problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
450. For what values of $x$ is the equality $|x+6|=$ $=-(x+6)$ true? | x\leqslant-6 |
10.239. Find the radius of the circle, in the segment of which, corresponding to a chord of length $6 \mathrm{~cm}$, a square with a side of $2 \mathrm{~cm}$ is inscribed. | \sqrt{10} |
Let $n$ be an integer greater than two, and let $A_1,A_2, \cdots , A_{2n}$ be pairwise distinct subsets of $\{1, 2, ,n\}$. Determine the maximum value of
\[\sum_{i=1}^{2n} \dfrac{|A_i \cap A_{i+1}|}{|A_i| \cdot |A_{i+1}|}\]
Where $A_{2n+1}=A_1$ and $|X|$ denote the number of elements in $X.$ | n |
Two dice are thrown one after the other, and the numbers obtained are denoted as $a$ and $b$. If the lengths of three line segments are $a$, $b$, and $5$, respectively, find the probability that these three line segments can form an isosceles triangle (including an equilateral triangle).
A: $ \dfrac {7}{36}$
B: $ \... | D |
Example: Given $\overrightarrow{O A}=(4,3)$, the graph of the function $y=x^{2}+b x+c$ is translated by the vector $\overrightarrow{O A}$ to obtain a graph that is tangent to the line $4 x+y-8=0$ at point $T(1,4)$. Find the original function's equation. | x^{2}+2x-2 |
Given $a, b \in \mathbb{R}$, and $i$ is the imaginary unit. If $a+i=2-bi$, then $(a+bi)^{2}=$ \_\_\_\_\_\_. | 3-4i |
2. Given that point $E$ is the midpoint of side $B C$ of rhombus $A B C D$, $\angle A B C=30^{\circ}$, and $P$ is a point on diagonal $B D$ such that $P C + P E = \sqrt{13}$. Then the maximum area of rhombus $A B C D$ is $\qquad$ | 10+4 \sqrt{3} |
1. Vasya, Petya, and Kolya study in the same class. Vasya lies in response to any question, Petya alternates between lying and telling the truth, and Kolya lies in response to every third question, and tells the truth in all other cases. One day, each of them was asked six times in a row how many people are in their cl... | 27 |
1. $\left(20 \frac{94}{95} \times 1.65-20 \frac{94}{95}+\frac{7}{20} \times 20 \frac{94}{95}\right) \times 47.5 \times 0.8 \times 2.5$. | 1994 |
14. Given that $c$ is a positive integer not exceeding 2017, and there exist positive integers $a$ and $b$ such that
$$
2^{a} \times 5^{b}=\left(a^{3}+a^{2}+a+1\right) c \text {. }
$$
Find the maximum value of $c$. | 1000 |
Anton writes down all positive integers in sequence that are divisible by 2. Berta writes down all positive integers in sequence that are divisible by 3. Clara writes down all positive integers in sequence that are divisible by 4. The orderly Dora notes the numbers written down by the others. She orders the numbers by ... | 3026 |
6. Triangle $L O M$ with angle $\angle L O M=\alpha^{\circ}$ is rotated by some acute angle around point $O$. As a result, point $L$ moves to point $N$, which lies on side $L M$, and point $M$ moves to point $K$ such that $O M \perp N K$. Find the angle of rotation (in degrees).
Note: In different versions, different ... | \frac{2}{3}\alpha |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, and if $a_2=3$, $a_1+a_6=12$, then the value of $a_7+a_8+a_9=$ （ ）
A: 27
B: 36
C: 45
D: 63 | C |
Problem 1. A student started working on an exam between 9 and 10 o'clock and finished between 13 and 14 o'clock. Find the exact time spent by the applicant on completing the task, given that at the beginning and the end of the work, the hour and minute hands, having swapped places, occupied the same positions on the cl... | \frac{60}{13} |
The coordinates of the vertex of the quadratic function $y=-2\left(x+3\right)^{2}-5$ are:
A: $(-3,5)$
B: $(-3,-5)$
C: $(3,5)$
D: $(3,-5)$ | B |
# Problem 1. 15 points
Find the minimum value of the expression
$$
\sqrt{x^{2}-2 \sqrt{3} \cdot|x|+4}+\sqrt{x^{2}+2 \sqrt{3} \cdot|x|+12}
$$
as well as the values of $x$ at which it is achieved.
# | 2\sqrt{7} |
The 20th term of the arithmetic sequence 8, 5, 2, ... is ______. | -49 |
$2 \cdot 30$ Given the equation $(x-m)^{2}-(x-n)^{2}=(m-n)^{2}, m$ is a positive number, $n$ is a negative number, then the set of $x$ values that satisfy the equation is
(A) $x \geqslant 0$.
(B) $x \leqslant n$.
(C) $x=0$.
(D) the set of all real numbers.
(E) none of the above.
(10th American High School Mathematics E... | n |
## Task A-4.3.
Determine all natural numbers $m$ and $n$ for which $2^{n}+5 \cdot 3^{m}$ is a square of some natural number. | (2,2) |
Let the function $f$ be defined on the set $\mathbb{N}$ such that
$\text{(i)}\ \ \quad f(1)=1$
$\text{(ii)}\ \quad f(2n+1)=f(2n)+1$
$\text{(iii)}\quad f(2n)=3f(n)$
Determine the set of values taken $f$. | \{0, 1, 3, 4, 9, 10, 12, 13, 27, \ldots\} |
The size of an exterior angle of a regular pentagon is ____ degrees. | 72^\circ |
11.3. Solve the system of equations $\left\{\begin{array}{l}\sqrt{1-3 x}-1=\sqrt{5 y-3 x} \\ \sqrt{5-5 y}+\sqrt{5 y-3 x}=5 .\end{array}\right.$ | (-\frac{8}{3},-\frac{4}{5}) |
12. If the three sides of a right triangle are pairwise coprime positive integers, then such a right triangle is called a primitive right triangle.
(1) Find the number of primitive right triangles in which the area is exactly $n$ times the perimeter, and denote this number as $a_{n}$;
(2) Find $\max _{1 \leqslant n \le... | 16 |
How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ?
(A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above. | 18 |
Given $A(2,3)$, $B(4,-3)$, point $P$ lies on the extension of segment $AB$, and $|\overrightarrow{AP}|=2|\overrightarrow{PB}|$. What are the coordinates of point $P$?
A: $(0,9)$
B: $(6,-9)$
C: $(\frac{10}{3},-1)$
D: $(6,-9)$ or $(\frac{10}{3},-1)$ | \text{B} |
In the sequence $\{a_n\}$, $a_{n+1} = a_n + 2$ ($n \in \mathbb{N}^*$), then the points $A_1(1, a_1)$, $A_2(2, a_2)$, ..., $A_n(n, a_n)$ are distributed on ( )
A: A straight line, and the slope of the line is -2
B: A parabola, and the parabola opens downwards
C: A straight line, and the slope of the line is 2
D... | \text{C} |
Given the sequence $\{a_n\}$ with its first $n$ terms sum $S_n$, where $a_1=1$ and $S_{n+1} = 2S_n + 1$, find the expression for $S_n$. | S_n = 2^n - 1 |
In triangle $\triangle ABC$, $\angle A=\angle B=60^{\circ}$, and $AB=10$ cm, then $BC=$____$cm$. | 10 \text{ cm} |
Petrov booked an apartment in a new building with five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment had the number 636. Then the developer changed the numbering to the opposite direction (from right to left, see the figure). As a result, Petrov's apartment becam... | 985 |
13. It is given that $k$ is a positive integer not exceeding 99. There are no natural numbers $x$ and $y$ such that $x^{2}-k y^{2}=8$. Find the difference between the maximum and minimum possible values of $k$. | 96 |
Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon. | 1506 |
Given that $α$ is an angle in the third quadrant and $f(α)= \frac{\sin (α- \frac{π}{2})\cos ( \frac{3π}{2}+α)\tan (π-α)}{\tan (-α-π)\sin (-α-3π)}$.
(1) Simplify $f(α)$;
(2) If $\cos (α- \frac{3}{2}π)= \frac{1}{5}$, find the value of $f(α)$. | f(α)=-\cos α= \frac{2 \sqrt{6}}{5} |
2.92 The largest even number that cannot be expressed as the sum of two odd composite numbers is?
Will the text be translated into English, keeping the original text's line breaks and format, and output the translation result directly.
Note: The note above is not part of the translation but is provided to clarify th... | 38 |
Given set $A=\{x|\frac{x-2}{x+1}≤0\}$, set $B=\{x|2m+3 \lt x \lt m^{2}\}$, $m\in R$.$(1)$ When $m=-2$, find $A\cup B$;$(2)$ If $A\cap B=B$, find the range of real number $m$. | [-\sqrt{2}, 3] |
Example 27. In a regular triangular pyramid, a plane is drawn through the side of the base, perpendicular to the opposite lateral edge. Let's find the area of the resulting section, if the base is equal to $a$, and the height of the pyramid is equal to $h$. | \frac{3^{2}}{4\sqrt{^{2}+3^{2}}} |
1. Among the four functions $y=2^{x}, y=\log _{2} x, y=x^{2}, y=\cos 2 x$, the number of functions for which $\frac{f\left(x_{1}\right)+f\left(x_{2}\right)}{2} < f\left(\frac{x_{1}+x_{2}}{2}\right)$ always holds when $0 < x_{1} < x_{2} < \frac{\pi}{2}$ is
A. 0
B. 1
C. 2
D. 3 | B |
Problem 10.3. Find all positive integers $t, x, y, z$ such that
$$
2^{t}=3^{x} 5^{y}+7^{z}
$$ | 6,1,1,2 |
Given the functions $f(x)=2\sin x$ and $g(x)=2 \sqrt {3}\cos x$, the straight line $x=m$ intersects with the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$, respectively. Find the maximum value of $|MN|$.
A: $3$
B: $4$
C: $2 \sqrt {2}$
D: $2$ | B |
If $C_{20}^{2x-1}=C_{20}^{x+3}$, then the value of $x$ is $(\ )$
A: $4$
B: $4$ or $5$
C: $6$
D: $4$ or $6$ | D |
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours? | 33\% |
$\sin 60^\circ = $ （ ）
A: $\frac { \sqrt {3}}{2}$
B: $-\frac { \sqrt {3}}{2}$
C: $-\frac {1}{2}$
D: $\frac {1}{2}$ | A |
In the process of selecting test points using the 0.618 method, if the test interval is \[2, 4\], and the result at the first test point $x_1$ is better than at the second test point $x_2$, where $x_1 > x_2$, then the third test point $x_3$ is __ . | 3.528 |
## Zadatak B-1.3.
U trokutu $A B C$ mjera kuta pri vrhu $A$ iznosi $60^{\circ}$ a mjera kuta pri vrhu $B$ iznosi $45^{\circ}$. Ako duljina stranice $\overline{A C}$ iznosi $8 \mathrm{~cm}$, izračunajte opseg trokuta $A B C$.
## Rjesenje.
Neka je $|B C|=a,|A C|=b=8,|A B|=c$, a neka je $|C D|=v$ (visina iz vrha $C$ ).... | 502 |
Given two vectors $\overrightarrow{a}=(2,m,3)$ and $\overrightarrow{b}=(4,-1,n)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, then the value of $mn$ is:
A: $1$
B: $-2$
C: $-3$
D: $6$ | C |
13. Given $\sin ^{3} \theta+\cos ^{3} \theta=\frac{11}{16}$, find the value of $\sin \theta+\cos \theta$. | \frac{1}{2} |
# Problem №4
Two ferries simultaneously depart from opposite banks of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant but not equal. The ferries meet at a distance of 720 meters from one bank, after which they continue their journey. On the return trip, they meet 400 meters fr... | 1760 |
Given set $A=\{0, 1, 2\}$, and set $B=\{x|x=2m, m\in\mathbb{N}\}$, then $A\cap B=$ ( )
A: $\{0\}$
B: $\{0, 2\}$
C: $\{0, 4\}$
D: $\{0, 2, 4\}$ | B |
## Task 2 - 070922
For two rational numbers $a$ and $b$, the following four inequalities hold:
$$
a+b \neq 3 ; \quad a-b \neq 10 ; \quad a \cdot b \neq 5 ; \quad a: b \neq 18.75
$$
However, the numbers $3$, $10$, $5$, and $18.75$ match (in a different order) with each of the numbers $a+b$, $a-b$, $a \cdot b$, and $a... | (\frac{15}{2},\frac{5}{2}) |
Given functions $f(x)$ and $g(x)$ with derivatives ${f'}(x)$ and ${g'}(x)$ on $R$, if $f(x+2)$ is an even function, $y=g(x+1)-2$ is an odd function, and $f(3-x)+g(x-1)=2$, then which of the following conclusions is correct?
A: ${f'}(2022)=0$
B: $g(2023)=0$
C: $f(x)$ is an odd function on $R$
D: ${g'}(x)$ is an odd ... | A \text{ and } D |
Ms. Carroll promised that anyone who got all the multiple choice questions right on the upcoming exam would receive an A on the exam. Which one of these statements necessarily follows logically?
$\textbf{(A)}\ \text{If Lewis did not receive an A, then he got all of the multiple choice questions wrong.}\\\textbf{(B)}\ \... | \textbf{(B)} |
# Problem 4. (3 points)
Three runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.
At some point, the first runner meets the second. Twenty minutes later, the second runner meets the third for the first time. An... | 100 |
2. Given an equilateral $\triangle A B C$ with side length $2, P$ is a point inside $\triangle A B C$, and the distances from point $P$ to the three sides $B C, A C, A B$ are $x, y, z$ respectively, and their product is $\frac{\sqrt{3}}{9}$. Then the sum of the squares of $x, y, z$ is | 1 |
12. (5 points) A group of students participates in a tree-planting activity. If 1 girl and 2 boys form a group, there are 15 boys left over; if 1 girl and 3 boys form a group, there are 6 girls left over. Therefore, the number of boys participating in the tree-planting activity is $\qquad$, and the number of girls is $... | 81,33 |
The foci of an ellipse are $F_{1}$ and $F_{2}$, and there exists a point $P$ on the ellipse such that $\angle F_{1}PF_{2}= \frac {2\pi}{3}$. The range of the eccentricity $e$ of the ellipse is ( )
A: $\left[ \frac {\sqrt {3}}{2},1\right)$
B: $\left[ \frac {1}{2},1\right)$
C: $(0, \frac {\sqrt {3}}{2}]$
D: $(0, \frac {... | A |
If the base radius of a cylinder is 2 and the generatrix length is 3, then the lateral surface area of the cylinder is (The answer can retain $\pi$). | 12\pi |
Let $a, b \in \mathbb{R}$. Then "$2^{a}+2^{b}=2^{a+b}$" is a ______ condition for "$a+b \geqslant 2$".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Necessary and sufficient
D: Neither sufficient nor necessary | \text{A} |
If the line $l_{1}$: $ax+y+2=0$ is perpendicular to the line $l_{2}$: $x+(a-2)y+1=0$, then $a=\_\_\_\_\_\_$. | 1 |
10. Let positive numbers $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ satisfy: $a_{1}, a_{2}, a_{3}$ form an arithmetic sequence with a common difference of $b_{1}$, $b_{1}, b_{2}, b_{3}$ form a geometric sequence with a common ratio of $a_{1}$, and $a_{3}=b_{3}$. Find the minimum value of $a_{3}$, and determine the valu... | \frac{15\sqrt{6}}{8} |
41. The ratio of the number of stamps owned by A and B is 5:3. If A gives 10 stamps to B, then the ratio of the number of stamps owned by A and B becomes 7:5. The total number of stamps owned by the two is $\qquad$.
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 240 |
Example 3. Calculate the area of the part of the surface of the paraboloid of revolution $2z = x^2 + y^2$, enclosed within the cylinder $x^2 + y^2 = R^2$. | \frac{2\pi}{3}(\sqrt{(1+R^{2})^{3}}-1) |
4. (3 points) Prove that for $n=12002$ the sum of binomial coefficients with a step of 6, i.e., $C_{n}^{1}+C_{n}^{7}+\ldots+C_{n}^{n-1}$, gives a remainder of 1 when divided by 3.
$C_{n}^{k}-$ the number of ways to choose $k$ items from $n$, which is $\frac{n!}{k!(n-k)!}$ if $0 \leqslant k \leqslant n$ and 0 in all ot... | 1 |
2. Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies
$$
\begin{array}{l}
f(x+1)=f(-x), \\
f(x)=\left\{\begin{array}{ll}
1, & -1<x \leqslant 0 \\
-1, & 0<x \leqslant 1 .
\end{array}\right.
\end{array}
$$
Then $f(f(3.5))=$ $\qquad$ | -1 |
## 120. Math Puzzle $5 / 75$
A reconnaissance boat of the People's Navy, which belongs to a fleet, is ordered to explore a sea area of 70 nautical miles in the direction of the fleet's movement. The speed of the fleet is 15 knots, and the speed of the reconnaissance boat is 28 knots.
After what time can the boat be e... | 3\frac{11}{43} |
6. 22 For a given positive integer $k$, define $f_{1}(k)$ as the square of the sum of the digits of $k$, and let
$$f_{n+1}(k)=f_{1}\left(f_{n}(k)\right),$$
Find: $f_{1991}\left(2^{1990}\right)$.
| 256 |
92. A five-digit license plate consists of English letters and numbers, with exactly two of the first four positions being English letters (letters $I, O$ are not allowed), and the last position must be a number. Xiao Li likes the number 18 and hopes that his license plate has two adjacent digits 1 and 8, with 1 preced... | 34560 |
If planes $\alpha$ and $\beta$ satisfy $\alpha \perp \beta$ and $\alpha \cap \beta = l$, and $P \in \alpha$, $P \notin l$, then among the following statements, the false one is ( )
A: A line through point P and perpendicular to plane $\alpha$ is parallel to plane $\beta$
B: A line through point P within plane $\alpha$ ... | \text{D} |
Here are four propositions about the complex number $z=2-i$: $p_{1}$: $|z|=5$; $p_{2}$: The conjugate of $z$ is $2+i$; $p_{3}$: $z^{2}=3-4i$; $p_{4}$: $ \dfrac {1}{z}= \dfrac {2}{3}+ \dfrac {1}{3}i$. Among these, the true propositions are ( ).
A: $p_{1}$, $p_{2}$
B: $p_{2}$, $p_{3}$
C: $p_{2}$, $p_{4}$
D: $p_{3}$, ... | B |
8.2. Which of the numbers is greater: $2^{2021}$ or $5^{864}$? | 2^{2021}>5^{864} |
[Inscribed, circumscribed, and exscribed circles; their radii] Pythagorean Theorem (direct and inverse).
Find the radius of the circumscribed circle of a right triangle if the radius of its inscribed circle is 3, and one of the legs is 10. | \frac{29}{4} |
3. Petya thought of five numbers. On the board, he wrote down their pairwise sums: $7, 9, 12, 16, 17, 19, 20, 21, 22$, 29. What numbers did Petya think of? | 2,5,7,14,15 |
Among the following sets of numbers, the pair that are opposite in sign is:
A: $131$ and $1-31$
B: $\sqrt{7}$ and $-\sqrt{7}$
C: $\frac{1}{3}$ and $\sqrt{\frac{1}{9}}$
D: $5^{2}$ and $\left(-5\right)^{2}$ | B |
"$x$ is an integer" is ( ) "$2x+1$ is an integer".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | A |
The coordinates of the foci of the hyperbola $4y^2-25x^2=100$ are ( )
A: $(-5, 0)$, $(5, 0)$
B: $(0, -5)$, $(0, 5)$
C: $(-\sqrt{29}, 0)$, $(\sqrt{29}, 0)$
D: $(0, -\sqrt{29})$, $(0, \sqrt{29})$ | \text{D} |
Define an odd function $f(x)$ on $\mathbb{R}$ that satisfies $f(2-x) = f(x)$, and is a decreasing function on the interval $[-3, -2]$. If $\alpha$ and $\beta$ are the acute angles of an obtuse triangle, then the relationship between $f(\sin\alpha)$ and $f(\cos\beta)$ is ( )
A: $f(\sin\alpha) > f(\cos\beta)$
B: $f(\s... | \text{B} |
6. (2003 Bulgarian Mathematical Competition) The sequence $\left\{y_{n}\right\}$ is defined as follows:
$$
y_{1}=y_{2}=1, y_{n+2}=(4 k-5) y_{n+1}-y_{n}+4-2 k, n=1,2, \cdots
$$
Find all integers $k$ such that every term in the sequence $\left\{y_{n}\right\}$ is a perfect square. | k=1ork=3 |
Solve the system
$$
\begin{aligned}
& y^{2}=4 x^{3}+x-4 \\
& z^{2}=4 y^{3}+y-4 \\
& x^{2}=4 z^{3}+z-4
\end{aligned}
$$ | (1,1,1) |
Given $f(x)=2^{x}+\\log \_{2}x$, find $f{{'}}(1)=$ _____. | 2\\ln 2+ \\dfrac {1}{\\ln 2} |
40th Swedish 2000 Problem 1 Each of the numbers 1, 2, ... , 10 is colored red or blue. 5 is red and at least one number is blue. If m, n are different colors and m+n ≤ 10, then m+n is blue. If m, n are different colors and mn ≤ 10, then mn is red. Find all the colors. | allblueexcept510 |
174. Find the derivative of the function $y=x^{2}-3 x+5$. | 2x-3 |
In the arithmetic sequence $\left\{ a_n \right\}$, it is known that $a_3+a_8=10$, then $3a_5+a_7=$ ______________. | 20 |
Given the equation $x^{3}-13 x^{2}+24 x-22=0$ has a complex root $1+i$, then the real root of this equation is | 11 |
Three, the sequence $\left\{a_{n}\right\}$ satisfies
$$
(n-1) a_{n+1}=(n+1) a_{n}-2(n-1) \text {, }
$$
$n=1,2, \cdots$, and $a_{100}=10098$. Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=(n-1)(n+2) |
4. (8 points) Tongtong and Linlin each have several cards: If Tongtong gives 6 cards to Linlin, Linlin will have 3 times as many as Tongtong; if Linlin gives 2 cards to Tongtong, then Linlin will have 2 times as many as Tongtong. How many cards did Linlin originally have. | 66 |
Let's take a three-digit number, for example, 346. By writing the digits in reverse order, we get 643. Subtracting the smaller number from the larger one $(643-346)$, we get 297. Writing the digits of this number in reverse order again: 792. The sum of these two numbers $297+792=1089$. Is it true that we always get 108... | 1089 |
During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side?
[img]https://cdn.artofproblemsolving.com/attachments/d/2/f... | 60^\circ |
A donkey suffers an attack of hiccups and the first hiccup happens at $4:00$ one afternoon. Suppose that
the donkey hiccups regularly every $5$ seconds. At what time does the donkey’s $700$th hiccup occur?
$\textbf{(A) }15 \text{ seconds after } 4:58$
$\textbf{(B) }20 \text{ seconds after } 4:58$
$\textbf{(C) }25 \text... | \textbf{(A)}15after4:58 |
Given that $a = \frac{1}{4}\log_{2}{3}$, $b = \frac{1}{2}$, and $c = \frac{1}{2}\log_{5}{3}$, then ( )
A: $c < a < b$
B: $a < b < c$
C: $b < c < a$
D: $b < a < c$ | A |
The maximum value of the function $y=x+2\cos x$ in the interval $\left[0,\frac{\pi}{2}\right]$ is _______. | \frac{\pi}{6}+ \sqrt{3} |
Given the hyperbola $$\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0，b＞0)$$ whose asymptotes are both tangent to the circle C: $x^{2}+y^{2}-6x+5=0$, and the right focus of the hyperbola is the center of circle C, find the equation of the hyperbola. | \frac {x^{2}}{5}- \frac {y^{2}}{4}=1 |
Calculate: $-\left(-1\right)+3^{2}\div \left(1-4\right)\times 2$. | -5 |
2. In $\triangle A B C$, if $\sin A+\cos A=-\frac{1}{3}$, then $\cos 2 A=$ . $\qquad$ | \frac{\sqrt{17}}{9} |
If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of "quasi-geometric progression." Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The probability that these $4$ numbers have the pro... | \frac{13}{35} |
14. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=0, a_{n+1}=\frac{n+2}{n} a_{n}+\frac{1}{n}$. Find the general term formula of the sequence $\left\{a_{n}\right\}$. | a_{n}=\frac{(n-1)(n+2)}{4} |
[Theorem of Three Perpendiculars]
At the base of a quadrilateral pyramid lies a rhombus $ABCD$, in which $\angle BAD=60^{\circ}$. It is known that $SD=SB$, $SA=SC=AB$. A point $E$ is taken on the edge $DC$ such that the area of triangle $BSE$ is the smallest among the areas of all sections of the pyramid containing th... | 2:5 |
Roman likes magic and mathematics. The last time he cast a spell with three-digit or four-digit numbers like this:
- from a given number, he created two new numbers by splitting it between the hundreds and tens digits (for example, from the number 581, he would get 5 and 81),
- he added the new numbers and wrote down ... | 18810 |
## Task A-4.7.
Let $n$ be a natural number. Each of the numbers $n, n+1, n+2, \ldots, 2 n-1$ has a greatest odd divisor. Determine the sum of these greatest odd divisors. | n^2 |
What are the conditions for the terms of a three-term geometric progression to be considered the side lengths from which a triangle can be constructed?
Can any of these triangles be right-angled? | 1\leq\frac{\sqrt{5}+1}{2} |
11.3. For what smallest $n$ is the following condition satisfied: if $n$ crosses are placed in some cells of a $6 \times 6$ table in any order (no more than one per cell), then there will definitely be three cells forming a strip of length 3, either vertical or horizontal, each containing a cross? | 25 |
6. As shown in the figure, in the convex quadrilateral $A B C D$, $A B=2$, $P$ is the midpoint of side $A B$. If $\angle D A B=\angle A B C$ $=\angle P D C=90^{\circ}$, then the minimum possible value of the area of quadrilateral $A B C D$ is ( ).
(A) 4
(B) $3 \sqrt{2}$
(C) $\frac{9}{2}$
(D) $2+2 \sqrt{21}$ | A |
On the diagonals $A C$ and $B D$ of trapezoid $A B C D$ with bases $B C=a$ and $A D=b$, points $K$ and $L$ are located respectively, such that
$C K: K A=B L: L D=7: 4$. Find $K L$.
# | \frac{1}{11}|7b-4a| |
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