problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
479. Matchstick Puzzle. Taking a box of matches, I found that I could form any pair of regular polygons shown in our diagram using all the matches each time. So, if I had 11 matches, I could use them to form, as shown, either a triangle and a pentagon, or a pentagon and a hexagon, or a square and a triangle (using only... | 36 |
The pair of equations $3^{x+y}=81$ and $81^{x-y}=3$ has:
$\textbf{(A)}\ \text{no common solution} \qquad \\ \textbf{(B)}\ \text{the solution} \text{ } x=2, y=2\qquad \\ \textbf{(C)}\ \text{the solution} \text{ } x=2\frac{1}{2}, y=1\frac{1}{2} \qquad \\ \textbf{(D)}\text{ a common solution in positive and negative integ... | \frac{17}{8},\frac{15}{8} |
6. Odd numbers from 1 to 49 are written in the form of a table
| 1 | 3 | 5 | 7 | 9 |
| ---: | ---: | ---: | ---: | ---: |
| 11 | 13 | 15 | 17 | 19 |
| 21 | 23 | 25 | 27 | 29 |
| 31 | 33 | 35 | 37 | 39 |
| 41 | 43 | 45 | 47 | 49 |
Five numbers are chosen such that no two of them are in the same row or column. What is ... | 125 |
In the arithmetic sequence $\{a_n\}$, if $a_2+a_4+a_6+a_8+a_10=80$, then the value of $S_{11}$ is ______. | 176 |
Problem 3. We will call a composite natural number $n$ "interesting" if all its natural divisors can be written in ascending order, and each subsequent divisor is divisible by the previous one. Find all "interesting" natural numbers from 20 to 90 (inclusive). | 25,27,32,49,64,81 |
3. If $\log _{a} 3>\log _{b} 3>0$, then the following equation that holds is $(\quad)$.
A. $a>b>1$
B. $b>a>1$
C. $a<b<1$
D. $b<a<1$ | B |
Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them? | \frac{\sqrt{6} + 2}{2} |
4. Find all natural numbers $a$ for which the number
$$
\frac{a+1+\sqrt{a^{5}+2 a^{2}+1}}{a^{2}+1}
$$
is also a natural number. | 1 |
(2) If $z \in \mathbf{C}$, for the quadratic equation in $x$, $x^{2}-z x+4+3 \mathrm{i}=0$ to have real roots, find the complex number $z$ that minimizes the modulus of $z$. | \(\frac{9\sqrt{5}}{5}+\frac{3\sqrt{5}}{5}\mathrm{i}) |
## 96. Math Puzzle 5/73
A copper wire that is $l_{1}=50 \mathrm{~m}$ long and $d_{1}=1 \mathrm{~mm}$ thick is drawn to a length of $l_{2}=1800 \mathrm{~m}$. What is the new diameter $d_{2}$? | 0.17 |
1. Given the set $M$ satisfies $M \subseteq\{0,1,2,3,4\}$ and $M \subseteq\{0,2,4,8\}$, the maximum number of elements in set $M$ is ( ).
A. 1
B. 2
C. 3
D. 6 | 3 |
Problem 9.4. A prime number $p$ is such that the number $p+25$ is the seventh power of a prime number. What can $p$ be? List all possible options. | 103 |
61. A bundle of wire, the first time used half of the total length plus 3 meters, the second time used half of the remaining plus 3 meters, at this point there are still 18 meters left, the original length of the wire is $\qquad$ meters. | 90 |
Given sets $A=\{1, 2, 3, 4\}$ and $B=\{1, 2\}$, find the number of sets $C$ that satisfy the condition $B \subseteq C \subseteq A$. | 4 |
Given two sequences $\{a_n\}$ and $\{b_n\}$ satisfying $a_1=1$, $a_2=2$, $b_1=2$, and for any positive integers $i$, $j$, $k$, $l$, when $i+j=k+l$, it holds that $a_i+b_j=a_k+b_l$. Then the value of $$\frac {1}{2013} \sum\limits_{i=1}^{2013}(a_{i}+b_{i})$$ is ( )
A: 2012
B: 2013
C: 2014
D: 2015 | \text{D} |
Given that the solution set of the inequality $\left(a+1\right)x \gt a+1$ is $x \lt 1$, then the range of values for $a$ is ______. | a \lt -1 |
5. Let $f(x)=x^{2}-\pi x, \alpha=\arcsin \frac{1}{3}, \beta=\arctan \frac{5}{4}, \gamma=\arccos \left(-\frac{1}{3}\right), \delta=\operatorname{arccot}\left(-\frac{5}{4}\right)$, then ( ).
A. $f(\alpha)>f(\beta)>f(\delta)>f(\gamma)$
B. $f(\alpha)>f(\delta)>f(\beta)>f(\gamma)$
C. $f(\delta)>f(\alpha)>f(\beta)>f(\gamma)$... | B |
Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola. | y = \frac{1}{4} |
1. Find all integers $n$ for which $0 \leqslant n \leqslant 90$ and
$$
\sin 80^{\circ}+\sin 50^{\circ}-\sin 20^{\circ}=\sqrt{2} \sin n^{\circ}
$$ | 85 |
Positive integers $a$ and $b$ satisfy $a b=2010$. If $a>b$, the smallest possible value of $a-b$ is
(A) 37
(B) 119
(C) 191
(D) 1
(E) 397 | 37 |
4. As shown in Figure $1, O$ is the center of a sphere with radius 1, points $A, B, C$ are on the surface of the sphere, $OA, OB, OC$ are pairwise perpendicular, $E, F$ are the midpoints of the arcs $\overparen{AB}, \overparen{AC}$, respectively. Then the spherical distance between points $E, F$ on the surface of the s... | \frac{\pi}{3} |
A number $x$ is randomly selected from the interval $[-1, 2]$. Determine the probability that $x$ belongs to the interval $[0, 1]$. | \frac{1}{3} |
3. From 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What is the maximum value that its perimeter can take? | 184 |
A computing device has a data input A and an operation output B, performing a certain operation program.
(1) When the natural number 1 is input from port A, the real number $\frac{1}{3}$ is obtained from port B, denoted as $f(1) = \frac{1}{3}$;
(2) When the natural number $n$ ($n \geq 2$) is input from port A, the resu... | 19 |
13.226. A manuscript of 80 pages was given to two typists. If the first typist starts typing the manuscript 3 hours after the second, then each of them will type half of the manuscript. If both typists start working simultaneously, then after 5 hours, 15 pages will remain untyped. How long will it take for each typist ... | 16 |
Given that function $f(x)$ satisfies $f(x) = f'(1)e^{x-1} - f(0)x + \frac{1}{2}x^2$.
(1) Find the analytical expression of $f(x)$;
(2) If $f(x) \geq m$ holds true in the interval $[-1, 2]$, find the range of values for the real number $m$. | m \leq 1 |
11. The lengths of the three altitudes of a triangle are in the ratio $3: 4: 6$. If the radius of the inscribed circle of the triangle is 1 , find the radius of the circumcircle of the triangle.
某三角形三條高的長度之比為 $3: 4: 6$ 。若該三角形內切圓的半徑是 1 , 求它的外接圓半徑。 | \frac{16}{5} |
Given that vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ are not parallel. If the vector $λ \overrightarrow {a}+ \overrightarrow {b}$ is parallel to $\overrightarrow {a}+2 \overrightarrow {b}$, then find the value of the real number $λ$. | \frac{1}{2} |
8.2. If a class of 30 people is seated in a cinema hall, then in any case, at least two classmates will end up in the same row. If the same is done with a class of 26 people, then at least three rows will be empty. How many rows are there in the hall? | 29 |
The equation about $x$, $\left(a-1\right)x^{2}+4x-3=0$, is a quadratic equation in one variable. Then, which of the following is true?
A: $a \gt 1$
B: $a=1$
C: $a\neq 1$
D: $a\geqslant 0$ | C |
A KML airline operates flights between several cities in such a way that from one city, you cannot directly reach more than three other cities. However, with at most one layover, you can get from anywhere to anywhere. What is the maximum number of cities between which the planes operate? | 10 |
1・32 Given $2 x-3 y-z=0, \quad x+3 y-14 z=0, z \neq 0$, then the value of $\frac{x^{2}+3 x y}{y^{2}+z^{2}}$ is
(A) 7.
(B) 2.
(C) 0.
(D) $-\frac{20}{17}$.
(E) -2.
(15th American High School Mathematics Examination, 1964) | 7 |
Example 1. Factorize:
$$
\left(x^{2}-x+15\right)\left(x^{2}-x-5\right)+51
$$ | (x-2)(x+1)\left(x^{2}-x+12\right) |
How many ways are there to rearrange the letters of the word RAVEN such that no two vowels are consecutive?
[i]2015 CCA Math Bonanza Team Round #2[/i] | 72 |
Calculate the following expressions (use simplification methods where possible):
0.11×1.8+8.2×0.11
0.8×(3.2-2.99÷2.3)
3.5-3.5×0.98
12.5×2.5×3.2
(8.1-5.4)÷3.6+85.7． | 86.45 |
In $\triangle ABC$, if $\sin (2A+B)=2\sin B$, find the maximum value of $\tan B$. | \dfrac{\sqrt{3}}{3} |
## Task A-2.1.
Let $x_{1}$ and $x_{2}$ be distinct solutions of the equation $x^{2}+5 x+3=0$. Calculate $\frac{x_{1}^{3} x_{2}+x_{1} x_{2}^{3}}{x_{1}+x_{2}}$. | -\frac{57}{5} |
What could be the largest prime divisor of a number in the decimal system of the form $\overline{x y x y x y}$? | 97 |
In the process of proving the inequality $\frac{1}{n+1}+\frac{1}{n+2}+…+\frac{1}{n+n}>\frac{13}{14}$ using mathematical induction, when going from $n=k$ to $n=k+1$, the left side of the inequality ( )
A: increases by $\frac{1}{2(k+1)}$
B: increases by $\frac{1}{2k+1}+\frac{1}{2k+2}$
C: increases by $\frac{1}{2(k+1... | \text{D} |
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 |
Given the equation $x^{2}+(4+i)x+4+ai=0$ ($a\in\mathbb{R}$) has a real root $b$, and $z=a+bi$, then the complex number $z=$\_\_\_\_\_\_. | 2-2i |
Find the term containing $x^2y^7$ in the expansion of $(x-y)^9$. | -36 x^2 y^7 |
Given the function $f(x)=\sin(2x+\frac{4π}{3})$, then ()
A: The smallest positive period of the function $f(x)$ is $\pi$
B: The line $x=\frac{7π}{12}$ is a symmetry axis of the graph of the function $f(x)$
C: The function $f(x+\frac{π}{3})$ is an even function
D: The decreasing interval of the function $f(x)$ is $[... | ABD |
The [sequence](https://artofproblemsolving.com/wiki/index.php/Sequence) $a_1, a_2, \ldots$ is [geometric](https://artofproblemsolving.com/wiki/index.php/Geometric_sequence) with $a_1=a$ and common [ratio](https://artofproblemsolving.com/wiki/index.php/Ratio) $r,$ where $a$ and $r$ are positive integers. Given that $\lo... | 46 |
## Task 1
In every 100 m preliminary race, 8 athletes start. The two best runners will participate in the final. How many athletes participated in the preliminaries if 8 of them competed in the final. | 32 |
Solve the following equation:
$$
\sqrt[3]{x}+\sqrt[3]{9-x}=3
$$ | x_1=1,x_2=8 |
Given $f(x)=a x^{2}+b x+c(a, b, c$ are real numbers) and its absolute value on $[-1,1]$ is $\leqslant 1$, find the maximum value of $|a|+|b|+|c|$. | 3 |
(4) Arrange all the simplest proper fractions with a denominator of 24 in ascending order, denoted as $\left.a_{1}\right) a_{2}, \cdots, a_{n}$, then the value of $\sum_{i=1}^{n} \cos \left(a_{i} \pi\right)$ is ( ).
(A) 1
(B) $\frac{1}{2}$
(C) 0
(D) $-\frac{1}{2}$ | 0 |
On a $16 \times 16$ torus as shown all 512 edges are colored red or blue. A coloring is good if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the 4 edges of an arbitrary cell. What is the largest number of good colorings such that none of them can be conve... | 4 |
If a positive number $m$ has two distinct square roots of $1-2a$ and $a-5$, then $m=\_\_\_\_\_\_.$ | 81 |
61. To pave a 112-meter road, Team A started and paved for several days, paving 12 meters per day. Team A had to leave due to some matters, and Team B took over, paving 20 meters per day. When the 112-meter road was completed, it was found that the average daily paving by Teams A and B was 14 meters. How many days did ... | 6 |
Given a point $P(x,y)$ on the ellipse $\frac{x^{2}}{4}+y^{2}=1$, find the maximum and minimum values of $u=2x+y$. | -\sqrt{17} |
## Task Condition
Derive the equation of the tangent line to the given curve at the point with abscissa \( x_{0} \).
$$
y=\frac{-2\left(x^{8}+2\right)}{3\left(x^{4}+1\right)}, x_{0}=1
$$ | -\frac{2}{3}\cdotx-\frac{1}{3} |
$14 \cdot 74$ For a real number $x$, $[x]$ denotes the greatest integer not exceeding $x$. Find the positive integer $n$ such that
$$
\left[\log _{2} 1\right]+\left[\log _{2} 2\right]+\left[\log _{2} 3\right]+\cdots+\left[\log _{2} n\right]=1994
$$
holds.
(12th American Invitational Mathematics Examination, 1994) | 312 |
The number sequence $1,4,5,7,12,15,16,18,23, \ldots$ was written based on a certain rule. Continue the number sequence until it exceeds the number 50.
Continue the number sequence until it exceeds the number 50. | 51 |
Find the greatest common divisor (GCD) of 204 and 85 using both the Euclidean algorithm and the method of successive subtraction. | 17 |
12.54. Find the angle $B$ of triangle $A B C$, if the length of the height $C H$ is half the length of side $A B$, and $\angle B A C=75^{\circ}$. | 30 |
If the one-variable quadratic equation $x^{2}-x+m=0$ has two equal real roots with respect to $x$, then the value of $m$ is ____. | \frac{1}{4} |
Simplify the following expressions of vectors:$(1)\overrightarrow{AB}+\overrightarrow{BC}-\overrightarrow{AD}$;$(2)(\overrightarrow{AB}-\overrightarrow{CD})-(\overrightarrow{AC}-\overrightarrow{BD})$;$(3)(\overrightarrow{AC}+\overrightarrow{BO}+\overrightarrow{OA})-(\overrightarrow{DC}-\overrightarrow{DO}-\overrightarr... | \overrightarrow{0} |
Example 1 Find the positive integer solutions of the equation $2 x^{2}+5 y^{2}=11(x y-11)$. | (14,27) |
Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$(P(x)+1)^2=P(x^2+1).$$ | P(x) = x - 1 |
Let $m$ be a natural number. Calculate $\sum_{k=1}^{m} \frac{1}{\sin ^{2} \frac{k \pi}{2 m+1}}$ and $\sum_{k=1}^{m} \cot ^{2} \frac{k \pi}{2 m+1}$.
Deduce $\sum_{k=1}^{\infty} \frac{1}{k^{2}}=\frac{\pi^{2}}{6}$. (Use $\cot \theta \leqslant \frac{1}{\theta} \leqslant \frac{1}{\sin \theta}$ when $0<\theta<\frac{\pi}{2}$... | \frac{\pi^2}{6} |
Given the function $f(x)= \frac {a+x^{2}}{x}$, $(a > 0)$, $x\in(0,b)$, then the correct judgment among the following is $(\quad)$
A: When $b > \sqrt {a}$, the minimum value of $f(x)$ is $2 \sqrt {a}$
B: When $0 0$, the minimum value of $f(x)$ is always $2 \sqrt {a}$ | A |
Given a parabola $Ω$: $y^{2}=2px$ ($0 < p \leq 8$) with focus $F$, tangent lines from $F$ to the circle $C$: $(x-3)^{2}+y^{2}=1$ intersect at point $T$ (the tangent point), and the length of the tangent $FT$ is $\sqrt{3}$.
(I) Find the equation of the parabola $C$.
(II) Construct a tangent line $l$ to the circle $C$:... | 9 |
9.199. $\frac{\log _{2}(\sqrt{4 x+5}-1)}{\log _{2}(\sqrt{4 x+5}+11)}>\frac{1}{2}$. | x\in(5;\infty) |
1. How many four-digit numbers exist that are divisible by 17 and end in 17? | 5 |
Tomonvo A.K.
Petya's bank account contains 500 dollars. The bank allows only two types of transactions: withdrawing 300
dollars or adding 198 dollars.
What is the maximum amount Petya can withdraw from his account if he has no other money? | 498 |
7. (1976 Polish Mathematical Olympiad) A fishing boat is fishing in the territorial waters of a foreign country without permission, and each time it casts a net, it causes a loss of equal value to the country's fishing yield. The probability that the boat is detained by the foreign coast guard during each net casting i... | k-1 |
From the numbers $1$, $2$, $3$, $4$, $5$, two different numbers are randomly selected. The probability that the sum of these two numbers is an odd number is ( )
A: $0.6$
B: $0.5$
C: $0.4$
D: $0.3$ | A |
Given that the function $f(x)$ is defined on $\mathbb{R}$ and is an even function with a period of 2. When $0 \leq x \leq 1$, $f(x) = x^2$. If the line $y = x + a$ has exactly two distinct intersection points with the curve $y = f(x)$, then the value of the real number $a$ is
A. $2k$ ($k \in \mathbb{Z}$)
B. $2k$ or ... | \text{D} |
If \\((3-2x)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{5}x^{5}\\), then \\(a_{0}+a_{1}+2a_{2}+3a_{3}+4a_{4}+5a_{5}=\\) \_\_\_\_\_\_. | 233 |
15. How many 3-digit numbers are there with the property that the 2-digit number obtained by deleting the middle digit is equal to one ninth of the original 3-digit number?
A 1
B 2
C 3
D 4
E 5 | 4 |
1. (10 points) Calculate: $84 \frac{4}{19} \times 1.375 + 105 \frac{5}{19} \times 0.8=$ | 200 |
70. One year, the ages of Xiao Ming's family of four were: Father 30 years old. Mother is two years younger than Father, Brother's age is three times that of Xiao Ming, this year the total age of the four family members is 114, then Xiao Ming's age this year is $\qquad$ | 14 |
4. Let $O$ be the incenter of $\triangle A B C$, $A B=5, A C=6, B C=7, O P=x O A+y O B+z O C$, $0 \leq x, y, z \leq 1$, then the area of the plane region covered by the trajectory of the moving point $P$ is equal to | 12\sqrt{6} |
If $|x-1|=2x$, then $x=$ . | \frac{1}{3} |
90. Find the number of all trees planted at five-foot intervals from each other on a rectangular plot of land, the sides of which are 120 ft. and 70 ft. | 375 |
Given $f\left(x\right)=x^{2}+2xf'\left(1\right)$, then $f\left(1\right)=\left(\ \ \right)$
A: $0$
B: $-4$
C: $-2$
D: $-3$ | \text{D} |
If vector $\overrightarrow{a}=(1,1)$, $\overrightarrow{b}=(1,-1)$, $\overrightarrow{c}=(-1,-2)$, then $\overrightarrow{c}$ equals to ( ).
A: $- \dfrac {1}{2} \overrightarrow{a}- \dfrac {3}{2} \overrightarrow{b}$
B: $- \dfrac {1}{2} \overrightarrow{a}+ \dfrac {3}{2} \overrightarrow{b}$
C: $\dfrac {3}{2} \overrightarr... | D |
If $a$ and $b$ are rational numbers, then among the following statements:
(1) If $|a| = b$, then $a = b$ must hold; (2) If $|a| > |b|$, then $a > b$ must hold; (3) If $|a| > b$, then $|a| > |b|$ must hold; (4) If $|a| = b$, then $a^2 = (-b)^2$ must hold. The correct one is (fill in the serial number). | 4 |
6. (3 points) A quadrilateral is divided into 1000 triangles. What is the maximum number of different points at which the vertices of these triangles can be located? | 1002 |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $\begin{cases}x= \sqrt{3}\cos \theta \\ y=\sin \theta\end{cases}$ (where $\theta$ is the parameter), and in the polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation ... | \left( \frac {3}{2}, \frac {1}{2}\right) |
## 2. Clock
At the moment, according to my watch, it is more than 3:20 but less than 3:25. I observe the exact position of the hands on my watch, and then I move the hands so that the minute hand is in the position previously occupied by the hour hand, and the hour hand is in the position previously occupied by the mi... | 3 |
Given the sets $A = \{2, 5, 6\}$ and $B = \{3, 5\}$, then the union of sets $A \cup B = \_\_\_\_$. | \{2, 3, 5, 6\} |
Given an arithmetic sequence {an} with the sum of its first n terms denoted as Sn, it satisfies the condition a4•x2\-S3•x+2＜0, and the solution set for this inequality is $$( \frac {2}{7}，1)$$.
(1) Find the general term formula for the sequence {an}.
(2) If sequence {cn} satisfies $$c_{n}=a_{2n}+2^{a_{n}}$$, find the s... | T_n = 2n^2 + n + \frac{2}{3}(4^n - 1) |
Problem 11.5. Quadrilateral $ABCD$ is inscribed in a circle. It is known that $BC=CD, \angle BCA=$ $64^{\circ}, \angle ACD=70^{\circ}$. A point $O$ is marked on segment $AC$ such that $\angle ADO=32^{\circ}$. How many degrees does the angle $BOC$ measure?
^{3}-k^{3}$ then find the formula for $\sum_{k=1}^{n} k^{2}$.
## Other Counting Tools
Example 3. There are $d^{n}$ different rolls of $\mathrm{n}$ colored dice with $\mathrm{d}$ faces. | \frac{n(n+1)(2n+1)}{6} |
If x=1 is the extreme point of the function f(x) = (e^x + a)lnx, then the real number a is equal to \_\_\_\_\_\_. | -e |
13. When Clara calculated the total score of her test results in each subject, she unconsciously swapped the tens and units digits of a certain subject's score. Then the score that is most likely to be the difference between the incorrect total score and the correct total score is ( ).
(A) 45
(B) 46
(C) 47
(D) 48
(E) 4... | A |
In a speech competition at school, there were 1, 2, and 3 students from grades 10, 11, and 12 respectively who won awards. If these six students are to be arranged in a row for a group photo, with the requirement that students of the same grade must stand next to each other, then the total number of different arrangeme... | C |
11. January 1, 2015 was a Thursday, based on this information, we can calculate that February 1, 2015 was a Saturday $\qquad$ | Sunday |
1. Given the sets
$$
A=\{x \mid 1 \leqslant x \leqslant 2\}, B=\left\{x \mid x^{2}-a x+4 \geqslant 0\right\} \text {. }
$$
If $A \subseteq B$, then the range of real number $a$ is
$\qquad$ | a \leqslant 4 |
Let $ a$ and $ b$ be integers such that the equation $ x^3\minus{}ax^2\minus{}b\equal{}0$ has three integer roots. Prove that $ b\equal{}dk^2$, where $ d$ and $ k$ are integers and $ d$ divides $ a$. | b = dk^2 |
Define the quotient set operation of sets as $\dfrac{A}{B}=\{x|x= \dfrac{m}{n},m\in A,n\in B\}$. Given the set $A=\{2,4,6\}$, $B=\{x|x= \dfrac{k}{2}-1,k\in A\}$, then the number of elements in the set $\dfrac{B}{A}\cup B$ is ( ).
A: $6$
B: $7$
C: $8$
D: $9$ | \text{B} |
4. In trapezoid $A B C D$, $A B / / D C, A B=40, C D=60, A C$ intersects $B D$ at point $P$, a line through $P$ parallel to $A B$ intersects $A D$ at $M$ and $B C$ at $N$, then $M N=$ $\qquad$ | 48 |
5. Given that $a, b, c$ are all positive numbers, then $\min \left\{\frac{1}{a}, \frac{2}{b}, \frac{4}{c}, \sqrt[3]{a b c}\right\}$ has the maximum value of | \sqrt{2} |
For how many integers $x$ will the following polynomial be a perfect square?
$$
4 x^{4}-12 x^{3}+17 x^{2}-6 x-14
$$ | 2 |
14. $f(x)$ is a function defined on $(0,+\infty)$ that is increasing, and $f\left(\frac{x}{y}\right)=f(x)-f(y)$.
(1) Find the value of $f(1)$; (2) If $f(6)=1$, solve the inequality $f(x+3)-f\left(\frac{1}{x}\right)<2$. | 0<x<\frac{-3+3\sqrt{17}}{2} |
Find the smallest natural number nonzero n so that it exists in real numbers $x_1, x_2,..., x_n$ which simultaneously check the conditions:
1) $x_i \in [1/2 , 2]$ , $i = 1, 2,... , n$
2) $x_1+x_2+...+x_n \ge \frac{7n}{6}$
3) $\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}\ge \frac{4n}{3}$ | 9 |
(2) Given the function $f(x)=\sqrt{3} \sin 2 x+2 \cos ^{2} x+a$, if the minimum value of $f(x)$ on the interval $\left[0, \frac{\pi}{2}\right]$ is -1, then the value of $a$ is $\qquad$. | -1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.