problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
2. Given $a, b, c > 0$, find the maximum value of the expression
$$
A=\frac{a^{4}+b^{4}+c^{4}}{(a+b+c)^{4}-80(a b c)^{4 / 3}}
$$ | 3 |
After a rightward translation of one unit length, the coordinates of the point $\left(4,-3\right)$ become ( ).
A: $(5,-3)$
B: $(3,-3)$
C: $(4,-2)$
D: $(4,-4)$ | A |
Given that $f(x)$ is an odd function, and $x_0$ is a root of $y = f(x) + e^x$, then $-x_0$ is definitely a root of which of the following functions?
A: $y = f(-x)e^x - 1$
B: $y = f(x)e^{-x} + 1$
C: $y = e^x f(x) - 1$
D: $y = e^x f(x) + 1$ | \text{C} |
In the mapping $f$: $A→B$, $A=B=(x,y)|x$, $y∈R$ and $f$: $((x,y)→(x-y,x+y))$ the element in set $B$ that corresponds to the element $(-1,2)$ in set $A$ is _____. | (-3,1) |
Among the following functions, the one that is increasing in the interval $(0,1)$ is $(\quad)$
A: $y=|x|$
B: $y=3-x$
C: $y= \dfrac {1}{x}$
D: $y=-x^{2}+4$ | A |
Given the inequality $|2x+5|+|2x-1|-t\geqslant 0$ with the solution set $R$.
1. Find the maximum value $s$ of the real number $t$.
2. If positive real numbers $a$ and $b$ satisfy $4a+5b=s$, find the minimum value of $y= \frac {1}{a+2b}+ \frac {4}{3a+3b}$. | y= \frac {3}{2} |
Given: The two square roots of a positive number $x$ are $a+3$ and $2a-15$, $\sqrt{2b-1}=13$.
$(1)$ Find the value of $x$;
$(2)$ Find the cube root of $a+b-1$. | 2\sqrt[3]{11} |
If $\log 2 = .3010$ and $\log 3 = .4771$, the value of $x$ when $3^{x+3} = 135$ is approximately
$\textbf{(A) \ } 5 \qquad \textbf{(B) \ } 1.47 \qquad \textbf{(C) \ } 1.67 \qquad \textbf{(D) \ } 1.78 \qquad \textbf{(E) \ } 1.63$ | 1.47 |
Given that $a, b \in R^{+}$, determine the relationship in size between $\frac{1}{a} + \frac{1}{b}$ and $\frac{1}{a+b}$. | \frac{1}{a} + \frac{1}{b} > \frac{1}{a+b} |
A3. Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function such that
$$
f(x)= \begin{cases}x ; & x \geq 2 \\ f(4-x) ; & 0 \leq x<2 \\ f(x+2) ; & x<0\end{cases}
$$
What is $f(-5)$?
(A) -3
(B) -1
(C) 1
(D) 3
(E) 5
Problems for 3rd grade | 3 |
Calculate: $(-1)^{2023}+|1-\sqrt{3}|-\frac{2}{\sqrt{3}-1}-\left(-\frac{1}{2}\right)^{-2}$. | -7 |
From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows:
75, 80, 85, 65, 95, 100, 70, 55, 65, 75, 85, 110, 120, 80, 85, 80, 75, 90, 90, 95, 70, 60, 60, 75, 90, 95, 65, 75, 80, 80. The number of students in this ... | 630 |
Given $$z= \frac {2+i}{-2i+1}$$ (where $i$ is the imaginary unit), the real part of the complex number $z$ is ( )
A: 0
B: -1
C: 1
D: 2 | \text{A} |
[ Examples and counterexamples. Constructions $\quad]$ [ Linear inequalities and systems of inequalities $]$
Can 20 numbers be written in a row so that the sum of any three consecutive numbers is positive, while the sum of all 20 numbers is negative? | =-25,b=13 |
【Example 8】 How many terms are there in the expansion of an $n$-order determinant?
| n! |
A small ball with a radius of $1$ can move freely in all directions inside a regular tetrahedron container with an edge length of $4\sqrt{6}$. The area of the container's inner wall that the small ball will never touch is ____. | 72\sqrt{3} |
Calculate the value of $(\log 2)^2 + \log 2 \cdot \log 50 + \log 25$. | 2 |
In the right triangle $ABC$, the foot of the altitude from $C$ to the hypotenuse $AB$ is $D$. The angle bisectors of $\angle ACD$ and $\angle BCD$ intersect the hypotenuse $AB$ at points $E$ and $F$, respectively. Determine the ratio of the radius of the incircle of triangle $ABC$ to the radius of the circumcircle of t... | \frac{\sqrt{2}}{2} |
8. Given that $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ are both arithmetic sequences, and $a_{11}=32, b_{21}=43$. Let $c_{n}=(-1)^{n} \cdot\left(a_{n}-b_{n}\right)$, the sum of the first 10 terms of the sequence $\left\{c_{n}\right\}$ is 5, and the sum of the first 13 terms is -5, then the sum of the first 100... | 10200 |
Find the equation of the circle that passes through points A(4, 2) and B(-1, 3), and the sum of its intercepts on the two coordinate axes is 2. | x^2+y^2-2x-12=0 |
Four (18 points) As shown in Figure 3, $ABCD$ is a square with a side length of 1, $U$ and $V$ are points on $AB$ and $CD$ respectively, $AV$ intersects with $DU$ at point $P$, and $BV$ intersects with $CU$ at point $Q$. Find the maximum area of quadrilateral $PUQV$. | \frac{1}{4} |
7. Let the four vertices of a regular tetrahedron be $A, B, C, D$, with each edge length being 1 meter. A small insect starts from point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to that vertex with equal probability, and crawls all the way to the end of ... | \frac{182}{729} |
The eight corners of a cube are cut off, yielding a polyhedron with $6$ octagonal faces and $8$ triangular faces. Given that all polyhedron's edges have length $2$, compute the volume of the polyhedron. | 56 + \frac{112\sqrt{2}}{3} |
4. In the cells of a $3 \times 3$ square, the numbers $1,2,3, \ldots, 9$ are arranged. It is known that any two consecutive numbers are located in adjacent (by side) cells. Which number can be in the central cell if the sum of the numbers in the corner cells is $18?$
# | 7 |
1. When one of two integers was increased 1996 times, and the other was reduced 96 times, their sum did not change. What can their quotient be | 2016 |
Use the short division method to find the greatest common divisor (GCD) and the least common multiple (LCM) of the following groups of numbers:
(1) 42 and 63; (2) 8 and 20. | 40 |
Given that {an} is an arithmetic sequence, a1 + a3 + a5 = 18, a2 + a4 + a6 = 24, find the value of a20 ( ).
A: 10
B: 20
C: 40
D: 80 | C |
Given the function $f(x)=-\sin^2x+m\cos x-1$, $x\in [-\frac{π}{3},\frac{2π}{3}]$.
$(1)$ If the minimum value of $f(x)$ is $-4$, find the value of $m$.
$(2)$ When $m=2$, if $|f(x_{1})-f(x_{2})|≤2a-\frac{1}{4}$ holds for any $x_{1}$, $x_{2}\in [-\frac{π}{3},\frac{2π}{3}]$, find the range of real number $a$. | [2,+\infty) |
3. Person A tosses a fair coin twice, and Person B tosses the same coin three times. If the probability that they end up with the same number of heads is written as a simplified fraction, the sum of the numerator and the denominator is $\qquad$ . (Romania) | 21 |
Which of the following propositions is correct? ( )
A: If two lines form equal angles with the same plane, then these two lines are parallel.
B: If a line is perpendicular to two lines within a plane, then this line is perpendicular to this plane.
C: If a line is parallel to two intersecting planes, then this line is p... | \text{C} |
4. Given the sequence $a_{0}, a_{1}, \cdots, a_{n}, \cdots$ satisfies the relation $\left(3-a_{n+1}\right)\left(6+a_{n}\right)=18$, and $a_{0}=3$. Then $\sum_{i=0}^{n} \frac{1}{a_{i}}=$ $\qquad$ . | \frac{1}{3}\left(2^{n+2}-n-3\right) |
Given the function $f(x)=x^{3}+ax$, if $f(2)=10$, then $a=$ ______. | 1 |
3. Given positive real numbers $a, b, c$ satisfying $a+b+c=1$. Then the maximum value of $a+\sqrt{b}+\sqrt[3]{c}$ is $\qquad$ . | \frac{5}{4}+\frac{2 \sqrt{3}}{9} |
An $a \times b \times c$ rectangular box is built from $a \cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \times b \times c$ parallel to the $(b \times c)$ faces of the box contains exactly $9$ red cubes, exactly $12$ green cubes, and some yellow cubes. Ea... | 180 |
4. In $\triangle A B C$, $A B=1, B C=2, \angle B-\angle C$ $=90^{\circ}$. Then the radius of the circumcircle of $\triangle A B C$ is $\qquad$ | \frac{\sqrt{3}+1}{2} |
5. A. A fair cube die has the numbers $1, 2, 3, 4, 5, 6$ on its six faces. When the die is rolled twice, let the remainders when the sum of the numbers on the top faces is divided by 4 be $0, 1, 2, 3$ with probabilities $P_{0}, P_{1}, P_{2}, P_{3}$, respectively. Then the largest among $P_{0}, P_{1}, P_{2}, P_{3}$ is $... | D |
Let's determine
$$
1 \cdot 4 + 3 \cdot 6 + 5 \cdot 8 + 7 \cdot 10 + \ldots
$$
the $n$-th term of the series, as well as the sum of the first $n$ terms. | \frac{n}{3}(4n^2+9n-1) |
Problem 5.5. A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of the square is 24, and the perimeter of the small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a figure is the sum of the lengths of all its sides.
 of a certain firework and the flight time $t$ (in seconds) is given by the equation $h=-\frac{3}{4}t^2+12t-21$. If this type of firework explodes at the highest point of ascent, then the time required from ignition to explosion is ______ seconds. | 8 \text{ seconds} |
Given the functions $f(x) = \frac{e^x - a}{x}$ and $g(x) = a \ln x + a$.
1. When $a=1$, determine the intervals of monotonicity for $F(x) = f(x) - g(x)$.
2. If $y=f(x)$ is always above $y=g(x)$ for $x>1$, find the range of the real number $a$. | a \leq \frac{1}{2}e |
Which of the following calculations is correct?
A: $a^{8}\div a^{2}=a^{4}$
B: $5ab-2ab=3$
C: $(a-b)^{2}=a^{2}-b^{2}$
D: $(-ab^{3})^{2}=a^{2}b^{6}$ | D |
7. The function $f(x)=a x^{2}+b x+3 a+b$ is an even function, with the domain $[a-1,2 a](a$ 、 $b \in \mathbf{R})$, its range is $\qquad$ . | [1,\frac{31}{27}] |
Given a circle C that passes through the three intersection points of the parabola $y=x^2-4x+3$ and the coordinate axes.
(1) Find the equation of circle C;
(2) Suppose the line $2x-y+2=0$ intersects circle C at points A and B, find $|AB|$. | \frac{6\sqrt{5}}{5} |
57. There are two triangles with respectively parallel sides and areas $S_{1}$ and $S_{2}$, where one of them is inscribed in triangle $A B C$, and the other
is circumscribed around it. Find the area of triangle $A B C$. | S_{ABC}=\sqrt{S_{1}S_{2}} |
$1.3,4$ ** Find all real numbers $a$ such that for any real number $x$, the value of the function $f(x)=x^{2}-2 x-|x-1-a|-|x-2|+4$ is a non-negative real number. | \mid-2\leqslant\leqslant1 |
A new language uses only the letters A, B, C, D, and E. The letters A and E are called vowels, while the letters B, C and D are called consonants. A sequence of letters is called a word if it does not include the same letter twice in a row, and it does not include two vowels in a row. How many words are there in this l... | 199776 |
2.4. The base of a quadrilateral pyramid is a rectangle with a diagonal equal to $b$, and the angle between the diagonals is $60^{\circ}$. Each of the lateral edges forms an angle of $45^{\circ}$ with the base plane. Find the volume of the pyramid. | \frac{b^{3}\sqrt{3}}{24} |
Let's determine the digits $A, B, C, D$ if
$$
\overline{A B^{2}}=\overline{A C D B}, \quad \text { and } \quad \overline{C D^{3}}=\overline{A C B D}
$$ | A=9,C=2,B=6,D=1 |
Solve the following inequality:
$$
\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}>\frac{4}{3} \sqrt{\frac{x}{x+\sqrt{x}}}
$$ | 1\leqx<\frac{25}{9} |
The line $x+y+2=0$ intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. If point $P$ is on the circle $(x-2)^{2}+y^{2}=2$, then the range of the area of $\triangle ABP$ is $\quad$.
A: $[2,6]$
B: $[4,8]$
C: $[\sqrt {2},3\sqrt {2}]$
D: $[2\sqrt {2},3\sqrt {2}]$ | (A) |
Given that the positive numbers $a$ and $b$ satisfy $a + b = 1$, find the minimum value of $$\frac{b}{a} + \frac{1}{b}$$, and the value of $a$ when the minimum value is achieved. | \frac{1}{2} |
(1) Use the "Euclidean algorithm" to find the greatest common divisor of 459 and 357.
(2) Use Horner's method to calculate the value of the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$ when $x = -4$. | 3392 |
69. A reconnaissance aircraft is flying in a circle centered at point $A$. The radius of the circle is $10 \mathrm{km}$, and the speed of the aircraft is 1000 km/h. At some moment, a missile is launched from point $A$ with the same speed as the aircraft, and it is controlled such that it always remains on the line conn... | \frac{\pi}{200} |
In a study group, there are four students: A, B, C, and D. In a math test, one student did not pass. When the other students were asked about the situation, A said, "The one who did not pass is among A, C, and D"; B said, "It is C who did not pass"; C said, "It is either A or B who did not pass"; D said, "What B said i... | A |
3. Triangle $A B C$ with angle $\angle A B C=135^{\circ}$ is inscribed in circle $\omega$. The lines tangent to $\omega$ at points $A$ and $C$ intersect at point $D$. Find $\angle A B D$, given that $A B$ bisects segment $C D$. Answer: $90^{\circ}$
... | B |
Calculate:
1. $$\sqrt {9}-(3-π)^{0}+( \frac {1}{5})^{-1}$$
2. $$(1- \sqrt {3})^{2}+ \sqrt {12}$$ | 4 |
Example 4. In a group of 21 students, including 5 excellent students, 10 well-performing students, and 6 poorly performing students. On the upcoming exam, excellent students can only receive excellent grades. Well-performing students can receive good or excellent grades with equal probability. Poorly performing student... | \frac{17}{21} |
Given $\sin 2\alpha = \frac{2}{3}$, then $\sin^2\left(\alpha + \frac{\pi}{4}\right) = $ （ ）
A: $\frac{1}{3}$
B: $\frac{1}{2}$
C: $\frac{3}{4}$
D: $\frac{5}{6}$ | \text{D} |
Given that the real numbers $x$ and $y$ satisfy the condition $\ln (x+y)=0$, find the minimum value of $\dfrac {1}{x}+ \dfrac {1}{y}$. | 4 |
The terms of the arithmetic sequence $\{a_n\}$ are all positive numbers, and it is given that $a_{3}^{2} + a_{8}^{2} + 2a_{3}a_{8} = 9$. Find the sum of the first 10 terms, $s_{10}$. | 15 |
Among $-2$, $\sqrt{4}$, $\frac{\sqrt{2}}{2}$, $3.14$, $\frac{22}{3}$, the number of rational numbers is ( )
A: $5$
B: $4$
C: $3$
D: $2$ | B |
Task 3. A team of workers was working on pouring the rink on the large and small fields, with the area of the large field being twice the area of the small field. In the part of the team that worked on the large field, there were 4 more workers than in the part that worked on the small field. When the pouring of the la... | 10 |
Exercise 6. Let $A B C D E F$ be a regular hexagon and $M \in[A, C], N \in[C, E]$. Suppose that $\frac{A M}{A C}$ and $\frac{C N}{C E}$ are equal to a number $r>0$, and that $B, M, N$ are collinear. Determine the value of $r$.
$
A. $[1,6]$
B. $[7,12]$
C. $[13,18]$
D. $[19,24]$ | B |
If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by
\[
F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q).
\]
Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$? | p = \frac{7}{12} |
Given $\{a_n\}$ is an arithmetic sequence, and $a_2 + a_8 = 12$, find the value of $a_5$.
A: 4
B: 5
C: 6
D: 7 | C |
Problem 2. In Grandfather Frost's bag, there are fewer than a hundred gifts for Petya, Vasya, Borya, and Lёsha. Grandfather Frost gave half of the gifts to Petya, a fifth to Vasya, and a seventh to Borya. How many gifts did Lёsha receive? | 11 |
Example 1 (Fill in the blank Question 1) Given $\frac{1}{4}(b-c)^{2}=$ $(a-b)(c-a)$ and $a \neq 0$. Then $\frac{b+c}{a}=$ $\qquad$ . | 2 |
There are 5 people standing in a row. If there is exactly one person between person A and person B, then the number of different arrangements is . | 36 |
One of the boxes that Joshua and Wendy unpack has Joshua's collection of board games. Michael, Wendy, Alexis, and Joshua decide to play one of them, a game called $\textit{Risk}$ that involves rolling ordinary six-sided dice to determine the outcomes of strategic battles. Wendy has never played before, so early on Mi... | 17 |
6. In the sequence of natural numbers starting from 1, certain numbers are colored red according to the following rules. First, color 1; then color two even numbers 2, 4; then color the three consecutive odd numbers closest to 4, which are $5, 7, 9$; then color the four consecutive even numbers closest to 9, which are ... | 3929 |
In a class with 35 students, a survey was conducted on preferences for mathematics and literature, and it was found that:
- 7 men like mathematics;
- 6 men like literature;
- 5 men and 8 women said they do not like either;
- there are 16 men in the class;
- 5 students like both; and
- 11 students like only mathematics... | 2 |
4. Philatelist Andrey decided to distribute all his stamps equally into 3 envelopes, but it turned out that one stamp was extra. When he distributed them equally into 5 envelopes, 3 stamps were extra; finally, when he distributed them equally into 7 envelopes, 5 stamps remained. How many stamps does Andrey have in tota... | 208 |
If the function $f(x) = x\ln(x-1) + a(x-1)$ is monotonic on $(0, +\infty)$, then the range of values for $a$ is ()
A: $(-\infty, 2)$
B: $(-\infty, 2]$
C: $(-2, +\infty)$
D: $[-2, +\infty)$ | D |
17. How many pairs $(m, n)$ of non-negative integers are there such that $m \neq n$ and $\frac{50688}{m+n}$ is an odd positive power of 2 ?
(2 marks)
有多少對非負整數 $(m, n)$ 滿足 $m \neq n$ 且 $\frac{50688}{m+n}$ 是 2 的正奇數暮?
（２分） | 33760 |
The output of a factory in December last year was $a$. If the average monthly growth rate is $p$, then the output in December this year is (__).
A: $ap$
B: $a(1+p)$
C: $a(1+p)^{11}$
D: $a(1+p)^{12}$ | D |
2. (10 points) In the following four equations: $\overline{\mathrm{AB}} \div \overline{\mathrm{CD}}=2, E \times F=0, G-H=1, I+J=4, A \sim J$ represent different digits from $0 \sim$ 9, then the two-digit number $\overline{\mathrm{AB}}$ cannot be ( )
A. 54
B. 58
C. 92
D. 96 | D |
# Task № 3.4
## Condition:
Anton makes watches for a jewelry store on order. Each watch consists of a bracelet, a precious stone, and a clasp.
The bracelet can be silver, gold, or steel. Anton has precious stones: zircon, emerald, quartz, diamond, and agate, and clasps: classic, butterfly, and buckle. Anton is only ... | 48 |
5. (20 points) The clock shows the time as fifteen minutes past five. Determine the angle between the minute and hour hands at this moment.
---
Note: The translation maintains the original text's formatting and structure. | 67.5 |
Given the function $f(x)= \sqrt {2}\cos (x- \frac {\pi}{12})$, $x\in\mathbb{R}$.
(Ⅰ) Find the value of $f(- \frac {\pi}{6})$;
(Ⅱ) If $\cos \theta= \frac {3}{5}$, $\theta\in( \frac {3\pi}{2},2\pi)$, find $f(2\theta+ \frac {\pi}{3})$. | \frac {17}{25} |
## T-1
Let $\mathbb{Z}$ denote the set of all integers and $\mathbb{Z}_{>0}$ denote the set of all positive integers.
(a) A function $f: \mathbb{Z} \rightarrow \mathbb{Z}$ is called $\mathbb{Z}$-good if it satisfies $f\left(a^{2}+b\right)=f\left(b^{2}+a\right)$ for all $a, b \in \mathbb{Z}$. Determine the largest pos... | 2 |
Problem 5.3. By September 1st, Vlad bought himself several ballpoint and gel pens. He noticed that if all the pens he bought were gel pens, he would have paid 4 times more than he actually did. And if all the pens were ballpoint pens, the purchase would have cost him 2 times less than the actual price. How many times m... | 8 |
Given a sequence $\{a_n\}$ where $a_1=3$, and for $n\geq2$, $a_n=4a_{n-1}+3$, find the general formula for $a_n$. | 4^n-1 |
Which is the smallest natural number ending in 4, such that when its last digit is written at the beginning of the number, we get four times the original number? | 102564 |
5. The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=p, a_{n+1}=a_{n}^{2}+2 a_{n}$. Then the general term $a_{n}=$ $\qquad$ . | a_{n}=(p+1)^{2^{n-1}}-1 |
Given $\sin \alpha = -\frac{2\sqrt{5}}{5}$ and $\tan \alpha < 0$
$(1)$ Find the value of $\tan \alpha$;
$(2)$ Calculate the value of $\frac{2\sin (\alpha+\pi)+\cos (2\pi-\alpha)}{\cos (\alpha- \frac{\pi}{2})-\sin ( \frac{3\pi}{2}+\alpha)}$. | -5 |
2. If $m>n>1$, then it must be true that ( ).
(A) $\log _{m} n>\log _{n} m>\log _{n}^{1} m$
(B) $\log _{\frac{1}{n}} m>\log _{n} m>\log _{m} n$
(C) $\log _{n} m>\log _{m} n>\log _{\frac{1}{n}} m$
(D) $\log _{n} m>\log _{\frac{1}{n}} m>\log _{m} n$ | C |
A math teacher randomly selects 3 questions from 6 for students to check, specifying that at least 2 questions must be answered correctly to pass. If a student can only solve 4 of these questions, then the probability of him passing is __________. | \dfrac{4}{5} |
2. There is a fraction, when the numerator is reduced by 2021 and the denominator is increased by 2021, it simplifies to $\frac{1}{2021}$. Then the minimum value of the sum of the original fraction's numerator and denominator is | 4044 |
(20 (12 points) Given the function $f(x)=\frac{2 x+3}{3 x}$, the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=1, a_{n+1}=$ $f\left(\frac{1}{a_{n}}\right), n \in \mathbf{N}^{*}$.
(1) Find the general term formula of the sequence $\left\{a_{n}\right\}$;
(2) Let $T_{n}=a_{1} a_{2}-a_{2} a_{3}+a_{3} a_{4}-a_{4} a_{5}+\... | -\frac{4}{9}(2n^{2}+3n) |
5. Given that $x, y, z$ are real numbers. If $x^{2}+y^{2}=1$, $y^{2}+z^{2}=2, x^{2}+z^{2}=2$, then the minimum value of $x y+y z+x z$ is ( ).
(A) $\frac{5}{2}$
(B) $\frac{1}{2}+\sqrt{3}$
(C) $-\frac{1}{2}$
(D) $\frac{1}{2}-\sqrt{3}$ | D |
The opposite of $-\frac{1}{2023}$ is ______. | \frac{1}{2023} |
Given $f(x) = a\sin{x} + b\tan{x} + 1$, and it satisfies $f(5) = 7$, then $f(-5) = \_\_\_\_\_\_$. | -5 |
In a Cartesian coordinate system in a plane, a point is chosen such that both its x-coordinate and y-coordinate are from the set A={0,1,2,3,4,5}. What is the probability that the chosen point lies exactly on the line y=x? | \frac{1}{6} |
Determine the coordinates of the vertex of the following quadratic functions by completing the square and using the vertex formula, respectively.
① $y=2x^2-4x-1$ (by completing the square)
② $y=-3x^2+6x-2$ (by using the vertex formula) | (1, 1) |
In triangle $\triangle ABC$ with the following conditions, the one that is not a right triangle is:
A: $\angle A=\angle B=3\angle C$
B: $\angle A+\angle B=\angle C$
C: $\angle A=\angle B=\frac{1}{2}\angle C$
D: $\angle A:\angle B:\angle C=1:2:3$ | \text{A} |
From among $2^{1/2}, 3^{1/3}, 8^{1/8}, 9^{1/9}$ those which have the greatest and the next to the greatest values, in that order, are
$\textbf{(A) } 3^{1/3},\ 2^{1/2}\quad \textbf{(B) } 3^{1/3},\ 8^{1/8}\quad \textbf{(C) } 3^{1/3},\ 9^{1/9}\quad \textbf{(D) } 8^{1/8},\ 9^{1/9}\quad \\ \text{(E) None of these}$ | \textbf{(A)}3^{1/3},\2^{1/2} |
3. Write down all three-digit numbers whose sum of digits is equal to 9. How many are there? | 45 |
Given that the slant height of a certain cone is $3$, denoting its lateral surface area as $S$ and volume as $V$, the sine value of the angle between the slant height and the base when $\frac{V}{S}$ reaches its maximum value is ( ).
A: $\frac{\sqrt{2}}{2}$
B: $\frac{\sqrt{3}}{2}$
C: $\frac{1}{2}$
D: $\frac{3}{4}$ | A |
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