problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
[Example 1.4.4] For a set of $n$ points $M$ in a plane, if every three points in $M$ form the vertices of an equilateral triangle, find the maximum value of $n$.
| 3 |
2. Given a large regular tetrahedron with an edge length of 6, a smaller regular tetrahedron is placed inside it. If the smaller tetrahedron can rotate freely within the larger one, the maximum edge length of the smaller tetrahedron is . $\qquad$ | 2 |
In $\triangle ABC$, if $\tan A= \frac {1}{3}$, $\angle C=150^{\circ}$, and $BC=1$, then the value of $AB$ is ______. | \frac { \sqrt {10}}{2} |
4. Let $f(x)=a x^{2}+b x+c$ be a quadratic trinomial with integer coefficients. If integers $m, n$ satisfy $f(m)-f(n)=1$. Then $|m-n|=$ $\qquad$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 1 |
Given sets $A=\{x|e^{x} \gt 1,x\in R\}$ and $B=\{x|x^{2}-x-2 \lt 0,x\in R\}$, then $A\cup B=\left(\ \ \right)$
A: $(0,1)$
B: $(0,2)$
C: $(-1,+\infty )$
D: $(-2,+\infty )$ | C |
Use mathematical induction to prove that $2^n > 2n + 1$, the first value of $n$ should be ( )
A: 1
B: 2
C: 3
D: 4 | C |
4. On an island, there live knights who always tell the truth, and liars who always lie. One day, 9 natives, among whom were both knights and liars, stood in a circle, and each said: "The two people standing opposite me are liars." Opposite a native stand the fourth native from him in a clockwise direction and the four... | 3or4 |
3. Let $f(x)$ be a polynomial with integer coefficients, for which
$$
f(0)=23, f\left(x_{1}\right)=f\left(x_{2}\right)=f\left(x_{3}\right)=\ldots=f\left(x_{n}\right)=2014
$$
for some distinct $x_{1}, x_{2}, \ldots, x_{n}$. Find the maximum value of $n$. | 4 |
## Task B-4.5.
A piece of paper has the shape of an equilateral triangle $ABC$ with side length $15 \, \text{cm}$. We fold the paper so that vertex $A$ comes to point $D$ on side $\overline{BC}$, with $|BD| = 3 \, \text{cm}$. This creates a fold $\overline{EF}$, where point $E$ is on $\overline{AB}$ and point $F$ is o... | \frac{7\sqrt{7}}{2} |
Find the solution set of the inequality $x(x+3)\geqslant 0$.
A: $\{x|-3\leqslant x\leqslant 0\}$
B: $\{x|x\geqslant 0 \text{ or } x\leqslant -3\}$
C: $\{x|0\leqslant x\leqslant 3\}$
D: $\{x|x\geqslant 3 \text{ or } x\leqslant 0\}$ | \text{B} |
In the diagram, the line with equation $y=2 x-8$ crosses the $x$-axis at $A$ and the $y$-axis at $B$. The area of $\triangle A O B$ is
(A) 8
(D) 32
(B) 16
(C) 12
(E) 4
 | 16 |
8,9
[ Area of a circle, sector, and segment ]
In the acute angles of a right triangle, two equal touching circles are inscribed. The sum of the areas of these circles is equal to the area of the circle inscribed in the triangle. Find the angles of the triangle. | \alpha=45 |
If there exists a line passing through the point (1, 0) that is tangent to both the curve $y=x^3$ and the curve $y=ax^2 + \frac{15}{4}x - 9$, then the value of $a$ is ( )
A: $-1$ or $-\frac{25}{64}$
B: $-1$ or $\frac{21}{4}$
C: $-\frac{7}{4}$ or $-\frac{25}{64}$
D: $-\frac{7}{4}$ or 7 | A |
Find the number of positive integers $k \le 2018$ for which there exist integers $m$ and $n$ so that $k = 2^m + 2^n$.
For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$. | 65 |
3. A five-digit number ABCDE, consisting of different digits, is divisible by both the three-digit number CDE and the two-digit number AB. Find the smallest possible value of ABCDE. | 12480 |
Form a 4-digit number without repeating digits using the numbers 1, 2, 3, 4. How many of these 4-digit numbers are odd? | 12 |
5. Bear Big and Bear Small jog on a circular track that is 1500 meters long every day.
On the first day, both start from the starting point at the same time and run the entire way. When Bear Big completes 4 laps and returns to the starting point, Bear Small has completed 3 laps and an additional 300 meters.
On the seco... | 600 |
Given the function $f(x)=-\frac{1}{2}x^{2}+b\ln x$ is a decreasing function in the interval $[\sqrt{2},+\infty)$, determine the range of values for $b$. | (-\infty, 2] |
If $a(x+b)=3 x+12$ for all values of $x$, then $a+b$ equals
(A) 12
(B) 15
(C) 8
(D) 7
(E) 13 | 7 |
Using a set of student triangles with internal angles (45°, 45°, 90° and 30°, 60°, 90°), the number of different angles that can be drawn greater than 0° and less than 176° is ( )
A: 8
B: 9
C: 10
D: 11 | D |
(7) Nine positive real numbers $a_{1}, a_{2}, \cdots, a_{9}$ form a geometric sequence, and $a_{1}+a_{2}=\frac{3}{4}$, $a_{3}+a_{4}+a_{5}+a_{6}=15$. Then $a_{7}+a_{8}+a_{9}=$ $\qquad$ . | 112 |
Problem 8.5. A car left the city for the village at the same time a cyclist left the village for the city. When the car and the cyclist met, the car immediately turned around and headed back to the city. As a result, the cyclist arrived in the city 35 minutes after the car. How many minutes did the cyclist spend on the... | 55 |
2. Given that \( i \) is the imaginary unit. Then
$$
\mathrm{i}+\mathrm{i}^{2}+\mathrm{i}^{3}+\mathrm{i}^{4}+\cdots+\mathrm{i}^{2013}=(\quad) \text {. }
$$
(A) \(\mathrm{i}\)
(B) \(-\mathrm{i}\)
(C) 0
(D) 1 | A |
7. Let $A B C D E$ be a square pyramid of height $\frac{1}{2}$ with square base $A B C D$ of side length $A B=12$ (so $E$ is the vertex of the pyramid, and the foot of the altitude from $E$ to $A B C D$ is the center of square $A B C D)$. The faces $A D E$ and $C D E$ meet at an acute angle of measure $\alpha$ (so that... | \frac{17}{144} |
18. A water tank is $\frac{5}{6}$ full. When 30 litres of water are removed from the tank, the tank is $\frac{4}{5}$ full. How much water does the tank hold when full?
A 180 litres
B 360 litres
C 540 litres
D 720 litres
E 900 litres | 900 |
A positive integer is called fancy if it can be expressed in the form
$$
2^{a_{1}}+2^{a_{2}}+\cdots+2^{a_{100}},
$$
where $a_{1}, a_{2}, \ldots, a_{100}$ are non-negative integers that are not necessarily distinct.
Find the smallest positive integer $n$ such that no multiple of $n$ is a fancy number.
Answer: The an... | 2^{101}-1 |
1. Class A and Class B have a total of 83 people, Class B and Class C have a total of 86 people, Class C and Class D have a total of 88 people. How many people are there in Class A and Class D? | 85 |
To improve the construction of urban rail transit and enhance the level of urban public transportation services, the total mileage of the second phase of the urban rail transit construction plan in Jinan City from 2020 to 2025 is approximately $159600$ meters. Express the number "$159600$" in scientific notation as ( )... | C |
A regular 3-sided pyramid has a height of $m=11 \mathrm{~cm}$, and the area of one side face is $210 \mathrm{~cm}^{2}$. Let's calculate the volume of the pyramid. | 825\sqrt{3}\approx1429\mathrm{~}^{3} |
The proposition "There exists an $x \in \mathbb{R}$ such that $\lambda x^2 - \lambda x + 1 < 0$ holds" is false. Determine the range of values for $\lambda$. | 0 \leq \lambda \leq 4 |
4. Find all positive real numbers $x, y, z$ such that
$$
2 x-2 y+\frac{1}{z}=\frac{1}{2014}, \quad 2 y-2 z+\frac{1}{x}=\frac{1}{2014}, \quad 2 z-2 x+\frac{1}{y}=\frac{1}{2014}
$$
| 2014,\quad2014,\quad2014 |
What is the last two digits of the number $(11^2 + 15^2 + 19^2 + ... + 2007^2)^2$? | 0 |
Calculate: $(-1)^{2023}-|1-\sqrt{3}|+\sqrt{6} \times \sqrt{\frac{1}{2}}$. | 0 |
3. (7-8 grade) Vasya thought of 5 natural numbers and told Petya all their pairwise sums (in some order): $122,124,126,127,128,129,130,131,132,135$. Help Petya determine the smallest of the numbers Vasya thought of. Answer: 60. | 60 |
If the graphs of several functions can coincide after translation, then these functions are called "functions of the same cluster". Given the following functions:
1. $f(x)=\sin x\cos x$;
2. $f(x)=2\sin (x+ \frac {\pi}{4})$;
3. $f(x)=\sin x+ \sqrt {3}\cos x$;
4. $f(x)= \sqrt {2}\sin 2x+1$.
Among them, the "f... | C |
Example 5. Find the largest integer $A$, such that for any permutation of all natural numbers from 1 to 100, there are 10 consecutive numbers whose sum is greater than or equal to $A$.
[22nd Polish Mathematical Olympiad] | 505 |
(4 points) Among the functions defined in their domain that satisfy $f(x) \cdot f(y) = f(x+y)$, the correct one is ()
A: $f(x) = kx$ ($k \neq 0$)
B: $f(x) = ax$ ($a > 0$ and $a \neq 1$)
C: $f(x) = \log_a x$ ($a > 0$ and $a \neq 1$)
D: $f(x) = ax^2 + bx + c$ ($a \neq 0$) | \text{B} |
The chord length obtained by the intersection of the line $x-3y+3=0$ and the circle $(x-1)^{2}+(y-3)^{2}=10$ is ( ).
A: $\sqrt {30}$
B: $\dfrac {5 \sqrt {3}}{2}$
C: $4 \sqrt {2}$
D: $3 \sqrt {3}$ | A |
I2.3 Let $R=(\sqrt{\sqrt{3}+\sqrt{2}})^{Q}+(\sqrt{\sqrt{3}-\sqrt{2}})^{Q}$. Find the value of $R$. | 10 |
Given the sets $M=\{x \mid -1 \leqslant x \leqslant 2\}$ and $N=\{y \mid y=2^{x}\}$, then the intersection $M \cap N=$ ()
A: $(0,2]$
B: $(0,2)$
C: $[0,2]$
D: $[2,+\infty)$ | A |
The chord length cut by the line $ax+2by+4=0$ and the circle $x^2+y^2+4x+2y+1=0$ is $4$. What is the minimum value of $a^2+b^2$?
A: $3$
B: $2$
C: $\sqrt{2}$
D: $1$ | \text{B} |
Marie Curie discovered radium, a radioactive element. When 1 kilogram of radium completely decays, the heat released is equivalent to the heat released by the combustion of 375,000 kilograms of coal. It is estimated that the Earth's crust contains 10 billion kilograms of radium. The heat released by the complete decay ... | 3.75 \times 10^{15} |
Find the values of $ t\in{[0,\ 2]}$ for which $ \int_{t\minus{}3}^{2t} 2^{x^2}\ dx$ is maximal and minimal. | t = -1 + \frac{\sqrt{33}}{3} |
Define a new operation: $a*b=a+b$, $a\otimes b=a-b$, where $a, b$ are rational numbers. For example, if $a=2$, $b=1$, then $a*b=2+1=3$, $a\otimes b=2-1=1$. Based on the above operation rules, simplify: $a^2b*3ab+5a^2b\otimes4ab$, and calculate the value of the polynomial when $a=5$, $b=3$. | 435 |
6. If real numbers $x, y$ satisfy $x \geqslant 0$, and
$$
\max \{1-x, x-1\} \leqslant y \leqslant x+2 \text {, }
$$
then the minimum value of the bivariate function $u(x, y)=2 x+y$ is
$\qquad$ . | 1 |
11th USAMO 1982 Problem 1 A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points? Solution | 1979 |
4. Given the equation $|x|=a x+1$ has one negative root and no positive root, then the range of values for $a$ is
(A) $a>-1$;
(B) $a=1$;
(C) $a \geqslant 1$;
(D) None of the above. | \geqslant1 |
A certain brand of flour is labeled with a mass of $\left(35\pm 0.25\right)kg$. Among the following $4$ bags of flour, the one with an acceptable mass is:
A: $34.70kg$
B: $34.80kg$
C: $35.30kg$
D: $35.51kg$ | B |
Find the value of $a$ such that in the expansion of $\left( \frac {a}{x} - \sqrt[3]{x^2}\right)^7$, the coefficient of the $x^3$ term is 14.
A: $-2$
B: $14$
C: $2$
D: $-14$ | C |
Calculate the value of $\sin 36^\circ \cos 24^\circ + \cos 36^\circ \sin 156^\circ = \_\_\_\_\_\_$. | \frac {\sqrt {3}}{2} |
A cube and the planes containing its faces divide the space into __________ parts. | 27 |
377. It is known that the number $\boldsymbol{e}$ (the base of natural logarithms) is defined as the limit of the expression $\left(1+\frac{1}{n}\right)^{n}$ as $n$ increases without bound:
$$
e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}
$$
But one might reason as follows: if the exponent of a base sl... | 2.71828\ldots |
[ Cubic Polynomials]
Solve the equation $x^{3}+x^{2}+x=-1 / 3$.
# | -\frac{1}{\sqrt[3]{2}+1} |
[ Projections of the bases, sides, or vertices of a trapezoid ] The Law of Sines
A quadrilateral $K L M N$ is inscribed in a circle of radius $R$, $Q$ is the point of intersection of its diagonals, $K L = M N$. The height dropped from point $L$ to side $K N$ is 6, $K N + L M = 24$, and the area of triangle $L M Q$ is ... | LM=4,KN=20,KL=MN=10;R=5\sqrt{5} |
Solve for real $x$ : \[ (x^2 + x -2 )^3 + (2x^2 - x -1)^3 = 27(x^2 -1 )^3. \] | x = 1, x = -1, x = -2, x = -\frac{1}{2} |
8. Given that the positive integer $n$ is a common multiple of 7 and 8, all the digits of $n$ are either 7 or 8, and both digits 7 and 8 appear at least once. Then the smallest value of $n$ that satisfies the above conditions is $\qquad$ . | 7888888 |
Boodanov I.I.
On the board, nine quadratic trinomials are written: $x^{2}+a_{1} x+b_{1}, x^{2}+a_{2} x+b_{2}, \ldots, x^{2}+a_{9} x+b_{9}$. It is known that the sequences $a_{1}, a_{2}, \ldots, a_{9}$ and $b_{1}, b_{2}, \ldots, b_{9}$ are arithmetic progressions. It turned out that the sum of all nine trinomials has a... | 4 |
1. Find all real numbers $x$ for which the inequality
$$
|||2-x|-x|-8| \leq 2008
$$
holds. | x\geq-1007 |
The degree of the monomial $-\frac{1}{4}π\;{x^2}{y^3}$ is ( ).
A: $-\frac{1}{4}$
B: $-\frac{1}{4}π$
C: $5$
D: $6$ | C |
The probability of a giant panda living to 10 years old is 0.8, and the probability of living to 15 years old is 0.6. If there is a giant panda that is already 10 years old, then the probability of it living to 15 years old is ( )
A: 0.8
B: 0.75
C: 0.6
D: 0.48 | B |
Given that the parametric equation of line $l$ is
$$
\begin{cases}
x=m+ \frac{\sqrt{2}}{2}t \\
y= \frac{\sqrt{2}}{2}t
\end{cases}
(t \text{ is the parameter}),
$$
with the origin as the pole and the positive $x$-axis as the initial line of a polar coordinate system, the polar equation of curve $C$ is $\rho^{2}\cos^{2... | 16 |
If the lateral surface of a cone is the area of a semicircle with an area of $2\pi$, then the volume of the cone is ____. | \frac{\sqrt{3}}{3}\pi |
3. Find the number of natural numbers $k$, not exceeding 485000, such that $k^{2}-1$ is divisible by 485. | 4000 |
When using the completing the square method to solve the equation $x^{2}+2x-3=0$, the correct result is:
A: $(x+2)^{2}=3$
B: $(x+1)^{2}=3$
C: $(x+2)^{2}=4$
D: $(x+1)^{2}=4$ | D |
5. Solve the system of equations:
$$
\left\{\begin{array}{l}
x y=1 \\
x+y+\cos ^{2} z=2
\end{array}\right.
$$
Answer: $x=y=1 ; z=\frac{\pi}{2}+\pi n$, where $n \in Z$. | 1;\frac{\pi}{2}+\pin, |
3. $a, b, c$ are non-negative real numbers, and satisfy $3a+2b+c=$ $5, 2a+b-3c=1$. Let $m=3a+b-7c$, and let $x$ be the minimum value of $m$, $y$ be the maximum value of $m$. Then $xy=$ $\qquad$ | \frac{5}{77} |
The graph of the odd function $f(x)=ax^{3}+bx^{2}+cx$ passes through points $A(-\sqrt{2}, \sqrt{2})$ and $B(2\sqrt{2},10\sqrt{2})$.
(1) Find the expression for $f(x)$;
(2) Determine the intervals of monotonicity for $f(x)$;
(3) If the equation $f(x)+m=0$ has three distinct real roots, find the range of values for $m... | (-2,2) |
Given point $A(1,-3)$ and vector $a=(3,4)$, if $\overrightarrow{AB}=2a$, find the coordinates of point $B$. | (7,5) |
Given an ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$ with its foci at $F_1$ and $F_2$, and the point $A$ has coordinates $(0,b)$. The line $AF_2$ intersects the ellipse at another point $B$. The perimeter of $\Delta AF_1B$ is $4\sqrt{3}$, the distance from $F_1$ to the line $AF_2$ is $\dfrac{2\sqrt{6}... | -\dfrac{7}{6} |
Three individuals, A, B, and C, work in the same office. There is only one telephone in the office. The probabilities that incoming calls through this phone are for A, B, and C are $\frac{1}{6}$, $\frac{1}{3}$, and $\frac{1}{2}$, respectively. If during a certain period there are three incoming calls, and each call is ... | \frac{5}{72} |
## Task 4 - 210734
Given a circle $k$ with center $M$. On $k$ lie the points $A$ and $B$ such that the angle $\angle B M A$ is a right angle. Furthermore, a point $C$ is defined by the following conditions:
(1) $C$ lies on $k$.
(2) It holds that $M B = B C$.
(3) The line through $A$ and $C$ intersects the segment $... | 75 |
The solution set of the inequality $x-1 \gt 0$ is ______. | x \gt 1 |
Given the function $f(x)=x^{2}-6x-9$, then the range of the function $f(x)$ for $x\in(1,4)$ is \_\_\_\_\_\_. | [-18,-14) |
Shooting 3 times, the event $A_i$ represents "hitting i times", where $i=0, 1, 2, 3$. Then $A=A_1 \cup A_2 \cup A_3$ represents ( )
A: All hits
B: At least 1 hit
C: Certain hit
D: 3 hits | \text{B} |
9. In one line, 19 numbers were written. The sum of any three consecutive numbers is positive. Can the sum of all 19 numbers be negative? | -1 |
10. Given the sequence $\left\{x_{n}\right\}, x_{1}=1$, and $x_{n+1}=\frac{\sqrt{3} x_{n}+1}{\sqrt{3}-x_{n}}, x_{1999}-x_{601}=$ | 0 |
The sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n=2n^2-3n$ $(n\in\mathbb{N}^*)$. Find $a_n$. | 4n-5 |
$2 \cdot 32$ Let $A$ be a set of integers, where the smallest element is 1 and the largest element is 100. Except for 1, each element is equal to the sum of two elements in $A$ (which can be twice a single element). Find the minimum number of elements in set $A$. | 9 |
19. 2017 minus its $\frac{1}{2}$, then minus the remaining $\frac{1}{3}$, then minus the remaining $\frac{1}{4}$, and so on, until finally minus the remaining $\frac{1}{1000}$, the final result is $m$, then the range of $m$ is ( ).
(A). $03$ | 2<\leqslant3 |
Given a sector with a radius of 2 and an area of 4, the absolute value of the central angle of this sector is \_\_\_\_\_\_ radians. | 2 |
$n$ family couples came to the ball. In each couple, the husband and wife are of exactly the same height, but no two couples have the same height. A waltz begins, and all the attendees randomly form pairs: each gentleman dances with a randomly chosen lady. Find the mathematical expectation of the random variable $X$ "T... | \frac{1}{2}(n-1) |
## 172. Math Puzzle $9 / 79$
The Becker family from Halle has three children. They want to travel by train to visit relatives in Bitterfeld.
For Anja, this trip is still free of charge, as she is only two years old. Silke and Frank each have to pay half the adult fare. However, the family receives a one-third discoun... | 9.60\mathrm{M} |
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$ | 3 |
The number that is 17 more than 5 times X is , and the number that is less than 5 times 22 by Y is . | 22 \times 5 - Y |
9.1. Chords $A A^{\prime}, B B^{\prime}$, and $C C^{\prime}$ of a sphere intersect at a common point $S$. Find the sum $S A^{\prime}+S B^{\prime}+S C^{\prime}$, if $A S=6, B S=3, C S=2$, and the volumes of pyramids $S A B C$ and $S A^{\prime} B^{\prime} C^{\prime}$ are in the ratio $2: 9$. If the answer is not an integ... | 18 |
Problem 8.7. Given an isosceles triangle $A B C$, where $A B=A C$ and $\angle A B C=53^{\circ}$. Point $K$ is such that $C$ is the midpoint of segment $A K$. Point $M$ is chosen such that:
- $B$ and $M$ are on the same side of line $A C$;
- $K M=A B$
- angle $M A K$ is the maximum possible.
How many degrees does angl... | 44 |
8.3.2. Seven numbers form an increasing arithmetic progression. The sum of their cubes is zero, and the sum of their squares is -224. Find the largest of these numbers. | 6\sqrt{2} |
8.2. It is known that $70 \%$ of mathematicians who have moved to IT regret their change of activity. At the same time, only $7 \%$ of all people who have moved to IT regret the change. What percentage of those who have moved to IT are mathematicians, if only they regret the change of activity? | 10 |
The solution set of the inequality $8-x^{2} \gt 2x$ is ______. | (-4,2) |
Determine the coefficient of $x^{2}y^{7}$ in the expansion of \\((x-2y)(x+y)^{8}\\). (Answer with a number.) | -48 |
87. The natural numbers $1,2,3,4,5 \cdots \cdots$ are arranged in sequence, and the multiples of 2 and 3 are crossed out, with all multiples of 7 being retained. The 2007th remaining number is $\qquad$. | 4683 |
Given that the terminal side of angle $\alpha$ intersects the unit circle at the coordinates $(\frac{1}{2}, \frac{\sqrt{3}}{2})$, find the value of $\cos \alpha$. | \frac{1}{2} |
Among the following four propositions:
(1) If $\alpha > \beta$, then $\sin\alpha > \sin\beta$
(2) The negation of the proposition "For all $x > 1$, $x^2 > 1$" is "There exists $x \leq 1$, $x^2 \leq 1$"
(3) The necessary and sufficient condition for the lines $ax + y + 2 = 0$ and $ax - y + 4 = 0$ to be perpendicul... | (3) |
Let the modulus of the complex number $a+bi$ ($a,b \in \mathbb{R}$) be $\sqrt{2018}$. Then, $(a+bi)(a-bi)=$ ______. | 2018 |
8.3. The height AH of isosceles triangle $\mathrm{ABC}$ with base $\mathrm{AC}$ is equal to the length of the perpendicular MK dropped from the midpoint M of side AB to the base of the triangle. Find the perimeter of triangle $\mathrm{ABC}$, if $\mathrm{AK}=\mathrm{a}$. | 20a |
5. The center of a circle with radius 2 lies on the circumference of a circle with radius 3. Find the area of the intersection of the circles bounded by these circumferences. | 9\pi-14\arccos\frac{1}{3}-4\sqrt{2} |
A sequence $a_1, a_2, a_3, \ldots$ of positive integers satisfies $a_1 > 5$ and $a_{n+1} = 5 + 6 + \cdots + a_n$ for all positive integers $n$. Determine all prime numbers $p$ such that, regardless of the value of $a_1$, this sequence must contain a multiple of $p$. | p = 2 |
At a national math contest, students are being housed in single rooms and double rooms; it is known that $75\%$ of the students are housed in double rooms. What percentage of the rooms occupied are double rooms?
[i]Proposed by Evan Chen[/i] | 60\% |
Which of the following events is a certain event?
A: Xiao Wang scores full marks in a math test
B: A certain shooter hits the bullseye in one shot
C: The Chinese soccer team beats the Brazilian team $3:0$ in a World Cup match
D: There are $2$ red balls and $1$ white ball in a pocket. If $2$ balls are drawn, at leas... | \text{D} |
Given $a$, $b$, $c$ are the sides opposite to the angles of $\triangle ABC$.
1. If the condition $a\sin A = b\sin B$ is satisfied, prove that $\triangle ABC$ is an isosceles triangle.
2. If $a+b=ab$, the length of side $c=2$, and angle $C= \frac{\pi}{3}$, find the area of $\triangle ABC$. | \sqrt{3} |
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