problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
81. Solve the equation $y^{\prime}+y \tan x=\cos ^{2} x$. | (\sinx+C)\cosx |
$\sin 210^\circ =\_\_\_\_\_\_\_\_\_.$ | - \dfrac{1}{2} |
1. [5] Determine the remainder when $1+2+\cdots+2014$ is divided by 2012 . | 1009 |
Let $A$ and $B$ be two sets. For $A \subseteq B$, which of the following statements is correct? ( )
A: There exists $x \in A$ such that $x \in B$.
B: $B \subseteq A$ must not hold.
C: $B$ cannot be an empty set.
D: $x \in A$ is a sufficient condition for $x \in B$. | \text{D} |
The area of the triangle formed by the common chord of the circles $x^{2}+y^{2}=1$ and $x^{2}+y^{2}-2x+2y=2$, and the two coordinate axes is ( ).
A: $\frac{1}{2}$
B: $\frac{1}{4}$
C: $\frac{1}{8}$
D: $1$ | C |
In the Cartesian coordinate system $xOy$, the parametric equation of curve $C_1$ is $\begin{cases} x=1+2\cos \alpha \\ y= \sqrt {3}\sin \alpha \end{cases}$ ($\alpha$ is the parameter). By reducing all the x-coordinates of points on curve $C_1$ to half of their original values and all the y-coordinates to $\frac{\sqrt{3... | \frac{2 \sqrt{15}+3 \sqrt{3}}{4} |
5 positive real numbers $x, y, z$ satisfy $xyz=1$. Prove: $\frac{x^{3}+y^{3}}{x^{2}+xy+y^{2}}+\frac{y^{3}+z^{3}}{y^{2}+yz+z^{2}}+$ $\frac{z^{3}+x^{3}}{z^{2}+zx+x^{2}} \geqslant 2$. | 2 |
2. In $\triangle A B C$, it is known that the three interior angles $\angle A$, $\angle B$, $\angle C$ are opposite to the sides $a$, $b$, $c$ respectively, and satisfy $a \sin A \cdot \sin B + b \cos ^{2} A = \sqrt{2} a$. Then $\frac{b}{a}=$ . $\qquad$ | \sqrt{2} |
A triangle of sides $\frac{3}{2}, \frac{\sqrt{5}}{2}, \sqrt{2}$ is folded along a variable line perpendicular to the side of $\frac{3}{2}.$ Find the maximum value of the coincident area. | \frac{9}{28} |
$\underline{\text { Gordrin P.K. }}$.
Between two parallel lines, a circle of radius 1 is placed, touching both lines, and an isosceles triangle, the base of which lies on one of the lines, and the vertex on the other. It is known that the triangle and the circle have exactly one common point and that this point lies ... | \frac{1}{2} |
8. [7] Alice rolls two octahedral dice with the numbers $2,3,4,5,6,7,8,9$. What's the probability the two dice sum to 11 ? | \frac{1}{8} |
Given that $\{a_{n}\}$ is an arithmetic sequence, $\{b_{n}\}$ is a monotonically increasing geometric sequence, $a_{1}=b_{1}=1$, $a_{2}+a_{4}=6$, $a_{3}b_{3}=12$.
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.
$(2)$ Find the sum $S_{n}$ of the first $n$ terms of the sequence $\{a_{n}+b_{n}\}$. | \frac{1}{2}n(n+1) + 2^{n} - 1 |
13. Given the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$, the endpoints of the real axis are $A_{1}$ and $A_{2}$, and $P$ is a moving point on the hyperbola different from points $A_{1}$ and $A_{2}$. The lines $P A_{1}$ and $P A_{2}$ intersect the line $x=1$ at points $M_{1}$ and $M_{2}$, respectively. Prove: The ci... | \left(-\frac{1}{2}, 0\right),\left(\frac{5}{2}, 0\right) |
$13 \cdot 10$ On the coordinate plane, a point with both coordinates as integers is called an integer point. For any natural number $n$, connect the origin $O$ with the point $A_{n}(n, n+3)$. Let $f(n)$ denote the number of integer points on the line segment $O A_{n}$, excluding the endpoints. Find the value of $f(1)+f... | 1326 |
438. Escape across the river. During the flight of the Turkish troops at Treise, a small detachment found itself on the bank of a wide and deep river. Here they found a boat in which two boys were boating. The boat was so small that it could only hold two children or one adult.
How did the officer manage to cross the ... | 1432 |
Given lines $a$, $b$ and plane $\alpha$, the following four statements are made:
① If $a \parallel \alpha$ and $b \subset \alpha$, then $a \parallel b$;
② If $a \cap \alpha = P$ and $b \subset \alpha$, then $a$ is not parallel to $b$;
③ If $a \parallel b$ and $b \perp \alpha$, then $a \perp \alpha$;
④ If $a \pa... | B |
If the equation $x^2+y^2-2x-4y+m=0$ represents a circle, then the range of the real number $m$ is \_\_\_\_\_\_. | (-\infty, 5) |
Given the function $f(x)=\frac{1}{3}x^{3}-4x+4$ defined on the interval $[0,3]$, let $M$ be its maximum value and $m$ be its minimum value. Determine the value of $M-m$. | \frac{16}{3} |
Question 29, Given $\theta \in\left[0, \frac{\pi}{2}\right]$, if $\theta$ satisfies the inequality $\sin ^{3} \theta-\cos ^{3} \theta \geq \ln \cot \theta$, find the range of values for $\theta$. | \theta\in[\frac{\pi}{4},\frac{\pi}{2}) |
$1 \cdot 35$ If $\frac{m}{n}=\frac{4}{3}$ and $\frac{r}{t}=\frac{9}{14}, \frac{3 m r-n t}{4 n t-7 m r}$ is
(A) $-5 \frac{1}{2}$.
(B) $-\frac{11}{14}$.
(C) $-1 \frac{1}{4}$.
(D) $\frac{11}{14}$.
(E) $-\frac{2}{3}$.
(5th American High School Mathematics Examination, 1954) | -\frac{11}{14} |
Example 3 Let $f: \mathbf{R} \rightarrow \mathbf{R}$ satisfy
$$
\begin{array}{l}
f(x f(x)+f(x) f(y)+y-1) \\
=f(x f(x)+x y)+y-1 .
\end{array}
$$
Find the analytical expression of the function $f(x)$. | f(x)=x |
When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it? | 9:36 |
9. (16 points) Let the range of the function $f(x)$ be $[1,2]$, and let $g(x)=f(x)+\frac{a}{f(x)}$. Find the minimum value of $p(a)=\max g(x)-\min g(x)$. | 3-2\sqrt{2} |
Given the line $ax+y-1=0$ intersects the circle $C$: $(x-1)^{2}+(y+a)^{2}=1$ at points $A$ and $B$, and $\triangle ABC$ is an isosceles right triangle, the value of the real number $a$ is $\boxed{\text{( )}}$
A: $\dfrac{1}{7}$ or $-1$
B: $-1$
C: $1$ or $-1$
D: $1$ | C |
In the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length $2$, $E$ is the midpoint of edge $CC_{1}$, and points $P$, $Q$ are moving points on the face $A_{1}B_{1}C_{1}D_{1}$ and the line segment $B_{1}C$, respectively. Find the minimum value of the perimeter of $\triangle PEQ$. | \sqrt{10} |
8. Given $a, b, c \in R, a^{2}+b^{2}+c^{2}=1$, then the minimum value of $3 a b-3 b c+2 c^{2}$ is $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | -\frac{1+\sqrt{7}}{2} |
6. Rebecca went swimming yesterday. After a while she had covered one fifth of her intended distance. After swimming six more lengths of the pool, she had covered one quarter of her intended distance.
How many lengths of the pool did she intend to complete?
A 40
B 72
C 80
D 100
E 120 | 120 |
Example. Find the probability that among six cards randomly taken from a deck of 36 cards, there will be exactly three black figure cards. | P(3)=\frac{546}{5797}\approx0.0942 |
Problem 4. Find all solutions to the puzzle
$$
\mathrm{G}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}+\mathrm{OH}=\mathrm{Mb} .
$$
(Same letters encode the same digits, different letters encode different digits.)
[5 points] (D. E. Schnol) | 0+12+12+12+12+12+12+12+12=96 |
The sum of the digits of a five-digit positive integer is 2 . (A five-digit integer cannot start with zero.) The number of such integers is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Part B: Each question is worth 6 credits. | 5 |
During a buoyancy experiment, Xiao Hua tied a cubic iron block with a thin string and completely submerged it in a cylindrical beaker filled with water, causing 40cm3 of water to overflow. Xiao Hua then lifted the iron block out of the beaker, and the water level in the beaker dropped by 0.6cm. What are the radius of t... | 3.42\text{cm} |
Given that $O$ is the coordinate origin, the focus of the parabola $C:y^{2}=2px\left(p \gt 0\right)$ is $F$, $P$ is a point on $C$, $PF$ is perpendicular to the $x$-axis, $Q$ is a point on the $x$-axis, and $PQ\bot OP$. If $|FQ|=6$, then the directrix equation of $C$ is ____. | x=-\frac{3}{2} |
## Task 2 - 310722
Susann wants to calculate the sum $s$ of all four-digit natural numbers that are divisible by 4.
Tamara determines the sum $t$ of all four-digit natural numbers that are divisible by 2 but not by 4.
a) Are $s$ and $t$ equal, or if not, which of the two numbers is the larger one?
b) What is the am... | 4500 |
# 6. Solution:
Let's find the range of the function $z=g(x)=14 \cos 2 x+28 \sin x+15$. The function $g(x)$ is defined for all real numbers. We will make a substitution. Let $t=\sin x$. Then $z=14\left(1-2 t^{2}\right)+28 t+15=29-28\left(t^{2}-t\right)=36-28(t-0.5)^{2}$ for $t \in[-1 ; 1]$, and $E_{g}=[-27 ; 36]$. The ... | E_{f}=[0.5;1] |
The sequence $\{a_n\}$ satisfies $a_1=1$, $na_{n+1}=(n+1)a_n+n(n+1)$, and $b_n=a_n\cos \frac {2n\pi}{3}$. Let $S_n$ be the sum of the first $n$ terms of the sequence $\{b_n\}$. Then $S_{24}=$ ?
A: $294$
B: $174$
C: $470$
D: $304$ | D |
12. Let $p(x)$ be a polynomial of degree 2010. What is the maximum degree that the polynomial $p(x-1)-3 p(x)+3 p(x+1)-p(x+2)$ can have?
(A) It is always the zero polynomial
(B) 0
(C) 1
(D) 2007
(E) 2010
## Numerical answer problems -5 points | 2007 |
5. Person A and Person B simultaneously solve the same math competition problem. If they solve it correctly, the teacher will give a reward of 190 yuan, with the rule that the reward goes to whoever solves it correctly. If both solve it correctly, the reward is split equally. It is known that the probability of A solvi... | 90 |
Given that line $l$ passes through points $A(1,2)$ and $B(4, 2+\sqrt{3})$, find the slope of line $l$ and its angle of inclination. | \frac{\pi}{6} |
A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is:
$\textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad \textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad \textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad \\ \textbf{(D)}\ 2Tx-a^2y-2aT=0 \qquad \textbf{(E)}\ \text{none of these}$ | \textbf{(B)} |
Compute the value of the following expressions:
(1) $\frac{\tan 53^{\circ} + \tan 7^{\circ} + \tan 120^{\circ}}{\tan 53^{\circ} \cdot \tan 7^{\circ}}$;
(2) $\left[2\sin 50^{\circ} + \sin 10^{\circ}(1 + \sqrt{3}\tan 10^{\circ})\right]\sqrt{1 - \cos 160^{\circ}}$. | \sqrt{6} |
Given that the sum of the first $n$ terms of the sequence $\{a_{n}\}$ is $S_{n}=\frac{n}{n+1}$ for $n\in \mathbb{N}_{+}$, and the sequence $\{b_{n}\}$ satisfies $b_{1}=1$ and $b_{n+1}=\frac{b_{n}}{b_{n}+2}$ for $n\in \mathbb{N}_{+}$. $(1)$ Find the general formula for the sequence $\{a_{n}\}$; $(2)$ Find the general fo... | b_{n+1} < a_{n} |
Let $P$ be a regular polygon with 2026 sides. A diagonal of $P$ is said to be good if its endpoints divide the perimeter of $P$ into two parts, each having an odd number of sides of $P$. The sides of $P$ are also considered good. Suppose that $P$ has been subdivided into triangles by 2023 diagonals that do not intersec... | 1013 |
If the function $f(x)$ satisfies $f(x) + 2f\left(\frac{1}{x}\right) = 2x + 1$, then $f(2) = \_\_\_$
A: $-\frac{1}{3}$
B: $\frac{2}{3}$
C: $\frac{8}{3}$
D: $\frac{1}{2}$ | A |
$1 \cdot 103$ Given the equation $\sqrt{a(x-a)}+\sqrt{a(y-a)}=\sqrt{x-a}-\sqrt{a-y}$ holds in the real number range, where $a, x, y$ are distinct real numbers, then the value of $\frac{3 x^{2}+x y-y^{2}}{x^{2}-x y+y^{2}}$ is
(A) 3.
(B) $\frac{1}{3}$.
(C) 2.
(D) $\frac{5}{3}$.
(China Junior High School Mathematics Leagu... | \frac{1}{3} |
9. In a right-angled tetrahedron $A B C D$, $D A, D B, D C$ are mutually perpendicular. Let $S, R$ represent its surface area and the radius of the circumscribed sphere, respectively. Then the maximum value of $\frac{S}{R^{2}}$ is $\qquad$. | \frac{2}{3}(3+\sqrt{3}) |
Let's determine the value of $\sqrt{3(7+4 \sqrt{3})}$ as the sum of two simple square roots! | 2\sqrt{3}+3 |
If the base radius of a cone is $\sqrt{2}$ and its height is 2, then its lateral surface area is ( )
A: $4 \sqrt{3}\pi$
B: $2 \sqrt{2}\pi$
C: $2 \sqrt{3}\pi$
D: $4 \sqrt{2}\pi$ | C |
In the Cartesian coordinate system xOy, given circles O: $x^{2}+y^{2}=1$ and $O_{1}: (x-4)^{2}+y^{2}=4$, there is a moving point P on the line $x+\sqrt{3}y+b=0$. Tangent lines from P to circles O and $O_{1}$ respectively meet the circles at points A and B. If there are exactly two points P such that $PB=2PA$, then the ... | (-4, \frac{20}{3}) |
A die is a cube with its faces numbered 1 through 6 . One red die and one blue die are rolled. The sum of the numbers on the top two faces is determined. What is the probability that this sum is a perfect square? | \frac{7}{36} |
Given an exponential sequence $\{b_{n}\}$ with the first term being $2$ and a common ratio of $\frac{1}{3}$, the sum of all terms in this sequence is ______. | 3 |
1. In the Cartesian coordinate system $x O y$, the line $l: y=$ $-\frac{4}{3} x+4$ intersects the $x$-axis and the $y$-axis at points $A$ and $B$, respectively. If the line $l$ is rotated counterclockwise by $90^{\circ}$ around point $O$, it intersects the $y$-axis at point $A^{\prime}$ and the line $l$ at point $C$. T... | B |
2. (2 points) The sequence $x_{n}$ is defined by the following conditions: $x_{1}=2, x_{1}+x_{2}+\ldots+$ $x_{n-1}+\frac{3}{2} x_{n}=3$. Find $x_{1000}$. | \frac{2}{3^{999}} |
Problem 6.3. Three merchants: Foma, Yerema, and Julius met in Novgorod. If Foma gives Yerema 70 gold coins, then Yerema and Julius will have the same amount of money. If Foma gives Yerema 40 gold coins, then Foma and Julius will have the same amount of money. How many gold coins should Foma give Yerema so that they bot... | 55 |
Given that O is the circumcenter of acute triangle ABC, A is the largest angle, and $$\frac {cosB}{sinC} \overrightarrow {AB}+ \frac {cosC}{sinB} \overrightarrow {AC}=m \overrightarrow {AO}$$, find the range of m. | [\sqrt {3},2) |
Points $E$ and $F$ lie on the sides $AB$ and $AC$ of a triangle $ABC$. Lines $EF$ and $BC$ meet at point $S$. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. The line passing through $A$ and parallel to $MN$ meets $BC$ at point $K$. Prove that $\frac{BK}{CK}=\frac{FS}{ES}$ .
. | \frac{BK}{CK} = \frac{FS}{ES} |
10. Given the sequence $\left\{a_{n}\right\}$ satisfies: $a_{1}=1, a_{n+1}=a_{n}+a_{n}^{2}\left(n \in \mathbf{N}^{*}\right)$. Let
$$
S_{n}=\frac{1}{\left(1+a_{1}\right)\left(1+a_{2}\right) \cdots\left(1+a_{n}\right)}, \quad T_{n}=\sum_{k=1}^{n} \frac{1}{1+a_{k}}
$$
Find the value of $S_{n}+T_{n}$. | 1 |
14. Calculate the product $p=1 \cdot 2^{\frac{1}{2}} \cdot 4^{\frac{1}{4}} \cdot 8^{\frac{1}{8}} \cdot 16^{\frac{1}{16}} \cdot \ldots$
In № $15-20$ find the sums: | 4 |
5. Given that the projection of an equilateral triangle $ABC$ on plane $\alpha$ is a triangle with side lengths of $2, 3, 2\sqrt{3}$, then the side length of the equilateral triangle $ABC$ is
A. $2\sqrt{3}$
B. $\sqrt{13}$
C. $\sqrt{15}$
D. 4 | \sqrt{13} |
Given $$f(x)= \begin{cases} x+2，x≤-1 \\ x^{2}-5，-1＜x＜3 \\ log_{2}x，x≥3\end{cases}$$, find the value of $f{f[f(4)]}$ ( ).
A: 1
B: -1
C: 2
D: -2 | A |
Three cubes have edges of lengths 4,5 and 6 .
The average (mean) of their volumes is
(A) 120
(B) 125
(C) 1125
(D) 261
(E) 135
 | 135 |
66. Can a quadrilateral have three obtuse angles? | Yes |
SG. 4 The average of $7,9, x, y, 17$ is 10 . If the average of $x+3, x+5, y+2,8$ and $y+18$ is $m$, find $m$. | 14 |
Given the function $f(x) = \frac{x}{x+4}e^{x+2}$.
(i) Discuss the monotonicity of the function and prove that when $x > -2$, $xe^{x+2} + x + 4 > 0$;
(ii) Prove that when $a \in [0,1)$, the function $g(x) = \frac{e^{x+2} - ax - 3a}{(x+2)^2}$ $(x > -2)$ has a minimum value. Let the minimum value of $g(x)$ be $h(a)$, find... | \left(\frac{1}{2}, \frac{e^2}{4}\right] |
8. Let's call the efficiency of a natural number $n$ the fraction of all natural numbers from 1 to $n$ inclusive that have a common divisor with $n$ greater than 1. For example, the efficiency of the number 6 is $\frac{2}{3}$.
a) Does there exist a number with an efficiency greater than $80\%$? If so, find the smalles... | 30030 |
The graph of the linear function $y=mx+3$ intersects with the graphs of the linear function $y=x+1$ and the direct proportion function $y=-x$ at the same point. Find the value of $m$. | 5 |
7. A. As shown in Figure 4, the side length of square $ABCD$ is $2 \sqrt{15}, E, F$ are the midpoints of sides $AB, BC$ respectively, $AF$ intersects $DE, DB$ at points $M, N$. Then the area of $\triangle DMN$ is $\qquad$. | 8 |
Given $f(x)=a{e^x}+\frac{1}{2}{x^2}-ax$ is monotonically decreasing on $(0,+\infty)$, the range of real number $a$ is ()
A: $(-\infty ,-1]$
B: $(-\infty ,-1)$
C: $(0,+\infty )$
D: $[0,+\infty)$ | A |
Given that $x$ and $y$ are positive numbers, and $xy + x - y - 10 = 0$, find the minimum value of $x + y$. | 6 |
Let $a$ and $b$ be distinct positive numbers. Solve the equation
$$
\sqrt{x-b^{2}}-\sqrt{x-a^{2}}=a-b
$$ | ^{2}+b^{2} |
$[\underline{\text { Modular Arithmetic (other) }}]$
For which $n$ is the number $n^{2001}-n^{4}$ divisible by 11? | n\equiv0,1\pmod{11} |
8. The system of equations $\left\{\begin{array}{l}x+x y+y=1, \\ x^{2}+x^{2} y^{2}+y^{2}=17\end{array}\right.$ has the real solution $(x, y)=$ | \left(\frac{3+\sqrt{17}}{2}, \frac{3-\sqrt{17}}{2}\right) \text{ or } \left(\frac{3-\sqrt{17}}{2}, \frac{3+\sqrt{17}}{2}\right) |
48. Uncle Li processes a batch of products. For each qualified item he processes, he gets a reward of 2.5 yuan; for each defective item, he is fined 20 yuan. One day, the number of qualified items he processed is 49 times the number of defective items, and he received a total of 205 yuan in rewards. How many defective ... | 2 |
Which of the following operations is correct?
A: $3a+2a=5a^{2}$
B: $ a^{6}\div a^{2}=a^{3}$
C: $ \left(-3a^{3}\right)^{2}=9a^{6}$
D: $ \left(a+2\right)^{2}=a^{2}+4$ | \text{C} |
1286. Find the area of the figure bounded by the line
$$
y^{2}=x(x-1)^{2}
$$ | \frac{8}{15} |
Given circle $C: x^{2}+y^{2}-2x-2\sqrt{3}y+3=0$, and point $A(0,m)$ ($m > 0$), where point $A$ and point $B$ are symmetric about the $x$-axis. If there exists a point $M$ on circle $C$ such that $\vec{AM} \cdot \vec{BM}=0$, then when $m$ is at its maximum value, the coordinates of point $M$ are
A: $\left(\frac{3}{2}, \... | C |
Given lines $l_{1}: 3x+2y+1=0$ and $l_{2}: x-2y-5=0$, let the intersection point of lines $l_{1}$ and $l_{2}$ be $A$. Then, the distance from point $A$ to the line $l_{0}: y=- \dfrac {3}{4}x- \dfrac {5}{2}$ is ( ).
A: $1$
B: $3$
C: $ \dfrac {5 \sqrt {7}}{7}$
D: $ \dfrac {15 \sqrt {7}}{7}$ | A |
The solutions to the equations $z^2=4+4\sqrt{15}i$ and $z^2=2+2\sqrt 3i,$ where $i=\sqrt{-1},$ form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form $p\sqrt q-r\sqrt s,$ where $p,$ $q,$ $r,$ and $s$ are positive integers and neither $q$ nor $s$ is divisible... | 20 |
The angle between plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $\overrightarrow{a}=(2,0)$, $|\overrightarrow{b}|=1$, then $|\overrightarrow{a}+2\overrightarrow{b}|$ equals to. | 2\sqrt{3} |
\section*{Problem 1 - 261011}
What is the last digit of the number
\[
z=4444^{444^{44^{4}}} ?
\] | 6 |
In $\triangle ABC$, $AB > BC > CA, AB = 6, \angle C - \angle B = 90^{\circ}$, circle $O$ is the incircle, $E$ is the tangency point on side $BC$. $EF$ is a diameter of circle $O$. Ray $AF$ intersects side $BC$ at point $D$. If $DE$ equals the circumradius of $\triangle ABC$, find the lengths of sides $BC$ and $AC$.
--... | 8-2\sqrt{7},7-\sqrt{7} |
Given proposition P: $x_1$ and $x_2$ are the two real roots of the equation $x^2 - mx - 2 = 0$, the inequality $a^2 - 5a - 3 \geq |x_1 - x_2|$ holds for any real number $m \in [-1,1]$; Proposition Q: There is only one real number $x$ that satisfies the inequality $x^2 + 2\sqrt{2}ax + 11a \leq 0$. If proposition P is fa... | a = 0 \text{ or } a = \frac{11}{2} |
A group of students go rowing. If they take the large boat, besides one boat that seats 6 people, the rest each seat 17 people; if they take the small boat, besides one boat that seats 2 people, the rest each seat 10 people. If the number of students is more than 100 but less than 200, what is the number of students? | 142 |
Given the set $A=\{x\in Z|x^{2}-2x-3 \lt 0\}$, the number of subsets of set $A$ is ( )
A: $3$
B: $4$
C: $8$
D: $16$ | C |
Given the proposition p: There exists $x_0 \geq 0$ such that $2x_0 = 3$, the negation of p is ( )
A: $\forall x < 0$, such that $2x \neq 3$
B: $\exists x_0 < 0$, such that $2x_0 \neq 3$
C: $\exists x_0 \geq 0$, such that $2x_0 \neq 3$
D: $\forall x \geq 0$, such that $2x \neq 3$ | \text{D} |
Problem 4. One mole of an ideal gas undergoes a closed cycle, in which:
$1-2$ - isobaric process, during which the volume increases by 4 times;
$2-3$ - isothermal process, during which the pressure increases;
$3-1$ - a process in which the gas is compressed according to the law $T=\gamma V^{2}$.
Find how many times... | 2 |
2. Given $n+2$ real numbers
$$
a_{1}, a_{2}, \cdots, a_{n}, 16, a_{n+2} \text {, }
$$
where the average of the first $n$ numbers is 8, the average of the first $n+1$ numbers is 9, and the average of these $n+2$ numbers is 10. Then the value of $a_{n+2}$ is . $\qquad$ | 18 |
# Task 11.5. (12 points)
On the first step, a unit circle was drawn on a sheet of paper, and the circle bounded by it was colored black. On each subsequent step, for each of the circles drawn on the previous step, four new internally tangent circles of equal radii are drawn. These four circles are externally tangent t... | \frac{4\pi}{(1+\sqrt{2})^{2}+4} |
Five. (20 points) Find all positive integers $n$ such that $n=$ $p_{1}^{2}+p_{2}^{2}+p_{3}^{2}+p_{4}^{2}$, where $p_{1}, p_{2}, p_{3}, p_{4}$ are the four smallest distinct positive divisors of $n$. | 130 |
5. If prime numbers $p, q$ satisfy $p^{2}-5 q-36=0$, then the median of the sample $p, q, 9, 16$ is $(\quad$.
(A) 16
(B) 13
(C) 11
(D) 13.5 | D |
8. Find all non-negative integers $a$ and $b$ that satisfy the equation
$$
a^{2}+b^{2}=841 \cdot(a b+1)
$$ | Thesolutions(,b)(underthecondition\leqb)thoseonlythosepairsof(a_{n},b_{n}),whichforeachn\in\mathbb{N}calculatedtheformulas:a_{n}=b_{n-1},b_{n}=k^{2}b_{n-1}-a_{n-1},a_{0}=0,b_{0}=k;herek= |
4. In the Cartesian coordinate plane, the number of integer points (i.e., points with both coordinates as integers) that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \leqslant 3 x \\
y \geqslant \frac{1}{3} x \\
x+y \leqslant 100
\end{array}\right.
$$
is $\qquad$ . (as shown in the figure) | 2551 |
## Zadatak B-4.4.
Zadani su geometrijski likovi $L_{1}, L_{2}, L_{3}, \ldots$ Lik $L_{1}$ jednakostranični je trokut površine 1. Svaki sljedeći lik nastaje tako da se svaka od stranica prethodnog lika podijeli na tri jednaka dijela i iznad srednjeg dijela konstruira novi jednakostranični trokut. Nakon konstrukcije jed... | \frac{8}{5}-\frac{3}{5}\cdot(\frac{4}{9})^{n-1} |
4. Vasya remembers that his friend Petya lives on Kurchatovskaya Street, house number 8, but he forgot the apartment number. In response to a request to clarify the address, Petya replied: “My apartment number is a three-digit number. If you rearrange the digits, you can get five other three-digit numbers. So, the sum ... | 425 |
Meg started with the number 100. She increased this number by $20 \%$ and then increased the resulting number by $50 \%$. Her final result was
(A) 120
(B) 240
(C) 187.5
(D) 200
(E) 180 | 180 |
6. If $A B C D$ is a convex quadrilateral, and $E$, $F, G$, $H$ are the centroids of $\triangle A B C, \triangle B C D, \triangle C D A, \triangle D A B$ respectively, then the ratio of the area of quadrilateral $E F G H$ to the area of quadrilateral $A B C D$ is $\qquad$ | \frac{1}{9} |
6. The sequence $\left\{x_{n}\right\}$ satisfies $x_{1}=\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$. Let $[x]$ denote the greatest integer not exceeding $x$, then $\left[\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2009}}\right]$ is $\qquad$ . | 1 |
$x_1$ and $x_2$ are the two real roots of the equation $x^2 - 2mx + (m^2 + 2m + 3) = 0$. Find the minimum value of $x_1^2 + x_2^2$. | \frac{9}{2} |
Given that $\left[x\right]$ represents the greatest integer not exceeding $x$, for example, $\left[3.2\right]=3$, $\left[-0.7\right]=-1$. Now, define $\{x\}=\left[x\right]-x$, for example, $\{1.5\}=\left[1.5\right]-1.5=-0.5$. Find the value of $\{2.9\}+\{-\frac{5}{3}\}$. | -\frac{37}{30} |
In triangle $\triangle ABC$, $\cos A= \frac {3}{5}$, and $\cos B= \frac {5}{13}$. The value of $\cos C$ is \_\_\_\_\_\_. | \frac {33}{65} |
3. Find the value of the expression $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$, if it is known that $a, b, c$ are three distinct real numbers satisfying the conditions:
$a^{3}-2020 a+1010=0, \quad b^{3}-2020 b+1010=0, c^{3}-2020 c+1010=0$. | 2 |
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