that, she subtracted the smaller sum from the larger sum. What result did she get?
|
439
| 0.5 |
Find the maximum value of the expression for \( a, b > 0 \):
$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$
|
2 \sqrt{87}
| 0.375 |
Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller.
|
512820
| 0.5 |
Toward the end of a game of Fish, the 2 through 7 of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players' hands be distributed?
|
450
| 0.75 |
The coefficients \( p \) and \( q \) of the quadratic equation \( x^{2}+p x+q=0 \) are chosen at random in the interval \( (0, 2) \). What is the probability that the roots of this equation will be real numbers?
|
\frac{1}{6}
| 0.875 |
What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square?
|
337
| 0.75 |
Find the number of solutions to $\sin x = \lg x$.
|
3
| 0.125 |
Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and heads towards point B. Half an hour later, it meets Vehicle B again. How many hours does it take for Vehicle B to travel from A to B?
|
7.2
| 0.375 |
A point inside an equilateral triangle is at distances of 3, 4, and 5 units from the sides. What is the length of the side of the triangle?
|
8 \sqrt{3}
| 0.75 |
There is a cube placed on stands and six different paints. How many ways can you paint all the faces of the cube (one color for each face, not necessarily using all the paints) so that adjacent faces (sharing an edge) are painted in different colors? (16 points)
|
4080
| 0.25 |
Let $PA$, $PB$, and $PC$ be three non-coplanar rays originating from point $P$, with each pair of rays forming a $60^\circ$ angle. A sphere with a radius of 1 is tangent to each of these three rays. Find the distance from the center of the sphere $O$ to point $P$.
|
\sqrt{3}
| 0.5 |
a) In a three-digit number, the first digit on the left was erased. Then, the resulting two-digit number was multiplied by 7, and the original three-digit number was obtained. Find such a number.
b) In a three-digit number, the middle digit was erased, and the resulting number is 6 times smaller than the original. Find such a three-digit number.
|
108
| 0.875 |
Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly chosen inside the quadrilateral \( AA'B'B \), what is the probability that \( M \) lies inside the triangle \( FA'B' \)?
|
\frac{1}{3}
| 0.75 |
A set of distinct positive integers has a sum of 1987. What is the maximum possible value for three times the total number of integers plus the number of odd integers?
|
221
| 0.25 |
A rectangular grid is given. Let's call two cells neighboring if they share a common side. Count the number of cells that have exactly four neighboring cells. There are 23 such cells. How many cells have three neighboring cells?
|
48
| 0.75 |
In the side face $A A^{\prime} B^{\prime} B$ of a unit cube $A B C D - A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ such that its distances to the two lines $A B$ and $B^{\prime} C^{\prime}$ are equal. What is the minimum distance from a point on the trajectory of $M$ to $C^{\prime}$?
|
\frac{\sqrt{5}}{2}
| 0.625 |
Given a sequence $\left\{a_{n}\right\}$, where $a_{1}=a_{2}=1$, $a_{3}=-1$, and $a_{n}=a_{n-1} a_{n-3}$, find $a_{1964}$.
|
-1
| 0.25 |
In trapezoid \(ABCD\), the side \(BC\) is equal to the diagonal \(BD\). On the smaller arc \(AB\) of the circumcircle of triangle \(ABC\), a point \(E\) is chosen such that \(BC = BE\). Find the angle \(\angle AED\).
|
90^\circ
| 0.5 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{x}\right)^{\cos ^{2}\left(\frac{\pi}{4}+x\right)}
$$
|
\sqrt{3}
| 0.75 |
Out of 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What can be the maximum perimeter of this rectangle?
|
184
| 0.375 |
The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_1 = 1\), and for each \(n \in \mathbf{N}^{*}\), \(a_n\) and \(a_{n+1}\) are the roots of the equation \(x^2 + 3nx + b_n = 0\). Find \(\sum_{k=1}^{20} b_k\).
|
6385
| 0.375 |
There are 5 black and white pieces arranged in a circle. The rule is: place a white piece between two adjacent pieces of the same color, and place a black piece between two adjacent pieces of different colors, then remove the original 5 pieces. Starting from the initial state as shown in figure (1), and following the above rule, determine the maximum number of black pieces that can be present on the circle.
|
4
| 0.75 |
Find the sum of the squares of the roots of the equation \(\left(x^{2}+6x\right)^{2}-1580\left(x^{2}+6x\right)+1581=0\).
|
3232
| 0.75 |
There are 4 balls of different masses. How many weighings on a balance scale without weights are needed to arrange these balls in descending order of mass?
|
5
| 0.625 |
In the cube \( A C_{1} \), determine the magnitude of the dihedral angle \( C_{1}-D_{1} B-C \).
|
60^\circ
| 0.75 |
A deck of cards contains 52 cards, with each suit ("Diamonds", "Clubs", "Hearts", "Spades") having 13 cards. The ranks are $2, 3, \ldots, 10, J, Q, K, A$. Two cards of the same suit with consecutive ranks are called "straight flush" cards, and $A$ with 2 is also considered as consecutive (i.e., $A$ can be used as 1). Determine the number of ways to select 13 cards from this deck such that every rank appears and no "straight flush" cards are included.
|
1594320
| 0.375 |
Find the number of ordered integer pairs \((m, n)\), where \(1 \leqslant m \leqslant 99\), \(1 \leqslant n \leqslant 99\), such that \((m+n)^2 + 3m + n\) is a perfect square.
|
98
| 0.875 |
In $\triangle ABC$ shown in Figure 2-13, points $D$ and $E$ lie on the extensions of $AC$ and $AB$ respectively. $BD$ and $CE$ intersect at $P$, and $BD = CE$. If point $P$ satisfies $\angle AEP - \angle ADP = k^2(\angle PED - \angle PDE)$ where $k$ is a constant, then $AB = AC$.
|
AB = AC
| 0.875 |
What is the maximum value of the sum of the cosines of all the angles of an isosceles triangle?
|
1.5
| 0.25 |
Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(3) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the gcd of $a + b$ and $c$ is greater than 1.
|
50
| 0.375 |
The box contains 120 colored pencils: 35 red, 23 green, 14 yellow, 26 blue, 11 brown, and 11 black. What is the minimum number of pencils one needs to take from the box in the dark (without seeing the pencils) to ensure that there are at least 18 pencils of one color among them?
|
88
| 0.75 |
How many pairs of positive integers $(x, y)$ are solutions to the equation $3x + 5y = 501?$
|
33
| 0.75 |
Inflation over two years will be:
$$
\left((1+0,025)^{\wedge 2-1}\right)^{*} 100 \%=5,0625 \%
$$
The real interest rate of a bank deposit with reinvestment for the second year will be $(1.06 * 1.06 /(1+0,050625)-1) * 100=6,95 \%$
|
6.95\%
| 0.625 |
Let \( m = 76^{2006} - 76 \). Find the remainder when \( m \) is divided by 100.
|
0
| 0.875 |
Let \( O \) be the vertex of the parabola and \( F \) be the focus, and let \( PQ \) be a chord passing through \( F \). Given \( |OF| = a \) and \( |PQ| = b \), find the area of triangle \( OPQ \).
|
a \sqrt{ab}
| 0.125 |
For any positive integers \( x \), \( y \), and \( z \), if \( x \mid y^3 \), \( y \mid z^3 \), \( z \mid x^3 \), and \( xyz \mid (x + y + z)^n \), find the smallest positive integer \( n \) that satisfies these conditions.
|
13
| 0.25 |
Find the sum of all three-digit natural numbers that do not contain the digits 0 or 5.
|
284160
| 0.75 |
In the square \( ABCD \) with side length 1, point \( F \) is the midpoint of side \( BC \), and \( E \) is the foot of the perpendicular dropped from vertex \( A \) to \( DF \).
Find the length of \( BE \).
|
1
| 0.75 |
Find the number of consecutive 0's at the end of the base 10 representation of 2006!.
|
500
| 0.75 |
Vasya needs to write one digit on each face of several cubes in such a way that any ordered combination of three digits from 000 to 999 inclusive can be obtained by selecting some three different cubes and placing them with suitable faces up in the correct order. Note that the digits 6 and 9 are not considered interchangeable when rotated by 180 degrees. What is the minimum number of cubes that Vasya must use?
|
5
| 0.875 |
What is the largest area that a figure in the $xy$-plane can have, located between the lines $x = -3$ and $x = 1$, and bounded below by the $x$-axis, and above by a tangent to the graph of the function $y = x^2 + 16$ at the point of tangency with the abscissa $x_0$ in the interval $-3 \leq x_0 \leq 1$?
|
68
| 0.875 |
Let \(ABC\) be a triangle with area 1. Let points \(D\) and \(E\) lie on \(AB\) and \(AC\), respectively, such that \(DE\) is parallel to \(BC\) and \(\frac{DE}{BC} = \frac{1}{3}\). If \(F\) is the reflection of \(A\) across \(DE\), find the area of triangle \(FBC\).
|
\frac{1}{3}
| 0.75 |
Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3}+y^{2}=1 \). A line \( l \) with a positive slope passes through point \( F \) and intersects \( E \) at points \( A \) and \( B \). Through points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn such that \( AM \perp l \) and \( BN \perp l \), each intersecting the x-axis at points \( M \) and \( N \), respectively. Find the minimum value of \( |MN| \).
|
\sqrt{6}
| 0.75 |
\[ 2016 \times 2014 - 2013 \times 2015 + 2012 \times 2015 - 2013 \times 2016 = \]
|
1
| 0.875 |
Let \( x * y = x + y - xy \), where \( x \) and \( y \) are real numbers. If \( a = 1 * (0 * 1) \), find the value of \( a \).
|
1
| 0.625 |
In how many ways can you place two knights - one white and one black - on a $16 \times 16$ chessboard so that they threaten each other? (A knight moves in an "L" shape, i.e., it moves to one of the nearest squares that is not on the same rank, file, or diagonal as its current position.)
|
1680
| 0.125 |
Given $\left\{\begin{array}{l}\sin \alpha+\sin \beta=1 \\ \cos \alpha+\cos \beta=0\end{array}\right.$, find $\cos 2 \alpha+\cos 2 \beta$.
|
1
| 0.875 |
Non-negative real numbers \( x_{1}, x_{2}, \cdots, x_{2016} \) and real numbers \( y_{1}, y_{2}, \cdots, y_{2016} \) satisfy:
(1) \( x_{k}^{2}+y_{k}^{2}=1 \) for \( k=1,2, \cdots, 2016 \);
(2) \( y_{1}+y_{2}+\cdots+y_{2016} \) is an odd number.
Find the minimum value of \( x_{1}+x_{2}+\cdots+x_{2016} \).
|
1
| 0.75 |
An isosceles trapezoid with a $30^{\circ}$ angle is circumscribed around a circle. Its midline is equal to 10. Find the radius of the circle.
|
2.5
| 0.875 |
Given 10 distinct points on a plane, consider the midpoints of all segments connecting all pairs of points. What is the minimum number of such midpoints that could result?
|
17
| 0.125 |
Two vertical chords are drawn in a circle, dividing the circle into 3 distinct regions. Two horizontal chords are added in such a way that there are now 9 regions in the circle. A fifth chord is added that does not lie on top of one of the previous four chords. The maximum possible number of resulting regions is \( M \) and the minimum possible number of resulting regions is \( m \). What is \( M^{2} + m^{2} \)?
|
296
| 0.375 |
In triangle ABC with sides AB = 5, BC = √17, and AC = 4, point M is taken on side AC such that CM = 1. Find the distance between the centers of the circumcircles of triangles ABM and BCM.
|
2
| 0.75 |
Solve the following equation:
$$
\frac{3+2 x}{1+2 x}-\frac{5+2 x}{7+2 x}=1-\frac{4 x^{2}-2}{7+16 x+4 x^{2}}
$$
|
\frac{7}{8}
| 0.5 |
On the sides $AB$ and $CD$ of rectangle $ABCD$, points $E$ and $F$ are marked such that $AFCE$ forms a rhombus. It is known that $AB = 16$ and $BC = 12$. Find $EF$.
|
15
| 0.625 |
In a swamp, there are 64 tufts of grass arranged in an \(8 \times 8\) square. On each tuft, there is either a frog or a toad. Frogs always tell the truth, while toads always lie. Each of them, both the frogs and the toads, proclaimed: "At least one of the neighboring tufts has a toad." What is the maximum number of toads that could be on these tufts?
Tufts are considered neighbors if they are horizontally or vertically adjacent in the rows and columns of the square; diagonal tufts are not considered neighbors.
|
32
| 0.875 |
Let the set \( M = \{1, 2, \cdots, 2020\} \). For any non-empty subset \( X \) of \( M \), let \( \alpha_X \) represent the sum of the maximum and minimum numbers in \( X \). Find the arithmetic mean of all such \( \alpha_X \).
|
2021
| 0.75 |
Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters?
|
28 \text{ meters}
| 0.25 |
Find the sum of all fractions in lowest terms with value greater than 10 but smaller than 100 and with the denominator equal to 3.
|
9900
| 0.625 |
Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - 4f(x) = 0 \) has exactly three solutions. How many solutions does the equation \( (f(x))^2 = 1 \) have?
|
2
| 0.875 |
A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left.
|
\frac{2}{3}
| 0.875 |
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