problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
12. For a residential building, the construction investment is 250 yuan per square meter, considering a lifespan of 50 years, and an annual interest rate of $5 \%$, then the monthly rent per square meter should be $\qquad$ yuan to recover the entire investment. | 1.14 |
Example 2. Given that 1 is a root of the equation $a x^{2}+b x+c=0$, find the value of $\frac{a^{2}+b^{2}+c^{2}}{a^{3}+b^{3}+c^{3}}+\frac{2}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$. | 0 |
## [ Properties and characteristics of an isosceles triangle. ]
[ The inscribed angle is half of the central angle ] [ The product of the lengths of the segments of the chords and the lengths of the segments of the secants ] [ Inscribed quadrilaterals (other) ]
[ Auxiliary similar triangles ]
From the vertex of the ... | \frac{mc}{n} |
2.128 The solution situation of the equation $\sqrt{x-1} \cdot \sqrt{x+1}=-\sqrt{x^{2}-1}$ is
(A) Unique solution.
(B) Two solutions.
(C) Infinitely many solutions.
(D) No real solutions.
(6th "Jinyun Cup" Junior High School Mathematics Invitational, 1989) | A |
It is well-known that the $n^{\text{th}}$ triangular number can be given by the formula $n(n+1)/2$. A Pythagorean triple of $\textit{square numbers}$ is an ordered triple $(a,b,c)$ such that $a^2+b^2=c^2$. Let a Pythagorean triple of $\textit{triangular numbers}$ (a PTTN) be an ordered triple of positive integers $(a,b... | 14 |
1. A line parallel to the selected side of a triangle with an area of 27 cuts off a smaller triangle with an area of 3. Find the area of the quadrilateral, three vertices of which coincide with the vertices of the smaller triangle, and the fourth lies on the selected side. Choose the answer option with the number close... | 9 |
A student types the following pattern on a computer (○ represents an empty circle, ● represents a solid circle) ○●○○●○○○●○○○○●… If this pattern of circles continues in the same manner, forming a series of circles, then the number of solid circles among the first 2010 circles is ____. | 61 |
A book has 500 pages, numbered 1, 2, 3, ..., how many times does the digit 1 appear in the page numbers? | 200 |
33. [18] Compute
$$
\int_{1}^{2} \frac{9 x+4}{x^{5}+3 x^{2}+x} d x .
$$
(No, your TI-89 doesn't know how to do this one. Yes, the end is near.) | \ln\frac{80}{23} |
Factorization: $a^{2}-1=$____. | (a+1)(a-1) |
Let $ABCD$ be a trapezoid with $AB\parallel DC$. Let $M$ be the midpoint of $CD$. If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$, find $\frac{AB}{CD}$.
[i]Proposed by Nathan Ramesh | \frac{\sqrt{2}}{2} |
Given a set of data $2$, $-2$, $6$, $4$, $-1$, the range of this data set is ____. | 8 |
## Task 1
Out of 33 pioneers in a pioneer group, 8 pioneers were able to win medals at the pioneer sports festival because they took first, second, or third place. All other pioneers received certificates for participation.
How many pioneers in this group received certificates? | 25 |
The solution to the fractional equation $\frac{1}{x}=\frac{2}{x+1}$ is $x=$____. | x=1 |
Marie does three equally time-consuming tasks in a row without taking breaks. She begins the first task at $1\!:\!00$ PM and finishes the second task at $2\!:\!40$ PM. When does she finish the third task?
$\textbf{(A) }\text{3:10 PM}\qquad\textbf{(B) }\text{3:30 PM}\qquad\textbf{(C) }\text{4:00 PM}\qquad\textbf{(D) }\t... | \textbf{(B)} |
For the function $f(x)$, if there exists $x_0 \neq 0$ such that $f(x_0) = -f(-x_0)$, then the points $(x_0, f(x_0))$ and $(-x_0, f(-x_0))$ are called a pair of odd symmetry points of the function graph. If $f(x) = e^x - a$ (where $e$ is the base of the natural logarithm) has odd symmetry points, then the range of value... | (1, +\infty) |
$2 \cdot 118$ If the equation with respect to $x$
$$
\frac{1}{x^{2}-x}+\frac{k-5}{x^{2}+x}=\frac{k-1}{x^{2}-1}
$$
has an extraneous root $x=1$, then the value of $k$ is
(A) 1 .
(B) 2 .
(C) 3 .
(D) 6
(China Beijing Junior High School Mathematics Competition, 1985) | 3 |
3. In the set of integers, solve the equation
$$
x^{5}+2=3 \cdot 101^{y}
$$ | (1,0) |
25th BMO 1989 Problem 3 f(x) is a polynomial of degree n such that f(0) = 0, f(1) = 1/2, f(2) = 2/3, f(3) = 3/4, ... , f(n) = n/(n+1). Find f(n+1). | f(n+1)=\begin{cases}1&ifnisodd\\\frac |
7. What is the value of $\left(4-\frac{1}{4}\right) \div\left(2-\frac{1}{2}\right)$ ?
A $1 \frac{1}{2}$
B 2
C $2 \frac{1}{2}$
D 3
E $4 \frac{1}{4}$ | 2\frac{1}{2} |
If $m,~n,~p$, and $q$ are real numbers and $f(x)=mx+n$ and $g(x)=px+q$, then the equation $f(g(x))=g(f(x))$ has a solution
$\textbf{(A) }\text{for all choices of }m,~n,~p, \text{ and } q\qquad\\ \textbf{(B) }\text{if and only if }m=p\text{ and }n=q\qquad\\ \textbf{(C) }\text{if and only if }mq-np=0\qquad\\ \textbf{(D) ... | \textbf{(D)} |
At a party, there are 25 children. We know that any two of them have at least one grandparent in common (grandfather or grandmother). Explain why at least 17 children have either the same grandfather or the same grandmother in this family. | 17 |
103. Digits and Squares. What is the smallest square of an integer that ends with the longest sequence of identical digits?
If the longest sequence of identical digits were five, for example, the number 24677777 would fit (if it were the smallest square, but this is not true). Zero is not considered a valid digit. | 1444 |
10.416 Determine the area of a triangle given its three heights $h_{1}$, $h_{2}, h_{3} \cdot$ | \sqrt{(\frac{1}{h_{1}}+\frac{1}{h_{2}}+\frac{1}{h_{3}})(\frac{1}{h_{1}}+\frac{1}{h_{2}}-\frac{1}{h_{3}})(\frac{1}{h_{1}}+\frac{1}{h_{3}}-\frac{1}{h_{2}})(} |
In triangle $ABC$, if angle $A$, $B$, $C$ correspond to sides $a$, $b$, $c$, respectively, and it is known that $\cos 2A - 3\cos(B+C) = 1$.
(1) Determine the size of angle $A$.
(2) If the area of $\triangle ABC$, $S = 5\sqrt{3}$ and $b = 5$, find the value of $\sin B \sin C$.
(3) If $a = 1$, determine the range of pos... | (2, 3] |
Variance is a measure of the magnitude of fluctuations in a set of data. For a set of data $x_{1}$, $x_{2}$, $x_{3}$, $\ldots$, $x_{n}$, the variance can be calculated using the following formula: $S^{2}=\frac{1}{n}[(x_{1}-3)^{2}+(x_{2}-3)^{2}+(x_{3}-3)^{2}+\ldots +(x_{n}-3)^{2}]$, where "3" represents the ( ).
A: min... | B |
3. Find the numerical value of the expression
$$
\frac{1}{x^{2}+1}+\frac{1}{y^{2}+1}+\frac{2}{x y+1}
$$
if it is known that $x$ is not equal to $y$ and the sum of the first two terms is equal to the third. | 2 |
79. Person A and Person B are in a 100-meter race. When A has run 68 meters, B is 15 meters away from the finish line. When B reaches the finish line, A is $\qquad$ meters away from the finish line. | 20 |
$25 \cdot 25$ If $a, b>0$, the line $a x+b y=6$ and the coordinate axes form a triangle in the first quadrant with an area of 6, then $a b$ equals
(A) 3 .
(B) 6 .
(C) 12 .
(D) 108 .
(E) 432 .
(40th American High School Mathematics Examination, 1989) | 3 |
Given that the chord length intercepted by line $l$: $y=x+a$ on the circle $(x-2)^{2}+y^{2}=1$ is $2$, find the value of $a$. | a=-2 |
8. Fill the circles with the numbers $0, 1, 2, 3, 4, 5, 6$ in the grid, with each number appearing exactly once, to form an integer equation consisting of single-digit and two-digit numbers. What number is filled in the grid?
$$
O \times O=\square=O \div O
$$ | 12 |
A study group obtained some data online about the relationship between the speed of sound in air and the air temperature, and compiled it into the following table:
| Air Temperature ($^{\circ}C$) | $-20$ | $-10$ | $0$ | $10$ | $20$ | $30$ |
|-------------------------------|-------|-------|-----|------|------|------|
|... | D |
Problem 6.8. There are exactly 120 ways to color five cells in a $5 \times 5$ table so that each column and each row contains exactly one colored cell.
There are exactly 96 ways to color five cells in a $5 \times 5$ table without a corner cell so that each column and each row contains exactly one colored cell.
How ma... | 78 |
Find all positive integers $k$, for which the product of some consecutive $k$ positive integers ends with $k$.
[i]Proposed by Oleksiy Masalitin[/i] | 1, 2, 4 |
Given the function $f(x)= \begin{cases} x^2+x, & -2\leqslant x\leqslant c \\ \frac{1}{x}, & c < x\leqslant 3 \end{cases}$, if $c=0$, then the range of $f(x)$ is _______; if the range of $f(x)$ is $[-\frac{1}{4},2]$, then the range of the real number $c$ is _______. | [\frac{1}{2},1] |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(x,-2)$, if $\overrightarrow{a} \parallel \overrightarrow{b}$, then the real number $x=$ ______. | -1 |
Xiao Ming went to the store and bought 9 loaves of bread, each weighing 3 liang. He should pay with 2 jin and 7 liang of grain coupons. However, he only brought 10 half-jin grain coupons, and the store only had 2-liang grain coupons, without any 1-liang grain coupons. Please calculate how many half-jin grain coupons Xi... | A |
Problem 9.2. Vanya thought of two natural numbers, the product of which equals 7200. What is the greatest value that the GCD of these numbers can take? | 60 |
If the decreasing interval of the function $y=a(x^{3}-x)$ is $\left(- \frac{ \sqrt{3}}{3}, \frac{ \sqrt{3}}{3}\right)$, then the range of $a$ is ______. | a > 0 |
2.6. (GDR, 74). Solve the equation
$$
(x+2)^{4}-x^{4}=y^{3}
$$
in integers. | -1,0 |
If line $a \perp b$ and line $a \parallel$ plane $\alpha$, then the positional relationship between line $b$ and plane $\alpha$ is ( )
A: $b \subset \alpha$
B: $b \parallel \alpha$
C: $b \subset \alpha$ or $b \parallel \alpha$
D: $b$ intersects $\alpha$ or $b \subset \alpha$ or $b \parallel \alpha$ | \text{D} |
Which of the following fractions has the greatest value?
(A) $\frac{44444}{55555}$
(B) $\frac{5555}{6666}$
(C) $\frac{666}{777}$
(D) $\frac{77}{88}$
(E) $\frac{8}{9}$ | \frac{8}{9} |
8 The line $\sqrt{3} x-y-\sqrt{3}=0$ intersects the parabola $y^{2}=4 x$ at points $A$ and $B$ ($A$ is above the x-axis), and intersects the x-axis at point $F$. If $\overrightarrow{O F}=\lambda \overrightarrow{O A}+\mu \overrightarrow{O B}$, then $\mu^{2}-$ $\lambda^{2}=$ $\qquad$ . | \frac{1}{2} |
Evaluate:
$(1)\ [(-1+i)\cdot i^{100}+( \frac {1-i}{1+i})^{5}]^{2017}-( \frac {1+i}{ \sqrt {2}})^{20}$
$(2) \ \int_{-1}^{1}\[3\tan x+\sin x-2x^{3}+ \sqrt {16-(x-1)^{2}}\]dx\cdot$ | \frac{4\pi}{3}+2\sqrt{3} |
1. If $f(x)=x^{6}-2 \sqrt{2006} x^{5}-x^{4}+x^{3}-$ $2 \sqrt{2007} x^{2}+2 x-\sqrt{2006}$, then $f(\sqrt{2006}+\sqrt{2007})=$ $\qquad$ | \sqrt{2007} |
11. Let real numbers $x, y$ satisfy the equation $(x+2)^{2}+y^{2}=1$. Then the maximum value of $\frac{y}{x}$ is $\qquad$ | \frac{\sqrt{3}}{3} |
If angle $\alpha = 2$, then the quadrant of angle $\alpha$ is ( )
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{B} |
Person A and person B practice shooting, with the probabilities of hitting the target being \\( \dfrac{1}{2} \\) and \\( \dfrac{1}{3} \\) respectively. Each person shoots once, and the following statements are made: \\(①\\) The probability of the target being hit exactly once is \\( \dfrac{1}{2} + \dfrac{1}{3} \\); \\(... | \text{C} |
The absolute value of the reciprocal of $-6$ is ______. | \frac{1}{6} |
5. Determine the sum of all four-digit numbers in which every two consecutive digits form a square of a natural number. (For example, 164 is such a three-digit number because 16 and 64 are squares of the numbers 4 and 8).
## Tasks worth 10 points: | 13462 |
7. When asked how old he was, a mathematician replied: "If you subtract 5 from my age, divide the result by 5, and then subtract 5 again, you will get the number 5." How old is he? | 55 |
17.200 Let $S$ be the set of points on two rays that intersect at an angle of $120^{\circ}$, and let $P$ be a fixed point on the angle bisector and inside the angle. Consider all distinct equilateral triangles $\triangle PQR$, where $Q$ and $R$ are points in $S$. (Points $Q$ and $R$ can be on the same ray; also, in thi... | E |
Given the function $f(x)=2\cos^2x-\sin^2x+2$, then
A: The minimum positive period of $f(x)$ is $\pi$, and the maximum value is $3$
B: The minimum positive period of $f(x)$ is $\pi$, and the maximum value is $4$
C: The minimum positive period of $f(x)$ is $2\pi$, and the maximum value is $3$
D: The minimum positive ... | \text{B} |
If the smallest positive period of the function $y=\cos^2\omega x-\sin^2\omega x$ ($\omega > 0$) is $\pi$, then $\omega=$ ______. | 1 |
If $|z+3+4i|\leq2$, then the maximum value of $|z|$ is ( )
A: 3
B: 7
C: 9
D: 5 | \text{B} |
The function $f(x) = ax + b\sin x + 1$, if $f(5) = 7$, then $f(-5) = \quad$. | -5 |
3. Find all positive integers $n$, such that the numbers in the set $S=$ $\{1,2, \cdots, n\}$ are colored red or blue, and satisfy the following condition: the set $S \times S \times S$ contains exactly 2007 ordered triples $(x, y, z)$, such that
(1) $x, y, z$ are of the same color;
(2) $x+y+z$ is divisible by $n$. | 69 \text{ and } 84 |
In a regular n-sided pyramid, the range of values for the dihedral angle formed by two adjacent lateral faces is
(A) $(0, \pi)$ (B) $(\pi, 2\pi)$ (C) $(0, 2\pi)$ (D) $(\pi, 3\pi)$ | \text{A} |
G9.3 If $c=1996 \times 19971997-1995 \times 19961996$, find the value of $c$. | 39923992 |
Given that $\triangle ABC \cong \triangle A'B'C'$, and $AB = A'B'$, $\angle B = 50^\circ$, $\angle C' = 70^\circ$, then $\angle A' =$ ____ degrees. | 60 |
8. Find $f: \mathbf{N}^{\top} \rightarrow \mathbf{N}^{\top}, f(m+n) f(m-n)=f\left(m^{2}\right)$ for all $m, n \in \mathbf{N}^{\top}$. | f(n)=1,n\in{N}^{*} |
(given to Ambroise Marigot). Solve $5 x^{3}+11 y^{3}+13 z^{3}=0$ in relative integers. | (0,0,0) |
Laila writes a list of numbers. Her first number is 4 . Each number after the first is 7 more than the previous number. Which of the following numbers appears in Laila's list?
(A) 45
(B) 46
(C) 47
(D) 48
(E) 49 | 46 |
Given $$\overrightarrow {AB} = (3, 4)$$, and $$\overrightarrow {AC} = (1, 3)$$, then $$\overrightarrow {BC} =$$ （ ）
A: (2, 1)
B: (4, 7)
C: (-2, -1)
D: (-4, -7) | C |
Xiao Ming and Xiao Liang went to a fruit supermarket together to buy "99 strawberries". Xiao Ming bought 1kg, Xiao Liang bought 2kg. They both liked the strawberries and agreed to buy the same kind of strawberries again. The second time they went to the supermarket, Xiao Ming spent the same amount of money as last time... | \text{money} |
Translate the graph of the function $y=x^{2}$ up by one unit to obtain the graph of which function?
A: $y=x^{2}-1$
B: $y=\left(x+1\right)^{2}$
C: $y=x^{2}+1$
D: $y=\left(x-1\right)^{2}$ | C |
2. (8 points) Dad's birthday is on May 1st, and Chunchun's birthday is on July 1st. Starting from December 26, 2012 (Day 1), until Day 2013, Dad and Chunchun have had $\qquad$ birthdays in total. | 11 |
Which of the following statements is correct?
A: $1$ is the smallest natural number.
B: The only numbers whose square is equal to themselves are $1$.
C: Every rational number has a reciprocal.
D: The number with the smallest absolute value is $0$. | D |
If the function $f(x) = kx^2 + (k-1)x + 2$ is an even function, then the value of $k$ is ______. | 1 |
Given the function $f(x)= \sin (wx + \frac{\pi}{6}) (w > 0)$, if the minimum distance from a symmetry center of the graph of the function $f(x)$ to the axis of symmetry is $\frac{\pi}{3}$, find the value of $w$. | \frac{3}{2} |
【Example 4】A person's matchbox contains $n$ red, green, and black-tipped matches each. Each time, one is taken at random. When the black-tipped matches are all used up, it is found that there are $r$ red and $g$ green matches left. Find the number of different ways this outcome can occur. | \frac{(3n-r--1)!}{(n-1)!(n-r)!(n-)!} |
51.6 ** Given the complex numbers $z=\cos \alpha+\mathrm{i} \sin \alpha, u=\cos \beta+\mathrm{i} \sin \beta$, and $z+u=\frac{4}{5}+\frac{3}{5} \mathrm{i}$.
(1) Find the value of $\tan (\alpha+\beta)$; (2) Prove that $z^{2}+u^{2}+z u=0$. | \tan(\alpha+\beta)=\frac{24}{7},\quadz^{2}+u^{2}+zu=0 |
A vertical-axis, rotational paraboloid-shaped tall container has its focus 0.05 meters above the axis point. We pour a little water into the container. At what angular velocity $\omega$ should we rotate the container around its axis so that the water flows out at the top of the container? | 9.9\mathrm{~}^{-1} |
Problem 2. (2 points)
The sum of the sines of five angles from the interval $\left[0 ; \frac{\pi}{2}\right]$ is 3. What are the greatest and least integer values that the sum of their cosines can take? | 2;4 |
4. For the convex pentagon $A B C D E$, the side lengths are sequentially $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$. It is known that a quadratic trinomial in $x$ satisfies:
When $x=a_{1}$ and $x=a_{2}+a_{3}+a_{4}+a_{5}$, the value of the quadratic trinomial is 5;
When $x=a_{1}+a_{2}$, the value of the quadratic trinomial i... | 0 |
Given $i$ is the imaginary unit, the imaginary part of the conjugate of the complex number $1-2i$ is ( )
A: 1
B: $-1$
C: 2
D: $-2$ | C |
The function $f(x)=\log_{2}x+x^{2}+m$ has a root in the interval $(2,4)$. The range of real number $m$ is ()
A: $(-\infty ,-18)$
B: $(5,+\infty )$
C: $(5,18)$
D: $(-18,-5)$ | \text{D} |
Given that the vertex of angle $\theta$ coincides with the origin, the initial side coincides with the positive semi-axis of the $x$-axis, and the terminal side passes through point $(1,2)$, find $\tan 2\theta =$ ___. | -\frac{4}{3} |
## Task 1
Calculate in your head:
$5627895+6 ; \quad 10001-9 ; \quad 332+407 ; \quad 105-96 ; \quad 5 \cdot 25 ; \quad 78: 3$ | 5627901;\quad9992;\quad739;\quad9;\quad125;\quad26 |
9.3. On the board, there are $N$ prime numbers (not necessarily distinct). It turns out that the sum of any three numbers on the board is also a prime number. For what largest $N$ is this possible | 4 |
Among the following groups of data, the one with the smallest range is ( )
A: 0, 1, 2, 3, 4
B: -2, -1, -2, 3
C: 110, 111, 112, 110, 109
D: -100, -200, -300, -400 | C |
There are some red, white, and black balls of the same size in a bag. If one ball is drawn from the bag, the probability of drawing a red ball is 0.42, and the probability of drawing a white ball is 0.28. If there are 21 red balls, then there are $\boxed{\text{black balls}}$. | 15 |
Problem 2. In a square of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if 209 cells remained after the cutout? | 225 |
I am thinking of three natural numbers. The product of the first and second is 24, the product of the second and third is 32, the product of the first and third is 48.
Which numbers am I thinking of?
(E. Novotná) | 6,4,8 |
4. On the table, there are 100 weights of different masses. A weight is called successful if its mass is equal to the sum of the masses of some two other weights on the table. What is the smallest number of successful weights that would allow us to assert with certainty that the masses of some two weights differ by at ... | 87 |
If a positive four-digit number's thousand digit \\(a\\), hundred digit \\(b\\), ten digit \\(c\\), and unit digit \\(d\\) satisfy the relation \\((a-b)(c-d) < 0\\), then it is called a "Rainbow Four-Digit Number", for example, \\(2012\\) is a "Rainbow Four-Digit Number". How many "Rainbow Four-Digit Numbers" are there... | 3645 |
Four. (25 points) Let the linear function $y=k x+b(k \neq 0)$ intersect the positive half-axes of the $x$-axis and $y$-axis at points $E$ and $F$, respectively, and intersect the linear function $y=2 x-1$ at point $A(m, 2)$, with $A$ being the midpoint of $E F$.
(1) Find the expression of the linear function;
(2) If th... | \frac{5}{2} |
(5 points) (2011•Chongqing) In the arithmetic sequence $\{a_n\}$, $a_2=2$, $a_3=4$, then $a_{10}=$ （ ）
A: 12
B: 14
C: 16
D: 18 | \text{D} |
Problem 4. In triangle $P Q R$, the lengths of the sides are 4, 7, and 9. Find the area of the figure consisting of those and only those points $M$ inside triangle $P Q R$ for which the condition $M P^{2}+$ $M Q^{2}+M R^{2} \leqslant 50$ is satisfied. | \frac{4\pi}{9} |
A1. At the reception with the mayor, 7 awardees were invited. The mayor shook hands with each of them, and all the awardees also shook hands with each other. How many handshakes were there in total?
(A) 21
(B) 28
(C) 36
(D) 42
(E) 56 | 28 |
89. If prime numbers $p, q$ satisfy $p q+p^{2}+q^{2}=199(p<q)$, find the value of the algebraic expression $\frac{2}{p(p+1)}+\frac{2}{(p+1)(p+2)}+\cdots+\frac{2}{(q-1) q}$. | \frac{11}{13} |
Given a triangle $ \triangle ABC $ in a coordinate plane, where $ A(-2, 0) $, $ B(0, 2) $, and $ C(\cos\theta, -1+\sin\theta) $ ($ \theta $ is a parameter), find the maximum area of $ \triangle ABC $. | 3 + \sqrt{2} |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{4}}(\sin x+\cos x)^{\frac{1}{\tan x}}$ | \sqrt{2} |
In the equation "$a^{2}+b^{2}=2ab$", "$a=b$" is a ( ).
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | C |
Problem 8.4. Given a square $A B C D$. Point $L$ is on side $C D$ and point $K$ is on the extension of side $D A$ beyond point $A$ such that $\angle K B L=90^{\circ}$. Find the length of segment $L D$, if $K D=19$ and $C L=6$.
$ denote the sum of the digits of a natural number $n$. Solve the equation $n+s(n)=2018$. | 2008 |
Given that $\left\{\begin{array}{l}x=1\\ y=2\end{array}\right.$ is a solution to the system of equations $\left\{\begin{array}{l}ax+y=-1\\ 2x-by=0\end{array}\right.$, then $a+b$ equals
A: $2$
B: $-2$
C: $4$
D: $-4$ | \text{B} |
Given that point $P$ lies on the line $y=2x$, if there exist two points $A$ and $B$ on the circle $C(x-3)^2+y^2=4$ such that $PA \perp PB$, then the range of the abscissa $x_{0}$ of point $P$ is _______. | \left[ \frac{1}{5},1\right] |
․ There is a conical container with its vertex at the bottom and the base horizontal, and its axial section is an equilateral triangle with a side length of 6. The container is filled with water. Now, a square prism with a base side length of $a(a<6)$ is vertically immersed in the container. To make the water overflow ... | 2 \sqrt{2} |
Integers from 1 to 100 are placed in a row in some order. Let us call a number [i]large-right[/i], if it is greater than each number to the right of it; let us call a number [i]large-left[/i], is it is greater than each number to the left of it. It appears that in the row there are exactly $k$ large-right numbers and e... | k = 50 |
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