problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$. If for $x \geq 0$, we have $f(x+2) = f(x)$, and when $x \in [0, 2)$, $f(x) = \log_{2}(x+1)$, then $f(2012) + f(-2013) = $ （ ）
A: 1
B: 2
C: -1
D: -2 | \text{C} |
Find all pairs $(x,y)$ with $x,y$ real, satisfying the equations $$\sin\bigg(\frac{x+y}{2}\bigg)=0~,~\vert x\vert+\vert y\vert=1$$ | \left(\frac{1}{2}, -\frac{1}{2}\right), \left(-\frac{1}{2}, \frac{1}{2}\right) |
If the sum of the maximum and minimum values of the function $y=a^x$ ($a>1$) on the interval $[0,1]$ is 3, then $a=\ $. | 2 |
In the diagram, $P T$ and $P S$ are tangents to a circle with centre $O$. The point $Y$ lies on the circumference of the circle; and the point $Z$ is where the line $P Y$ meets the radius $O S$.
Also, $\angle S P Z=10^{\circ}$ and $\angle T O S=150^{\circ}$.
How many degrees are there in the sum of $\angle P T Y$ and... | 160 |
2. Let the set $M=\left\{x \mid x=2^{n}-2^{k}\right.$, where $n, k \in$ $\mathbf{N}$, and $n>k\}, P=\{x \mid 1912 \leqslant x \leqslant 2004$, and $x$ $\in \mathbf{N}\}$. Then, the sum of all elements in the set $M \cap P$ is $\qquad$. | 3904 |
Calculate: $\sqrt{3}\tan30°-{({\frac{1}{2}})^{-1}}+\sqrt{8}-|{1-\sqrt{2}}|$. | \sqrt{2} |
Example 3 Find the range of $y=\sqrt{x^{2}-2 x+5}+\sqrt{x^{2}-6 x+13}$.
Analysis: If points $A$ and $B$ are placed above the $x$-axis, then $AB$ is parallel to the $x$-axis, so we can solve the problem by placing the points below the $x$-axis. | [2 \sqrt{5},+\infty) |
3. For arbitrary real numbers $a$ and $b (a \neq 0)$, find the minimum value of the expression $\frac{1}{a^{2}}+2 a^{2}+3 b^{2}+4 a b$. | \sqrt{\frac{8}{3}} |
3. Let $A B C D$ be a quadrilateral for which $\angle B A C=\angle A C B=20^{\circ}, \angle D C A=30^{\circ}$ and $\angle C A D=40^{\circ}$. Determine the measure of the angle $\angle C B D$. | 80 |
6. Among the 99 positive integers $1,2, \cdots, 99$, if $k$ numbers are arbitrarily taken, such that there must be two numbers $a$ and $b (a \neq b)$ satisfying $\frac{1}{2} \leqslant \frac{b}{a} \leqslant 2$. Then the smallest possible value of $k$ is ( ).
(A) 6
(B) 7
(C) 8
(D) 9 | B |
## Problem Statement
Calculate the definite integral:
$$
\int_{-1}^{0}(x+2)^{3} \cdot \ln ^{2}(x+2) d x
$$ | 4\ln^{2}2-2\ln2+\frac{15}{32} |
7.106 Given $f(x)=a \sin x+b \sqrt[3]{x}+4$ (where $a, b$ are real numbers) and $f\left(\lg _{\left.\log _{3} 10\right)} 10\right)$. Then the value of $f(\operatorname{lglg} 3)$ is
(A) -5.
(B) -3.
(C) 3.
(D) It takes different values depending on the values of $a, b$.
(China High School Mathematics League, 1993) | 3 |
Given the function $f(x) = \frac{2}{x-1}$, where $x \in [2, 6]$
(1) Prove that the function $f(x)$ is a decreasing function on the interval $[2, 6]$;
(2) Find the maximum and minimum values of the function $f(x)$ on the interval $[2, 6]$. | \frac{2}{5} |
Example 12. There are three urns with balls. The first contains 5 blue and 3 red balls, the second - 4 blue and 4 red, and the third - 8 blue. One of the urns is randomly chosen, and a ball is randomly drawn from it. What is the probability that it will be red (event $A$). | \frac{7}{24} |
The condition "$-3 \lt m \lt 1$" is ( ) for the inequality $\left(m-1\right)x^{2}+\left(m-1\right)x-1 \lt 0$ to hold for any $x\in R$.
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither necessary nor sufficient condition | \text{A} |
Which of the following expressions represents $y$ as a direct proportion to $x$?
A: $y=x^{2}$
B: $y=\frac{2}{x}$
C: $y=\frac{x}{3}$
D: $y^{2}=3x$ | C |
$\left.\begin{array}{l}{[\quad \text { Properties of sections }} \\ {[\text { Theorems of Ceva and Menelaus }]}\end{array}\right]$
Construct the section of the triangular pyramid $A B C D$ by a plane passing through the midpoints $M$ and $N$ of edges $A C$ and $B D$ and point $K$ on edge $C D$ such that $C K: K D=1: 2... | 1:2 |
5. Given the set $\{1,2,3, \cdots, 3 n-1,3 n\}$, it can be divided into $n$ mutually disjoint triples $|x, y, z|$, where $x+y=3 z$. The two smallest positive integers $n$ that satisfy the above requirement are $\qquad$. | 5,8 |
The function $f(x) = x^3 + ax - 2$ is increasing in the interval $[1, +\infty)$, then the range of the real number $a$ is ( )
A: $[3, +\infty)$
B: $[-3, +\infty)$
C: $(-3, +\infty)$
D: $(-\infty, -3)$ | \text{B} |
$i$ is the imaginary unit. If a complex number $z$ satisfies $\dfrac{1+z}{1-z}=i$, then the value of the complex number $z^{2019}$ is:
A: $i$
B: $-1$
C: $-i$
D: $1$ | \text{C} |
8. In a right trapezoid, the upper base is shorter than the lower base. If the volume of the solid of revolution obtained by rotating it around its lower base is $80 \pi$,
the volume of the solid of revolution obtained by rotating it around its upper base is $112 \pi$, and the volume of the solid of revolution obtaine... | 16+2\sqrt{13} |
8.1. Find any four different natural numbers that have the following property: if the product of any two of them is added to the product of the other two numbers, the result is a prime number. | 1,2,3,5 |
Example 13 Given that $a$, $b$, and $c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$. If the distances from $A$ and $B$ to the origin are both less than 1, find the minimum value of $a+b+c$.
(1996, National Junior High School Mathematics League... | 11 |
Given an ellipse Γ: $$\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}$$\=1 (a＞b＞0) with its left vertex M (-2, 0) and an eccentricity of $$\frac { \sqrt {2}}{2}$$.
1. Find the equation of the ellipse Γ.
2. A line AB passing through N (1, 0) intersects the ellipse Γ at points A and B. When $$\overrightarrow {MA}\cdot \overr... | \frac {3 \sqrt {6}}{2} |
In $\triangle ABC$, $A = \frac{\pi}{4}$, $\cos B = \frac{\sqrt{10}}{10}$, then $\sin C$ equals ( )
A: $\frac{2 \sqrt{5}}{5}$
B: $-\frac{2 \sqrt{5}}{5}$
C: $\frac{\sqrt{5}}{5}$
D: $-\frac{\sqrt{5}}{5}$ | A |
Which of the following statements about $\sqrt[3]{-8}$ is incorrect?
A: Represents the cube root of $-8$
B: Results in $-2$
C: Is equal to $-\sqrt[3]{8}$
D: Is meaningless | \text{D} |
1. Given $a>b>2$, and $a+b$, $a-b$, $a b$, $\frac{b}{a}$ form a geometric sequence in a certain order. Then $a=$ $\qquad$ | 7+5\sqrt{2} |
4. Given that $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$ are both equilateral triangles, and the bases of the two triangles $B C=B^{\prime} C^{\prime}$, and $A B>A^{\prime} B^{\prime}$. The orthocenters of $\triangle A B C$ and $\triangle A^{\prime} B^{\prime} C^{\prime}$ are denoted as $H$ and ... | B |
The number of integer solutions to the equation $|xyz| = 4$ is ( )
A: 64
B: 48
C: 8
D: 6 | \text{B} |
Given $\left\{\begin{array}{l}2x+4y=5\\ x-y=10\end{array}\right.$, the value of $x+y$ is ( )
A: $4$
B: $5$
C: $6$
D: $7$ | B |
**Text**: Among the following random variable $X$'s distribution, which does not belong to a binomial distribution is __________.
1. According to the CCTV News, the probability of a computer being infected by a certain virus after downloading data from a certain website within the next week is 0.65. Suppose that a co... | 2 |
1. Find the smallest natural number ending with the four-digit number 2018, which is a multiple of the number 2017. | 8782018 |
## Task Condition
Approximately calculate using the differential.
$y=\sqrt{4 x-1}, x=2.56$ | 3.04 |
Problem 13. In triangle $A B C$, side $A C=$ $=5 \text{ cm}, B C-A B=2 \text{ cm}, \angle A: \angle C=2$. Find the lengths of sides $A B$ and $B C$. | AB=4 |
## Zadatak B-2.5.
Jednadžbe $x^{2}-45 x+4 a+4=0$ i $x^{2}-47 x+5 a-4=0, a \in \mathbb{R}, a<20$ imaju jedno zajedničko realno rješenje. Koliko iznosi umnožak preostalih dvaju rješenja tih jednadžbi?
| 2024 |
4. Let $p(x)$ be a polynomial of degree $2n$, $p(0)=p(2)=\cdots=p(2n)=0, p(1)=p(3)=\cdots=$ $p(2n-1)=2, p(2n+1)=-30$. Find $n$. | 2 |
Given three points $A(1,0)$, $B(0,\sqrt{3})$, $C(2,\sqrt{3})$, find the distance from the center of the circumcircle of $\triangle ABC$ to the origin. | \frac{\sqrt{21}}{3} |
11. (10 points) There are 20 piles of stones, each containing 2006 stones. The rule is: taking one stone from each of any 19 piles and placing them into another pile is considered one operation. After fewer than 20 such operations, one pile has 1990 stones, and another pile has between 2080 and 2100 stones. How many st... | 2090 |
(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$.
(b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple... | G_n = 0 |
8. Let $f(x)=\cos \frac{x}{5}, a=f\left(\log _{0} \frac{1}{\pi}\right)$,
$$
b=f\left(\log _{\pi} \frac{1}{\mathrm{e}}\right), c=f\left(\log _{\frac{1}{4}} \frac{1}{\pi^{2}}\right) .
$$
Then the correct relationship among the following is ( ).
(A) $a>b>c$
(B) $b>c>a$
(C) $c>a>b$
(D) $b>a>c$ | D |
A rectangular plot of land with an area of $800 \mathrm{~m}^{2}$ is to be fenced along a river using $100 \mathrm{~m}$ of fencing. What are the dimensions of the plot? | =10b=40or=20b=40 |
Given that $F\_1$ and $F\_2$ are the left and right foci of the ellipse $\frac{x^{2}}{4} + \frac{y^{2}}{3} = 1$, and point $P(x, y)$ is on the line $y - x - 3 = 0$ $(x \neq -3$ and $x \neq \pm \sqrt{3})$, the slopes of lines $PF\_1$ and $PF\_2$ are $k\_1$ and $k\_2$ respectively. Find the value of $\frac{1}{k\_2} - \fr... | D |
Solve the equations:
(1) $(x+1)^2-5=0$;
(2) $2x^2-4x+1=0$;
(3) $(2x+1)(x-3)=-7$;
(4) $3(x-2)^2=x(x-2)$. | x_1=2, x_2=3 |
At a bus stop where buses numbered $1$, $3$, $5$, and $8$ all stop (assuming that only one bus can stop at this station at a time), a passenger is waiting for either bus number $1$ or $3$. Assuming that the probability of each bus arriving first is equal, the probability that the first bus to arrive is the one the pass... | \frac{1}{2} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and it satisfies $(2a-c)\cos B = b\cos C$;
(1) Find the magnitude of angle $B$;
(2) Let $\overrightarrow{m}=(\sin A, \cos 2A), \overrightarrow{n}=(4k,1)$ ($k>1$), and the maximum value of $\overrightarr... | \frac{3}{2} |
9. Three pairwise coprime positive integers, all greater than 1, have a product of 27000. Then their sum is ( ).
(A) 100
(B) 137
(C) 156
(D) 160
(E) 165 | D |
2. Let $n_{1}, n_{2} \in \mathbb{R}$ and $f\left(n_{1}\right)=f\left(n_{2}\right)$. Then
$6 n_{1}+5=(f \circ f)\left(n_{1}\right)=f\left(f\left(n_{1}\right)\right)=f\left(f\left(n_{2}\right)\right)=(f \circ f)\left(n_{2}\right)=6 n_{2}+5 \Rightarrow n_{1}=n_{2}$
In the given relation, we substitute $n$ with $f(n)$ an... | 431 |
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous. | 0 |
Example 6 For any subset $A$ of $X_{n}=\{1,2, \cdots, n\}$, denote the sum of all elements in $A$ as $S(A), S(\varnothing)=0$, and the number of elements in $A$ as $|A|$. Find $\sum_{A \subseteq X_{n}} \frac{S(A)}{|A|}$. | \frac{1}{2}(n+1)\left(2^{n}-1\right) |
The three logical structures of an algorithm are ( )
A: Sequential structure; Flow structure; Loop structure
B: Sequential structure; Conditional structure; Nested structure
C: Sequential structure; Conditional structure; Loop structure
D: Flow structure; Conditional structure; Loop structure | C |
Suppose that the number $a$ satisfies the equation $4 = a + a^{ - 1}$. What is the value of $a^{4} + a^{ - 4}$?
$\textbf{(A)}\ 164 \qquad \textbf{(B)}\ 172 \qquad \textbf{(C)}\ 192 \qquad \textbf{(D)}\ 194 \qquad \textbf{(E)}\ 212$ | 194 |
If the solution to the equation $3x + (2k-1) = x - 6(3k+2)$ with respect to $x$ is $x=1$, then the value of $k$ is ( )
A: $-\dfrac{13}{20}$
B: $\dfrac{13}{20}$
C: $\dfrac{9}{20}$
D: $-\dfrac{9}{20}$ | A |
If the sequence $\{a_{n}\}$ satisfies $(a_{i}-a_{i+2})(a_{i+2}-a_{i+1}) \gt 0$ for any three consecutive terms $a_{i}$, $a_{i+1}$, $a_{i+2}$, then the sequence is called a "jump sequence." Which of the following statements is correct?
A: There exists an arithmetic sequence $\{a_{n}\}$ that is a "jump sequence."
B: Th... | \text{C} |
In the three-dimensional Cartesian coordinate system $Oxyz$, the point symmetric to $(-1,2,1)$ with respect to the plane $xOz$ is ( ).
A: $(-1,-2,1)$
B: $(-1,2,1)$
C: $(-1,-2,-1)$
D: $(1,-2,-1)$ | A |
2. Let $A=[-2,4), B=\left\{x \mid x^{2}-a x-4 \leqslant 0\right\}$. If $B \subseteq A$, then the range of real number $a$ is ( ).
(A) $[-1,2)$
(B) $[-1,2]$
(C) $[0,3]$
(D) $[0,3)$ | D |
Observe the following equations: $2^3 = 3 + 5$, $3^3 = 7 + 9 + 11$, $4^3 = 13 + 15 + 17 + 19$, $5^3 = 21 + 23 + 25 + 27 + 29$. If, in a similar manner, $m^3$ is decomposed and the last number on the right side of the equation is 131, then the positive integer $m$ equals \_\_\_\_\_\_\_\_\_. | 11 |
The equation of the circle with center at (-1, 2) and tangent to the line $2x-y-1=0$ is \_\_\_\_\_\_. | (x+1)^2+(y-2)^2=5 |
What is the integer equal to $2023^{4}-(2022)(2024)\left(1+2023^{2}\right)$ ? | 1 |
478. Two pedestrians set out simultaneously from point A in the same direction. The first one met a tourist walking towards A, 20 minutes after leaving A, while the second one met the tourist 5 minutes later than the first. Ten minutes after the second meeting, the tourist arrived at A. Find the ratio of the speeds of ... | \frac{15}{8} |
Given vectors $|\vec{a}| = \sqrt{3}$, $|\vec{b}| = 2$, and $\vec{a} \cdot (\vec{a} - \vec{b}) = 0$, find the magnitude of $\vec{a} - \vec{b}$. | 1 |
Given the functions $f(x)=ax-\ln x$ and $g(x)=\ln (x^{2}-2x+a)$,
(1) If $a=0$, find the zeros of the function $F(x)=f(x)+g(x)$;
(2) Let proposition $P$: $f(x)$ is monotonically decreasing on $\left[ \frac{1}{4}, \frac{1}{2}\right]$, and proposition $q$: the domain of $g(x)$ is $\mathbb{R}$. If the compound proposition ... | a \in (1, 2] |
14. As shown in Figure 3, given that point $P$ is an external point of $\odot O$, $P S$ and $P T$ are two tangents to $\odot O$, a secant line $P A B$ through point $P$ intersects $\odot O$ at points $A$ and $B$, and intersects $S T$ at point $C$. Prove:
$$
\frac{1}{P C}=\frac{1}{2}\left(\frac{1}{P A}+\frac{1}{P B}\rig... | \frac{1}{P C}=\frac{1}{2}\left(\frac{1}{P A}+\frac{1}{P B}\right) |
Determine the maximum value of the function $f(x) = \sqrt{x^4 - 3x^2 - 6x + 13} - \sqrt{x^4 - x^2 + 1}$. | \sqrt{10} |
4. Mary found that when a certain three-digit number is squared, the last three digits of the product are the same as the original three-digit number. What is the sum of all different three-digit numbers that satisfy this property? | 1001 |
Find the height and the radius of the base of the cylinder of the largest volume inscribed in a sphere of radius $R$.
# | \frac{2R}{\sqrt{3}};R\sqrt{\frac{2}{3}} |
Let the set $M=\{x|x > 1\}$, and $P=\{x|x^{2}-6x+9=0\}$. Among the following relationships, the correct one is $\quad$
A: $M=P$
B: $P \subsetneq M$
C: $M \subsetneq P$
D: $M \cup P = \mathbb{R}$ | B |
Given that the eccentricity of the hyperbola $C$ is $2$, with left and right foci $F\_1$ and $F\_2$ respectively, and point $A$ is on $C$. If $|F\_1A|=2|F\_2A|$, then $\cos ∠AF\_2F\_1=$ _____ . | \\frac{1}{4} |
Given a quadratic function $f(x)=ax^2+bx+c$ $(a > 0, b, c \in \mathbb{R})$, and $f\left( \frac{1}{a} \right) < 0$, then the number of zeros of the function $y=f\left( f\left( x \right) \right)$ is $(\quad)$
A: $1$
B: $2$
C: $3$
D: $4$ | D |
Given the function $y=a^{1-x}$ ($a > 0$ and $a \neq 1$), its graph always passes through point $A$. If point $A$ lies on the line $mx+ny-1=0$ ($mn > 0$), then the minimum value of $\dfrac{1}{m} + \dfrac{2}{n}$ is \_\_\_\_\_\_. | 3+2\sqrt{2} |
Fill in the corresponding sets with the following numbers: $8$, $-1$, $-0.4$, $\frac{3}{5}$, $0$, $\frac{1}{3}$, $-1\frac{3}{7}$, $-\left(-5\right)$, $-|-\frac{20}{7}|$.
Positive number set $\{ \_\_\_\_\_\_\ldots \}$;
Negative number set $\{ \_\_\_\_\_\_\ldots \}$;
Integer set $\{ \_\_\_\_\_\_\ldots \}$;
Fracti... | \{8, \frac{3}{5}, 0, \frac{1}{3}, -(-5)\} |
If a natural number $n$ makes the addition operation $n + (n + 1) + (n + 2)$ without any carry-over, then $n$ is called a "powerful number". For example, $32$ is a "powerful number" because $32 + 33 + 34$ does not produce any carry-over; $23$ is not a "powerful number" because $23 + 24 + 25$ produces a carry-over. Supp... | 6 |
Given that $x, y$ are positive numbers, $\theta \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right)$, and satisfying $\frac{\cos\theta}{x} = \frac{\sin\theta}{y}$ and $\frac{\sin^2\theta}{x^2} + \frac{\cos^2\theta}{y^2} = \frac{10}{3(x^2+y^2)}$, find the value of $\frac{(x+y)^2}{x^2+y^2}$. | \frac{2 + \sqrt{3}}{2} |
Given the function $f(x) = \ln(ax + 1) + \frac{1-x}{1+x}$, where $a > 0$.
(1) If $f(x)$ has an extremum at $x=1$, find the value of $a$;
(2) If the minimum value of $f(x)$ is 1, determine the range of values for $a$. | a \in [2, \infty) |
What is the area of the figure in the complex plane enclosed by the origin and the set of all points $\tfrac{1}{z}$ such that $(1-2i)z+(-2i-1)\overline{z}=6i$? | \frac{5\pi}{36} |
1. (17 points) Three cyclists, Dima, Misha, and Petya, started a 1 km race simultaneously. At the moment Misha finished, Dima still had to cover one tenth of the distance, and at the moment Dima finished, Petya still had to cover one tenth of the distance. How far apart (in meters) were Petya and Dima when Misha finish... | 90 |
12.117. The base of a pyramid inscribed in a cone is a quadrilateral with adjacent sides equal in pairs, and the angle between one pair of adjacent sides is $\alpha$. Find the ratio of the volume of the pyramid to the volume of the cone. | \frac{2\sin\alpha}{\pi} |
Given a sequence $\{a_n\}$ that satisfies: $a_n = \log_{n+1}(n+2)$, define $k$ (where $k \in \mathbb{N}^*$) such that the product $a_1 \cdot a_2 \cdot a_3 \cdots a_k$ is an integer as the "hope number". Then, the sum $M$ of all hope numbers in the interval $[1, 2010]$ is ( )
A: 2026
B: 2036
C: 2046
D: 2048 | A |
Task 4/84 All rational solutions $(x ; y)$ of the equation
$$
4 x^{2} y^{2}-4 x^{2} y-x^{2}+4 x=2
$$ | (1;0.5) |
Rotate the line $x-y+\sqrt{3}-1=0$ counterclockwise by 15° around the point $(1, \sqrt{3})$. What is the equation of the resulting line $l$? | y=\sqrt{3}x |
Task 1. 15 points
Find the minimum value of the expression
$$
\sqrt{x^{2}-\sqrt{3} \cdot|x|+1}+\sqrt{x^{2}+\sqrt{3} \cdot|x|+3}
$$
as well as the values of $x$ at which it is achieved. | \sqrt{7} |
7. (10 points) The difference in area between two squares is 2016 square centimeters. If the side lengths of such pairs of squares are both integer centimeters, then how many pairs of squares satisfy the above condition. | 12 |
2. $[\mathbf{2 5}]$ Is the number
$$
\left(1+\frac{1}{2}\right)\left(1+\frac{1}{4}\right)\left(1+\frac{1}{6}\right) \ldots\left(1+\frac{1}{2018}\right)
$$
greater than, less than, or equal to 50 ? | S<50 |
Given that $\tan \alpha = -\frac{1}{2}$, find the value of $\frac{1 + 2\sin \alpha \cos \alpha}{\sin^2 \alpha - \cos^2 \alpha}$. | -\frac{1}{3} |
1. What is the value of $2021-2223+2425$ ?
A 2122
B 2223
C 2324
D 2425
E 2526 | 2223 |
In the Cartesian coordinate system, there is a sequence of points $P_1, P_2, \ldots, P_{n-1}, P_n, \ldots$ (where $n \in \mathbb{N}^*$). The coordinates of point $P_n$ are $\left( n,\frac{2}{n} \right)$. The line passing through points $P_n$ and $P_{n+1}$ forms a triangle with the coordinate axes, and the area of this ... | 4n+\frac{n}{n+1} |
Given the function $f(\sin x)=\cos 2x$, find the value of $f(\frac{1}{2})$ as $(\ \ \ )$.
A: $-\frac{1}{2}$
B: $-\frac{\sqrt{3}}{2}$
C: $\frac{1}{2}$
D: $\frac{\sqrt{3}}{2}$ | C |
Problem 6.7. Anya places pebbles on the sand. First, she placed one stone, then added pebbles to form a pentagon, then made a larger outer pentagon with pebbles, then another outer pentagon, and so on, as shown in the picture. The number of stones she had arranged on the first four pictures: 1, 5, 12, and 22. If she co... | 145 |
Given in \\(\triangle ABC\\), \\(a=2\\), \\(b=1\\), and \\(C=60^{\circ}\\), find the length of side \\(c=\\) ______. | \sqrt{3} |
Example 9. Find all integer values of $x$ that make $y=\frac{x^{2}-2 x+4}{x^{2}-3 x+3}$ an integer. | x = -1, 1, 2 |
Triangle $ABC$ has side lengths $AB=18$, $BC=36$, and $CA=24$. The circle $\Gamma$ passes through point $C$ and is tangent to segment $AB$ at point $A$.
Let $X$, distinct from $C$, be the second intersection of $\Gamma$ with $BC$. Moreover, let $Y$ be the point on $\Gamma$ such that segment $AY$ is an angle bisector o... | 69 |
Given that the sequence $a_{n}$ is an arithmetic sequence with a non-zero common difference, and $a_{3}=5$, $a_{2}$, $a_{4}$, $a_{12}$ form a geometric sequence. The sequence $\{b_{n}\}$ consists of positive real numbers, its sum of the first $n$ terms is $S_{n}$, and it satisfies $4S_{n}=b_{n}^{2}+2b_{n}-3$ $(n\in\mat... | 6 |
Niki has $15$ dollars more than twice as much money as her sister Amy. If Niki gives Amy $30$ dollars, then Niki will have hals as much money as her sister. How many dollars does Niki have? | 55 |
Given $a, b \in \mathbb{R}$, and $i$ is the imaginary unit. If $(a+i)(1+i)=bi$, then $a+bi=$ ______. | 1+2i |
3. In $\triangle A B C$, $B D$ bisects $\angle A B C$ and intersects $A C$ at point $D, C E$ bisects $\angle A C B$ and intersects $A B$ at point $E$. If $B E+$ $C D=B C$, then the degree measure of $\angle A$ is ( ).
(A) $30^{\circ}$
(B) $45^{\circ}$
(C) $60^{\circ}$
(D) $90^{\circ}$ | C |
Calculate: $-{1}^{2023}-{(-\frac{1}{2})}^{-3}+{(π-3.14)}^{0}-|2-4|$. | 6 |
The sequence $\{a_n\}$ consists of positive numbers, and $S_n$ is the sum of its first $n$ terms. For any $n \in \mathbf{N}^*$, $2a_n$, $2S_n$, and $a_n^2$ form an arithmetic sequence. The general formula for $\{a_n\}$ is ______________. | a_n = 2n |
Given a sequence $\{a\_n\}$ that satisfies $a\_1 + 2a\_2 = 3$ and for any $n \in \mathbb{N}^*$, the points $P\_n(n, a\_n)$ in the sequence always satisfy $P\_n P\_{n+1} = (1, 2)$. Determine the sum of the first $n$ terms of the sequence, denoted as $S\_n$. | n\left(n - \frac{4}{3}\right) |
2. Ingots of highly enriched uranium must be stored in such a way that the distance to the nearest ingot is as large as possible. If there are several suitable places, any one of them can be chosen. Unfortunately, no one knows how many ingots will need to be stored. Ingots arrive one by one. Once an ingot is placed in ... | 12,34,56,78 |
Problem 7.8. Given an isosceles triangle $ABC (AB = BC)$. On the ray $BA$ beyond point $A$, point $E$ is marked, and on side $BC$, point $D$ is marked. It is known that
$$
\angle ADC = \angle AEC = 60^{\circ}, AD = CE = 13.
$$
Find the length of segment $AE$, if $DC = 9$.
![](https://cdn.mathpix.com/cropped/2024_05_... | 4 |
In a theater, the setting for Acts II-V of King Lear is determined by a draw before the acts. The auditorium is divided into four sections, and the audience moves to another area with their seats if the section they are sitting in is chosen as the next setting. Let's assume that all four areas are large enough that if ... | \frac{1}{2} |
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