problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that $y=f(x)$ is an odd function, when $x \in (0,2)$, $f(x)=\ln x - ax (a > \frac{1}{2})$, and when $x \in (-2,0)$, the minimum value of $f(x)$ is $1$. Determine the value of $a=$ ( ).
A: $\frac{1}{4}$
B: $\frac{1}{3}$
C: $\frac{1}{2}$
D: $1$ | D: 1 |
4. How many different totals can be obtained by rolling three standard dice and adding up the scores?
A 14
B 15
C 16
D 17
E 18 | 16 |
4. In the sequence $\left\{a_{n}\right\}$, $a_{0}=2, a_{n}=a_{n-1}-n+3, n \geqslant 1$, then $a_{n}=$ | -\frac{n(n+1)}{2}+3n+2 |
Compose the equation of the line passing through the point $A(0 ; 7)$ and tangent to the circle $(x-15)^{2}+(y-2)^{2}=25$.
# | 7or-\frac{3}{4}x+7 |
1. Given the function $f: \mathbf{R} \rightarrow \mathbf{R}$, for any $x, y \in$ $\mathbf{R}$, we have
$$
f(2 x+f(y))=x+y+f(x) .
$$
Find $f(x)$. | f(x)=x(x\in{R}) |
Let $0^{\circ}\leq\alpha,\beta,\gamma\leq90^{\circ}$ be angles such that \[\sin\alpha-\cos\beta=\tan\gamma\] \[\sin\beta-\cos\alpha=\cot\gamma\]
Compute the sum of all possible values of $\gamma$ in degrees.
[i]Proposed by Michael Ren[/i] | 45^\circ |
Given that the constant term of the expansion of the binomial (a$\sqrt {x}$ - $\frac {1}{ \sqrt {x}}$)$^6$ is -160, find the value of a. | 2 |
In the Cartesian coordinate system $xOy$, it is known that $P$ is a moving point on the graph of the function $f(x)=\ln x$ ($x > 0$). The tangent line $l$ at point $P$ intersects the $x$-axis at point $E$. A perpendicular line to $l$ through point $P$ intersects the $x$-axis at point $F$. If the midpoint of the line se... | \dfrac {1}{2}(e+ \dfrac {1}{e}) |
G8.4 If $m n p, n m p, m m p$ and $n n p$ are numbers in base 10 composed of the digits $m, n$ and $p$, such that: $m n p-n m p=180$ and $m m p-n n p=d$. Find $d$. | 220 |
A group of boys and girls waits at a bus stop. In the first bus that passes, only 15 girls board, and 2 boys remain for each girl at the bus stop. In the second bus that passes, only 45 boys board, and 5 girls remain for each boy at the bus stop. Determine the number of boys and girls that were at the stop before the f... | M=40,H=50 |
14. Let the function $f(x)=a x^{2}+8 x+3(a<0)$. For a given negative number $a$, there is a largest positive number $l(a)$, such that the inequality $|f(x)| \leqslant 5$ holds for the entire interval $[0, l(a)]$. Question: For what value of $a$ is $l(a)$ the largest? Find this largest $l(a)$. Prove your conclusion. | \frac{\sqrt{5}+1}{2} |
Given $a > 1$. If the area of the closed figure formed by the curve $y= \frac {1}{x}$ and the lines $y=0$, $x=1$, and $x=a$ is 2, then $a= \_\_\_\_\_\_$. | e^{2} |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the parametric equations.
$$
\begin{aligned}
& \left\{\begin{array}{l}
x=4 \cos ^{3} t \\
y=4 \sin ^{3} t
\end{array}\right. \\
& \frac{\pi}{6} \leq t \leq \frac{\pi}{4}
\end{aligned}
$$ | \frac{3}{2} |
12.49 Find all integer solutions $(x, y, z)$ of the equation $2^{x}+3^{y}=z^{2}$.
(German Federal Mathematics Competition, 1987) | (0,1,2),(3,0,3),(4,2,5),(0,1,-2),(3,0,-3),(4,2,-5) |
7. As shown in the figure, a cross-section of the cube $A B C D-E F G H$ passes through vertices $A, C$ and a point $K$ on edge $E F$, dividing the cube into two parts with a volume ratio of $3: 1$. Then the value of $\frac{E K}{K F}$ is $\qquad$ . | \sqrt{3} |
Given $m$ and $n$ are two different lines, $\alpha$ and $\beta$ are two different planes, which of the following statements is correct?
A: If $m \perp \alpha$, $\alpha \perp \beta$, then $m$ is parallel to $\beta$.
B: If $m$ is parallel to $\alpha$, $\alpha$ is parallel to $\beta$, then $m$ is parallel to $\beta$.
C... | \text{D} |
Which of the following statements is false?
A: The sum of any two sides of a triangle is greater than the third side.
B: A parallelogram is both an axisymmetric figure and a centrally symmetric figure.
C: The diagonals of a rectangle are equal.
D: The median of a triangle is parallel to the third side of the triang... | \text{B} |
A line $l$ with a slope of $1$ passes through the focus $F$ of the parabola $y^{2}=2px\left(p \gt 0\right)$. If $l$ is tangent to the circle $\left(x-5\right)^{2}+y^{2}=8$, then $p$ is ____. | 2 \text{ or } 18 |
Let $i$ be the imaginary unit, and consider the sets $A=\{1, i\}$ and $B=\left\{-\frac{1}{i}, \frac{(1-i)^2}{2}\right\}$. What is $A \cup B$? | \{1, i, -i\} |
$\left[\begin{array}{l}\text { Factorization } \\ {[\underline{\text { Modular Arithmetic (other) })]}]}\end{array}\right.$
What remainders can be obtained when dividing $n^{3}+3$ by $n+1$ for a natural number $n>2$? | 2 |
Solve the following inequality:
$$
x^{3}+1>x^{2}+x
$$ | x\in(-1,1)\cup(1,\infty) |
14. $[\mathbf{8}]$ Compute
$$
\sum_{k=1}^{2009} k\left(\left\lfloor\frac{2009}{k}\right\rfloor-\left\lfloor\frac{2008}{k}\right\rfloor\right) .
$$ | 2394 |
9. Given positive numbers $a, b, c, d$ satisfying $a+b+c+d=1$, find the minimum value of $M=\sqrt{a^{2}+\frac{1}{8 a}}+\sqrt{b^{2}+\frac{1}{8 b}}+\sqrt{c^{2}+\frac{1}{8 c}}+\sqrt{d^{2}+\frac{1}{8 d}}$. | 3 |
Find all positive prime numbers $p$ and $q$ such that $p^{2}-q^{3}=1$.
Remark. $p$ is prime if it has only two divisors: 1 and itself. The numbers $2,3,5,7,11,13$ are prime, but $1,4,6,8,9$ are not. | p=3,q=2 |
(5 points) (2011•Guangdong) In a regular pentagonal prism, a line connecting two vertices that are not on the same side face and not on the same base is called its diagonal. Then, the total number of diagonals in a regular pentagonal prism is ( )
A: 20
B: 15
C: 12
D: 10 | D |
A store owner sells a product, which he must sell at a price no less than 20% above the cost price. However, to gain more profit, he marks up the price by 80% above the cost price. If you want to buy this product marked at 360 yuan, the maximum discount the owner can offer is ___ yuan. | 120 |
5.17. In triangle $A B C$, point $N$ lies on side $A B$ and $A N=$ $=3 N B$; median $A M$ intersects $C N$ at point $O$. Find $A B$, if $A M=C N=7$ cm and $\angle N O M=60^{\circ}$. | 4\sqrt{7} |
1. Let $A B C D$ be a unit square. Draw a quadrant of a circle with $A$ as centre and $B, D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A, C$ as end points of the arc. Inscribe a circle $\Gamma$ touching the arc $A C$ internally, the arc $B D$ internally and also touching ... | \frac{3}{8} |
The term with the largest coefficient in the expansion of $(1-x)^7$ is the ( ) term.
A: 4
B: 5
C: 7
D: 8 | B |
If $3 < a < 5$, the complex number $z = (a^2 - 8a + 15) + (a^2 - 5a - 14)i$ corresponds to a point in the ______ quadrant on the complex plane. | \text{third} |
Given the function $f(x)=\sin x+5x$, where $x\in(-1,1)$, if $f(1-\alpha)+f(1-\alpha^{2}) < 0$, then the range of $\alpha$ is __________. | (1, \sqrt{2}) |
18. A five-digit number $\overline{A B C D E}$ is a multiple of 2014, and $\overline{C D E}$ has exactly 16 factors. What is the minimum value of $\overline{A B C D E}$? | 24168 |
6. Determine all ordered triples $(x, y, z)$ of distinct real numbers that satisfy the set equation
$$
\{x, y, z\}=\left\{\frac{x-y}{y-z}, \frac{y-z}{z-x}, \frac{z-x}{x-y}\right\} \text {. }
$$
(J. Šimša) | (,-\frac{1}{1+},-\frac{1+}{}),(1,-2,-\frac{1}{2}),(-\frac{1}{2},1,-2),(-2,-\frac{1}{2},1) |
Example 11. Find the number of consecutive zeros at the end of 1987!.
untranslated text:
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。
translated text:
Example 11. Find the number of consecutive zeros at the end of 1987!.
Note: The note at the end is not part of the original text and is provided for context. | 494 |
If point $P$ is on the ellipse $\frac{x^2}{2} + y^2 = 1$, and $F_1$, $F_2$ are the two foci of the ellipse, and the angle $\angle F_1PF_2 = 90^\circ$, then the area of $\triangle F_1PF_2$ is ( ).
A: $\frac{1}{2}$
B: $\frac{\sqrt{3}}{2}$
C: $2$
D: $1$ | D: 1 |
8. Six people, all of different weights, are trying to build a human pyramid: that is, they get into the formation
$$
\begin{array}{c}
\text { A } \\
\text { B C } \\
\text { D E F }
\end{array}
$$
We say that someone not in the bottom row is "supported by" each of the two closest people beneath her or him. How many d... | 16 |
3.360. $\frac{\cos 67^{\circ} \cos 7^{\circ}-\cos 83^{\circ} \cos 23^{\circ}}{\cos 128^{\circ} \cos 68^{\circ}-\cos 38^{\circ} \cos 22^{\circ}}-\operatorname{tg} 164^{\circ}$.
| 0 |
4. In a regular tetrahedron $A B C D$, $M, N$ are the midpoints of $A B, C D$ respectively, then the angle formed by $M N$ and $A D$ is
A. $30^{\circ}$
B. $45^{\circ}$
C. $60^{\circ}$
D. $90^{\circ}$ | 45 |
Calculate the imaginary part of the complex number $z= \frac {4+3i}{1+2i}$. | -1 |
Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. The incircle of $ABC$ meets $BC$ at $D$. Line $AD$ meets the circle through $B$, $D$, and the reflection of $C$ over $AD$ at a point $P\neq D$. Compute $AP$.
[i]2020 CCA Math Bonanza Tiebreaker Round #4[/i] | 2\sqrt{145} |
4. The continuation of the height $B H$ of triangle $A B C$ intersects the circumscribed circle around it at point $D$ (points $B$ and $D$ lie on opposite sides of line $A C$). The degree measures of arcs $A D$ and $C D$, not containing point $B$, are $60^{\circ}$ and $90^{\circ}$, respectively. Determine in what ratio... | \sqrt{3}:1 |
Evaluate the sum \[ 11^2 - 1^1 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2. \] | 2100 |
In 2003, the average monthly rainfall in Mathborough was $41.5 \mathrm{~mm}$. In 2004, the average monthly rainfall in Mathborough was $2 \mathrm{~mm}$ more than in 2003. The total amount of rain that fell in Mathborough in 2004 was
(A) $500 \mathrm{~mm}$
(B) $522 \mathrm{~mm}$
(C) $496 \mathrm{~mm}$
(D) $498 \mathrm{~... | 522\mathrm{~} |
In how many different ways can 22 be written as the sum of 3 different prime numbers? That is, determine the number of triples $(a, b, c)$ of prime numbers with $1<a<b<c$ and $a+b+c=22$. | 2 |
9.2. In a kindergarten, each child was given three cards, each of which had either "MA" or "NYA" written on it. It turned out that 20 children could form the word "MAMA" from their cards, 30 children could form the word "NYANYA," and 40 children could form the word "MANYA." How many children had all three cards the sam... | 10 |
If $a$, $b$, $c$ are all non-zero real numbers, and $a+b+c=abc=a^{3}$, then the minimum value of $ab+bc+ca$ is ( )
A: $6$
B: $8$
C: $9$
D: $13$ | C |
Given the function $f(x)=\log({\sqrt{{x^2}+1}+x})$, where $a$ and $b$ are positive real numbers satisfying $f\left(2a-2\right)+f\left(b\right)=0$, then the minimum value of $\frac{{2a+b}}{{ab}}$ is ( )
A: $2$
B: $4$
C: $6$
D: $8$ | \text{B} |
Given $\overrightarrow{a}+ \overrightarrow{b}+ \overrightarrow{c}=0$, $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=3$, $|\overrightarrow{c}|= \sqrt{19}$, then the angle between vector $\overrightarrow{a}$ and $\overrightarrow{b}$ is ( ).
A: $60^{\circ}$
B: $45^{\circ}$
C: $30^{\circ}$
D: None of the above | A |
4・169 Solve the equation in the set of natural numbers
$$
x^{y}=y^{x}(x \neq y) .
$$ | 2,4or4,2 |
The graph of the power function $f(x) = kx^{\alpha}$ (where $k, \alpha \in \mathbb{R}$) passes through the point $\left( \frac{1}{3}, 9 \right)$. Find the value of $k+\alpha$. | k + \alpha = -1 |
Coordinate Systems and Parametric Equations
In the polar coordinate system, a circle has its center at $C(3, \frac{\pi}{6})$ and a radius of $r=1$.
(1) Find the rectangular coordinate equation of the circle.
(2) If the line $\begin{cases} x=-1+\frac{\sqrt{3}}{2}t \\ y=\frac{1}{2}t \end{cases} (t \text{ is a parameter})... | \sqrt{3} |
On the meadow, there were 45 sheep and several shepherds. After half of the shepherds and a third of the sheep left, the remaining shepherds and sheep together had 126 legs. All the sheep and shepherds had the usual number of legs.
How many shepherds were originally on the meadow?
(L. Hozová)
Hint. How many sheep we... | 6 |
$6 \cdot 28 x$ is any real number, which of the following equations can be true?
(A) $\sqrt{x^{2}-1}=\sqrt{x+1} \cdot \sqrt{x-1}$.
(B) $10^{\lg x}=x$.
(C) $x^{0}=1$.
(D) $\sqrt[2 n+1]{x}=-\sqrt[2 n+1]{-x}$ ( $n$ is a natural number $)$.
(China Guangzhou, Wuhan, Fuzhou and other five cities junior high school mathematic... | D |
If the graph of the linear function $y=kx+b$ does not pass through the third quadrant, then which of the following is true?
A: $k \gt 0$, $b \gt 0$
B: $k \gt 0$, $b \lt 0$
C: $k \lt 0$, $b\geqslant 0$
D: $k \lt 0$, $b\leqslant 0$ | C |
Given the circle with center at $C(4,-6)$ and a radius of $4$, find its equation. | (x-4)^2+(y+6)^2=16 |
Given a function $f(x) = 2x^2 - 2ax + b$, with $f(-1) = -8$. For all $x \in \mathbb{R}$, $f(x) \geq f(-1)$ holds; let $A = \{ x \mid f(x) > 0 \}$, $B = \{ x \mid |x - t| \leq 1 \}$.
(I) When $t = 1$, find $(C_R A) \cup B$.
(II) Let proposition $P$ be: $A \cap B \neq \emptyset$. If $\neg P$ is true, determine the range ... | [-2, 0] |
A moving circle has its center on the parabola $y^2=8x$, and it is always tangent to the line $x+2=0$. Then, the moving circle must pass through the point __. | (2,0) |
The common difference of the arithmetic sequence $\{a_n\}$ is 2, and $a_1$, $a_1+a_2$, $2(a_1+a_4)$ form a geometric sequence. Find the sum $S_n$ of the first $n$ terms of the sequence $\{\frac{a_n}{2^{n-1}}\}$. | S_n = 6 - \frac{4n+6}{2^n} |
8. Find the smallest positive integer $k$ such that the equation $2 x y-3 x-5 y=k$ has an odd number of positive integral solutions.
(1 mark)
求最小的正整數 $k$, 使方程 $2 x y-3 x-5 y=k$ 的正整數解數目為奇數。 | 5 |
The ellipse with its center at the origin and one focus at $F_1(0,c)$ is intersected by the line $y=3x-2$. The x-coordinate of the midpoint of the chord cut by this line is $\frac{1}{2}$. The eccentricity of this ellipse is ( ).
A: $\frac{1}{2}$
B: $\frac{\sqrt{6}}{3}$
C: $\frac{\sqrt{5}}{3}$
D: $\frac{2}{3}$ | \text{B} |
5. Solve the equation:
$$
\log \left(\frac{1}{x}\right) \cdot \log \left(\frac{4}{x}\right)=\frac{3}{4} \cdot(\log 4)^{2}
$$
Solve each problem independently on the attached paper, and each problem on its own page.
Do not sign the sheets, write only your code.
Write your answers with a pen clearly and neatly. Each ... | x_1=8,x_2=\frac{1}{2} |
Typing a document on a computer, if A types for 30 minutes and then B continues for another 25 minutes, the document can be completed; if B starts typing and types for 30 minutes, followed by A continuing for 24 minutes, the document can also be completed. The time it takes for A and B to complete the document individu... | \text{B} |
Given that $\sin \alpha= \frac {4}{5}$, and $0 < \alpha < \frac {\pi}{2}$, find the values of $\cos \alpha$ and $\sin (\alpha+ \frac {\pi}{4})$. | \frac{7\sqrt {2}}{10} |
4. Given the sequence $\left\{x_{n}\right\}$ :
$$
1,3,3,3,5,5,5,5,5, \cdots
$$
formed by all positive odd numbers arranged from smallest to largest, and each odd number $k(k=1,3,5, \cdots)$ appears consecutively $k$ times. If the general term formula of this sequence is $x_{n}=a[\sqrt{b n+c}]+d$, then $a+b+c+d=$ $\qqu... | 3 |
5. Real numbers $x, y, a$ satisfy $x+y=a+1$ and $xy=a^2-7a+16$. Then the maximum value of $x^2+y^2$ is $\qquad$ . | 32 |
Given that the normal vector of plane \\(\alpha\\) is \\((2,3,-1)\\) and the normal vector of plane \\(\beta\\) is \\((4,\lambda,-2)\\), if \\(\alpha \perp \beta\\), then the value of \\(\lambda\\) is _____.
A: \\(-6\\)
B: \\(6\\)
C: \\(- \frac{10}{3}\\)
D: \\( \frac{10}{3}\\) | C |
Among the numbers 2004, 2005, 2006, 2007, the one that cannot be expressed as the difference of squares of two integers is ( )
A: 2005
B: 2006
C: 2007
D: 2004 | \text{B} |
## Task 4 - 060734
The number $\frac{16}{15}$ is to be represented in the form $\frac{16}{15}=\frac{a}{m}+\frac{b}{n}$. Here, $a, b, m, n$ should be natural numbers such that the fractions $\frac{a}{m}$ and $\frac{b}{n}$ are not reducible and are not whole numbers.
Give three examples of such a representation, where
... | \frac{16}{15}=\frac{2}{15}+\frac{14}{15},\frac{16}{15}=\frac{2}{3}+\frac{2}{5},\frac{16}{15}=\frac{8}{15}+\frac{8}{15} |
1. $\left(12\right.$ points) Solve the equation $\left(x^{3}-3\right)\left(2^{\operatorname{ctg} x}-1\right)+\left(5^{x^{3}}-125\right) \operatorname{ctg} x=0$. | \sqrt[3]{3};\frac{\pi}{2}+\pin,\quadn\inZ |
6.001. $\frac{x^{2}+1}{x-4}-\frac{x^{2}-1}{x+3}=23$. | x_{1}=-\frac{55}{16},x_{2}=5 |
3. If the inequality $0 \leqslant x^{2}+p x+5 \leqslant 1$ has exactly one real solution, then the range of values for $p$ is $\qquad$ . | \4 |
If line $l$ passes through point $P(-4,6)$, and intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively, then when $P$ is the midpoint of $AB$, the equation of line $l$ is _______. | 3x - 2y + 24 = 0 |
Let $f(x)=\left\{\begin{array}{l}\sqrt{x-1}(x\geq1)\\ 1,(x<1)\end{array}\right.$, then $f\left(f\left(f\left(2\right)\right)\right)=\_\_\_\_\_\_.$ | 1 |
12・20 Assume $z=a+b i$ is a solution to the polynomial equation
$$
c_{4} z^{4}+i c_{3} z^{3}+c_{2} z^{2}+i c_{1} z+c_{0}=0
$$
where $c_{0}, c_{1}, c_{2}, c_{3}, a$ and $b$ are real constants, and $i^{2}=-1$. Then which of the following numbers is also a solution to the equation?
(A) $-a-b i$.
(B) $a-b i$.
(C) $-a+b i$... | C |
Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:
$(a)$ $P(x)>x$ for all positive integers $x$.
$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some ... | P(x) = x + 1 |
【Question 12】
There are 45 workers. If each person can produce 30 part A per hour, or 25 part B per hour, or 20 part C per hour. Now, 3 part A, 5 part B, and 4 part C are used to assemble a certain machine. Therefore, the number of people assigned to produce part A, part B, and part C should be $\qquad$ people, $\qquad... | 9,18,18 |
11.2. Graph on the coordinate plane the set of solutions to the system of equations $\left\{\begin{array}{l}x-2 y=1 \\ x^{3}-6 x y-8 y^{3}=1\end{array}\right.$. | \frac{x-1}{2} |
If digits $A$, $B$, and $C$ (between $0$ and $9$ inclusive) satisfy
\begin{tabular}{c@{\,}c@{\,}c@{\,}c}
& $C$ & $C$ & $A$ \\
+ & $B$ & $2$ & $B$ \\\hline
& $A$ & $8$ & $8$ \\
\end{tabular}
what is $A \cdot B \cdot C$?
[i]2021 CCA Math Bonanza Individual Round #5[/i] | 42 |
If the graph of the function $y=f(x)$ is symmetric to the graph of $y=2^x$ with respect to the line $y=x$, what is the interval of increase for the function $y=f(4x-x^2)$? | (0, 2) |
18. Let the variable $x$ satisfy the inequality: $x^{2}+b x \leqslant-x(b<-1)$, and the minimum value of $f(x)=x^{2}+b x$ is $-\frac{1}{2}$, find the value of the real number $b$. | -\frac{3}{2} |
Among the following options, the counterexample that can prove the proposition "the sum of two acute angles is an acute angle" to be false is ( ).
A: $\angle A=20^{\circ}$, $\angle B=60^{\circ}$
B: $\angle A=50^{\circ}$, $\angle B=90^{\circ}$
C: $\angle A=40^{\circ}$, $\angle B=50^{\circ}$
D: $\angle A=40^{\circ}$,... | C |
## 5.6. Find the limit of the function
$$
\lim _{x \rightarrow \infty} \frac{3 x^{4}-2 x^{3}+6 x^{2}+x-2}{9 x^{4}+5 x^{3}+7 x^{2}+x+1}
$$ | \frac{1}{3} |
A2. Let $a$ and $b$ be positive real numbers such that $3 a^{2}+2 b^{2}=3 a+2 b$. Find the minimum value of
$$
A=\sqrt{\frac{a}{b(3 a+2)}}+\sqrt{\frac{b}{a(2 b+3)}}
$$
| \frac{2}{\sqrt{5}} |
3. Let $A B C D-A_{1} B_{1} C_{1} D_{1}$ be a cube, and point $M$ be the midpoint of edge $A A_{1}$. Then the plane angle $\theta$ (acute angle) of the dihedral angle $B_{1}-M C-A_{1}$ is $\qquad$ (express in radians or inverse trigonometric functions). | \frac{\pi}{4} |
1. (5 points) Calculate: $(70 \div 4+90 \div 4) \div 4=$ | 10 |
Given the function $f(x)=a\ln x+ \frac{2x+1}{x} \ (a\in \mathbb{R})$ has its tangent at $x = 2$ perpendicular to the line $4x+y=0$.
(Ⅰ) Find the intervals of monotonicity for the function $f(x)$;
(Ⅱ) If there exists $x\in(1,+\infty)$ such that $f(x) < \frac{m(x-1)+2}{x} \ (m\in \mathbb{Z})$ holds, find the minimum va... | 5 |
Find all natural numbers $n$ such that the sum of the three largest divisors of $n$ is $1457$. | 987, 1023, 1085, 1175 |
4. Evaluate the sum
$$
\frac{1}{2\lfloor\sqrt{1}\rfloor+1}+\frac{1}{2\lfloor\sqrt{2}\rfloor+1}+\frac{1}{2\lfloor\sqrt{3}\rfloor+1}+\cdots+\frac{1}{2\lfloor\sqrt{100}\rfloor+1} .
$$ | \frac{190}{21} |
IMO 1990 Problem A2 Take n ≥ 3 and consider a set E of 2n-1 distinct points on a circle. Suppose that exactly k of these points are to be colored black. Such a coloring is "good" if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly n points from E. Find t... | n |
For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing
$$\frac{x^3y+y^3z+z^3x}{x+y+z}+\frac{4c}{xyz}\ge2c+2$$ | c = 1 |
Which of the following equations is a factorization?
A: $m\left(a-2\right)=am-2m$
B: $(x+3)\left(x-3\right)=x^{2}-9$
C: $x^{2}+3x-5=x\left(x+3\right)-5$
D: $4x^{2}-1=\left(2x+1\right)\left(2x-1\right)$ | D |
Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last ... | 7 |
In the book "Strange Stories from a Chinese Studio," there is a poem that goes like this: "Fetching water and chopping wood is unbearable, please come back and learn the art of passing through walls. The secret claims to have no obstacles, but the tomb on the forehead remains unenlightened." Here, we call equations in ... | D |
Three, find the smallest positive integer $n$ (where $n>1$) such that the square mean of the first $n$ natural numbers is an integer.
(Note: The square mean of $n$ numbers $a_{1}, a_{2}, \cdots, a_{i}$ is defined as $\left.\left[\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}\right]^{\frac{1}{2}}\right)$ | 337 |
13. Rohan wants to cut a piece of string into nine pieces of equal length. He marks his cutting points on the string. Jai wants to cut the same piece of string into only eight pieces of equal length. He marks his cutting points on the string. Yuvraj then cuts the string at all the cutting points that are marked. How ma... | 16 |
Let [set](https://artofproblemsolving.com/wiki/index.php/Set) $\mathcal{A}$ be a 90-[element](https://artofproblemsolving.com/wiki/index.php/Element) [subset](https://artofproblemsolving.com/wiki/index.php/Subset) of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}.$ Find the number of pos... | 901 |
13.2 設 $n$ 為整數。求 $n^{a}-n$ 除以 30 的稌值 $b$ 。
Let $n$ be an integer. Determine the remainder $b$ of $n^{a}-n$ divided by 30 . | 0 |
Find the integers $n$ such that $n-1$ divides $n^{2}+2$. | n\in{-2,0,2,4} |
4・145 Solve the system of equations
$$\left\{\begin{array}{l}
x_{1}+x_{2}+x_{3}=6, \\
x_{2}+x_{3}+x_{4}=9, \\
x_{3}+x_{4}+x_{5}=3, \\
x_{4}+x_{5}+x_{6}=-3, \\
x_{5}+x_{6}+x_{7}=-9, \\
x_{6}+x_{7}+x_{8}=-6, \\
x_{7}+x_{8}+x_{1}=-2, \\
x_{8}+x_{1}+x_{2}=2 .
\end{array}\right.$$ | x_{1}=1, x_{2}=2, x_{3}=3, x_{4}=4, x_{5}=-4, x_{6}=-3, x_{7}=-2, x_{8}=-1 |
8. Square $A B C D$ has sides of length $3 \mathrm{~cm}$. The points $M$ and $N$ lie on $A D$ and $A B$ so that $C M$ and $C N$ split the square into three pieces of the same area. What is the length of $D M$ ?
A $0.5 \mathrm{~cm}$
B $1 \mathrm{~cm}$
C $1.5 \mathrm{~cm}$
D $2 \mathrm{~cm}$
E $2.5 \mathrm{~cm}$ | 2\mathrm{~} |
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