problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Problem 4. The Dursleys are hiding Harry Potter on an island that is 9 km from the shore. The shore is straight. On the shore, 15 kilometers from the point on the shore closest to the island, Hagrid is on a magical motorcycle and wants to reach Harry as quickly as possible. The motorcycle travels along the shore at a s... | 3 |
1. Define an operation “ * ”: For any pair of real numbers $(x, y)$, it always holds that
$$
(x, y) *(x, y)=\left(x+y+1, x^{2}-y-1\right) \text {. }
$$
If real numbers $a$ and $b$ satisfy $(a, b) *(a, b)=(b, a)$, then $a=$ $\qquad$ ,$b=$ $\qquad$ . | a=-1, b=1 |
Let $L_1$ and $L_2$ be perpendicular lines, and let $F$ be a point at a distance $18$ from line $L_1$ and a distance $25$ from line $L_2$. There are two distinct points, $P$ and $Q$, that are each equidistant from $F$, from line $L_1$, and from line $L_2$. Find the area of $\triangle{FPQ}$. | 210 |
Given point $M\left(2 \sqrt{2}, \frac{2 \sqrt{3}}{3}\right)$ on the ellipse $G$: $\frac{{x}^{2}}{{a}^{2}}+ \frac{{y}^{2}}{{b}^{2}}=1 (a > b > 0)$, and the sum of the distances from point $M$ to the two foci is $4 \sqrt{3}$.
(1) Find the equation of the ellipse $G$;
(2) If the line $l$ with slope $1$ intersects the elli... | \frac {9}{2} |
[ Area of a triangle (through semiperimeter and radius of inscribed or exscribed circle) [ Similar auxiliary triangles [ Trapezoids (other).
In trapezoid $ABCD$, the bases $AD=4, BC=1$ and the angles $A$ and $D$ at the base are given, equal to $\operatorname{arctg} 2$ and $\operatorname{arctg} 3$, respectively.
Find ... | \frac{18}{25+2\sqrt{130}+\sqrt{445}} |
Let $A B C D$ be a convex quadrilateral with $A B=A D$. Let $T$ be a point on the diagonal $A C$ such that $\angle A B T+\angle A D T=\angle B C D$. Prove that $A T+A C \geq A B+A D$. | AT+AC\geqAB+AD |
Let \[T_0=2, T_1=3, T_2=6,\] and for $n\ge 3$, \[T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.\] The first few terms are \[2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.\] Find a formula for $T_n$ of the form \[T_n=A_n+B_n,\] where $\{A_n\}$ and $\{B_n\}$ are well known sequences. | T_n = n! + 2^n |
The glass gauge on a [cylindrical](https://artofproblemsolving.com/wiki/index.php/Cylinder) coffee maker shows that there are $45$ cups left when the coffee maker is $36\%$ full. How many cups of coffee does it hold when it is full?
$\text{(A)}\ 80 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 125 \qquad \text{(D)}\ 130 \... | 125 |
Among the following functions, the one that is neither an odd function nor an even function is ( ).
A: $y=x^{2}+|x|$
B: $y=2^{x}-2^{-x}$
C: $y=x^{2}-3^{x}$
D: $y= \dfrac {1}{x+1}+ \dfrac {1}{x-1}$ | C |
1. Replace the asterisk $(*)$ in the expression $\left(x^{3}-2\right)^{2}+\left(x^{2}+*\right)^{2}$ with a monomial so that after squaring and combining like terms, there are four terms. | x^{6}+x^{4}+4x^{2}+4 |
17. Let $f(x)$ be a polynomial in $x$ of degree 5. When $f(x)$ is divided by $x-1, x-2, x-3$, $x-4$ and $x^{2}-x-1, \mathrm{f}(x)$ leaves a remainder of $3,1,7,36$ and $x-1$ respectively. Find the square of the remainder when $f(x)$ is divided by $x+1$. | 841 |
55. Apply the binomial theorem of Newton to find the value of the following sums:
a) $C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\ldots+C_{n}^{n}$
b) $C_{n}^{0}-C_{n}^{1}+C_{n}^{2}-\ldots+(-1)^{n} C_{n}^{n}$
c) $C_{n}^{0}+\frac{1}{2} C_{n}^{1}+\frac{1}{3} C_{n}^{2}+\ldots+\frac{1}{n+1} C_{n}^{n}$
d) $C_{n}^{1}+2 C_{n}^{2}+3 C_{... | 2^n |
Given three points on a plane \\(A(2,-4)\\), \\(B(0,6)\\), and \\(C(-8,10)\\), find the coordinates of \\( \frac {1}{2} \overrightarrow{AC} - \frac {1}{4} \overrightarrow{BC} \\). | (-3, 6) |
4. Considering $\boldsymbol{x}$ and $\boldsymbol{y}$ as integers, solve the system of equations (11 points):
$$
\left\{\begin{array}{l}
8^{x^{2}-2 x y+1}=(z+4) 5^{|y|-1} \\
\sin \frac{3 \pi z}{2}=-1
\end{array}\right.
$$ | (-1;-1;-3),(1;1;-3) |
In the figure on the right, $O$ is the center of the circle, $OK$ and $OA$ are perpendicular to one another, $M$ is the midpoint of $OK$, $BN$ is parallel to $OK$, and $\angle AMN=\angle NMO$. Determine the measure of $\angle A B N$ in degrees.
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=dir(90), K... | 36^\circ |
4. Given $\frac{x}{x^{2}+x+1}=\frac{1}{8}(0<x<1)$. Then the value of $\sqrt{x}-\frac{1}{\sqrt{x}}$ is ( ).
(A) $-\sqrt{7}$
(B) $\sqrt{7}$
(C) $-\sqrt{5}$
(D) $\sqrt{5}$ | C |
Given that $x+x^{-1}=2$, find the value of $x^2+x^{-2}$.
Simplify $$4x^{ \frac {1}{4}}(-3x^{ \frac {1}{4}}y^{- \frac {1}{3}})÷(-6x^{- \frac {1}{2}}y^{- \frac {2}{3}})$$. | 2xy^{\frac{1}{3}} |
If the proposition "$\exists x\in \left[-1,2\right]$, $x^{2}-m \gt 1$" is false, then the minimum value of the real number $m$ is ____. | 3 |
Calculate: $\left(\pi -2023\right)^{0}+|-9|-3^{2}$. | 1 |
Given set A = {x | 4x - 8 < 0} and set B = {x | 9^x < 3}, find A ∩ B.
A: ($$-∞, \frac{1}{2}$$)
B: ($$-∞, \frac{1}{3}$$)
C: ($$\frac{1}{2}, +∞$$)
D: ($$\frac{1}{3}, +∞$$) | \text{A} |
5. Let $0<a<1$. If $x_{1}=a, x_{2}=a^{x_{1}}, x_{3}=a^{x_{2}}, x_{4}$ $=a^{x_{3}}, \cdots, x_{n}=a^{x_{n}-1}, \cdots$, then the sequence $\left\{x_{n}\right\}(\quad$.
(A) increasing
(B) odd terms increase, even terms decrease
(C) decreasing
(D) even terms increase, odd terms decrease | B |
Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005.$ Find $\lfloor P\rfloor.$ | 45 |
Given a moving point $M(x_{0}, y_{0})$ on the parabola $y^{2}=x$, two lines passing through $M$ intersect the parabola at points $P$ and $Q$, and intersect the line $x=t$ at points $A$ and $B$.$(1)$ If the ordinate of point $M$ is $\sqrt{2}$, find the distance between $M$ and the focus;$(2)$ If $t=-1$, $P(1,1)$, $Q(1,-... | 1 |
If the random variable \\(X\\) follows a binomial distribution, where \\(P(x=0)=\dfrac{1}{3}\\), then the values of \\(E(3X+2)\\) and \\(D(3X+2)\\) are respectively ( )
A: \\(4\\) and \\(4\\)
B: \\(4\\) and \\(2\\)
C: \\(2\\) and \\(4\\)
D: \\(2\\) and \\(2\\) | \text{B} |
The country responded to the call of the World Health Organization (WHO) by issuing the "Guidelines for Physical Exercise and Sedentary Behavior," focusing on reducing sedentary time, strengthening physical exercise, and improving physical condition. It was proposed to engage in moderate-intensity aerobic exercise for ... | 13.6 |
In the Cartesian coordinate system, point $M(1,-2)$ is symmetric with respect to the origin to point $N$. Then the coordinates of point $N$ are ( ).
A: $(-2,1)$
B: $(1,-2)$
C: $(2,-1)$
D: $(-1,2)$ | \text{D} |
When Xiao Zhao, Xiao Qian, Xiao Sun, and Xiao Li were asked who had been to the Great Wall, Xiao Zhao said: "I haven't been"; Xiao Qian said: "Xiao Li has been"; Xiao Sun said: "Xiao Qian has been"; Xiao Li said: "I haven't been." Assuming that only one of the four students is lying, it can be determined that the one w... | D |
1. The cold water tap fills the bathtub in 17 minutes, and the hot water tap in 23 minutes. The hot water tap was opened. After how many minutes should the cold water tap be opened so that by the time the bathtub is completely filled, there is an equal amount of cold and hot water in it? | 3 |
In the geometric sequence $\{a_n\}$, the common ratio $q \neq 1$, and the arithmetic sequence $\{b_n\}$ satisfies $b_1=a_1=3$, $b_4=a_2$, $b_{13}=a_3$
(I) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
(II) Let $c_n=(-1)^{n}b_{n}+a_{n}$, calculate the sum of the first $2n$ terms of the sequence... | S_{2n}=\frac{3^{2n+1}-3}{2}+2n |
There are 12 students standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different ways to adjust is \_\_\_\_\_\_. | 840 |
Given sets $A=\{3, 4\}$, $B=\{0, 1, 2, 3, 4\}$ and $A \subseteq C \subseteq B$, list all the possible sets $C$ that satisfy the given condition. | \{0, 1, 2, 3, 4\} |
Given that the pole of the polar coordinate system coincides with the origin of the rectangular coordinate system, and the polar axis coincides with the positive semi-axis of the x-axis of the rectangular coordinate system. The parameteric equations of the line are \\(\begin{cases} x=-1+\frac{3}{5}t \\\\ y=-1+\frac{4}{... | \frac{\sqrt{41}}{5} |
Given the function $g(x)=x^{3}+3ax-2$.
$(1)$ For what value of $a$ is the $x$-axis a tangent to the curve $y=g(x)$;
$(2)$ Find the range of $a$ for which $g(x)$ has extreme values, and calculate the sum of the maximum and minimum values;
$(3)$ Let $f(x)=\left( \frac {1}{3}g′(x)-ax\right)e^{x}-x^{2}$, if the function $f... | (-\infty,0) |
5 Given a set of real numbers $A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\}$ and $B=\left\{b_{1}\right.$, $\left.b_{2}, \cdots, b_{50}\right\}$, if the mapping $f$ from $A$ to $B$ makes each element in $B$ have a preimage, and
$$
f\left(a_{1}\right) \leqslant f\left(a_{2}\right) \leqslant \cdots \leqslant f\left(a_{1... | \mathrm{C}_{99}^{49} |
Given a circle \\(⊙O\\) and a point \\(P\\) inside \\(⊙O\\), a line passing through \\(P\\) intersects \\(⊙O\\) at points \\(A\\) and \\(B\\). If \\(PA \cdot PB = 24\\) and \\(OP = 5\\), then the radius of \\(⊙O\\) is \_\_\_\_\_\_. | 7 |
Determine the relationship between the three numbers $a = \log_{2}0.4$, $b = 0.4^2$, and $c = 2^{0.4}$. | a < b < c |
6. Jack Sparrow needed to distribute 150 piastres into 10 purses. After placing a certain number of piastres in the first purse, he put more in each subsequent purse than in the previous one. As a result, it turned out that the number of piastres in the first purse was not less than half the number of piastres in the l... | 16 |
Suppose there are $2019$ distinct points in a plane and the distances between pairs of them attain $k$ different values. Prove that $k$ is at least $44$. | 44 |
The function is given by $y=\sin (2x+ \frac{π}{4})$.
(1) Find the values of $A$, $ω$, and $φ$;
(2) Find the range of $x∈[0, \frac{π}{2}]$. | [-\frac{\sqrt{2}}{2}, 1] |
Find the standard equation of the ellipse with foci on the coordinate axes and passing through points $A(\sqrt{3},-2)$ and $B(-2\sqrt{3},1)$. | \dfrac{x^2}{15} + \dfrac{y^2}{5} = 1 |
2. Given that the equations of the two asymptotes of a hyperbola are $x-\sqrt{3} y=0$ and $\sqrt{3} x+y=0$, then its eccentricity is ( )
A: $\sqrt{2}$
B: $\sqrt{3}$
C: $2 \sqrt{2}$
D: $\sqrt{3}+1$ | \sqrt{2} |
Example 1 As shown in Figure 1, the lengths of the sides $AB$, $BC$, $CD$, and $DA$ of quadrilateral $ABCD$ are 1, 9, 8, and 6, respectively. For the following statements:
(1) Quadrilateral $ABCD$ is circumscribed around a circle;
(2) Quadrilateral $ABCD$ is not inscribed in a circle;
(3) The diagonals are not perpendi... | C |
Example 8. Find $\int x^{2} \sin \left(x^{3}+1\right) d x$. | -\frac{1}{3}\cos(x^{3}+1)+C |
\section*{Problem 2 - 111032}
Determine all ordered pairs \((x ; y)\) of two-digit natural numbers \(x\) and \(y\) with \(x>y\) for which the following holds:
a) If the digits of the number \(x\) are written in reverse order, the number \(y\) is obtained.
b) If the digits of the number \(x^{2}\) are written in rever... | (21,12)(31,13) |
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more ... | 14 |
Aureliano writes a list containing five numbers, the first of which is 6 and the last of which is 08. The product of the first three numbers is 648, the product of the three central numbers is 432, and the product of the three last numbers is 288. What is Aureliano's list?
# | 6,12,9,4,8 |
Find all functions $f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that the equation
$$
f(x f(x)+y)=f(y)+x^{2}
$$
holds for all rational numbers $x$ and $y$.
Here, $\mathbb{Q}$ denotes the set of rational numbers.
(Slovakia, Patrik Bak)
Answer: $f(x)=x$ and $f(x)=-x$. | f(x)=x \text{ and } f(x)=-x |
37th IMO 1996 shortlist Problem 22 Find all positive integers m and n such that [m 2 /n] + [n 2 /m] = [m/n + n/m] + mn. Solution | (,n)=(k,k^2+1)or(k^2+1,k)foranypositiveintegerk |
2. By how much is the sum of the squares of the first hundred even numbers greater than the sum of the squares of the first hundred odd numbers | 20100 |
1. The solution set of the inequality $\frac{4 x^{2}}{(1-\sqrt{1+2 x})^{2}}<2 x+9$ is | \left[-\frac{1}{2}, 0\right) \cup\left(0, \frac{45}{8}\right) |
Convert the quinary number $444_{(5)}$ to an octal number.
A: $194_{(8)}$
B: $233_{(8)}$
C: $471_{(8)}$
D: $174_{(8)}$ | D |
Example 9 Given real numbers $x, y, z$ satisfy $x+y=5$, $z^{2}=x y+y-9$. Then, $x+2 y+3 z=$ $\qquad$
(1995, Zu Chongzhi Cup Junior High School Mathematics Invitational Competition) | 8 |
Given the parametric equation of curve $C_1$ as $$\begin{cases} x=2-t \\ y= \sqrt {3}t \end{cases}$$ (where $t$ is the parameter), when $t=-1$, it corresponds to a point A on curve $C_1$ and the symmetric point of A with respect to the origin is B. Establishing a coordinate system with the origin O as the pole and the ... | 36 |
7.268. $\left\{\begin{array}{l}9^{\sqrt[4]{x y^{2}}}-27 \cdot 3^{\sqrt{y}}=0, \\ \frac{1}{4} \lg x+\frac{1}{2} \lg y=\lg (4-\sqrt[4]{x}) .\end{array}\right.$
The system of equations is:
\[
\left\{\begin{array}{l}
9^{\sqrt[4]{x y^{2}}}-27 \cdot 3^{\sqrt{y}}=0, \\
\frac{1}{4} \lg x+\frac{1}{2} \lg y=\lg (4-\sqrt[4]{x}) ... | (1;9),(16;1) |
Let $n_0$ be the product of the first $25$ primes. Now, choose a random divisor $n_1$ of $n_0$, where a choice $n_1$ is taken with probability proportional to $\phi(n_1)$. ($\phi(m)$ is the number of integers less than $m$ which are relatively prime to $m$.) Given this $n_1$, we let $n_2$ be a random divisor of $n_1$, ... | \frac{256}{5929} |
The analytical expression of the distance $s$ (unit: $m$) traveled by a car after braking with respect to the time $t$ (unit: $s$) is $s=15t-6t^{2}$. How far did the car travel before coming to a stop after braking? | \frac{75}{8} \text{ m} |
Given that $p: -2 \leqslant x \leqslant 10$ and $q: 1 - m \leqslant x \leqslant 1 + m$. If $p$ is a sufficient but not necessary condition for $q$, find the range of values for the real number $m$. | [9, +\infty) |
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$ | 8 |
11. If from any $\mathrm{n}$ numbers, one can always select 4 numbers such that their sum is a multiple of 4, then what is the minimum value of $n$? | 7 |
The figure below is a map showing $12$ cities and $17$ roads connecting certain pairs of cities. Paula wishes to travel along exactly $13$ of those roads, starting at city $A$ and ending at city $L$, without traveling along any portion of a road more than once. (Paula is allowed to visit a city more than once.)
How ma... | 4 |
8. If one edge length of a tetrahedron is $x$, and the other edges are all 1, the volume is $F(x)$, then the function $F(x)$ on its domain
(A) is an increasing function but has no maximum value
(B) is an increasing function and has a maximum value
(C) is not an increasing function and has no maximum value
(D) is not an... | D |
Given the function $f(x)= \dfrac {x^{2}+ax+b}{x}(x\neq 0)$ is an odd function, and $f(1)=f(4)$
(Ⅰ) Find the values of the real numbers $a$ and $b$;
(Ⅱ) Prove that the function $f(x)$ is monotonically decreasing in the interval $(0,2]$ and monotonically increasing in the interval $(2,+\infty)$. | a=0, b=4 |
Determine the interval in which the root of the function $f(x) = \log_{3}x + x - 3$ is located.
A: $(0,2)$
B: $(1,2)$
C: $(2,3)$
D: $(3,4)$ | C |
Simplify $$\frac {1}{sin70 ^\circ }- \frac { \sqrt {3}}{cos70 ° }$$\=\_\_\_\_\_\_． | -4 |
A store sells a type of notebook. The retail price for each notebook is 0.30 yuan, a dozen (12 notebooks) is priced at 3.00 yuan, and for purchases of more than 10 dozen, each dozen can be paid for at 2.70 yuan.
(1) There are 57 students in the ninth grade class 1, and each student needs one notebook of this type. Wh... | 51.30 |
Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is $AE$?
$\textbf{(A)}\ \frac {9}{2}\qquad \textbf{(B)}\ \frac {50}{11}\qquad \textbf{(C)}\ \frac {21}{4}\qquad \textbf{(D)}\ \frac {17}{3}\qquad ... | 6 |
3. Let $n<100$. Then the largest integer $n$ such that the expansion of $(a+b)^{n}$ has three consecutive terms with coefficients in arithmetic progression is $\qquad$ . . . | 98 |
6. Given the set $M=\{1,2, \cdots, k\}$, for $A \subseteq M$, denote the sum of all elements in $A$ as $S(A)$. If $M$ can be divided into two disjoint subsets $A$ and $B$, and $A \cup B=M, S(A)=2 S(B)$. Find all values of $k$.
(1994, Sichuan Province High School Mathematics Competition) | k=3m \text{ or } k=3m-1 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow \pi} \frac{\sin \left(\frac{x^{2}}{\pi}\right)}{2^{\sqrt{\sin x+1}}-2}$ | \frac{2}{\ln2} |
Find the solutions of the equation
$$
\sqrt{x+\sqrt{4 x+\sqrt{16 x+\sqrt{\ldots+\sqrt{4^{n} x+3}}}}}=1+\sqrt{x}
$$ | 2^{-2n} |
Given the hyperbola $x^{2}-y^{2}=1$, points $F_{1}$ and $F_{2}$ are its two foci, and point $P$ is a point on the hyperbola. If $PF_{1} \perp PF_{2}$, then the eccentricity of the ellipse with foci $F_{1}$ and $F_{2}$ and passing through $P$ is equal to ( ).
A: $\dfrac{\sqrt{5}}{5}$
B: $\dfrac{\sqrt{6}}{3}$
C: $\d... | B |
Example 2 Suppose point $O$ is inside $\triangle A B C$, and $\overrightarrow{O A}+2 \overrightarrow{O B}+3 \overrightarrow{O C}=0$.
Then $S_{\triangle A B C}: S_{\triangle A O C}=(\quad$.
(A) 2
(B) $\frac{3}{2}$
(C) 3
(D) $\frac{5}{3}$
(2004, National High School Mathematics Competition) | C |
Example 3 For any positive integer $n$, prove:
$$
\frac{1}{1 \times 2+2^{2}}+\frac{1}{2 \times 3+3^{2}}+\cdots+\frac{1}{n(n+1)+(n+1)^{2}}<\frac{5}{12} .
$$ | \frac{5}{12} |
9. (GBR 5) ${ }^{1 \mathrm{M} 03}$ Let $\{f(n)\}$ be a strictly increasing sequence of positive integers: $0<f(1)<f(2)<f(3)<\cdots$. Of the positive integers not belonging to the sequence, the $n$th in order of magnitude is $f(f(n))+1$. Determine $f(240)$. | 388 |
10. As shown in Figure 4, in quadrilateral $A B C D$, $A B=B C, \angle A B C$ $=\angle C D A=90^{\circ}, B E \perp A D$ at $E, S_{\text {quadrilateral } A B C D}=8$. Then the length of $B E$ is ( ).
(A) 2
(B) 3
(C) $\sqrt{3}$
(D) $2 \sqrt{2}$ | D |
95. Xiao Ming puts several chess pieces into the small squares of a $3 \times 3$ grid, with each small square either empty or containing one or more chess pieces. Now, by counting the total number of chess pieces in each row and each column, 6 numbers are obtained, and these 6 numbers are all different. What is the min... | 8 |
Given vectors $\overrightarrow{a}=\left( \frac{1}{2},k\right)$, $\overrightarrow{b}=\left(k-1,4\right)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, then the value of the real number $k$ is ( )
A: $\frac{1}{9}$
B: $\frac{2}{9}$
C: $- \frac{1}{7}$
D: $2$ | A |
Problem 3. For what least natural $k$ is the expression $2019 \cdot 2020 \cdot 2021 \cdot 2022 + k$ a square of a natural number? | 1 |
I2.1 If $x, y$ and $z$ are positive real numbers such that $\frac{x+y-z}{z}=\frac{x-y+z}{y}=\frac{-x+y+z}{x}$ and $a=\frac{(x+y) \cdot(y+z) \cdot(z+x)}{x y z}$, find the value of $a$.
I2.2 Let $u$ and $t$ be positive integers such that $u+t+u t=4 a+2$. If $b=u+t$, find the value of $b$. | 10 |
【Question 2】
Xiaohuo, Xiaoyan, and Xiaodian love collecting books. The three of them have a total of 1248 books. Xiaohuo has 64 more books than Xiaoyan, and Xiaoyan has 32 fewer books than Xiaodian. Xiaohuo has
$\qquad$books. | 448 |
1. It is known that the quadratic trinomials $x^{2}+p x+q$ and $x^{2}+q x+p$
have different real roots. Consider all possible pairwise products of the roots of the first quadratic trinomial with the roots of the second (there are four such products in total). Prove that the sum of the reciprocals of these products doe... | 1 |
Given the function $f(x) = |x-2|$, the equation $a[f(x)]^2 - f(x) + 1 = 0$ has four different real solutions. The range of the real number $a$ is ______. | (0, \frac{1}{4}) |
The equation $4^{x}-2^{x-1}+a=0$ has negative roots. Determine the range of values for $a$.
A: $a\geqslant \frac{1}{8}$
B: $0 < a\leqslant \frac{1}{16}$
C: $-\frac{1}{8}\leqslant a < 0$
D: $-\frac{1}{2} < a\leqslant \frac{1}{16}$ | D |
6.228. $\left\{\begin{array}{l}x \sqrt{y}+y \sqrt{x}=30 \\ x \sqrt{x}+y \sqrt{y}=35\end{array}\right.$ | (4,9),(9,4) |
Given positive numbers $x$ and $y$ that satisfy $x + 4y = 40$, find the maximum value of $\lg x + \lg y$. | 2 |
Given the universal set $U=R$, the domain of the function $y=\ln \left(1-x\right)$ is $M$, and the set $N=\{x|x^{2}+5x-6 \lt 0\}$. Which of the following conclusions is correct?
A: $M\cap N=N$
B: $M\cup N=U$
C: $M\cap (C_{U}N)=0$
D: $M\subseteq (C_{U}N)$ | A |
If the graph of $f(x)$ is symmetric about the origin, and it is an increasing function in the interval $[3,7]$ with a maximum value of 5, then $f(x)$ in the interval $[-7,-3]$ is ( )
A: An increasing function and the minimum value is -5
B: An increasing function and the maximum value is -5
C: A decreasing function and... | \text{A} |
Given the curve $$y=f(x)= \frac {4}{x}$$
(1) Find the equation of the tangent line to the curve $y=f(x)$ at point A(2, 2);
(2) Find the equation of the line that is tangent to the curve $y=f(x)$ and passes through point B(2, 0). | 4x + y - 8 = 0 |
3. The equation $x^{2}+a x-2=0$ has two distinct roots $x_{1}$ and $x_{2}$; in this case,
$$
x_{1}^{3}+\frac{22}{x_{2}}=x_{2}^{3}+\frac{22}{x_{1}}
$$
Find all possible values of $a$. | \3 |
6. Among the 8 vertices, 12 midpoints of edges, 6 centers of faces, and the center of the cube, a total of 27 points, the number of groups of three collinear points is ( ).
(A) 57
(B) 49
(C) 43
(D) 37 | B |
In the past 10 years, China's land greening project has made significant progress, with an additional forest area exceeding 22,000,000 hectares. Express 22,000,000 in scientific notation as ( ).
A: $22\times 10^{6}$
B: $2.2\times 10^{6}$
C: $22\times 10^{7}$
D: $2.2\times 10^{7}$ | D |
Given that the polar coordinate equation of curve $C\_1$ is $ρ\cos θ-ρ\sin θ+2=0$, and the parametric equation of curve $C\_2$ is $ \begin{cases} x=\cos α \ y=2\sin α \end{cases}(α$ is the parameter$), the horizontal coordinates of all points on curve $C\_2$ are tripled and the vertical coordinates are multiplied by $\... | 2\sqrt{7} |
16.2.16 * Find the smallest natural number $n$ that satisfies the following properties: (1) $n$ ends with 6 in its decimal representation; (2) when the last digit 6 of $n$ is moved to the front, the resulting number is four times $n$. | 153846 |
# Problem 2. (2 points)
In a positive non-constant geometric progression, the arithmetic mean of the second, seventh, and ninth terms is equal to some term of this progression. What is the minimum possible number of this term? | 3 |
Given a function $f(x)$ with domain $\mathbb{R}$ and its derivative $f'(x)$, it satisfies $f'(x) - 2f(x) > 4$. If $f(0) = -1$, then the solution set of the inequality $f(x) + 2 > e^{2x}$ is ______. | (0, +\infty) |
## Problem Statement
Calculate the volumes of the bodies bounded by the surfaces.
$$
\frac{x^{2}}{27}+y^{2}=1, z=\frac{y}{\sqrt{3}}, z=0(y \geq 0)
$$ | 2 |
Problem 1. How many four-digit numbers have a sum of digits equal to 3? | 10 |
Example 1.7 Find the number of all permutations of $n$ distinct elements $a_{1}, a_{2}, \cdots, a_{n}$ in which $a_{1}$ and $a_{2}$ are not adjacent. | (n-2)(n-1)! |
In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}.$ Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB.$ Find the greatest integer that does not exceed $1000r.$ | 777 |
Due to the rainstorms, our basement filled with water. To remove the water, three pumps were set up. The first pump alone could pump out the water in 3 hours, the second in 4 hours, and the third in 6 hours. After 30 minutes of working together, the second pump broke down. The remaining water was pumped out by the firs... | 1\frac{3}{4} |
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