problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
157. The battery fired six shots at the object. The probability of hitting the object with one shot is 0.3. Find: a) the most probable number of hits; b) the probability of the most probable number of hits; c) the probability that the object will be destroyed, if for this at least two hits are required. | 0.58 |
3. What is the greatest value that the sum $\sin ^{2} a+\sin ^{2}\left(a+60^{\circ}\right)+\sin ^{2}\left(a+120^{\circ}\right)+\ldots+\sin ^{2}\left(a+300^{\circ}\right)$ can take? | 3 |
The numbers from $1$ to $8$ are placed at the vertices of a [cube](https://artofproblemsolving.com/wiki/index.php/Cube) in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
$\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}... | 18 |
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$. | 2 |
The following bar graph represents the length (in letters) of the names of 19 people. What is the median length of these names?
$\textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad \textbf{(E) }7$ | 4 |
4. (8 points) In another 12 days, it will be 2016, Hao Hao sighs: I have only experienced 2 leap years so far, and the year I was born is a multiple of 9. So how old will Hao Hao be in 2016? | 9 |
1. Given two groups of numerical sequences, each consisting of 15 arithmetic progressions containing 10 terms each. The first terms of the progressions in the first group are $1, 2, 3, \ldots, 15$, and their differences are respectively $2, 4, 6, \ldots, 30$. The second group of progressions has the same first terms $1... | \frac{160}{151} |
Given three automates that deal with the cards with the pairs of natural numbers. The first, having got the card with ($a,b)$, produces new card with $(a+1,b+1)$, the second, having got the card with $(a,b)$, produces new card with $(a/2,b/2)$, if both $a$ and $b$ are even and nothing in the opposite case; the third, h... | (1, 50) |
4. Given the quadratic equation $x^{2}-x+1-m=0$ with two real roots $\alpha, \beta$ satisfying $|\alpha|+|\beta| \leqslant 5$. Then the range of the real number $m$ is $\qquad$ | \frac{3}{4} \leqslant m \leqslant 7 |
For any natural number $n$, we denote $\mathbf{S}(n)$ as the sum of the digits of $n$. Calculate $\mathbf{S}^{5}\left(2018^{2018^{2018}}\right)$. | 7 |
Example 10. Solve the equation
$$
x \int_{0}^{x} y(t) d t=(x+1) \int_{0}^{x} t y(t) d t, \quad x>0
$$ | C\frac{1}{x^{3}}e^{-1/x} |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-3,4)$.
$(1)$ If $\overrightarrow{a}⊥(\overrightarrow{a}+λ\overrightarrow{b})$, find the value of the real number $\lambda$.
$(2)$ Find the angle between $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{a}-\overrightarrow{b}$. | \frac{3\pi}{4} |
3. The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown.
What is the sum of the other three numbers he will write... | 11 |
In a mathematics competition, there are 30 problems. Each correctly solved problem is worth 4 points, a wrong solution results in -1 point. If someone does not attempt a problem, they get 0 points for it. How many different total scores can a contestant achieve? | 145 |
1. If $-3<x<-1$, then simplifying $|2-| 1+x||$ yields ().
(A) $1-x$
(B) $-3+x$
(C) $3-x$
(D) $3+x$ | D |
4. Given $\alpha, \beta \in \mathbf{R}$, and $\cos \alpha+\cos \beta=\frac{1}{3}, \tan (\alpha+\beta)=\frac{24}{7}$, then $\sin \alpha+\sin \beta=$ | \frac{1}{4} |
1. Given that $f(x)$ is an odd function defined on $\mathbf{R}$, and for any $x \in \mathbf{R}$, we have
$$
f(2+x)+f(2-x)=0 .
$$
If for $x \in[-1,0)$,
$$
f(x)=\log _{2}(1-x) \text {, }
$$
then $f(1)+f(2)+\cdots+f(2021)=$ $\qquad$ | -1 |
# Task 2. (20 points)
Find the maximum possible value of the ratio of a three-digit number to the sum of its digits.
# | 100 |
A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine $ S_{1024}.$ | 1024 |
Calculate the result of $2m^{3}\cdot 3m^{4}$.
A: $5m^{7}$
B: $5m^{12}$
C: $6m^{7}$
D: $6m^{12}$ | C |
There are infinitely many ordered pairs $(m,n)$ of positive integers for which the sum
\[ m + ( m + 1) + ( m + 2) +... + ( n - 1 )+n\]
is equal to the product $mn$. The four pairs with the smallest values of $m$ are $(1, 1), (3, 6), (15, 35),$ and $(85, 204)$. Find three more $(m, n)$ pairs. | (493, 1189), (2871, 6930), (16731, 40391) |
The number of circular pipes with an inside diameter of $1$ inch which will carry the same amount of water as a pipe with an inside diameter of $6$ inches is:
$\textbf{(A)}\ 6\pi \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 36\pi$ | 36 |
3. (10 points) Person A and Person B start riding bicycles from place $A$ to place $B$ at the same time. Person A's speed is 1.2 times that of Person B. After riding 4 kilometers, Person B's bicycle breaks down, and the downtime is equivalent to the time it would take to ride $\frac{1}{6}$ of the entire journey. After ... | 36 |
3. Rewrite the original equation in the form
$$
16 z^{2}+4 x y z+\left(y^{2}-3\right)=0
$$
This is a quadratic equation in terms of z. It has a solution if $D / 4=4 x^{2} y^{2}-16\left(y^{2}-3\right) \geq 0$. After transformations, we obtain the inequality
$$
y^{2}\left(x^{2}-4\right)+12 \geq 0
$$
If $x^{2}-4 \geq ... | x\in(-\infty;-2]\cup[2;+\infty) |
5.5. Four dolls and five robots cost 4100 rubles, while five dolls and four robots cost 4000. How much does one doll cost? | 400 |
4. Arrange the eight numbers $-7, -5, -3, -2, 2, 4, 6, 13$ as $a, b, c, d, e, f, g, h$ such that
$$
(a+b+c+d)^{2}+(e+f+g+h)^{2}
$$
is minimized. Then this minimum value is $\qquad$
(2011, Xin Zhi Cup Shanghai Junior High School Mathematics Competition) | 34 |
In the plane, Point $ O$ is the center of the equilateral triangle $ABC$ , Points $P,Q$ such that $\overrightarrow{OQ}=2\overrightarrow{PO}$.
Prove that\[|PA|+|PB|+|PC|\le |QA|+|QB|+|QC|.\] | |PA| + |PB| + |PC| \leq |QA| + |QB| + |QC| |
Given the parabola $y^{2}=2px (p > 0)$ passes through point $A(1, \dfrac {1}{2})$, determine its latus rectum equation.
A: $x=- \dfrac {1}{32}$
B: $x=- \dfrac {1}{16}$
C: $y=- \dfrac {1}{32}$
D: $y=- \dfrac {1}{16}$ | B |
Given $(2- \sqrt{3}x)^{50} = a_0 + a_1x + a_2x^2 + \ldots + a_{50}x^{50}$, where $a_0, a_1, a_2, \ldots, a_{50}$ are constants, calculate $(a_0 + a_2 + a_4 + \ldots + a_{50})^2 - (a_1 + a_3 + a_5 + \ldots + a_{49})^2$. | 1 |
13.304. Two meshing gears $A$ and $B$ are mounted tightly: the first on shaft $O_{1}$, and the second on shaft $O_{2}$. Gear $A$ has 10 more teeth than $B$. At a certain rotational speed of shaft $O_{1}$, shaft $O_{2}$ makes 63 revolutions per minute. If the gears are swapped, then at the same speed of shaft $O_{1}$, s... | 2030 |
18. A plane intersects a right circular cylinder of radius 1 to form an ellipse. If the major axis is 50% longer than the minor axis, the length of the major axis is
(A) 1 ; (B) $\frac{3}{2}$;
(C) 2 ;
(D) $\frac{9}{4}$;
(E) 3 . | E |
1. Each athlete participating in the marathon can be categorized into at least one of three categories: cheerful, trained, or beginners. It is known that every sixteenth cheerful athlete is a beginner, and one-fifth of the beginners are cheerful. One-fifth of the trained athletes are cheerful, and only one-tenth of the... | 130 |
In some year, a certain number was not a Sunday in any month. Determine this number.
# | 31 |
Given the parabola $y^2 = 2px$ passes through point A(4, 4), find the equation of the directrix and the distance from point A to the focus. | 5 |
In a geometric sequence with a common ratio of $2$, if $a_1a_3 = 6a_2$, then $a_4$ equals $(\quad)$.
A: $4$
B: $8$
C: $12$
D: $24$ | D |
Let set $A=\{x|x^2-2x-3\leq0\}$, and $B=\{x|x^2-5x\geq0\}$, then find $A\cap(\complement_{\mathbb{R}}B)$. | (0,3] |
The minimum positive period of the function $y=\sin \left(\frac{\pi}{3}-2x\right)+\sin 2x$ is ______. | \pi |
Given the function $f(x)= \begin{cases} x\sin x, & 0 < x < \pi \\ \sqrt {x}, & x\geqslant \pi\end{cases}$, $g(x)=f(x)-kx(k\in\mathbb{R})$
(1) When $k=1$, the function $g(x)$ has \_\_\_\_\_\_ zero points;
(2) If the function $g(x)$ has three zero points, then the range of $k$ is \_\_\_\_\_\_. | (0, \frac { \sqrt {\pi}}{\pi}] |
1. Let the sum of the digits of the natural number $x$ be $S(x)$. Then the solution set of the equation $x+S(x)+S(S(x))+S(S(S(x)))=2016$ is $\qquad$ | 1980 |
The central distance of two circles is $c=714 \mathrm{~mm}$, the external common tangents intersect at an angle of $\alpha=36^{\circ} 8^{\prime}$, and the internal ones intersect at an angle of $\beta=104^{\circ} 12^{\prime}$. What are the radii of the circles? | 392.42 |
7. Given a regular prism $K L M N K_{1} L_{1} M_{1} N_{1}$ with base $K L M N$. Planes $\alpha$ and $\beta$ are perpendicular to $L_{1} N$ and pass through vertices $K$ and $N_{1}$, respectively. Let $A$ and $B$ be the points of intersection of planes $\alpha$ and $\beta$ with the diagonal $L_{1} N$, such that $A N < B... | L_{1}N=2(1+\sqrt{13}),V=32\sqrt{6+2\sqrt{13}} |
Given the function $f(x) = \sin[\cos x] + \cos[\sin x]$, where $[x]$ represents the greatest integer not exceeding the real number $x$, there are four conclusions about $f(x)$:
① One period of $f(x)$ is $2\pi$
② $f(x)$ is neither odd nor even
③ $f(x)$ is monotonically decreasing in the interval $(0, \pi)$
④ The max... | \text{A} |
G4.1 $x_{1}=2001$. When $n>1, x_{n}=\frac{n}{x_{n-1}}$. Given that $x_{1} x_{2} x_{3} \ldots x_{10}=a$, find the value of $a$. | 3840 |
Example 2. Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$. | 0.6956 |
59. Given $0<a<1$, and it satisfies $\left[a+\frac{1}{30}\right]+\left[a+\frac{2}{30}\right]+\cdots+\left[a+\frac{29}{30}\right]=18$, then the value of $[10 a]$ is
$\qquad$ . ( $[x]$ represents the greatest integer not exceeding $x$) | 6 |
On the circle $C:x^2+y^2=4$, there are two points $A(x_1,y_1)$ and $B(x_2,y_2)$. If $|\vec{AB}|=2\sqrt{3}$, then $x_1x_2+y_1y_2=$ ? | -2 |
3. If real numbers $x, y$ satisfy $4 x+3 y-2 x\left[\frac{x^{2}+y^{2}}{x^{2}}\right]=0$, then the value of $\frac{y}{x}$ is $\qquad$ (where $[x]$ denotes the greatest integer not exceeding $x$). | \frac{y}{x}=-\frac{2}{3} \text{ or } 2 |
1. Solve the equation $\frac{2 \sin 3 x}{\sin x}-\frac{3 \cos 3 x}{\cos x}=7|\sin x|$. | \\arcsin\frac{3}{4}+k\pi,k\inZ |
9.17 Let $\overline{a 679 b}$ be a five-digit number in decimal notation, divisible by 72. Determine the values of $a$ and $b$.
(12th Canadian Mathematics Competition, 1980) | =3,\quadb=2 |
In the arithmetic sequence $\{a_n\}$, where $d=1$ and $S_{98}=137$, find the sum of $a_2+a_4+a_6+\ldots+a_{98}$. | 93 |
6. The domain of the function $y=\sin x$ is $[a, b]$, and the range is $\left[-1, \frac{1}{2}\right]$. Then the maximum value of $b-a$ is ( ).
(A) $\pi$
(B) $2 \pi$
(C) $\frac{4 \pi}{3}$
(D) $\frac{5 \pi}{3}$ | C |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $a^{2}+b^{2}+\sqrt{2}ab=c^{2}$. Find the range of values for $\tan A\tan 2B$. | \left(0, \frac{1}{2}\right) |
\section*{Problem 1 - 201021}
Determine all natural numbers \(n\) for which the number \(1+4 \cdot 9^{2 n}\) is a prime number! | 0 |
Among the five types of shapes: isosceles triangle, equilateral triangle, rectangle, square, and circle, the number of shapes that are both axisymmetric and centrally symmetric is ( ).
A: 2
B: 3
C: 4
D: 5 | B |
In a Cartesian coordinate system, for rectangle ABCD, if line AB is defined by the equation $x - 2y - 2 = 0$ and the diagonal line of the rectangle is relative to the point $(0,1)$, then find the length of the chord intercepted on the circle $(x - 1)^2 + (y - 1)^2 = 9$ by line CD. | 4 |
On semicircle, with diameter $|AB|=d$, are given points $C$ and $D$ such that: $|BC|=|CD|=a$ and $|DA|=b$ where $a, b, d$ are different positive integers. Find minimum possible value of $d$ | 8 |
(Inequality Optional Question)
In the range of real numbers, the solution set of the inequality $||x-2|-1| \leq 1$ is \_\_\_\_\_\_. | [0, 4] |
$[$ Inclusion-Exclusion Principle $]$
Petya plans to spend all 90 days of his vacation in the village and during this time, he will go swimming every second day (i.e., every other day), go to the store for groceries every third day, and solve math problems every fifth day. (On the first day, Petya did the first, secon... | 24 |
2. Let $x, y, z$ be real numbers for which the following relations are true: $x=\sqrt{1-2 y z}, y=$ $\sqrt{1-2 z x}, z=\sqrt{1-2 x y}$. Calculate $x+y+z$.
Gazeta Matematică 11/2015. | \sqrt{3} |
108. Bus tickets have numbers from 000001 to 999999. A number is considered lucky if the first three digits are odd and different, the second three digits are even, and 7 and 8 do not stand next to each other. How many different lucky numbers exist? | 7200 |
8.3. Find the largest natural number, all digits of which are different, and the product of these digits is a cube of some natural number. | 984321 |
You plot weight $(y)$ against height $(x)$ for three of your friends and obtain the points
$(x_{1},y_{1}), (x_{2},y_{2}), (x_{3},y_{3})$. If $x_{1} < x_{2} < x_{3}$ and $x_{3} - x_{2} = x_{2} - x_{1}$,
which of the following is necessarily the slope of the line which best fits the data?
"Best fits" means that the su... | A |
A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered as shown, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wa... | 353 |
87. How many different divisors does the number have a) 800; b) 126000? | 18 |
Simplify first, then evaluate: $(1-\frac{1}{x+1})\div \frac{x}{{x}^{2}+2x+1}$, where $x=2$. | 3 |
Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_x w = 24$, $\log_y w = 40$ and $\log_{xyz} w = 12$. Find $\log_z w$. | 60 |
Which of the following expressions are not polynomials among $2x^{2}$, $1-2x=0$, $ab$, $a>0$, $\frac{1}{a}$, $\pi$, and $m=\frac{1}{2}$?
A: $1$ expression
B: $2$ expressions
C: $3$ expressions
D: $4$ expressions | D |
The solution of the equation $2^{x}+3x=k$ is within the interval $[1,2)$. Then, the range of $k$ is \_\_\_\_\_\_\_\_. | [5,10) |
If $2a^{3x}b^{y+5}$ and $5a^{2-4y}b^{2x}$ are like terms, then $xy=$ \_\_\_\_\_\_. | -2 |
Given that F is the focus of the parabola $C: y^2 = 8x$, and let M be a point on C. The extension line of FM intersects the y-axis at point N. If M is the midpoint of FN, then $|FN| =$ __________ . | 6 |
Given $\overrightarrow{AB}=(2,3),\overrightarrow{BC}=(3,t)$, where $t \gt 0$. If $|\overrightarrow{BC}|=\sqrt{10}$, then $\overrightarrow{AB}•\overrightarrow{AC}=\_\_\_\_\_\_$. | 22 |
Example 10 Arrange all positive integers that are coprime with 105 in ascending order. Try to find the 1000th term of this sequence. | 2186 |
In a chess tournament, sports masters and candidates for masters are participating. What is the smallest number of people that can participate in this tournament if it is known that among them, masters are less than half, but more than $45 \%$.
# | 11 |
17. Let $P, Q, R, S$ be four integer points on the circle $\odot O: x^{2}+y^{2}=25$, and $|P Q|, |R S|$ are both irrational numbers. Then the maximum possible value of $\frac{|P Q|}{|R S|}$ is ( ).
(A) 3
(B) 5
(C) $3 \sqrt{5}$
(D) 7
(E) $5 \sqrt{2}$ | 7 |
In a certain class, $25\%$ of students failed the math test, $10\%$ failed the Chinese test, and $5\%$ failed both. Given that a student failed the math test, the probability that they also failed the Chinese test is $\_\_\_\_\_\_$. | \frac{1}{5} |
In the inn U Temného hvozdu, giant twins Pravdoslav and Křivomil serve the guests. Pravdoslav is honest and always charges accurately, while Křivomil is dishonest and always adds two krejcars to the price of each pie and each jug of mead. One day, seven dwarfs visited this inn and sat at two tables. The dwarfs paid the... | k=6,=6 |
In triangle $ABC$, sides $a$, $b$, and $c$ are opposite to angles $A$, $B$, and $C$ respectively. Given that $2b\cos{C} = a\cos{C} + c\cos{A}$,
(I) Find the size of angle $C$;
(II) If $b=2$ and $c=\sqrt{7}$, find the value of $a$ and the area of triangle $ABC$. | \frac{3\sqrt{3}}{2} |
11.1. In each of the four volleyball teams, there are six players, including a captain and a setter, and these are different people. In how many ways can a team of six players be formed from these four teams, such that there is at least one player from each team and there must be a pair of captain and setter from at le... | 9720 |
Simplify the following ratios to the simplest form of integer ratios.
32:48
0.15:3
$$\frac {11}{12}$$: $$\frac {3}{8}$$. | 22:9 |
(1) Calculate: ${{\sin }^{2}}120^\circ +\cos 180^\circ +\tan 45^\circ -{{\cos }^{2}}(-330^\circ )+\sin (-210^\circ )$
(2) Simplify: $\dfrac{\sin (\alpha -2\pi )\cos (\alpha -\dfrac{\pi }{2})\cos (\pi +\alpha )}{\sin (3\pi -\alpha )\sin (-\pi -\alpha )}$ | -\cos \alpha |
9. (Another solution to problem 8)
Let $q$ be an odd prime, and $p=4q+1$ also be a prime.
(1) Prove that the congruence $x^{2} \equiv -1 \pmod{p}$ has exactly two solutions, and each solution is a quadratic non-residue modulo $p$.
(2) Prove that except for the above solution, all other quadratic non-residues modulo $p... | 2, 3, 8, 10, 11, 14, 15, 18, 19, 21, 26, 27 |
1. Solve the equation
$$
\sqrt{\log _{a} \sqrt[4]{a x}+\log _{x} \sqrt[4]{a x}}+\sqrt{\log _{a} \sqrt[4]{\frac{x}{a}+\log _{x} \sqrt[4]{\frac{a}{x}}}}=a
$$
where \( a \) is a positive real number different from 1. | x_{1}=^{^{2}},x_{2}=^{\frac{1}{^{2}}} |
Fourteen, in Figure 6, the radii of the two circles are 8 and 6, the distance between the two centers is 12, a line passing through one of the intersection points of the two circles intercepts equal chords $\mathrm{QP}$ and $\mathrm{PR}$ on the two circles, find the square of the length of $QP$. | 130 |
Given the function $f(x)=(x-1)\sin x+2\cos x+x$.
(I) Find the equation of the tangent line to the curve $y=f(x)$ at the point $(0,f(0))$.
(II) Find the maximum and minimum values of the function $f(x)$ on the interval $[0,\pi]$. | \pi-2 |
We define a new operation: $x⊗y=\frac{xy}{3}-2y$. For example, $2⊗3=\frac{2×3}{3}-2×3=-4$. Then, the smallest integer solution for the inequality $2\otimes a\leqslant 2$ with respect to $a$ is ______. | -1 |
With the changes in the market and the reduction in production costs, the price of computers decreases by $\frac{1}{3}$ every 4 years. If the price of a computer was 8100 yuan in 2000, what would its price be in 2016?
A: 3000 yuan
B: 2400 yuan
C: 1600 yuan
D: 1000 yuan | C |
Given that the line $y = \frac{1}{m}$ is a tangent to the curve $y = xe^x$, the value of the real number $m$ is ($\,$)
A: $$- \frac {1}{e}$$
B: $-e$
C: $$\frac {1}{e}$$
D: $e$ | B |
Given proposition $p$: $\forall x \in \mathbb{R}$, $\log_{2}(x^{2}+4) \geqslant 2$, and proposition $q$: $y=x^{\frac{1}{2}}$ is a decreasing function on its domain, then among the following propositions, the true one is ( ).
A: $p \lor (\neg q)$
B: $p \land q$
C: $(\neg p) \lor q$
D: $(\neg p) \land (\neg q)$ | A |
Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ . | (2, 6), (2, -6), (4, 18), (4, -18) |
Task 1. On a plate, there are various candies of three types: 2 lollipops, 3 chocolate candies, and 5 jelly candies. Svetlana sequentially ate all of them, choosing each subsequent candy at random. Find the probability that the first and last candies eaten were of the same type. | \frac{14}{45} |
The graph of the function $f(x)=kx-1-k$ always passes through a fixed point $A$. If point $A$ is on the line $mx-ny-1=0$, where $m > 0$, $n > 0$, then the minimum value of $\dfrac{1}{m}+\dfrac{2}{n}$ is ( )
A: $4$
B: $5$
C: $6$
D: $3+2\sqrt{2}$ | D |
1. In tetrahedron $ABCD$, $AD \perp$ plane $BCD$, $\angle ABD = \angle BDC = \theta < 45^{\circ}$. It is known that $E$ is a point on $BD$ such that $CE \perp BD$, and $BE = AD = 1$.
(1) Prove: $\angle BAC = \theta$;
(2) If the distance from point $D$ to plane $ABC$ is $\frac{4}{13}$, find the value of $\cos \theta$.
| \cos \theta = \frac{4}{5} |
28. Determine the length of the path over which the amount of grain in the seeder's box will decrease by $14 \%$, if the box has a working width of 4 m, it was filled with 250 kg of grain, and the sowing rate is 175 kg per 1 ha. | 500 |
In the Cartesian coordinate system, given points $A(2,m)$ and $B(n,-3)$ are symmetric about the $x$-axis, then the value of $m+n$ is ( )
A: $-1$
B: $1$
C: $5$
D: $-5$ | C |
Let the function $y = f(x)$. When the independent variable $x$ changes from $x_0$ to $x_0 + \Delta x$, the change in the function $\Delta y$ is ( )
A: $f(x_0 + \Delta x)$
B: $f(x_0) + \Delta x$
C: $f(x_0) \cdot \Delta x$
D: $f(x_0 + \Delta x) - f(x_0)$ | \text{D} |
Let's determine the multiples of 5 that, when divided by 11, leave a remainder of 4. | 15+55m |
. We color the non-zero rational numbers in two colors: white and black. We assume that 1 is colored white, that $x$ and $x+1$ are never colored the same color, and that $x$ and $\frac{1}{x}$ always have the same color. What is the color of $\frac{1543}{275}$? | white |
Simplify the expression $\frac{4}{{x+2}}+x-2$ to get:
A: $1$
B: $\frac{{{x^2}}}{{{x^2}-4}}$
C: $\frac{x}{{x+2}}$
D: $\frac{{{x^2}}}{{x+2}}$ | \text{D} |
Given that $C$ is an interior angle of $\triangle ABC$, and the vectors $\overrightarrow{m}=(2\cos C-1,-2)$, $\overrightarrow{n}=(\cos C,\cos C+1)$. If $\overrightarrow{m}\perp \overrightarrow{n}$, then $\angle C$ equals to ( ).
A: $ \dfrac {\pi}{6}$
B: $ \dfrac {\pi}{3}$
C: $ \dfrac {2\pi}{3}$
D: $ \dfrac {5\pi}{6... | C |
6. As shown in Figure $8, \odot O$ has a radius of $1, B$ is any point on the circumference. The vertex $A$ of $\triangle A B C$ is also on the circumference, such that $A B = B C, \angle A B C=90^{\circ}$. What is the maximum length of $O C$? | \sqrt{2}+1 |
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