problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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3rd CaMO 1971 Problem 1 A diameter and a chord of a circle intersect at a point inside the circle. The two parts of the chord are length 3 and 5 and one part of the diameter is length 1. What is the radius the circle? | 8 |
6.206. $\left\{\begin{array}{l}x+y+x y=7 \\ x^{2}+y^{2}+x y=13 .\end{array}\right.$ | (3;1),(1;3) |
Given $m$ and $n$ are two non-coincident lines, and $\alpha$ and $\beta$ are two non-coincident planes. The following four propositions are given:
① If $m \subset \alpha$, $n \subset \alpha$, $m \parallel \beta$, $n \parallel \beta$, then $\alpha \parallel \beta$;
② If $m \perp \alpha$, $n \perp \beta$, $m \paralle... | 2 |
What is the maximum number of acute angles that can be in an $n$-sided convex polygon? | 3 |
Example 4 Inequality
$$
-2<\sqrt{x^{2}-2 x+4}-\sqrt{x^{2}-10 x+28}<2
$$
The solution set is $\qquad$ .
(2006, National High School Mathematics League, Heilongjiang Province Preliminary Contest) | 3-\sqrt{2}<x<3+\sqrt{2} |
Let $f(x)$ be a polynomial of degree $n$ with real coefficients, having $n$ (not necessarily distinct) real roots. Prove that for all real $x$,
$$f(x)f''(x)\le f'(x)^2.$$ | f(x)f''(x) \leq f'(x)^2 |
3. Find the smallest distance from the point with coordinates $(10 ; 5 ; 10)$ to a point whose coordinates are positive and satisfy the inequality $(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right) \geq 9 \sqrt{1-(2 x+y)^{2}}$. In your answer, write the square of the found distance. | 115.2 |
Call a positive integer $n$ $k$-pretty if $n$ has exactly $k$ positive divisors and $n$ is divisible by $k$. For example, $18$ is $6$-pretty. Let $S$ be the sum of positive integers less than $2019$ that are $20$-pretty. Find $\tfrac{S}{20}$. | 472 |
Task 14. (8 points)
To attend the section, Mikhail needs to purchase a tennis racket and a set of tennis balls. Official store websites have product catalogs. Mikhail studied the offers and compiled a list of stores where the items of interest are available:
| Item | Store | |
| :--- | :---: | :---: |
| | Higher Le... | 5854 |
A circle with center $C$ passes through points $A(0,2)$ and $B(2,0)$. The center $C$ is located on the line $l_1: 2x-y-4=0$.
(I) Find the equation of circle $C$.
(II) Find the length of the chord intercepted by line $l_2: 3x+4y-8=0$ on circle $C$. | 4 |
In triangle $\triangle ABC$, $AB=3$, $AC=5$. If $O$ is the circumcenter of $\triangle ABC$, then the value of $\overrightarrow{AO} \cdot \overrightarrow{BC}$ is ____. | 8 |
8. In $\triangle A B C$, the three sides $a, b, c$ form an arithmetic sequence, and $\angle A=3 \angle C, \cos \angle C=$ | \frac{1+\sqrt{33}}{8} |
In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$, respectively. Given that $A=60^\circ$, $b=1$, and the area of triangle $ABC$ is $\sqrt{3}$, find the value of $c$. | 4 |
100. In the numbers $1, 2, 3, 4, \ldots, 100$, take two different numbers such that their sum is divisible by 5. The maximum number of different ways to do this is $\qquad$. | 990 |
183 If $a, b, c \in [0,1]$, then the maximum value of the ternary function
$$
f(a, b, c)=a(1-a+a \cdot b)(1-a b+a b c)(1-c)
$$
is = | \frac{8}{27} |
Euler's formula $e^{ix}=\cos x+i\sin x\left(x\in R\right)$ was discovered by the famous Swiss mathematician Euler, and this formula is known as the "mathematical bridge." If the complex numbers ${z_1}={e^{i\frac{π}{3}}},{z_2}={e^{i\frac{π}{6}}}$, then $z_{1}z_{2}=\left(\ \ \right)$
A: $-i$
B: $i$
C: $-\frac{{\sqrt{2... | B |
1. The remainder of the polynomial $x^{12}-x^{6}+1$ divided by $x^{2}-1$ is
(A) 1 ;
(B) -1 ;
(C) $x-1$;
(D) $x+1$. | 1 |
Given the function $f(x) = \log_a(x-1) + 4$ ($a>0$ and $a \neq 1$) always passes through a fixed point $P$. If point $P$ is also on the graph of the power function $g(x)$, then $g(4) = \boxed{16}$. | 16 |
3. As shown in Figure 2, quadrilateral $ABCD$ is inscribed in $\odot O$, with $BD$ being the diameter of $\odot O$, and $\overparen{AB}=\overparen{AD}$. If $BC + CD = 4$, then the area of quadrilateral $ABCD$ is $\qquad$ . | 4 |
6. In a three-digit number, the digit in the hundreds place is 2 more than the digit in the units place. Find the difference between this number and the number formed by the same digits but in reverse order. | 198 |
14. Solve the inequality $\log _{2}\left(x^{12}+3 x^{10}+5 x^{8}+3 x^{6}+1\right)<1+\log _{2}\left(x^{4}+1\right)$. | x\in(-\sqrt{\frac{\sqrt{5}-1}{2}},\sqrt{\frac{\sqrt{5}-1}{2}}) |
Let the linear functions of $x$ be $y=a_1x+b_1$ and $y=a_2x+b_2$. Then, the function $y=m(a_1x+b_1)+n(a_2x+b_2)$ (where $m+n=1$) is called the generating function of these two functions.
(1) When $x=1$, find the value of the generating function of $y=x+1$ and $y=2x$;
(2) If the intersection point of the graphs of $... | 2 |
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i] | \left( \frac{2n}{3} \right)^{3/2} |
Problem 6. In an acute-angled triangle $A B C$ with side $A C=1$, the altitude $B H$ is drawn, in triangle $B H C$ - the bisector $C L$, in triangle $B L C$ - the median $B M$. The line $A L$ intersects side $B C$ at point $K$, and $\angle B H K=\angle M H C=15^{\circ}$. Find the area of quadrilateral $K L H M$. | \frac{3(2-\sqrt{3})}{8} |
5. On the number line, points with integer coordinates are painted red and blue according to the following rules: a) points whose coordinate difference is 7 must be painted the same color; b) points with coordinates 20 and 14 should be painted red, and points with coordinates 71 and 143 - blue. In how many ways can all... | 8 |
Example 13. Given $4 x-3 y-6 z=0, x+2 y-7 z$ $=0, x y z \neq 0$. Then the value of $\frac{2 x^{2}+3 y^{2}+6 z^{2}}{x^{2}+5 y^{2}+7 z^{2}}$ is equal to | 1 |
1. Find the number of primitive roots of the following primes:
a) 7
d) 19
b) 13
e) 29
c) 17
f) 47 . | 2, 4, 8, 6, 12, 22 |
Given the sequence $\{a_n\}$ is a non-zero arithmetic sequence, $S_n$ denotes the sum of the first $n$ terms, and $S_{2n-1} = a_n^2$ for any $n \in \mathbb{N^*}$. If the inequality $\dfrac{1}{a_1a_2} + \dfrac{1}{a_2a_3} + \ldots + \dfrac{1}{a_na_{n+1}} \leqslant n\log_{\frac{1}{8}}\lambda$ holds for any $n \in \mathbb{... | \frac{1}{2} |
Given a point P on the line $2x-y+1=0$ and O being the coordinate origin, find the minimum value of $|OP|$. | \frac { \sqrt {5} }{ 5 } |
If the graph of the function $y=f(x)$ is symmetric about the point $(h, k)$, then the function $g(x)=f(x+h)-k$ is
A: An odd function
B: An even function
C: Both an odd and an even function
D: Neither odd nor even | \text{A} |
Let $m/n$, in lowest terms, be the [probability](https://artofproblemsolving.com/wiki/index.php/Probability) that a randomly chosen positive [divisor](https://artofproblemsolving.com/wiki/index.php/Divisor) of $10^{99}$ is an integer multiple of $10^{88}$. Find $m + n$. | 634 |
7.62 In the figure shown, $\theta$ is measured in radians, $C$ is the center of the circle, $BCD$ and $ACE$ are straight line segments, and $AB$ is the tangent to the circle at point $A$. Given $0<\theta<\frac{\pi}{2}$, the necessary and sufficient condition for the two shaded areas to be equal is
(A) $\operatorname{tg... | \operatorname{tg}\theta=2\theta |
Given a positive integer $n$, find the least $\lambda>0$ such that for any $x_1,\ldots x_n\in \left(0,\frac{\pi}{2}\right)$, the condition $\prod_{i=1}^{n}\tan x_i=2^{\frac{n}{2}}$ implies $\sum_{i=1}^{n}\cos x_i\le\lambda$.
[i]Huang Yumin[/i] | \lambda = \frac{n}{\sqrt{3}} |
The complex numbers $z$ and $w$ satisfy the system \[z + \frac{20i}w = 5+i\]
\[w+\frac{12i}z = -4+10i\] Find the smallest possible value of $\vert zw\vert^2$. | 40 |
Let \\(a=\frac{\sqrt{2}}{2}(\sin {17}^{\circ}+\cos {17}^{\circ}), b=2\cos^2{13}^{\circ}-1, c=\frac{\sqrt{3}}{2}\\), then ( )
A: \\(a < b < c\\)
B: \\(b < a < c\\)
C: \\(b < c < a\\)
D: \\(c < a < b\\) | \text{D} |
Given the inequality about $x$: $|2x-m| \leq 1$ has exactly one integer solution, which is $2$.
(Ⅰ) Find the value of the integer $m$;
(Ⅱ) Given $a$, $b$, $c \in \mathbb{R}$, if $4a^4 + 4b^4 + 4c^4 = m$, find the maximum value of $a^2 + b^2 + c^2$. | \sqrt{3} |
The line $ax-y+3=0$ intersects the circle $(x-1)^{2}+(y-2)^{2}=4$ at points $A$ and $B$, and $|AB|=2\sqrt{2}$. Then $a=$ .
A: $1$
B: $\sqrt{3}$
C: $2$
D: $3$ | A |
3.390. Derive the formula $\cos (n+1) \alpha=2 \cos \alpha \cos n \alpha-\cos (n-1) \alpha$, where $n$ is any real number, and use it to express $\cos 3 \alpha$ and $\cos 4 \alpha$ as polynomials in $\cos \alpha$. | \cos3\alpha=4\cos^{3}\alpha-3\cos\alpha;\cos4\alpha=8\cos^{4}\alpha-8\cos^{2}\alpha+1 |
3. Determine all triples of real numbers $a, b, c$ that satisfy the conditions
$$
a^{2}+b^{2}+c^{2}=26, \quad a+b=5 \quad \text { and } \quad b+c \geq 7
$$
The written part of the school round in category A takes place
on Tuesday, December 11, 2012
so that it starts in the morning and the contestants have 4 hours o... | =1,b=4,=3 |
## SUBJECT IV
Determine the functions $\quad f, g:(0, \infty) \rightarrow R$ with the property that $f\left(\frac{x}{3}\right)+2 \leq \log _{3} x \leq g(x)-1$ and $g\left(\frac{x}{3}\right) \leq \log _{3} x \leq f(x)+1$ for any $x \in(0, \infty)$.
NOTE:
All problems are mandatory.
Each problem is worth a maximum of... | f(x)=\log_3x-1,\quad(x)=\log_3x+1 |
Calculate:$(1)$ Calculate: $(-\frac{1}{2})^{0}+|1-\tan60°|+\sqrt{12} \times (-2)^{-1}$;$(2)$ Simplify first, then evaluate: $(\frac{x^{2}-3x+5}{x-1}+3-x)÷\frac{x^{2}+4x+4}{1-x}$, where $x=\sqrt{3}-2$. | -\frac{\sqrt{3}}{3} |
If the sum of the interior angles of a polygon is three times the sum of its exterior angles, then the number of sides of this polygon is ______. | 8 |
Observing $3^3 = 9 = 4 + 5$, we have $3^2 + 4^2 = 5^2$; $5^2 = 25 = 12 + 13$, thus we have $5^2 + 12^2 = 13^2$; $7^2 = 49 = 34 + 15$, thus we have $7^2 + 24^2 = 25^2$; continue this pattern and write down two more equations. | 11^2 + 60^2 = 61^2 |
Given $f(x) = a \ln x + \frac{1}{2x} + \frac{3}{2}x + 1$, where $a \in \mathbb{R}$, the tangent line of the curve $y = f(x)$ at the point $(1, f(1))$ is perpendicular to the y-axis.
1. Find the value of $a$.
2. Find the extreme values of function $f(x)$. | 3 |
Factorize: $(a^{2}+a)^{2}+4(a^{2}+a)-12$. | (a-1)(a+2)(a^{2}+a+6) |
Given a complex number $z=\frac{{1+3i}}{{1-2i}}$, the imaginary part of the conjugate of $z$ is ( ).
A: $1$
B: $i$
C: $-i$
D: $-1$ | \text{D} |
The value of $\sin 80^\circ \cos 20^\circ - \cos 80^\circ \sin 20^\circ$ is \_\_\_\_\_\_. | \frac{\sqrt{3}}{2} |
Example 7 Find the minimum perimeter of a convex pentagon with vertices at lattice points in the Cartesian coordinate plane.
untranslated text retained the original line breaks and format. | 2+3\sqrt{2} |
Given that the sequence $\{a_{n}\}$ is an arithmetic sequence with a first term of $1$ and a common difference of $2$, and the sequence $\{b_{n}\}$ is a geometric sequence with a first term of $1$ and a common ratio of $2$. Let ${c_n}={a_{{b_n}}}$, and let ${T_n}$ be the sum of the first $n$ terms of the sequence $\{c_... | AB |
In a geometric sequence $\{a_n\}$ where each term is positive, $a_2$, $\frac{1}{2}a_3$, and $a_1$ form an arithmetic sequence. The value of the common ratio $q$ is ( )
A: $\frac{\sqrt{5}-1}{2}$
B: $\frac{\sqrt{5}+1}{2}$
C: $\frac{1-\sqrt{5}}{2}$
D: $\frac{\sqrt{5}-1}{2}$ or $\frac{\sqrt{5}+1}{2}$ | \text{B} |
26. For a cube with edge length of $m$ cm, some of its faces are painted red, and then it is cut into small cubes with edge length of 1 cm. The ratio of the number of small cubes with at least one red face to the number of small cubes with no red surface is $13: 12$. What is the minimum value of $m$ at this time? $\qqu... | 5 |
18. Find the value of
$$
\prod_{k=1}^{45} \tan (2 k-1)^{\circ}
$$ | 1 |
4. Let $S=\{1,2, \cdots, 1990\}$. If the sum of the elements of a 31-element subset of $S$ is divisible by 5, it is called a good subset of $S$. Find the number of good subsets of $S$.
(IMO - 31 Preliminary Question) | \frac{1}{5}C_{1990}^{31} |
28.2.11 ** Let $M$ be a set of $n$ points in the plane, satisfying:
(1) $M$ contains 7 points which are the 7 vertices of a convex heptagon;
(2) For any 5 points in $M$, if these 5 points are the 5 vertices of a convex pentagon, then this convex pentagon contains at least one point from $M$ inside it.
Find the minimum ... | 11 |
For non-zero real numbers $x$ and $y$, if $x^{2}+2|y|=2xy$, which of the following statements is correct?
A: When $x \gt 0$, $x+y \gt 3$
B: When $x \lt 0$, $x+y \lt -3$
C: When $x \gt 0$, $x^{2}+2|y|\geqslant 8$
D: When $x \lt 0$, $x^{2}+2|y|\leqslant 8$ | AB |
Three, (13 points) As shown in Figure 4, two planes $m \parallel n$, line segment $AD$ intersects planes $m$ and $n$ at points $B$ and $C$, respectively. Another line through point $A$ intersects planes $m$ and $n$ at points $M$ and $P$, respectively. Another line through point $D$ intersects planes $m$ and $n$ at poin... | 3+\sqrt{3} |
Simplify first and then evaluate: When $m=-1$, find the value of $\left(2m+3\right)\left(2m-3\right)-\left(m-1\right)\left(m+5\right)$. | 3 |
In the Cartesian coordinate system, if the line $y=x$ is perpendicular to the line $\begin{cases}x=1+\cos \theta \\ y=t\sin \theta\end{cases}$, where $t$ is a parameter and $0 \leqslant \theta < \pi$, then $\theta=$ ( )
A: $\dfrac{\pi}{6}$
B: $\dfrac{\pi}{4}$
C: $\dfrac{2\pi}{3}$
D: $\dfrac{3\pi}{4}$ | D |
6. The volume of a regular $n$-sided pyramid $(n \geqslant 3)$ is $V$, and the area of each side triangular face is $S$. Point $O$ is located inside the base. Then the sum of the distances from point $O$ to each side face is $\qquad$ . | \frac{3 V}{S} |
10. (This question is worth 20 points) Let $a, b, c$ all be greater than 1, and satisfy
$$
\left\{\begin{array}{l}
\lg a+\log _{b} c=3, \\
\lg b+\log _{a} c=4 .
\end{array}\right.
$$
Find the maximum value of $\lg a \cdot \lg c$. | \frac{16}{3} |
1. Given $\sqrt{x}=\frac{1}{\sqrt{a}}-\sqrt{a}$. Then the value of $\sqrt{4 x+x^{2}}$ is ( ).
(A) $a-\frac{1}{a}$
(B) $\frac{1}{a}-a$
(C) $a+\frac{1}{a}$
(D) cannot be determined | B |
Simplify the expression $S_n = n + (n-1) \times 2 + (n-2) \times 2^2 + \ldots + 2 \times 2^{n-2} + 2^{n-1}$ to get ( )
A: $2^{n+1} + n - 2$
B: $2^{n+1} - n + 2$
C: $2^n - n - 2$
D: $2^{n+1} - n - 2$ | \text{D} |
Given $$\cos\left(\alpha+ \frac {\pi}{6}\right)=- \frac {1}{3}$$, the value of $$\sin\left(\alpha- \frac {\pi}{3}\right)$$ is ( )
A: $$\frac {1}{3}$$
B: $$- \frac {1}{3}$$
C: $$\frac {2 \sqrt {3}}{3}$$
D: $$- \frac {2 \sqrt {3}}{3}$$ | \text{A} |
Given the function $y=\cos (x+ \frac {π}{3})$, stretch the abscissa of each point on the graph to twice its original length (the ordinate remains unchanged), then shift it to the left by $\frac {π}{6}$ units. What is the smallest positive period of the resulting function?
A: $π$
B: $2π$
C: $4π$
D: $8π$ | \text{C} |
4. If $x$ and $y$ are real numbers for which $|x|+x+5 y=2$ and $|y|-y+x=7$, find the value of $x+y$.
(A) -3
(B) -1
(C) 1
(D) 3
(E) 5 | 3 |
If the set $A=\{x|x^2-2x+m=0\}=\emptyset$, then the range of the real number $m$ is ( )
A: $(-\infty, -1)$
B: $(-\infty, 1)$
C: $(1, +\infty)$
D: $[1, +\infty)$ | \text{C} |
2. How many six-digit numbers of the form $\overline{a b c a c b}$ are divisible by 23? | 42 |
Example 5 Given a quadrilateral pyramid $P-ABCD$, the base $ABCD$ is a rhombus with side length 2, $\angle ABC=60^{\circ}, E$ is the midpoint of $AB$, $PA \perp$ plane $ABCD$, and the sine of the angle formed by $PC$ and plane $PAD$ is $\frac{\sqrt{6}}{4}$.
(1) Find a point $F$ on edge $PD$ such that $AF \parallel$ pla... | \frac{4\sqrt{31}}{31} |
Given the proposition p: For all $x \in \mathbb{R}$, $x^2 - x + 1 > 0$, then $\neg p$ is
A: There exists $x \in \mathbb{R}$, $x^2 - x + 1 \leq 0$
B: For all $x \in \mathbb{R}$, $x^2 - x + 1 \leq 0$
C: There exists $x \in \mathbb{R}$, $x^2 - x + 1 > 0$
D: For all $x \in \mathbb{R}$, $x^2 - x + 1 \geq 0$ | \text{A} |
In the calendar, the duration of a year is taken as 365 days 5 hours 48 minutes 46 seconds. Express this in days, expand the measurement into a continued fraction, 1] then form the approximate fractions of the continued fraction. Explain the measures regarding leap years in the Julian and Gregorian calendars based on t... | 365+\frac{97}{400}=365+\frac{20952}{86400} |
Let $i$ be the imaginary unit. If the complex number $z$ satisfies $z(1+i) = (1-i)$, then the modulus of $z$, $|z|$, is
A: $-1$
B: $1$
C: $\sqrt{2}$
D: $2$ | B |
4. Inside parallelogram $A B C D$, a point $E$ is chosen such that $A E=D E$ and $\angle A B E=90^{\circ}$. Point $M$ is the midpoint of segment $BC$. Find the angle $DME$. (A. Kuznetsov) | 90 |
Which of the following propositions is true?
A: If $ab=0$, then $a=0$
B: The axis of symmetry of the parabola $y=x^{2}+x$ is $x=1$
C: The sum of the exterior angles of a hexagon is equal to that of a pentagon
D: A quadrilateral with equal diagonals is a rectangle | C |
Given functions $f\left(x\right)=2x^{2}-4x+1$ and $g\left(x\right)=2x+a$, if there exist $x_{1}$ and $x_{2}\in [\frac{1}{2}$, $1]$, such that $f(x_{1})=g(x_{2})$, then the range of $a$ is ____. | [-3, -\frac{3}{2}] |
Determine all polynomials $P(x) \in \mathbb R[x]$ satisfying the following two conditions :
(a) $P(2017) = 2016$ and
(b) $(P(x) + 1)^2 = P(x^2 + 1)$ for all real numbers $x$.
[i]proposed by Walther Janous[/i] | P(x) = x - 1 |
3. A dog2 was located at point A when it started to chase a fox at a distance of $30 \mathrm{~m}$ from $A$. The dog's jump is $2 \mathrm{~m}$, and the fox's jump is $1 \mathrm{~m}$. At the same time, while the dog makes two jumps, the fox makes three jumps. At what distance from $A$ will the dog catch the fox? | 120 |
A school plans to purchase 400 workbooks, with each workbook costing x yuan. The school should pay a total of __ yuan. | 400x |
$10 \cdot 46$ Find the sum of all digits of the numbers $1,2,3, \cdots, 10^{n}-2,10^{n}-1$. (Kiev Mathematical Olympiad, 1970) | \frac{1}{2}\cdot10^{n}\cdot9n |
6.1. How many triangles with integer sides have a perimeter equal to 27? (Triangles that differ only in the order of the sides - for example, $7,10,10$ and $10,10,7$ - are considered the same triangle.) | 19 |
3. Usually, schoolboy Gavriil takes a minute to go up a moving escalator by standing on its step. But if Gavriil is late, he runs up the working escalator and thus saves 36 seconds. Today, there are many people at the escalator, and Gavriil decides to run up the adjacent non-working escalator. How much time will such a... | 40 |
XXXVIII OM - II - Problem 5
Determine all prime numbers $ p $ and natural numbers $ x, y $, for which $ p^x - y^3 = 1 $. | (2,1,1)(3,2,2) |
Given the linear function $y = kx - k$, if $y$ decreases as $x$ increases, then the graph of this function passes through which quadrants?
A: Quadrants I, II, III
B: Quadrants I, II, IV
C: Quadrants II, III, IV
D: Quadrants I, III, IV | \text{B} |
In the expansion of $(x^{2}+x+y)^{6}$, the coefficient of $x^{5}y^{2}$ is ( ).
A: $60$
B: $15$
C: $120$
D: $30$ | A |
## 2. Square multiplication
If you increase one side of the square $K$ three times and the other side two times, you will get a rectangle $P$ whose area is $96 \mathrm{~cm}^{2}$. By how many millimeters is the perimeter of the rectangle $P$ greater than the perimeter of the square $K$?
## Result: 240 | 240 |
Let $p$: for real number $x$, $x^{2}-2ax-3a^{2} \lt 0\left(a \gt 0\right)$, $q: 2\leqslant x \lt 4$.
$(1)$ If $a=1$, and both $p$ and $q$ are true propositions, find the range of values for $x$.
$(2)$ If $q$ is a sufficient but not necessary condition for $p$, find the range of real numbers for $a$. | [\frac{4}{3}, +\infty) |
Which of the following is the result of factoring the polynomial $-(2x-y)(2x+y)$?
A: $4x^2-y^2$
B: $4x^2+y^2$
C: $-4x^2-y^2$
D: $-4x^2+y^2$ | D |
Given the specific proposition p: $\exists x \in \mathbb{R}, 2x+1 \leq 0$. The negation of proposition p is ( )
A: $\exists x \in \mathbb{R}, 2x+1 > 0$
B: $\forall x \in \mathbb{R}, 2x+1 > 0$
C: $\exists x \in \mathbb{R}, 2x+1 \geq 0$
D: $\forall x \in \mathbb{R}, 2x+1 \geq 0$ | \text{B} |
8. For a positive integer $x$, let $S(x)$ denote the sum of the digits of $x$. Then the maximum value of $S(x)-9[\lg x]$ is $\qquad$ ( $[x]$ denotes the greatest integer not exceeding the real number $x$). | 9 |
1. (10 points) The rules of the game are as follows: from 64 different items, on each turn, the player must form a set of items that has not been mentioned in the game before, with the number of items equal to the player's age in years. Players take turns; either player can start the game; the player who cannot make a ... | 34 |
Given an ellipse C: $$\frac {x^{2}}{a^{2}}$$+ $$\frac {y^{2}}{b^{2}}$$\=1 (a>b>0) with an eccentricity of $$\frac {1}{2}$$. The left and right vertices are A1 and A2, respectively, and B is one endpoint of the minor axis. The area of △A1BA2 is 2$$\sqrt {3}$$.
(1) Find the standard equation of the ellipse C;
(2) Line l:... | 3 |
1. What is the difference between the greatest and the least of the following five quantities?
$$
20+20 \quad 20 \times 20 \quad 202+0 \quad\left(2^{0}\right)^{\left(2^{0}\right)} \quad 20+2+0
$$ | 399 |
1. $A \subseteq B, A \subseteq C$, and $B=\{0,1,2,3,4\}, C=\{0, 2,4,8\}$. Then the maximum number of subsets of $A$ is ( ).
(A) 1
(B) 2
(C) 4
(D) 8 | D |
(5 points) Given $f(x-1) = x^2$, then $f(x) =$ ______. | (x+1)^2 |
The expression that represents factoring from left to right is ( ).
A: $a^{2}-5a-6=\left(a-6\right)\left(a+1\right)$
B: $ax+bx+c=\left(a+b\right)x+c$
C: $(a+b)^{2}=a^{2}+2ab+b^{2}$
D: $(a+b)\left(a-b\right)=a^{2}-b^{2}$ | A |
Given that events E and F are mutually exclusive, P(E) = 0.2, P(E ∪ F) = 0.8, find P(F) = ___. | 0.6 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, satisfying $b^{2}+c^{2}-a^{2}=bc$, $\overrightarrow{AB}\cdot \overrightarrow{BC} > 0$, $a= \frac { \sqrt {3}}{2}$. The range of values for $b+c$ is \_\_\_\_\_\_. | ( \frac { \sqrt {3}}{2}, \frac {3}{2}) |
No matter what value $\lambda$ takes, the line $(\lambda+2)x-(\lambda-1)y+6\lambda+3=0$ always passes through a fixed point \_\_\_\_\_\_. | (-3, 3) |
9. Given that point $P$ is a moving point on the line $x+2 y=4$, and through point $P$ two tangents are drawn to the ellipse $x^{2}+4 y^{2}=4$, with the points of tangency being $A$ and $B$, when point $P$ moves, the line $AB$ passes through a fixed point with coordinates | (1,\frac{1}{2}) |
Calculate the area of the figure enclosed by the vertical lines $x=1$ and $x=2$, the curve $y=\sqrt{x}$, and the x-axis. | \frac{4\sqrt{2} - 2}{3} |
13. Let $a_{1}=1, a_{2}=2$ and for all $n \geq 2, a_{n+1}=\frac{2 n}{n+1} a_{n}-\frac{n-1}{n+1} a_{n-1}$. It is known that $a_{n}>2+\frac{2009}{2010}$ for all $n \geq m$, where $m$ is a positive integer. Find the least value of $m$. | 4021 |
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