problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Children at the camp earned points for completing tasks. These points could be further exchanged: five points for a sticker, six points for a badge. Jára calculated that if he wanted only stickers, one point would remain unused. If he chose only badges, three points would remain unused. In the end, he managed to use al... | 51 |
In triangle $ABC^{}_{}$, $A'$, $B'$, and $C'$ are on the sides $BC$, $AC^{}_{}$, and $AB^{}_{}$, respectively. Given that $AA'$, $BB'$, and $CC'$ are concurrent at the point $O^{}_{}$, and that $\frac{AO^{}_{}}{OA'}+\frac{BO}{OB'}+\frac{CO}{OC'}=92$, find $\frac{AO}{OA'}\cdot \frac{BO}{OB'}\cdot \frac{CO}{OC'}$. | 94 |
5.1.7 ** Let the geometric sequence $\left\{z_{n}\right\}$ be such that $z_{1}=1, z_{2}=a+b \mathrm{i}, z_{3}=b+a \mathrm{i}(a, b \in \mathbf{R}$, $a>0$ ), find the smallest value of $n$ that satisfies $z_{1}+z_{2}+\cdots+z_{n}=0$, and calculate the value of $z_{1} z_{2} \cdots z_{n}$ at this time. | 12 |
35. What is the smallest positive integer $n$, where $n \neq 11$, such that the highest common factor of $n-11$ and $3 n+20$ is greater than 1 ? | 64 |
4. Let square $A B C D$. An equilateral triangle $\triangle A B E$ is constructed over side $\overline{A B}$, such that point $E$ is inside the square. Line $D E$ intersects side $\overline{B C}$ at point $F$. What is the measure of angle $\angle B E F$? | 45 |
$36 \cdot 67$ The 2nd "Chinese Youth" Cup Junior High School Mathematics Competition, if there are 148 students whose scores are between 100 and 120 points (scores are recorded as positive integers), and students with the same score are listed as the same rank, assuming the maximum number of students in the same rank i... | 8 |
## Problema 2.
Sea $\mathrm{P}$ un punto del lado $\mathrm{BC}$ de un triángulo $\mathrm{ABC}$. La paralela por $\mathrm{P}$ a $\mathrm{AB}$ corta al lado $\mathrm{AC}$ en el punto $\mathrm{Q}$ y la paralela por $\mathrm{P}$ a $\mathrm{AC}$ corta al lado $\mathrm{AB}$ en el punto $\mathrm{R}$. La razón entre las áreas... | \frac{k}{(1+k)^2} |
## Task 34/66
How many space diagonals does a rhombic dodecahedron have? | 43 |
Equilateral $\triangle ABC$ has side length $600$. Points $P$ and $Q$ lie outside of the plane of $\triangle ABC$ and are on the opposite sides of the plane. Furthermore, $PA=PB=PC$, and $QA=QB=QC$, and the planes of $\triangle PAB$ and $\triangle QAB$ form a $120^{\circ}$ dihedral angle (The angle between the two plan... | 450 |
16. Esther and Frida are supposed to fill a rectangular array of 16 columns and 10 rows, with the numbers 1 to 160 . Esther chose to do it row-wise so that the first row is numbered $1,2, \ldots, 16$ and the second row is $17,18, \ldots, 32$ and so on. Frida chose to do it columnwise, so that her first column has $1,2,... | 322 |
In $\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and $\sin A + \sin C = 2 \sin B$. Determine the range of values for angle $B$. | B \in (0, \frac{\pi}{3}] |
Consider two drums of sufficiently large capacity, one of them empty and the other full of liquid.
a) Determine if it is possible to place exactly one liter of liquid from the full drum into the empty one, using two buckets, one with a capacity of 5 liters and the other with a capacity of 7 liters.
b) Determine if it... | 1 |
## Problem Statement
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \frac{1+3+5+\ldots+(2 n-1)}{1+2+3+\ldots+n}
$$ | 2 |
12. Given that the center of a hyperbola is at the origin, and its foci are on the coordinate axes, the distance from point $P(-2,0)$ to its asymptotes is $\frac{2 \sqrt{6}}{3}$. If a line with a slope of $\frac{\sqrt{2}}{2}$ passing through point $P$ intersects the hyperbola at points $A$ and $B$, and intersects the $... | \sqrt{3} \text{ or } \sqrt{21} |
20. A scientist is walking through a forest, and she records the heights of 5 trees in a row using integers. She notices that the height of each tree is either twice or half the height of the tree to its right. Unfortunately, due to the rain, some of her data is lost, where the dashes represent the missing data (as sho... | 24.2 |
Given two lines $l_1: ax - y + 2a = 0$ and $l_2: (2a - 1)x + ay + a = 0$ are perpendicular to each other, find the value of the real number $a$. | 0 \text{ or } 1 |
If $x, y, z \in \mathbb{R}$ are solutions to the system of equations
$$\begin{cases}
x - y + z - 1 = 0\\
xy + 2z^2 - 6z + 1 = 0\\
\end{cases}$$
what is the greatest value of $(x - 1)^2 + (y + 1)^2$? | 11 |
1774. To determine the percentage of second-grade items in a batch, a random repeated sample of 100 units is taken. Determine the confidence limits for the percentage of second-grade items in the entire batch, which can be guaranteed with a confidence probability of \( P = 0.997 \), if 25 second-grade items were found ... | 11.95to38.05 |
A robot walks 1m in a certain direction, then turns left 45°, walks another 1m, then turns left 45° again, and repeats the instruction "walk 1m, then turn left 45°". When the robot returns to the starting point for the first time, it has walked m. | 8 |
Figures $I$, $II$, and $III$ are squares. The perimeter of $I$ is $12$ and the perimeter of $II$ is $24$. The perimeter of $III$ is
$\text{(A)}\ 9 \qquad \text{(B)}\ 18 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 72 \qquad \text{(D)}\ 81$ | 36 |
In a test with 84 questions, if you answer 58/84 correctly, then what is your percentage of correct answers? | 69 |
10. Disks (from 9th grade. 3 points). At a familiar factory, metal disks with a diameter of 1 m are cut out. It is known that a disk with a diameter of exactly 1 m weighs exactly 100 kg. During manufacturing, there is a measurement error, and therefore the standard deviation of the radius is 10 mm. Engineer Sidorov bel... | 4 |
Given that the angle between the unit vector $$\overrightarrow {a}$$ and the unit vector $$\overrightarrow {b}$$ is 120°, find the value of $$|2 \overrightarrow {a}- \overrightarrow {b}|$$. | \sqrt {7} |
1. The students of Class (1), Grade 3 in a certain school plan to go mountain climbing on Sunday. They plan to set off at 8:30 am, and try to climb the farthest mountain on the map. After reaching the summit, they will hold a 1.5-hour cultural and entertainment activity, and must return to the base before 3 pm. If the ... | D |
A particle is launched from the surface of a uniform, stationary spherical planet at an angle to the vertical. The particle travels in the absence of air resistance and eventually falls back onto the planet. Spaceman Fred describes the path of the particle as a parabola using the laws of projectile motion. Spacewoman K... | E |
Problem 18. Find the roots of the equation $F(x)=$ $=2^{2 x}-5 \cdot 3^{x}+29=0$ (to the nearest 0.01). | x_1=2,x_2\approx5.55 |
Let $T_k = \frac{k(k+1)}{2}$ be the $k$-th triangular number. The in finite series
$$\sum_{k=4}^{\infty}\frac{1}{(T_{k-1} - 1)(Tk - 1)(T_{k+1} - 1)}$$
has the value $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 451 |
Circle $C$ passes through the origin and its center is on the positive $x$-axis. If the chord intercepted by the line $x-y=0$ on circle $C$ has a length of $2\sqrt{2}$, then the equation of circle $C$ is \_\_\_\_\_\_\_\_\_. | (x-2)^{2}+y^{2}=4 |
Emil and Olivia ran a race. Their race times totalled 1 hour 52 minutes. If Emil's time was 4 minutes less than Olivia's time, how many minutes did it take Olivia to run the race?
(A) 78
(B) 56
(C) 58
(D) 74
(E) 55 | 58 |
Find the coefficient of the $x^3$ term in the expansion of $((x+1)(x-2)^{5})$. | -40 |
Given $\sin 2\alpha= \frac {24}{25}$, where $0 < \alpha < \frac {\pi}{2}$, find the value of $\sqrt {2}\cos \left( \frac {\pi}{4}-\alpha\right)$.
A: $- \frac {1}{5}$
B: $ \frac {1}{5}$
C: $- \frac {7}{5}$
D: $ \frac {7}{5}$ | D |
13.062. An apprentice turner is machining pawns for a certain number of chess sets. He wants to learn to produce 2 more pawns per day than he does now; then he would complete the same task 10 days faster. If he could learn to produce 4 more pawns per day than he does now, the time required to complete the same task wou... | 15 |
Given that the focus of the parabola $y^{2}=8x$ is $F$ and its directrix is $l$. Let $P$ be a point on the parabola and $PA \perp l$, where $A$ is the foot of the perpendicular. If the angle of inclination of $AF$ is $\frac{2π}{3}$, then $|PF|=$ _____ . | 8 |
Given that the diameter of the base and the height of the cylinder are equal to the diameter of a sphere, find the ratio of the surface area of the sphere to the surface area of the cylinder, and the ratio of the volume of the sphere to the volume of the cylinder. | \frac{2}{3} |
1. Some objects are in each of four rooms. Let $n \geqslant 2$ be an integer. We move one $n$-th of objects from the first room to the second one. Then we move one $n$-th of (the new number of) objects from the second room to the third one. Then we move similarly objects from the third room to the fourth one and from ... | 5 |
Let's determine the value of $c$ such that the ratio of the roots of the equation
$$
5 x^{2}-2 x+c=0
$$
is $-\frac{3}{5}$. | -3 |
8. Determine $\left(2017^{\frac{1}{\log _{2} 2017}} \times 2017^{\frac{1}{\log _{4} 2017}} \times 2017^{\frac{1}{\log _{8} 2017}} \times 2017^{\frac{1}{\log _{16} 2017}} \times 2017^{\frac{1}{\log _{32} 2017}}\right)^{\frac{1}{5}}$.
| 8 |
Given that the point $A$ on the number line represents the number $7$, when point $A$ is moved $3$ units to the right to point $B$, the number represented by point $B$ is ______. | 10 |
Given ${(1-2x)^{2021}}={a_0}+{a_1}x+{a_2}{x^2}+⋯+{a_{2021}}{x^{2021}}$, then ( )
A: The sum of coefficients of all terms in the expansion is $-1$
B: The coefficient of the largest term in the expansion is the $1010$th term
C: $\frac{{{a_1}}}{2}+\frac{{{a_2}}}{{{2^2}}}+\frac{{{a_3}}}{{{2^3}}}+⋯+\frac{{{a_{2021}}}}{{{... | ACD |
7. There are a total of 10040 numbers written on the blackboard, including 2006 ones, 2007 twos, 2008 threes, 2009 fours, and 2010 fives. Each operation involves erasing 4 different numbers and writing the fifth type of number (for example, erasing 1, 2, 3, and 4 each once and writing one 5; or erasing 2, 3, 4, and 5 e... | 8 |
Given that $\left\{a_n\right\}$ is an arithmetic sequence, and $a_4 + a_{14} = 2$, then $a_9 = \_\_\_\_\_\_\_\_\_$. | 1 |
Example 7. $\rho^{2}=a \cos 2 \varphi$. Find the orthogonal trajectories of the family of lemniscates | \rho^{2}=C\sin2\varphi |
Which of the following calculations is correct?
A: $3+2ab=5ab$
B: $6xy-6x=y$
C: $-5m^{2}n+5mn^{2}=0$
D: $-x^{2}y+3x^{2}y=2x^{2}y$ | D |
(4) Given that the domain of $y=f\left(2^{x}\right)$ is $[-1,1]$, find the domain of $y=f\left(\log _{2} x\right)$. | x\in[\sqrt{2},4] |
For all $n>1$ let $f(n)$ be the sum of the smallest factor of $n$ that is not 1 and $n$ . The computer prints $f(2),f(3),f(4),...$ with order:$4,6,6,...$ ( Because $f(2)=2+2=4,f(3)=3+3=6,f(4)=4+2=6$ etc.). In this infinite sequence, how many times will be $ 2015$ and $ 2016$ written? (Explain your answer) | 3 |
An arithmetic sequence with common difference $r$ is a sequence of numbers $s_{0}, s_{1}, s_{2}, \ldots$ that satisfies for all indices $k, s_{k+1}=s_{k}+r$. Determine an expression for $s_{n}$ in terms of $s_{0}, r$, and $n$. | s_{n}=s_{0}+nr |
1. Two sisters were picking currants: the elder one in a four-liter bucket, and the younger one in a bucket with a capacity of 3.5 liters. The elder sister always worked faster than the younger one. When the elder sister had collected three quarters of her bucket, and the younger one had collected more than half of her... | 1.5 |
(2) Given $3 \sin ^{2} \alpha+2 \sin ^{2} \beta=1,3(\sin \alpha+\cos \alpha)^{2}-2(\sin \beta+\cos \beta)^{2}=$ 1 , then $\cos 2(\alpha+\beta)=$ $\qquad$ . | -\frac{1}{3} |
2. The cities $A$ and $B$ are $106 \mathrm{~km}$ apart. At noon, from city $A$, Antonio set off on a bicycle heading towards $B$ at a constant speed of $30 \mathrm{~km} / \mathrm{h}$. Half an hour later, from city $B$, Bojan set off on foot heading towards $A$ at a constant speed of $5 \mathrm{~km} / \mathrm{h}$. At th... | 155 |
Given $z=\frac{{1-i}}{{2+2i}}$, then $z-\overline{z}=\left(\ \ \right)$
A: $-i$
B: $i$
C: $0$
D: $1$ | \text{A} |
In the geometric sequence $\{a_n\}$, $a_2a_3=32$, $a_5=32$.
(1) Find the general formula for the sequence $\{a_n\}$.
(2) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Find the value of $S_1+2S_2+\ldots+nS_n$. | (n-1)2^{n+2}+4-n(n+1) |
Given the function $f(x)= \sqrt {3}(\sin ^{2}x-\cos ^{2}x)+2\sin x\cos x$.
(I) Find the smallest positive period of $f(x)$;
(II) If $x\in\[- \frac {π}{3}, \frac {π}{3}\]$, find the range of $f(x)$ and the interval where it is strictly increasing. | \left[-\frac{\pi}{12},\frac{\pi}{3}\right] |
"m=2" is a condition for the function $f(x) = -3 + mx + x^2$ to have two distinct roots. This condition is ( )
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both necessary and sufficient
D: Neither sufficient nor necessary | A |
(7) Let $y=\sin x+\cos x+\tan x+\cot x+\sec x+\csc x$, then the minimum value of $|y|$ is $\qquad$ . | 2\sqrt{2}-1 |
Given that the solution set of the inequality $ax^{2}-x-b \gt 0\left(a,b\in R\right)$ with respect to $x$ is $\{x\left|\right.x \gt 2$ or $x \lt -1\}$.
$(1)$ Find the values of $a$ and $b$;
$(2)$ If $c\in R$, solve the inequality $ax^{2}-\left(ac+b-1\right)x+\left(b-1\right)c \lt 0$. | (c, 1) |
3. Let $[x]$ denote the greatest integer not exceeding the real number $x$. Set $A=\left[\frac{7}{8}\right]+\left[\frac{7^{2}}{8}\right]+\cdots+\left[\frac{7^{2016}}{8}\right]$. Then the remainder when $A$ is divided by 50 is $\qquad$ . | 42 |
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$
Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main di... | \sum_{i=1}^n \min(p_i, q_i) |
7. An ellipse has its major and minor axes parallel to the coordinate axes, and it is tangent to the line $2x + y = 11$ at point $P(4,3)$. It also passes through points $Q(0,-1)$ and $R(1, \sqrt{10} + 1)$. Find the equation of this ellipse. | 2(x-2)^{2}+(y-1)^{2}=12 |
Given that the sequence {an} is a geometric sequence with a common ratio q n. If a2 = 2 and S3 = 7:
1. Find the general term formula for {an}.
2. Let m ∈ Z. If Sn < m always holds true, find the minimum value of m. | 8 |
5. Let's introduce the function $g(x)=f(x)+(x-2021)^{2}-4$. For this function, the conditions $g(2019)=g(2020)=g(2021)=g(2022)=g(2023)=0$ are satisfied, meaning that the function $g(x)$ has 5 roots. Since it is a polynomial of the 5th degree, it has no other roots. Therefore,
$$
g(x)=(x-2019)(x-2020)(x-2021)(x-2022)(x... | -125 |
One. (20 points) As shown in Figure 5, $P$ is a point outside circle $\odot O$, $PA$ is tangent to $\odot O$ at point $A$, and $PBC$ is a secant of $\odot O$. $AD \perp PO$ at point $D$. If $PB=4$, $CD=5$, $BC=6$, find the length of $BD$. | 2 |
Given two points M(5, a) and N(b, -2) in the Cartesian coordinate system, if line MN is parallel to the x-axis, find the values of a and b. | 5 |
2. Divide the set of positive odd numbers in ascending order into groups, with the $n$-th group containing $n^{2}$ odd numbers. Then 2005 is in the ( )-th group.
(A) 14
(B) 15
(C) 16
(D) 17 | A |
B3. Let $\operatorname{tg} \alpha=\frac{a+2 b}{a \sqrt{3}}, \operatorname{tg} \beta=\frac{b+2 a}{b \sqrt{3}}, a>0, b>0, \alpha, \beta \in\left[0, \frac{\pi}{2}\right]$. Determine $\alpha+\beta$. | \frac{2\pi}{3} |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying for all $x, y \in \mathbb{R}$ :
$$
f(x+y)=f(x)-f(y)
$$ | f(x)=0 |
(7) Arrange the positive integers whose sum of digits is 5 in ascending order to form a sequence, then 2012 is the $\qquad$th term. | 38 |
Given the proposition "The incircle of an equilateral triangle touches the midpoints of its three sides," we can analogously conjecture that the insphere of a regular tetrahedron touches the faces that are all equilateral triangles at what position? ( )
A: The vertices of the equilateral triangles
B: The center of the ... | \text{B} |
Given the function $f(x)=\sin (x+θ)+ \sqrt {3}\cos (x+θ)$, where $θ∈\left[-\dfrac {π}{2}, \dfrac {π}{2}\right]$, and the function $f(x)$ is an even function, find the value of $θ$. | \dfrac {π}{6} |
Calculate the following expression:
$$\sqrt{(1)(2 \frac{1}{4})} - (-9.6)^{0} - (3 \frac{3}{8})^{- \frac{2}{3}} + 0.1^{-2} \tag{1}$$
Given that $x+x^{-1}=3$, find the value of:
$$\frac{x^{ \frac{1}{2}}+x^{- \frac{1}{2}}}{x^{2}+x^{-2}+3} \tag{2}$$ | \frac{\sqrt{5}}{10} |
2. Calculate: $\left[20.75+\left(3.74-2 \frac{1}{2}\right) \div 9 \frac{23}{25}\right] \div 41.75=$ | 0.5 |
Randomly select $2n$ numbers $x_{1}$, $x_{2}$, ..., $x_{n}$, $y_{1}$, $y_{2}$, ..., $y_{n}$ from the interval $[0,1]$ to form $n$ pairs $(x_{1},y_{1})$, $(x_{2},y_{2})$, ..., $(x_{n},y_{n})$. Among these, there are $m$ pairs whose sum of squares is less than $1$. Then, the approximate value of $\pi$ obtained by the met... | C |
Solve the equation $x+1-2(x-1)=1-3x$. Where did the solution process go wrong? Point out and correct it. | x=-1 |
$32 \cdot 18$ A positive integer like 34689 is called an "ascending number," where each digit is greater than the digit to its immediate left. There are $C_{9}^{5}\binom{9}{5}=126$ five-digit "ascending numbers." If these numbers are arranged in ascending order, the 97th "ascending number" will not contain the digit
(A... | B |
Example 8. Find the solution of the equation
$$
x^{3} \sin y \cdot y^{\prime}=2
$$
satisfying the condition
$$
y \rightarrow \frac{\pi}{2} \quad \text { as } \quad x \rightarrow \infty
$$ | \arccos\frac{1}{x^{2}} |
9. Let $x, y \in \mathbf{R}$ satisfy $x-6 \sqrt{y}-4 \sqrt{x-y}+12=0$, then the range of values for $x$ is | 14-2\sqrt{13}\leqx\leq14+2\sqrt{13} |
1.50 Given $a+b+c=6, a^{2}+b^{2}+c^{2}=10, a^{3}+b^{3}+c^{3}=$ 36. Then the value of $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ is
(A) $\frac{13}{18}$.
(B) $\frac{18}{13}$.
(C) $-\frac{13}{18}$.
(D) $-\frac{18}{13}$.
(Shenyang City, Liaoning Province, China "Yucai Cup" Junior High School Mathematics Competition, 1990) | \frac{13}{18} |
[ Measurement of segment lengths and angle measures. Adjacent angles.]
On a straight line, points $A, B$, and $C$ are given. It is known that $A B=5$, and segment $A C$ is one and a half times longer than $B C$. Find the segments $A C$ and $B C$. | BC=10,AC=15orBC=2,AC=3 |
The Fruit and Vegetable Kingdom is holding a king election, with each of the 205 people nationwide casting 1 vote, choosing 1 from Tomato Warrior, Banana Superman, and Carrot Doctor, and the one with the most votes will be elected. As of now, Tomato Warrior has 54 votes, Banana Superman has 46 votes, and Carrot Doctor ... | 31 |
Factorize: $a^2b(x-y)^3-ab^2(y-x)^2$. | ab(x-y)^2(ax-ay-b) |
10. As shown in Figure $3, A$ is a point on the positive half-axis of $x$, $\angle A O N=60^{\circ} ; B$ is a point on $O N$, and $A B=2$; $C$ is the midpoint of $A B$, and its symmetric point about $O B$ is $D$. Find the equation of the trajectory of point $D$, and draw a rough graph. | 3x^{2}+\frac{y^{2}}{3}=1(-\frac{1}{2}\leqslantx\leqslant\frac{1}{2},\frac{\sqrt{3}}{2}\leqslanty\leqslant\sqrt{3}) |
1. In the diagram, square $E F G H$ is centred inside square $A B C D$ and the region between the two squares is shaded.
(a) Determine the area of the shaded region if the length of $A B$ is $10 \mathrm{~cm}$ and the length of $E F$ is $8 \mathrm{~cm}$.
(b) Determine the length of $E F$ if the length of $A B$ is $13 \m... | \frac{}{b}=\frac{5}{3} |
Which of the following quadratic radicals belongs to the simplest quadratic radicals?
A: $\sqrt{32}$
B: $\sqrt{4x+8}$
C: $\sqrt{\frac{2}{a}}$
D: $\sqrt{{b}^{2}+4}$ | D |
2. Find all quadruples of positive real numbers $(a, b, c, d)$ such that
$$
\begin{array}{l}
\text { satisfy } a+b+c+d=1, \text { and } \\
\max \left\{\frac{a^{2}}{b}, \frac{b^{2}}{a}\right\} \max \left\{\frac{c^{2}}{d}, \frac{d^{2}}{c}\right\} \\
=(\min \{a+b, c+d\})^{4} .
\end{array}
$$ | (\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4}) |
Given that the terminal side of angle $\alpha$ intersects the unit circle at the point $\left(-\frac{\sqrt{3}}{2}, -\frac{1}{2}\right)$, the value of $\sin\alpha$ is ( )
A: $-\frac{\sqrt{3}}{2}$
B: $-\frac{1}{2}$
C: $\frac{\sqrt{3}}{2}$
D: $\frac{1}{2}$ | \text{B} |
Given $a, b \in \mathbb{R}$, among the following four conditions, the one that is a necessary but not sufficient condition for $a < b$ is ( ).
A: $|a| < |b|$
B: $2^a < 2^b$
C: $a < b - 1$
D: $a < b + 1$ | \text{D} |
4. The escalator on Fairy Island moves at a constant speed from the bottom to the top. Leti climbs up the escalator, covering 30 steps per minute, and reaches the top in 5 minutes; Meiqi climbs down the escalator, covering 35 steps per minute, and reaches the bottom in 8 minutes. Therefore, this escalator has $\qquad$ ... | 200 |
Given a triangle $ABC$ with the sides opposite to angles $A$, $B$, $C$ being $a$, $b$, $c$ respectively, if $a=\sqrt{3}$, $\sin B=\frac{1}{2}$, and $C=\frac{\pi}{6}$, then find $b=$ ______. | 1 |
6.159. $\sqrt{x+2}-\sqrt[3]{3 x+2}=0$.
6.159. $\sqrt{x+2}-\sqrt[3]{3 x+2}=0$. | 2 |
$7 \cdot 54$ In $\triangle A B C$, if $A B=2 B C, \angle B=2 \angle A$, then $\triangle A B C$ is
(A) an acute triangle.
(B) a right triangle.
(C) an obtuse triangle.
(D) cannot be determined.
(China Sichuan Province Junior High School Mathematics Competition, 1992) | B |
11.033. In a triangular pyramid, the lateral edges are mutually perpendicular and have lengths $\sqrt{70}, \sqrt{99}$, and $\sqrt{126}$ cm. Find the volume and the area of the base of the pyramid. | 21\sqrt{55}\mathrm{~}^{3},84\mathrm{~}^{2} |
3. Let the midpoint of the chord $AB$ of the circle $x^{2}+y^{2}=9$ be $(1,1)$, and the line $AB$ intersects the $x$-axis at point $P$, then $|PA| \cdot |PB|=$
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | 5 |
If the proposition "$p \wedge q$" is false, and "$\neg p$" is false, then $q$ is \_\_\_\_\_\_\_. (Fill in "true" or "false") | \text{false} |
Given that in triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. If $a^{2}\cos A\sin B=b^{2}\sin A\cos B$, then the shape of triangle $\triangle ABC$ is:
A: Isosceles triangle
B: Isosceles triangle or right triangle
C: Right triangle
D: Acute tr... | B |
Example 4 Given a unit cube $A B C D-$ $A_{1} B_{1} C_{1} D_{1}$, $E$ is a moving point on $B C$, and $F$ is the midpoint of $A B$. Determine the position of point $E$ such that $C_{1} F \perp A_{1} E$. | a=\frac{1}{2} |
Let the function $f(x) = xe^x$. Then ( ).
A: $x = 1$ is a local maximum point of $f(x)$
B: $x = 1$ is a local minimum point of $f(x)$
C: $x = -1$ is a local maximum point of $f(x)$
D: $x = -1$ is a local minimum point of $f(x)$ | \text{D} |
Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
[i]Proposed by Ray Li[/i] | 200 |
Given the function $f(x)=2\sqrt{3}\sin x\cos x+2\cos^{2}x-1$.
(I) Find the smallest positive period of $f(x)$;
(II) Find the maximum and minimum values of $f(x)$ on the interval $\left[-\frac{\pi }{6}, \frac{\pi }{4}\right]$. | -1 |
Given the function $f(x)=ax^{2}+2x+c$ has its lowest point at $(-1,-2)$.
$(1)$ Find the solution set of the inequality $f(x) > 7$;
$(2)$ If for any $x∈[2,4]$, the inequality $f(x-t)\leqslant x-2$ always holds, find the range of the real number $t$. | [3,3+ \sqrt {2}] |
Calculate the following expressions:
(1) $(1)(-2.48)+(+4.33)+(-7.52)+(-4.33)$
(2) $(2)(+3 \frac{5}{6})+(-5 \frac{1}{7})+(-2 \frac{1}{6})+(-32 \frac{6}{7})$
(3) $(3) \frac{4}{5}-(+\frac{5}{6})-(+\frac{3}{5})+\frac{1}{6}$
(4) $-1^{4}-\frac{1}{6}×[2-(-3)^{2}]$. | \frac{1}{6} |
## Problem Statement
Calculate the volume of the tetrahedron with vertices at points \( A_{1}, A_{2}, A_{3}, A_{4} \) and its height dropped from vertex \( A_{4} \) to the face \( A_{1} A_{2} A_{3} \).
\( A_{1}(1 ; 3 ; 6) \)
\( A_{2}(2 ; 2 ; 1) \)
\( A_{3}(-1 ; 0 ; 1) \)
\( A_{4}(-4 ; 6 ;-3) \) | 23\frac{1}{3},2\sqrt{14} |
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