problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2sin15°cos15°= \_\_\_\_\_\_ ． | \frac{1}{2} |
[ Product of lengths of chord segments and lengths of secant segments ] [ Auxiliary similar triangles Area of a triangle (using height and base). ]
Diameter $A B$ and chord $C D$ of a circle intersect at point $E$, and $C E=D E$. Tangents to the circle at points $B$ and $C$ intersect at point $K$. Segments $A K$ and $... | \frac{27}{4} |
9. Given that two cars, A and B, start from points $A$ and $B$ respectively at the same time, and travel back and forth between $A$ and $B$ at a constant speed. If after the first meeting, car A continues to drive for 4 hours to reach $B$, while car B only drives for 1 hour to reach $A$, then when the two cars meet for... | 86 |
Given the function $f(x)=2|x-1|+|x+2|$.
$(1)$ Find the solution set of $f(x)\leqslant 9$;
$(2)$ If the minimum value of the function $f(x)$ is $M$, and $a+b+c=M$, find the minimum value of $4a^{2}+b^{2}+c^{2}$. | 4 |
To improve the quality of drinking water, more and more residents are starting to purchase household water purifiers. One merchant seized the opportunity and purchased $160$ units of two models, $A$ and $B, from the manufacturer. The cost price of model $A$ is $150$ yuan per unit, and the cost price of model $B$ is $35... | 200 \text{ yuan} |
Example 20. The diameter of a part manufactured in the workshop is a random variable distributed according to the normal law with parameters $a=4.5$ cm and $\sigma=0.05$ cm. Find the probability that the diameter size of a randomly taken part differs from the mathematical expectation by no more than $1$ mm. | 0.9544 |
How many sequences of nonnegative integers $a_1,a_2,\ldots, a_n$ ($n\ge1$) are there such that $a_1\cdot a_n > 0$, $a_1+a_2+\cdots + a_n = 10$, and $\prod_{i=1}^{n-1}(a_i+a_{i+1}) > 0$?
[i]Ray Li.[/i]
[hide="Clarifications"][list=1][*]If you find the wording of the problem confusing, you can use the following, equiva... | 19683 |
Let $ k$ be a positive constant number. Denote $ \alpha ,\ \beta \ (0<\beta <\alpha)$ the $ x$ coordinates of the curve $ C: y\equal{}kx^2\ (x\geq 0)$ and two lines $ l: y\equal{}kx\plus{}\frac{1}{k},\ m: y\equal{}\minus{}kx\plus{}\frac{1}{k}$. Find the minimum area of the part bounded by the curve $ C$ and two lines $... | \sqrt{\frac{2}{3}} |
3. Let $p>3$ be a prime number. For an arbitrary set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{Z}$, let
$$
S_{a}=\left\{x \in\{0,1, \ldots, p-1\} \mid(\exists s \in S) \quad x \equiv_{p} a \cdot s\right\}
$$
(a) How many sets $S \subseteq\{1,2, \ldots, p-1\}$ are there such that the sequence $S_{1}, S_{2}, \ldots, ... | 2 |
7. In the Magic Land, there is a magic stone that grows uniformly upwards. To prevent it from piercing the sky, the Elders of the Immortal World decided to send plant warriors to consume the magic stone and suppress its growth. Each plant warrior consumes the same amount every day. If 14 plant warriors are dispatched, ... | 17 |
Find the value of the algebraic cofactor of the element $-3$ in the third-order determinant $$\begin{vmatrix} 1 & -3 & 5 \\ 4 & 0 & 3 \\ -1 & 2 & 1\end{vmatrix}$$ | -7 |
Example 1 Given $f(x)=\frac{a^{x}}{a^{x}+\sqrt{a}}$ (where $a$ is a constant greater than 0), find the value of $f\left(\frac{1}{1001}\right)+f\left(\frac{2}{1001}\right)+\cdots+$ $f\left(\frac{1000}{1001}\right)$. | 500 |
6. In the Cartesian coordinate system, line $l$ intersects the positive semi-axes of the $x$-axis and $y$-axis at points $N$ and $M$, respectively. A square $ABCD$ is inscribed in the right triangle $\triangle MON$, with points $A$ and $B$ on segments $MO$ and $NO$, respectively, and points $C$ and $D$ on segment $MN$.... | A |
4・16 A student attempts to find the average of three numbers $x, y$, and $z$ by first finding the average of $x$ and $y$, and then finding the average of that result and $z$. When $x<y<z$, the student's final result is
(A) correct.
(B) always less than $A$.
(C) always greater than $A$.
(D) sometimes less than $A$ and s... | C |
8-2. On a distant planet in the mangrove forests, there lives a population of frogs. The number of frogs born each year is one more than the difference between the number of frogs born in the previous two years (the larger number minus the smaller number).
For example, if 5 frogs were born last year and 2 in the year ... | 2033 |
. Determine all polynomials $p(x)$ with real coefficients such that
$$
p\left((x+1)^{3}\right)=(p(x)+1)^{3}
$$
and
$$
p(0)=0
$$
Answer: $p(x)=x$. | p(x)=x |
In the Cartesian coordinate system, the coordinates of the point $(-2,3)$ after moving $4$ units to the right are ( ).
A: $(2,3)$
B: $(-6,3)$
C: $(-2,7)$
D: $(-2,-1)$ | A |
Given the sets M={y|y=2^x, x>0} and N={x|y=log(2x-x^2)}, determine M∩N. | (1,2) |
8. (10 points) Five children $A, B, C, D,$ and $E$ participate in the "Happy Read Pinyin" competition. When they go on stage, the five stand in a row. They each have a contestant number on their chest, and the sum of the 5 numbers is 35. It is known that the sum of the numbers of the contestants standing to the right o... | 24 |
2. How many three-digit natural numbers are there such that the digit in the hundreds place is equal to the sum of the digits in the tens and units places? | 54 |
4. In the equality $a+b=c+d=e+f$ the letters represent different prime numbers less than 20. Determine at least one solution. | 5+19=7+17=11+13 |
The equation of a parabola is given: $y^{2}=18 x$ and the coordinates of a point: $x_{1}=4, y_{1}=3$. Determine the equation of the chord that is bisected by the point, as well as the area of the region bounded by the chord and the parabola. | 14\sqrt{7} |
A rectangular piece of paper $A E C D$ has dimensions $8 \mathrm{~cm}$ by $11 \mathrm{~cm}$. Corner $E$ is folded onto point $F$, which lies on $D C$, as shown. The perimeter of trapezoid $A B C D$ is closest to
(A) $33.3 \mathrm{~cm}$
(B) $30.3 \mathrm{~cm}$
(C) $30.0 \mathrm{~cm}$
(D) $41.3 \mathrm{~cm}$
(E) $35.6 \m... | 33.3\mathrm{~} |
7. The minimum value of the expression $|x-1|+2|2 x-1|+3|3 x-1|+4|4 x-1|+5|5 x-1|$ is $m$, then the largest integer not exceeding $m$ is $\qquad$ . | 3 |
Given two circles \\(C_{1}\\): \\(x^{2}+y^{2}-6x-7=0\\) and \\(C_{2}\\): \\(x^{2}+y^{2}-6y-27=0\\) intersect at points \\(A\\) and \\(B\\), the equation of the perpendicular bisector of segment \\(AB\\) is __________. | x+y-3=0 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $A - C = 90^{\circ}$ and $a + c = \sqrt{2}b$, find angle $C$. | 15^{\circ} |
Given a sequence of positive terms $\{a_{n}\}$ with the sum of the first $n$ terms denoted as $S_{n}$, and $2\sqrt{{S_n}}={a_n}+1$.
$(1)$ Prove that $\{a_{n}\}$ is an arithmetic sequence.
$(2)$ Let the sum of the first $n$ terms of the sequence $\left\{{\frac{{{S_n}}}{{{a_n}{a_{n+1}}}}}\right\}$ be $T_{n}$. If the ... | \left(\frac{6}{7}, \frac{10}{9}\right] |
Calculate the simplified result of $\sin {65^{\circ}}\sin {115^{\circ}}+\cos {65^{\circ}}\sin {25^{\circ}}$.
A: $\sin {40^{\circ}}$
B: $\cos {40^{\circ}}$
C: $1$
D: $0$ | C |
A2. The sum of the digits of a five-digit number is 44. What is the product of the digits of this five-digit number?
(A) $2^{3} \cdot 3^{8}$
(B) $2^{3} \cdot 9^{3}$
(C) $8 \cdot 4^{9}$
(D) $8 \cdot 3^{4}$
(E) None of the above. | 2^{3}\cdot3^{8} |
How many (non-degenerate) triangles with integer side lengths and a perimeter of 100 exist?
# | 208 |
Let the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$ be $S_n$, and it satisfies $S_{2016} > 0$, $S_{2017} < 0$. For any positive integer $n$, we have $|a_n| \geqslant |a_k|$. Determine the value of $k$.
(A) 1006
(B) 1007
(C) 1008
(D) 1009 | D |
In the mid-16th century, the English mathematician Robert Recorde first used "$=$" as an equal sign in his book "The Whetstone of Witte." Later, the English mathematician Thomas Harriot first used the symbols "$ \lt $" and "$ \gt $," which were gradually accepted by the mathematical community. The introduction of inequ... | \text{C} |
If a function satisfies $f(x) = -f(x+2)$, then the value that must be equal to $f(100)$ is ( )
A: $f(1)$
B: $f(2)$
C: $f(3)$
D: $f(4)$ | D |
17. Let $p(x)$ be a polynomial with integer coefficients such that $p(m)-p(n)$ divides $m^{2}-n^{2}$ for all integers $m$ and $n$. If $p(0)=1$ and $p(1)=2$, find the largest possible value of $p(100)$. | 10001 |
Given that $F_{1}$ and $F_{2}$ are the two foci of the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, and $P(1, \frac{\sqrt{2}}{2})$ is a point on the ellipse. It is also given that $\sqrt{2}|PF_{1}|$, $|F_{1}F_{2}|$, $\sqrt{2}|PF_{2}|$ form an arithmetic sequence.
$(1)$ Find the standard e... | \left(\frac{5}{4}, 0\right) |
S is 16 ** Find all real numbers $x$ in the interval $0 \leqslant x \leqslant 2 \pi$ that satisfy the inequality
$$
2 \cos x \leqslant|\sqrt{1+\sin 2 x}-\sqrt{1-\sin 2 x}| \leqslant \sqrt{2}
$$ | \frac{\pi}{4}\leqslantx\leqslant\frac{7}{4}\pi |
Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$? | 14 |
Problem 7.8. On a rectangular sheet of paper, a picture in the shape of a "cross" was drawn from two rectangles $A B C D$ and $E F G H$, the sides of which are parallel to the edges of the sheet. It is known that $A B=9, B C=5, E F=3, F G=10$. Find the area of the quadrilateral $A F C H$.
^{a k c}(a+b+c)^{a+b+c}$ is $\qquad$ . | -1 |
If $\overrightarrow{a}=(2,3,-1)$ and $\overrightarrow{b}=(-2,1,3)$, then the value of $|\overrightarrow{a}-\overrightarrow{b}|$ is ______. | 6 |
1. Seven dwarfs lined up by height, starting with the tallest. The first (i.e., the tallest) said: "My height is 60 cm." The second said: "My height is 61 cm." In order next: "My height is 62 cm," "My height is 63 cm," "My height is 64 cm," "My height is 65 cm," and finally the shortest said: "My height is 66 cm." What... | 1 |
In triangle $ABC$, if $\sin(A+B) = \sin(A-B)$, then triangle $ABC$ must be a(n) ( )
A: Isosceles triangle
B: Equilateral triangle
C: Right triangle
D: Acute triangle | \text{C} |
6. The sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1, \ldots$ is formed as follows: write down infinitely many ' 1 's, insert a ' 2 ' between the first and the second ' 1 's, insert two ' 2 's between the second and the third ' 1 's, insert three ' 2 's between the third and the fourth ' 1 's, and so on. If $a_{n}$ denotes t... | 7806 |
A coastal city plans to build 7 seaside theme parks along a coastal avenue. Due to financial reasons, it is planned to reduce 2 of the seaside theme parks. The theme parks at both ends are not included in the adjustment plan, and two adjacent theme parks cannot be adjusted at the same time. The number of adjustment pla... | C |
2.1. Vasya loves to collect mushrooms. He calculated that over the autumn he collected a number of mushrooms that is expressed by a three-digit number, the sum of the digits of which is 14. Then Vasya calculated that 8% of the collected mushrooms were white, and 14% were boletus. How many mushrooms did Vasya collect? | 950 |
Given that the slant height of a cone is $2$ and its lateral surface area is $2\pi$, what is the volume of the cone?
A: $\frac{{\sqrt{3}\pi}}{3}$
B: $\sqrt{3}\pi$
C: $\frac{\pi}{3}$
D: $\pi$ | A |
Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\]
the following inequality also holds
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{... | \left(\frac{3}{4}\right)^n |
[ Planar graphs. Euler's formula ]
In the country of Lake, there are seven lakes connected by ten non-intersecting channels, such that one can sail from any lake to any other. How many islands are there in this country?
# | 4 |
272. The Artist's Puzzle. An artist decided to purchase a canvas for a miniature, the area of which should be 72 cm $^{2}$. To stretch the miniature on a stretcher, there should be strips of clean canvas 4 cm wide at the top and bottom, and 2 cm wide on the sides.
What are the smallest dimensions of the required canva... | 10\times20 |
The equation of the circle with center at $(1,1)$ and passing through the origin is
A: $(x-1)^{2}+(y-1)^{2}=1$
B: $(x+1)^{2}+(y+1)^{2}=1$
C: $(x+1)^{2}+(y+1)^{2}=2$
D: $(x-1)^{2}+(y-1)^{2}=2$ | \text{D} |
5.1. From point $A$ to point $B$, which are 10 km apart, a car departed at 7:00. After traveling $2 / 3$ of the distance, the car passed point $C$, from which a cyclist immediately set off for point $A$. As soon as the car arrived at $B$, a bus immediately departed from $B$ in the opposite direction and arrived at $A$ ... | 6 |
16. $[\mathbf{9}]$ Solve for $x$ :
$$
x\lfloor x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor\rfloor=122 \text {. }
$$ | \frac{122}{41} |
4. Given in rectangle $A B C D$, $A B=72, A D=$ 56. If $A B$ is divided into 72 equal parts, and parallel lines to $A D$ are drawn through each division point; and if $A D$ is divided into 56 equal parts, and parallel lines to $A B$ are drawn through each division point, then these parallel lines divide the entire rect... | 120 |
Given an arithmetic sequence $a_1, a_2, \ldots, a_9$ with a common difference of 3, and a random variable $\xi$ that takes the values $a_1, a_2, \ldots, a_9$ with equal probability, find the variance $D\xi = \_\_\_\_\_\_$. | 60 |
2. Let the sequence $\left\{a_{n}\right\}$ satisfy
$$
a_{1}>1, a_{n+1}=a_{n}+\frac{1}{a_{n}^{2}} \text {. }
$$
Then the sequence $\left\{a_{n}\right\}()$.
(A) is bounded and decreasing
(B) is bounded and increasing
(C) is unbounded and increasing
(D) is unbounded and decreasing | C |
5. A product that initially contained $98\%$ water dried out over time and began to contain $97\%$ water. By what factor did it shrink (i.e., reduce its weight)? | 1.5 |
23. In $\triangle A B C$, $\angle A$ is the smallest angle, $\angle B$ is the largest angle, and $2 \angle B=5 \angle A$. If the maximum value of $\angle B$ is $m^{\circ}$, and the minimum value of $\angle B$ is $n^{\circ}$, then $m+n=(\quad)$.
(A) 155
(B) 165
(C) 175
(D) 185 | 175 |
4. The following infinite sequence composed of 1s and 2s has an interesting property: starting from the first term, group the same consecutive numbers together, and then write down the number of terms in each group in order. The infinite sequence formed by these numbers is exactly the sequence itself. This sequence is ... | 1,1,2,1,1,2,2,1 |
4. Calculate $\sqrt{6+\sqrt{20}}-\sqrt{6-\sqrt{20}}$. | 2 |
2. In triangle $A B C$, the median $B M$ is twice as short as side $A B$ and forms an angle of $40^{\circ}$ with it. Find the angle $A B C$. | 110 |
The equation of the tangent line to the function $f\left(x\right)=2\ln x+x^{2}$ at the point $\left(1,1\right)$ is ____. | 4x-y-3=0 |
One. (50 points) As shown in Figure 1, in $\triangle ABC$, $AB = AC = 3, BC = 2, AD \perp BC$ at point $D, BE \perp AC$ at point $E, EF \perp BC$ at point $F, AD$ intersects $BE$ at point $H$. A circle $\odot O$ is drawn with $HF$ as its diameter, intersecting $HE$ and $EF$ at points $P$ and $Q$ respectively. Extend $Q... | P Q=\frac{\sqrt{946}}{108}, K H=\frac{329 \sqrt{2}}{576} |
The probability of twins being born in Shvambria is $p$, and triplets are not born in Shvambria.
a) Estimate the probability that a Shvambrian met on the street is one of a pair of twins?
b) In a certain Shvambrian family, there are three children. What is the probability that there is a pair of twins among them?
c)... | \frac{Np}{p+1} |
$3, 7$ different positive integers arranged in ascending order form an arithmetic sequence. It is known that the average of the first three numbers is 20, and the average of the last three numbers is 24. Then, the average of the middle three numbers is $\qquad$ - | 22 |
10.3. In trapezoid $A B C D$, the bases $B C=3$ and $A D=9$, angle $\angle B A D=30$ degrees, and $\angle A D C=60$ degrees. A line through point $D$ divides the trapezoid into two equal areas. Find the length of the segment of this line that is inside the trapezoid. | \sqrt{39} |
Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$. | 5 |
Given that $a=4^{\log _{3}2}$, $b=4^{\log _{9}6}$, and $c=(\frac{1}{2})^{-\sqrt{5}}$, determine the correct order of $a$, $b$, and $c$.
A: $a > b > c$
B: $a > c > b$
C: $b > c > a$
D: $c > b > a$ | D |
Calculate:$(1) 3x^{2}\cdot x^{4}-(-x^{3})^{2}$;$(2) a^{3}\cdot a+(-a^{2})^{3}\div a^{2}$. | 0 |
Let $P$ be any point on the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{4} = 1$ and $Q(2,0)$. Determine the minimum length of the line segment $PQ$. | \frac{2\sqrt{6}}{3} |
Calculate: $16^{-\frac{1}{2}} - \left(\frac{27}{8}\right)^{\frac{2}{3}} + \sqrt[4]{(3-\pi)^{4}} = \_\_\_\_\_\_$. | \pi - 5 |
Three years ago, the mother's age was four times that of her son. The son is 12 years old this year. How old is the mother this year? | 39 |
(1) In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $A=60^{\circ}$, $b=1$, and the area of $\triangle ABC$ is $\sqrt{3}$, then $a=$ .
(2) The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=3n^2-2n$. Then, the general term formula of the sequ... | c\in( \sqrt{5},3) |
Randomly select $6$ movie fans to rate the movie "The Battle at Lake Changjin." The sample data obtained is as follows: $92$, $93$, $95$, $95$, $97$, $98$. Which of the following statements about the sample are correct?
A: The mean is $95$
B: The range is $6$
C: The variance is $26$
D: The $80$th percentile is $97$ | ABD |
7.42 In $\triangle A B C$, $\angle A B C=120^{\circ}, \quad A B=3, \quad B C=4$. If a perpendicular line from $A$ to $A B$ intersects with a perpendicular line from $C$ to $B C$ at $D$, then $C D$ equals
(A) 3 .
(B) $\frac{8}{\sqrt{3}}$.
(C) 5 .
(D) $\frac{11}{2}$.
(E) $\frac{10}{\sqrt{3}}$.
(43rd American High School ... | \frac{10}{\sqrt{3}} |
8、Let the line $l: y=k x+m$ (where $k, m$ are integers), intersect the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{12}=1$ at two distinct points $A, B$, and intersect the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{12}=1$ at two distinct points $C, D$, such that the vector $\overrightarrow{A C}+\overrightarrow{B D}=\overrightar... | 9 |
Archimedes has a two-pan balance with arms of different lengths. Aiming to weigh two kilograms of sugar, he proceeded as follows: he placed a one-kilogram weight on the left pan and sugar on the other side until the balance was level. Then, he emptied both pans, placed the one-kilogram weight on the right pan and sugar... | >2 |
Given that \\(f(x)\\) is an even function defined on \\(\mathbb{R}\\), and is monotonically increasing in the interval \\((-\infty,0)\\), if the real number \\(a\\) satisfies \\(f(2^{|a-1|}) > f(-\sqrt{2})\\), then the range of values for \\(a\\) is \\((\quad)\).
A: \\((-∞, \frac{1}{2} )\\)
B: \\((-∞, \frac{1}{2} )∪( \... | \text{C} |
Let the sum of the digits of a 1995-digit number divisible by 9 be $a$. We then add the digits of $a$, and let the result be $b$. What can the sum of the digits of $b$ be? | 9 |
Given that the focus of the parabola $y^2 = 4x$ is $F$, and the line $l$ passing through $F$ intersects the parabola at points $A$ and $B$. If $M$ is the midpoint of $AB$ and the vertical line through $M$ intersects the parabola at point $P$ in the first quadrant such that $|PF| = \frac{3}{2}$, find the equation of lin... | \sqrt{2}x - y - \sqrt{2} = 0 |
Among the following four propositions, the incorrect one is ( ).
A: If lines $a$ and $b$ are parallel to each other, then lines $a$ and $b$ determine a plane.
B: If four points are not coplanar, then any three points among them are not collinear.
C: If two lines have no common points, then these two lines are skew line... | C |
Given ${\left(2x-\sqrt{3}\right)}^{10}=a\_0+a\_1x+a\_2x^2+\cdots+a_{10}x^{10}$,
(1) Find $a\_0$;
(2) Find $\left(a\_0+a\_1+a\_2+a\_3+\cdots+a_{10}\right)\left(a\_0-a\_1+a\_2-a\_3+\cdots+a_{10}\right)$. | 1 |
How should the graph of the quadratic function $y = -2x^2 + 4x + 1$ be moved to obtain the graph of $y = -2x^2$?
A: Move 1 unit to the left and 3 units up
B: Move 1 unit to the right and 3 units up
C: Move 1 unit to the left and 3 units down
D: Move 1 unit to the right and 3 units down | \text{C} |
The equation of the circle with center at the point (0,1) and radius 2 is ( )
A: $x^2+(y-1)^2=2$
B: $(x-1)^2+y^2=2$
C: $x^2+(y-1)^2=4$
D: $(x-1)^2+y^2=4$ | C |
3. An electronic clock displays time from 00.00.00 to 23.59.59. How much time during a day does the clock display exactly four digit 3s? | 105 |
4. In the coordinate plane, points with both coordinates as integers are called integer points. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Then $f(1)+f(2)+$ $\cdots+f(1990)=$ $\... | 1326 |
Let $f(x)$ have a derivative, and satisfy $\lim_{\Delta x \to 0} \frac{f(1)-f(1-2\Delta x)}{2\Delta x}=-1$. Then, the slope of the tangent line at point $(1,f(1))$ on the curve $y=f(x)$ is $(\quad)$
A: $2$
B: $-1$
C: $1$
D: $-2$ | \text{B} |
Suppose $a,b,c,x,y,z$ are pairwisely different real numbers. How many terms in the following can be $1$ at most:
$$\begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,\\
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned}$$ | 2 |
1. Given $A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty)$, where $a<b$, if $A \cup B=\mathbf{R}$, then the minimum value of $a-b$ is $\qquad$ . | -1 |
Task B-4.7. The base of a triangular pyramid is a triangle with sides $a, b$, and $c$. The length of side $c$ is $7 \, \text{cm}$, $a - b = 5 \, \text{cm}$, and the angle opposite side $c, \gamma = 60^\circ$. The lateral face containing the longest base edge is perpendicular to the base plane and congruent to the base.... | 9 |
1. In the plane, there are points $A(-2,1), B(1,4), C(4,-3)$, point $D$ is on line $A B$, $\frac{A D}{B D}=\frac{1}{2}$, extend $D C$ to $E$, such that $\frac{|C E|}{|E D|}=\frac{1}{4}$, then the coordinates of point $E$ are
A. $(0,1)$
B. $(0,1)$ or $\left(2, \frac{11}{3}\right)$
C. $\left(-\frac{8}{3}, \frac{11}{3}\ri... | (\frac{17}{3},-\frac{14}{3}) |
The slope of the tangent line to the graph of the function $f(x)= \frac {2x-5}{x^{2}+1}$ at the point $(0,f(0))$ is ( )
A: $ \frac {1}{2}$
B: $- \frac {1}{2}$
C: $-2$
D: $2$ | D |
[Isosceles, inscribed and circumscribed trapezoids]
The perimeter of a trapezoid circumscribed about a circle is 40. Find the midline of the trapezoid. | 10 |
2. Let the set
$$
A=\left\{a_{1}, a_{2}, \cdots, a_{5}\right\}\left(a_{i} \in \mathbf{R}_{+}, i=1,2, \cdots, 5\right) .
$$
If the set of the products of the four elements in all four-element subsets of set $A$ is $B=\{2,3,4,6,9\}$, then the sum of the elements in set $A$ is $\qquad$ | \frac{49}{6} |
If the proposition "$\exists x \in \mathbb{R}$, such that $x^{2}+(a-1)x+1 < 0$" is false, then the range of the real number $a$ is. | -1 \leqslant a \leqslant 3 |
5. Given a sequence $x_{n}$ such that $x_{1}=1, x_{2}=2, x_{n+2}=\left|x_{n+1}\right|-x_{n}$. Find $x_{2015}$. | 0 |
In $\vartriangle ABC, PQ // BC$ where $P$ and $Q$ are points on $AB$ and $AC$ respectively. The lines $PC$ and $QB$ intersect at $G$. It is also given $EF//BC$, where $G \in EF, E \in AB$ and $F\in AC$ with $PQ = a$ and $EF = b$. Find value of $BC$.
| BC = \frac{ab}{2a - b} |
Find a costant $C$, such that $$ \frac{S}{ab+bc+ca}\le C$$ where $a,b,c$ are the side lengths of an arbitrary triangle, and $S$ is the area of the triangle.
(The maximal number of points is given for the best possible constant, with proof.) | \frac{1}{4\sqrt{3}} |
$\lg 4 + 2\lg 5 =$ __________. | 2 |
Find the largest integer $k$ ($k \ge 2$), for which there exists an integer $n$ ($n \ge k$) such that from any collection of $n$ consecutive positive integers one can always choose $k$ numbers, which verify the following conditions:
1. each chosen number is not divisible by $6$, by $7$, nor by $8$;
2. the positive dif... | 108 |
16. Person A drives 15 miles at a speed of 30 miles per hour, then increases the speed to 55 miles per hour. To increase the average speed to 50 miles per hour, he still needs to drive ( ) miles.
(A) 45
(B) 62
(C) 90
(D) 110
(E) 135 | 110 |
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