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64,974 | 3. The equation of the circle passing through the intersection points of the curves $y=\frac{1}{r}$ and $\mathrm{y}=\mathrm{x}^{2}+3 \mathrm{x}-7$ is $\qquad$ ; | (x+1)^{2}+(y+3)^{2}=20 |
64,993 | In a right-angled triangle $A B C$, from vertices $B$ and $C$, one point each starts moving simultaneously towards point $A$ along the hypotenuse $B A=c=85$ m and the leg $C A=b=75$ m, respectively. The points move at speeds of $8.5 \mathrm{~m} / \mathrm{sec}$ and $5 \mathrm{~m} / \mathrm{sec}$. At what time will the distance between the two points be $26 \mathrm{~m}$. (Without using trigonometry!) | 4 |
64,994 | 8. The school stage of the Magic and Wizardry Olympiad consists of 5 spells. Out of 100 young wizards who participated in the competition,
- 95 correctly performed the 1st spell
- 75 correctly performed the 2nd spell
- 97 correctly performed the 3rd spell
- 95 correctly performed the 4th spell
- 96 correctly performed the 5th spell.
What is the minimum number of students who could have correctly performed exactly 4 out of 5 spells under the described conditions? | 8 |
64,999 | 3. In triangle $A B C$, we have
$$
|\angle B A C|=40^{\circ},|\angle C B A|=20^{\circ} \text { and }|A B|-|B C|=10
$$
The angle bisector of $\angle A C B$ intersects the segment $\overline{A B}$ at point $M$. Determine the length of the segment $\overline{C M}$. | 10 |
65,005 | 17. 5555 children, numbered 1 to 5555 , sit around a circle in order. Each child has an integer in hand. The child numbered 1 has the integer 1 ; the child numbered 12 has the integer 21; the child numbered 123 has the integer 321 and the child numbered 1234 has the integer 4321. It is known that the sum of the integers of any 2005 consecutive children is equal to 2005 . What is the integer held by the child numbered 5555 ?
(2 marks)
有 5555 名小孩, 他們編號為 1 至 5555 , 並順序圍圈而坐。每人手上均有一個整數：編號為 1 的小孩手中的整數為 1 , 編號為 12 的小孩手中的整數為 21 , 編號為 123 的小孩手中的整數為 321 , 編號為 1234 的小孩手中的整數為 $4321 \circ$ 已知任意連續 2005 位小孩手上的整數之和均為 2005 。問編號為 5555 的小孩手中的整數是甚麼？ | -4659 |
65,029 | We choose $n$ points on a circle and draw all the associated chords (we make sure that 3 chords are never concurrent). Into how many parts is the circle divided? | \frac{n^{4}-6n^{3}+23n^{2}-18n+24}{24} |
65,055 | 6. Let the set $M=\{1,2, \cdots, 2020\}$. For any non-empty subset $X$ of $M$, let $\alpha_{X}$ denote the sum of the largest and smallest numbers in $X$. Then the arithmetic mean of all such $\alpha_{X}$ is $\qquad$ . | 2021 |
65,068 | 3. For any pair of numbers, a certain operation «*» is defined, satisfying the following properties: $a *(b * c)=(a * b) \cdot c$ and $a * a=1$, where the operation «$\cdot$» is the multiplication operation. Find the root $x$ of the equation: $\quad x * 2=2018$. | 4036 |
65,074 | Given the equation $x^{2}+p x+q=0$. Let's write down the quadratic equation in $y$ whose roots are
$$
y_{1}=\frac{x_{1}+x_{1}^{2}}{1-x_{2}} \quad \text { and } \quad y_{2}=\frac{x_{2}+x_{2}^{2}}{1-x_{1}}
$$
where $x_{1}$ and $x_{2}$ are the roots of the given equation. | (1+p+q)y^{2}+p(1+3q-p^{2})y+q(1-p+q)=0 |
65,079 | 5. Find the maximum value of the expression $(\sin 3 x+\sin 2 y+\sin z)(\cos 3 x+\cos 2 y+\cos z)$. $(15$ points) | 4.5 |
65,114 | 1. A sequence of ants walk from $(0,0)$ to $(1,0)$ in the plane. The $n$th ant walks along $n$ semicircles of radius $\frac{1}{n}$ with diameters lying along the line from $(0,0)$ to $(1,0)$. Let $L_{n}$ be the length of the path walked by the $n$th ant. Compute $\lim L_{n}$. | \pi |
65,127 | Let $ABC$ be an equilateral triangle with side length 16. Three circles of the same radius $r$ are tangent to each other in pairs, and each of these circles is tangent to two sides of the triangle. The radius $r$ can be written as $r=\sqrt{a}-b$ where $a$ and $b$ are integers. Determine $a+b$.
Soit $A B C$ un triangle équilatéral de côté 16. Trois cercles de même rayon $r$ sont tangents entre eux deux à deux, et chacun de ces cercles est tangent à deux côtés du triangle. Le rayon $r$ s'écrit $r=\sqrt{a}-b$ où $a$ et $b$ sont des entiers. Déterminer $a+b$.
Let $ABC$ be an equilateral triangle with side length 16. Three circles of the same radius $r$ are tangent to each other in pairs, and each of these circles is tangent to two sides of the triangle. The radius $r$ can be written as $r=\sqrt{a}-b$ where $a$ and $b$ are integers. Determine $a+b$. | 52 |
65,144 | 1. [3] Triangle $A B C$ is isosceles, and $\angle A B C=x^{\circ}$. If the sum of the possible measures of $\angle B A C$ is $240^{\circ}$, find $x$. | 20 |
65,151 | Call a positive integer [i]prime-simple[/i] if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple? | 6 |
65,156 | 115. Division by 37. I would like to know if the number 49129308213 is divisible by 37, and if not, what the remainder is. How can I do this without performing the division? It turns out that with a skilled approach, the answer to my question can be obtained in a few seconds. | 33 |
65,168 | 10. (20 points) Let the sequence of non-negative integers $\left\{a_{n}\right\}$ satisfy:
$$
a_{n} \leqslant n(n \geqslant 1) \text {, and } \sum_{k=1}^{n-1} \cos \frac{\pi a_{k}}{n}=0(n \geqslant 2) \text {. }
$$
Find all possible values of $a_{2021}$. | 2021 |
65,173 | 2. Points $X, Y$, and $Z$ lie on a circle with center $O$ such that $X Y=12$. Points $A$ and $B$ lie on segment $X Y$ such that $O A=A Z=Z B=B O=5$. Compute $A B$. | 2\sqrt{13} |
65,184 | $\underline{\text { Folklore }}$
Solve the equation: $2 \sqrt{x^{2}-16}+\sqrt{x^{2}-9}=\frac{10}{x-4}$. | 5 |
65,207 | Let $ABC$ be an isosceles right-angled triangle, having the right angle at vertex $C$. Let us consider the line through $C$ which is parallel to $AB$ and let $D$ be a point on this line such that $AB = BD$ and $D$ is closer to $B$ than to $A$. Find the angle $\angle CBD$. | 105^\circ |
65,210 | $1.$A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty space in the upper part of the cone that is 2 centimeters high.
Find the height of the bottle. | 10 |
65,229 | 11.2. Given different real numbers $a_{1}, a_{2}, a_{3}$ and $b$. It turned out that the equation $\left(x-a_{1}\right)\left(x-a_{2}\right)\left(x-a_{3}\right)=b$ has three different real roots $c_{1}, c_{2}, c_{3}$. Find the roots of the equation $(x+$ $\left.+c_{1}\right)\left(x+c_{2}\right)\left(x+c_{3}\right)=b$.
(A. Antropov, K. Sukhov) | -a_{1},-a_{2},-a_{3} |
65,230 | 5. In the spatial quadrilateral $ABCD$, $AB=2, BC=$ $3, CD=4, DA=5$. Then $\overrightarrow{AC} \cdot \overrightarrow{BD}=$ $\qquad$ | 7 |
65,245 | ## Task Condition
Find the derivative.
$$
y=\frac{3+x}{2} \cdot \sqrt{x(2-x)}+3 \arccos \sqrt{\frac{x}{2}}
$$ | -\frac{x^{2}}{\sqrt{x(2-x)}} |
65,246 | 72. A positive integer $x$, if it is appended to any two positive integers, the product of the two new numbers still ends with $x$, then $x$ is called a "lucky number". For example: 6 is a "lucky number"; but 16 is not, because $116 \times 216=25056$, the end is no longer 16. The sum of all "lucky numbers" with no more than 3 digits is . $\qquad$ | 1114 |
65,250 | 9.5. Find the maximum value of the expression $(\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot(3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y) \cdot$ If the answer is not an integer, round it to the nearest integer. | 27 |
65,275 | 4. Use $1,2,3,4,5,6$ to form a six-digit number without repeating digits. By arbitrarily extracting any two adjacent digits, 5 different two-digit numbers can be obtained. The maximum sum of these 5 two-digit numbers is $\qquad$ . | 219 |
65,283 | 1. Given the function $f(x)=\frac{\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+2 x^{2}+x}{2 x^{2}+\cos x}$ has a maximum value $M$, and a minimum value $m$, then the value of $M+m$ is | 2 |
65,288 | 7. A circle is drawn through two vertices of an equilateral triangle $A B C$ with an area of $21 \sqrt{3} \mathrm{~cm}^{2}$, for which two sides of the triangle are tangents. Find the radius of this circle. | 2\sqrt{7} |
65,299 | 5. (6 points) In the figure, $A B=A D, \angle D B C=21^{\circ}, \angle A C B=39^{\circ}$, then $\angle A B C=$ $\qquad$ degrees. | 81 |
65,320 | Point $ A$ lies at $ (0, 4)$ and point $ B$ lies at $ (3, 8)$. Find the $ x$-coordinate of the point $ X$ on the $ x$-axis maximizing $ \angle AXB$. | 5\sqrt{2} - 3 |
65,339 | 7. For any $x \in\left[-\frac{\pi}{6}, \frac{\pi}{2}\right]$, the inequality $\sin ^{2} x+a \sin x+a+3 \geqslant 0$ always holds. Then the range of the real number $a$ is $\qquad$ . | [-2,+\infty) |
65,366 | 1.3. At the Faculty of Mechanics and Mathematics of Moscow State University, there are $n$ female students and $2 n-1$ male students (both female and male students are numbered). It is known that the $k$-th female student is acquainted with male students numbered from 1 to $2 k-1$. How many ways are there to form $n$ pairs, each consisting of an acquainted male and female student?
## Second Round | n! |
65,380 | 11.2 What is the maximum value that the ratio of the radius of the inscribed circle to the radius of the circumscribed circle of a right triangle can reach? | \sqrt{2}-1 |
65,382 | 50. Given a triangle $ABC$. The tangent to the circumcircle of this triangle at point $B$ intersects the line $AC$ at point $M$. Find the ratio $|AM|:|MC|$, if $|AB|:|BC|=k$. | k^2 |
65,392 | Problem 3. How many natural numbers $n>1$ exist, for which there are $n$ consecutive natural numbers, the sum of which is equal to 2016? | 5 |
65,398 | Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$. | 901 |
65,399 | 20. Let $a, b$ and $c$ be real numbers such that $\frac{a b}{a+b}=\frac{1}{3}, \frac{b c}{b+c}=\frac{1}{4}$ and $\frac{a a}{c+a}=\frac{1}{5}$. Find the value of $\frac{24 a b c}{a b+b c+c a}$. | 4 |
65,402 | (13) Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{n+1}=1+a_{n}+\sqrt{1+4 a_{n}}(n \in$ $\mathbf{N}^{*}$ ), then the general term of the sequence $a_{n}=$ $\qquad$ . | 1+(n-1)(n+\sqrt{5}-1) |
65,418 | ## Task A-3.5.
For a natural number $n \geqslant 2$, let $D(n)$ be the greatest natural divisor of the number $n$ different from $n$. For example, $D(12)=6$ and $D(13)=1$.
Determine the greatest natural number $n$ such that $D(n)=35$. | 175 |
65,419 | 【16】There are four points $A, B, C, D$ arranged from left to right on a straight line. Using these four points as endpoints, 6 line segments can be formed. It is known that the lengths of these 6 line segments are $14, 21, 34, 35, 48, 69$ (unit: millimeters). Then the length of line segment $\mathrm{BC}$ is ( ) millimeters. | 14 |
65,456 | 3. Problem: Find the area of the region bounded by the graphs of $y=x^{2}, y=x$, and $x=2$. | 1 |
65,459 | 5. (10 points) The sum of all four-digit numbers composed of four non-zero digits without repetition is 73326, then the largest of these four-digit numbers is $\qquad$ . | 5321 |
65,462 | 5. Bear Big and Bear Small jog on a circular track that is 1500 meters long every day.
On the first day, both start from the starting point at the same time and run the entire way. When Bear Big completes 4 laps and returns to the starting point, Bear Small has completed 3 laps and an additional 300 meters.
On the second day, Bear Big runs the entire way, while Bear Small alternates between running and walking, with the running speed being twice the walking speed. Both start from the starting point at the same time, and when Bear Big completes 3 laps and returns to the starting point, Bear Small also returns to the starting point.
So, the distance Bear Small walked on the second day is $\qquad$ meters. | 600 |
65,465 | 4. Let the foci of an ellipse be $F_{1}(-1,0)$ and $F_{2}(1,0)$ with eccentricity $e$, and let the parabola with vertex at $F_{1}$ and focus at $F_{2}$ intersect the ellipse at point $P$. If $\frac{\left|P F_{1}\right|}{\left|P F_{2}\right|}=e$, then the value of $e$ is . $\qquad$ | \frac{\sqrt{3}}{3} |
65,478 | Problem 11.3. In a football tournament, 15 teams participated, each playing against each other exactly once. For a win, 3 points were awarded, for a draw - 1 point, and for a loss - 0 points.
After the tournament ended, it turned out that some 6 teams scored at least $N$ points each. What is the largest integer value that $N$ can take?
Fig. 13: to the solution of problem 11.2
# | 34 |
65,479 | 31. In the diagram below, $A B C D$ is a quadrilateral such that $\angle A B C=$ $135^{\circ}$ and $\angle B C D=120^{\circ}$. Moreover, it is given that $A B=2 \sqrt{3} \mathrm{~cm}$, $B C=4-2 \sqrt{2} \mathrm{~cm}, C D=4 \sqrt{2} \mathrm{~cm}$ and $A D=x \mathrm{~cm}$. Find the value of $x^{2}-4 x$. | 20 |
65,480 | 5.2. Find the sum of the digits in the decimal representation of the integer part of the number $\sqrt{\underbrace{11 \ldots 11}_{2018} \underbrace{55 \ldots 55}_{2017} 6}$. | 6055 |
65,482 | 11. The sequence $x_{1}=0, x_{n+1}=\frac{n+2}{n} x_{n}+\frac{1}{n}$, the general term formula of the sequence $\left\{a_{n}\right\}$ is | x_{n}=\frac{1}{4}(n-1)(n+2) |
65,484 | The gure below shows a large square divided into $9$ congruent smaller squares. A shaded square bounded by some of the diagonals of those smaller squares has area $14$. Find the area of the large square.
[img]https://cdn.artofproblemsolving.com/attachments/5/e/bad21be1b3993586c3860efa82ab27d340dbcb.png[/img] | 63 |
65,516 | 9. (14 points) Given the function $f(x)=a x^{2}+b x+c$ $(a, b, c \in \mathbf{R})$, when $x \in[-1,1]$, $|f(x)| \leqslant 1$.
(1) Prove: $|b| \leqslant 1$;
(2) If $f(0)=-1, f(1)=1$, find the value of $a$. | 2 |
65,531 | 8 Real numbers $x, y, z$ satisfy $x^{2}+y^{2}+z^{2}=1$, then the maximum value of $xy+yz$ is
$\qquad$ . | \frac{\sqrt{2}}{2} |
65,538 | 10. (20 points) Given various natural numbers $a, b, c, d$, for which the following conditions are satisfied: $a>d, a b=c d$ and $a+b+c+d=a c$. Find the sum of all four numbers. | 12 |
65,550 | [ Heron's Formula [ Area of a Triangle (through semiperimeter and radius of inscribed or exscribed circle).
In a triangle, the sides are in the ratio 2:3:4. A semicircle is inscribed in it with the diameter lying on the largest side. Find the ratio of the area of the semicircle to the area of the triangle.
# | \frac{9\pi}{10\sqrt{15}} |
65,557 | 3. Let the function be
$$
y(x)=(\sqrt{1+x}+\sqrt{1-x}+2)\left(\sqrt{1-x^{2}}+1\right) \text {, }
$$
where, $x \in[0,1]$. Then the minimum value of $y(x)$ is | 2+\sqrt{2} |
65,559 | 16. A square fits snugly between the horizontal line and two touching circles of radius 1000 , as shown. The line is tangent to the circles.
What is the side-length of the square? | 400 |
65,576 | H5. A two-digit number is divided by the sum of its digits. The result is a number between 2.6 and 2.7 .
Find all of the possible values of the original two-digit number. | 29 |
65,587 | $1 \cdot 30$ Try to find the approximate value of $\frac{2}{3!}+\frac{3}{4!}+\frac{4}{5!}+\cdots+\frac{19}{20!}$, accurate to the third decimal place. | 0.500 |
65,590 | At least how many groups do we need to divide the first 100 positive integers into so that no group contains two numbers where one is a multiple of the other? | 7 |
65,609 | 4. Given that $n$ is a positive integer, and
$$
n^{4}+2 n^{3}+6 n^{2}+12 n+25
$$
is a perfect square. Then $n=$ $\qquad$
(2014, National Junior High School Mathematics League (Grade 8)) | 8 |
65,612 | ## Task Condition
Find the derivative.
$$
y=\frac{x-3}{2} \sqrt{6 x-x^{2}-8}+\arcsin \sqrt{\frac{x}{2}-1}
$$ | \sqrt{6x-x^{2}-8} |
65,615 | 20 Find the real solutions of the equation $\left(x^{2008}+1\right)\left(1+x^{2}+x^{4}+\cdots+x^{2006}\right)=2008 x^{2007}$. | 1 |
65,621 | 4. In how many ways can two knights, two bishops, two rooks, a queen, and a king be arranged on the first row of a chessboard so that the following conditions are met:
1) The bishops stand on squares of the same color;
2) The queen and the king stand on adjacent squares. (20 points). | 504 |
65,631 | 2.016. $\left(\frac{\left(z^{2 / p}+z^{2 / q}\right)^{2}-4 z^{2 / p+2 / q}}{\left(z^{1 / p}-z^{1 / q}\right)^{2}+4 z^{1 / p+1 / q}}\right)^{1 / 2}$. | |z^{1/p}-z^{1/q}| |
65,636 | ## Task Condition
Find the derivative.
$$
y=\sqrt{1+x^{2}} \operatorname{arctg} x-\ln \left(x+\sqrt{1+x^{2}}\right)
$$ | \frac{x\cdot\arctanx}{\sqrt{1+x^{2}}} |
65,641 | 9. How many pairs of integers solve the system $|x y|+|x-y|=2$ if $-10 \leq x, y \leq 10 ?$ | 4 |
65,645 | 70. For which natural numbers $n$ is the sum $5^{n}+n^{5}$ divisible by 13? What is the smallest $n$ that satisfies this condition? | 12 |
65,652 | B1 Give the smallest positive integer that is divisible by 26, ends in 26, and for which the sum of the digits equals 26. | 46826 |
65,668 | 1. The sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=1, a_{2}=3, \text { and } \\
a_{n+2}=\left|a_{n+1}\right|-a_{n}
\end{array}\left(n \in \mathbf{N}_{+}\right) .
$$
Let $\left\{a_{n}\right\}$'s sum of the first $n$ terms be $S_{n}$. Then $S_{100}=$ | 89 |
65,687 | 3. (2000 National High School Competition Question) Given that $A$ is the left vertex of the hyperbola $x^{2}-y^{2}=1$, and points $B$ and $C$ are on the right branch of the hyperbola, $\triangle A B C$ is an equilateral triangle, then the area of $\triangle A B C$ is | 3\sqrt{3} |
65,691 | Compute the number of two digit positive integers that are divisible by both of their digits. For example, $36$ is one of these two digit positive integers because it is divisible by both $3$ and $6$.
[i]2021 CCA Math Bonanza Lightning Round #2.4[/i] | 14 |
65,700 | A bug crawls along a triangular iron ring. At each vertex, it has an equal chance of crawling to one of the other two vertices. Find the probability that it returns to the starting point after 10 crawls. | \frac{171}{2^{9}} |
65,723 | 2. Polynomials $P(x)$ and $Q(x)$ of equal degree are called similar if one can be obtained from the other by permuting the coefficients (for example, the polynomials $2 x^{3}+x+7$ and $x^{3}+2 x^{2}+7 x$ are similar). For what largest $k$ is it true that for any similar polynomials $P(x), Q(x)$, the number $P(2009)-Q(2009)$ is necessarily divisible by $k$? | 2008 |
65,731 | 11.171. A cube is inscribed in a hemisphere of radius $R$ such that four of its vertices lie on the base of the hemisphere, while the other four vertices are located on its spherical surface. Calculate the volume of the cube. | \frac{2R^{3}\sqrt{6}}{9} |
65,734 | Problem 5. A monkey becomes happy when it eats three different fruits. What is the maximum number of monkeys that can be made happy with 20 pears, 30 bananas, 40 peaches, and 50 tangerines? Justify your answer. $[8$ points] (A.V. Shapovalov) | 45 |
65,738 | [ [Evenness and Oddness]
All the dominoes were laid out in a chain. At one end, there were 5 dots. How many dots are at the other end?
# | 5 |
65,784 | 12. Two squares $A B C D$ and $B E F G$ with side lengths of $8 \mathrm{~cm}$ and $6 \mathrm{~cm}$ respectively are placed side by side as shown in the figure. Connecting $D E$ intersects $B G$ at $P$. What is the area of the shaded region $A P E G$ in the figure? | 18 |
65,822 | 15. (14th "Hope Cup" Mathematics Invitational Test) Let the function $f(x)=\sqrt[3]{1+x}-\lambda x$, where $\lambda>0$.
(1) Find the range of $\lambda$ such that $f(x)$ is a monotonic function on the interval $[0,+\infty)$;
(2) Can this monotonicity be extended to the entire domain $(-\infty,+\infty)$?
(3) Solve the inequality $2 x-\sqrt[3]{1+x}<12$. | (-\infty,7) |
65,855 | [ [ product rule ]
In the USA, the date is typically written as the month number, followed by the day number, and then the year. In Europe, however, the day comes first, followed by the month and the year. How many days in a year cannot be read unambiguously without knowing which format it is written in
# | 132 |
65,859 | 59 If three lines: $4 x+y=4, m x+y=0,2 x-3 m y=4$. cannot form a triangle, then the value of $m$ is . $\qquad$ | 4,-\frac{1}{6},-1,\frac{2}{3} |
65,864 | 3. Let $m>n \geqslant 1$, find the minimum value of $m+n$ such that $: 1000 \mid 1978^{m}-1978^{n}$. | 106 |
65,887 | 9. Given the function
$$
f(x)=\sin \left(\omega x+\frac{\pi}{2}\right) \cdot \sin \left(\omega x+\frac{\pi}{4}\right)(\omega>0)
$$
has the smallest positive period of $\pi$. Then the range of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ is . $\qquad$ | [0,\frac{2+\sqrt{2}}{4}] |
65,895 | B2. In a bag, there are 5 red and 5 green coins. Each coin has the number 0 on one side and the number 1 on the other side. All coins are fair, meaning the probability of landing on 0 is equal to the probability of landing on 1. We randomly select 6 coins from the bag and toss them. What is the probability that the sum of the numbers that land on the red coins is equal to the sum of the numbers that land on the green coins? | \frac{89}{336} |
65,917 | 12.007. In rectangle $A B C D$ ($A B \| C D$), triangle $A E F$ is inscribed. Point $E$ lies on side $B C$, point $F$ - on side $C D$. Find the tangent of angle $E A F$, if $A B: B C = B E: E C = C F: F D = k$. | \frac{k^{2}+k+1}{(1+k)^{2}} |
65,951 | 4. The four-digit number $\overline{a b c d}$ is divisible by 3, and $a, b, c$ are permutations of three consecutive integers. Then the number of such four-digit numbers is
$\qquad$ . | 184 |
65,952 | A $3,15,24,48, \ldots$ sequence consists of the multiples of 3 that are 1 less than a square number. What is the remainder when the 2001st term of the sequence is divided by 1000? | 3 |
65,986 | ## PROBLEM 1
Solve in $R$ the equation:
$$
54^{x}+27^{x}+9^{x}+3^{x}=2^{x}
$$ | -1 |
66,005 | Condition of the problem
Find the derivative of the specified order.
$y=e^{\frac{x}{2}} \cdot \sin 2 x, y^{IV}=?$ | \frac{161}{16}\cdote^{\frac{x}{2}}\cdot\sin2x-15\cdote^{\frac{x}{2}}\cdot\cos2x |
66,013 | 10.138. Three equal circles of radius $r$ touch each other pairwise. Calculate the area of the figure located outside the circles and bounded by their arcs between the points of tangency. | \frac{r^{2}(2\sqrt{3}-\pi)}{2} |
66,028 | 5. The solution set of the equation $\frac{\sqrt{x}+2^{x}}{\sqrt{x}+2^{x+1}}+\frac{\sqrt{x}+3^{x}}{\sqrt{x}+3^{x+1}}+\frac{\sqrt{x}+6^{x}}{\sqrt{x}+6^{x+1}}=1$ is | {0} |
66,031 | 12. (2004 High School League - Liaoning Preliminary) The sum of the maximum and minimum values of the function $f(x)=\frac{\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)+2 x^{2}+x}{2 x^{2}+\cos x}$ is $\qquad$ . | 2 |
66,041 | ## Task B-1.1.
Simplify the fraction $\frac{(x+1)^{4}-4\left(x+x^{2}\right)^{2}+4 x^{2}-(1+x)^{2}}{3 x^{2}+x}$ if $x \neq 0, x \neq-\frac{1}{3}$. | (x+2)(1-x) |
66,056 | 81. The square of an integer ends with four identical digits. Which ones? | 0 |
66,074 | 35. There are $k$ people and $n$ chairs in a row, where $2 \leq k<n$. There is a couple among the $k$ people. The number of ways in which all $k$ people can be seated such that the couple is seated together is equal to the number of ways in which the $(k-2)$ people, without the couple present, can be seated. Find the smallest value of $n$. | 12 |
66,079 | 2. When $m$ takes all real values from 0 to 5, the number of integer $n$ that satisfies $3 n=m(3 m-8)$ is $\qquad$ . | 13 |
66,080 | A square-based frustum has a base edge and every side edge of 4. The edge of its top face is 2. What is the maximum distance between two vertices of the frustum? | \sqrt{32} |
66,081 | 1.48 In the expression $x_{1}: x_{2}: \cdots: x_{n}$, use parentheses to indicate the order of operations, and the result can be written in fractional form:
$$
\frac{x_{i_{1}} x_{i_{2}} \cdots x_{i_{k}}}{x_{j_{1}} x_{j_{2}} \cdots x_{j_{n-k}}}
$$
(At the same time, each letter in $x_{1}, x_{2}, \cdots, x_{n}$ may appear in the numerator or in the denominator.) How many different fractions can be obtained by adding parentheses in all possible ways? | 2^{n-2} |
66,087 | 6. Let real numbers $a, b, c$ satisfy $a^{2}+b^{2}+c^{2}=1$, and let the maximum and minimum values of $ab+bc+ca$ be $M, m$ respectively. Then $M-m=$ $\qquad$ . | \frac{3}{2} |
66,088 | Task 66. Find the distance from the point of tangency of two circles with radii $R$ and $r$ to their common external tangent.
## Java 9. Lemma about the perpendicular from the point of tangency
##
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
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Task 66. Find the distance from the point of tangency of two circles with radii $R$ and $r$ to their common external tangent.
## Java 9. Lemma about the perpendicular from the point of tangency | \frac{2Rr}{R+r} |
66,107 | I2.2 If $f(x)=\frac{25^{x}}{25^{x}+P}$ and $Q=f\left(\frac{1}{25}\right)+f\left(\frac{2}{25}\right)+\cdots+f\left(\frac{24}{25}\right)$, find the value of $Q$. | 12 |
66,114 | 10.4. How many solutions in integers $x, y$ does the equation $|3 x+2 y|+|2 x+y|=100$ have? | 400 |
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