problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and $S_3 = 6$, then the value of $5a_1 + a_7$ is ( )
A: 12
B: 10
C: 24
D: 6 | A |
Given four points E, F, G, and H in space, consider the following propositions:
Proposition A: Points E, F, G, and H are not coplanar.
Proposition B: Line EF does not intersect with line GH.
Which of the following statements is true?
A: A $\Rightarrow$ B
B: B $\Rightarrow$ A
C: A $\Leftrightarrow$ B
D: Neither "A $\Rig... | \text{A} |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
y^{2} f(x)+x^{2} f(y)+x y=x y f(x+y)+x^{2}+y^{2}
$$
for all real numbers $x, y$. | f(x)=ax+1 |
Let $S_n$ be the sum of the first $n$ terms of the sequence $\{a_n\}$, and $a_1 = -2$, $S_n = 2a_n + 2$. Find $a_n$. | -2^n |
G3.3 If $c$ is the maximum value of $\log (\sin x)$, where $0<x<\pi$, find the value of $c$. | 0 |
Given the sequence ${a_n}$ that satisfies $$a_{1}=1，a_{n+1}= \frac {1}{2}a_{n}+1(n∈N^*)$$, by calculating $a_1$, $a_2$, $a_3$, $a_4$, we can conjecture the formula for $a_n$ as \_\_\_\_\_\_. | a_n = \frac {2^{n}-1}{2^{n-1}} |
A company has a total of 41 employees (including the manager), where the manager's salary is higher than that of the other employees. This year, the manager's salary increased from 200,000 yuan last year to 230,000 yuan, while the salaries of the other employees remained the same as last year. As a result, compared to ... | \text{B} |
8 The three sides of a triangle are $t^{2}+t+1$, $t^{2}-1$, and $2 t+1$, then the largest interior angle of the triangle is $\qquad$ . | 120 |
16. In an addition equation, each addend is composed of the digits 0 and 6, and the result of the calculation is 20222022. The addends in this equation must contain at least $\qquad$ digit 6. | 26 |
1. There are 800 marbles in a bag. Each marble is colored with one of 100 colors, and there are eight marbles of each color. Anna draws one marble at a time from the bag, without replacement, until she gets eight marbles of the same color, and then she immediately stops.
Suppose Anna has not stopped after drawing 699 m... | \frac{99}{101} |
Given the function $f(x)=\begin{cases} 2^x , & x > 0, \\ x+1, & x\leqslant 0. \end{cases}$ If $f(a)+f(1)=0$, then the value of the real number $a$ is __________. | -3 |
14. As shown in Figure 2, the area of rectangle $A B C D$ is equal to the area of right triangle $D C E$, $A B=5, A D=6$. Then $D E=(\quad)$.
(A) 12
(B) 13
(C) 14
(D) 15
(E) 16 | 13 |
Define a function $f(x)$ on $\mathbb{R}$ satisfying $f(x+2) + f(x) = 0$, and the function $f(x+1)$ is an odd function. Consider the following statements:
① The smallest positive period of the function $f(x)$ is 4;
② The graph of the function $f(x)$ is symmetric about the point $(1,0)$;
③ The graph of the function ... | \text{A} |
$31 \cdot 19$ The number of integer solutions to the equation $x^{2}-y^{2}=12$ is
(A) 2 pairs.
(B) 4 pairs.
(C) 8 pairs.
(D) 12 pairs.
(3rd "Jinyun Cup" Junior High School Mathematics Invitational, 1986) | 4 |
If the function $f(x) = x^2 - m\cos(x) + m^2 + 3m - 8$ has a unique zero, then the set of real numbers $m$ that satisfy this condition is __________. | \{2\} |
Three. (25 points) Select $k$ numbers from 1 to 2012, such that among the selected $k$ numbers, there are definitely three numbers that can form the lengths of the sides of a triangle (the lengths of the three sides of the triangle must be distinct). What is the minimum value of $k$ that satisfies the condition? | 17 |
Which of the following expressions is a perfect square? ( )
A: ${{x}^{2}}-x+\dfrac{1}{4}$
B: $1+{{x}^{2}}$
C: $x+xy+1$
D: ${{x}^{2}}+2x-1$ | A |
Given the proposition: "There exists $x \in [1,2]$ such that $x^{2}+2x+a \geqslant 0$" is true, the range of values for $a$ is _______ . | a \geqslant -8 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $\overrightarrow{a}=(\sqrt{3},1)$, $(\overrightarrow{a}-2\overrightarrow{b})\cdot(\overrightarrow{a}+2\overrightarrow{b})=-12$, and $\overrightarrow{a}\cdot\overrightarrow{b}=2$.
$(1)$ Find the size of the angle between vectors $\overrightarrow{a}$ a... | 2\sqrt{7} |
A calculator can accept a pair of integers for conversion, and it changes the pair of integers (a, b) by pressing a button according to the following rules: when the sum of the two numbers in the original pair of integers (a, b), a+b, is divided by 4, the remainders 0, 1, 2, 3 correspond to the outputs (a+1, b), (a, b+... | (4, 14) |
Let $a, b \in \mathbb{R}$, if $b - |a| > 0$, then among the following inequalities, the correct one is ( )
A: $a - b > 0$
B: $a + b > 0$
C: $a^2 - b^2 > 0$
D: $a^3 + b^3 < 0$ | \text{B} |
## Task 5
Mr. Müller has manufactured 16 chair legs.
How many chairs can he build? | 4 |
Three points $A,B,C$ are such that $B \in ]AC[$. On the side of $AC$ we draw the three semicircles with diameters $[AB], [BC]$ and $[AC]$. The common interior tangent at $B$ to the first two semi-circles meets the third circle in $E$. Let $U$ and $V$ be the points of contact of the common exterior tangent to the first ... | \frac{r_1 r_2}{(r_1 + r_2)^2} |
69. As shown in the figure, in $\triangle A B C$, $B C>A C, \angle A=60^{\circ}, D$ and $E$ are the midpoints of $A B$ and $A C$ respectively. If $P C$ bisects $\angle A C B$ and $P D$ bisects $\angle A D E$, then $\angle D P C=$ $\qquad$ ـ. | 120 |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, whose sums of the first $n$ terms are $A_n$ and $B_n$ respectively, and $$\frac {A_{n}}{B_{n}} = \frac {7n+1}{4n+27}$$, then the value of $$\frac {a_{6}}{b_{6}}$$ is ( )
A: $$\frac {4}{3}$$
B: $$\frac {7}{4}$$
C: $$\frac {3}{2}$$
D: $$\frac {78}{71}$$ | \text{D} |
12. A regular tetrahedron is a polyhedron with four faces, each of which is an equilateral triangle, as shown. A solid regular tetrahedron is cut into two pieces by a single plane cut.
Which of the following could not be the shape of the section formed by the cut?
A a pentagon
B a square
C a rectangle that is not a squ... | A |
For which non-negative integers $ a<2007$ the congruence $ x^2\plus{}a \equiv 0 \mod 2007$ has got exactly two different non-negative integer solutions?
That means, that there exist exactly two different non-negative integers $ u$ and $ v$ less than $ 2007$, such that $ u^2\plus{}a$ and $ v^2\plus{}a$ are both divisi... | a \equiv 0, 8, 5, 2 \pmod{9} |
Example 16 Find the product $T_{n}=\left(\cos \frac{A}{2}+\cos \frac{B}{2}\right) \cdot\left(\cos \frac{A}{2^{2}}+\cos \frac{B}{2^{2}}\right) \cdots \cdots\left(\cos \frac{A}{2^{n}}+\cos \frac{B}{2^{n}}\right)$. | \frac{\sin\frac{A+B}{2}\sin\frac{A-B}{2}}{2^{n}\sin\frac{A+B}{2^{n+1}}\sin\frac{A-B}{2^{n+1}}} |
393. Several people had dinner together and had to pay 175 shillings for the bill. It turned out that two of them had no money, so each of the rest had to pay 10 shillings more than their share. How many people had dinner? | 7 |
Let's simplify the following fractions:
a) $\frac{3 a^{4} c+2 a^{4} d-3 b^{4} c-2 b^{4} d}{\left[9 c^{2}(a-b)-4 d^{2}(a-b)\right]\left[(a+b)^{2}-2 a b\right]}$,
b) $\frac{a^{2}+b^{2}+c^{2}+2 a b+2 a c+2 b c}{a^{2}-b^{2}-c^{2}-2 b c}$,
c) $\frac{a^{2}-3 a b+a c+2 b^{2}-2 b c}{a^{2}-b^{2}+2 b c-c^{2}}$. | \frac{+b}{3-2},\frac{+b+}{-b-},\frac{-2b}{+b-} |
If the coefficient of the $x^{2}$ term in the expansion of $((1-ax)(1+2x)^{4})$ is $4$, then evaluate $\int_{\frac{e}{2}}^{a}{\frac{1}{x}dx}$ = \_\_\_\_\_\_. | \ln 5 - 1 |
In $\triangle ABC$, if $\sin (A+B-C)=\sin (A-B+C)$, then $\triangle ABC$ must be __( )__.
A: Isosceles triangle
B: Right triangle
C: Isosceles or right triangle
D: Isosceles right triangle | C |
1. (8 points) The calculation result of the expression $20.15 \div(1.2+1.9) \times\left(4^{2}-3^{2}+2^{2}-1^{2}\right)$ is | 65 |
## Task 4
Check the following statements:
a) The product of the smallest two-digit number and the smallest three-digit number is equal to the smallest four-digit number.
a) The product of the largest two-digit number and the largest three-digit number is equal to the largest four-digit number. | 1000 |
The "Nine Chapters on the Mathematical Art" is an ancient Chinese mathematical classic. It contains the following problem: "There are five people: a senior official, a Bu Geng, a Zan Guo, a Shang Zao, and a public officer, who together hunted five deer. They wish to divide the deer according to their ranks. How many do... | \text{B} |
3. Let $X=\{0,1,2,3,4,5,6,7,8,9\}$. Let $S \subseteq X$ be such that any nonnegative integer $n$ can be written as $p+q$ where the nonnegative integers $p, q$ have all their digits in $S$. Find the smallest possible number of elements in $S$.
| 5 |
10.3. In the quadrilateral $A B C D$, $A B=2, B C=4, C D=5$. Find its area, given that it is both inscribed and circumscribed. | 2\sqrt{30} |
Calculate the value of $\left(\log \frac{1}{4} - \log 25\right) \div 100^{-\frac{1}{2}}$. | -20 |
"x=1" is a ( ) condition for "x^2=1".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both sufficient and necessary
D: Neither sufficient nor necessary | A |
[Isosceles, Inscribed, and Circumscribed Trapezoids] [Pythagorean Theorem (Direct and Inverse)]
In an isosceles trapezoid \(ABCD\) with the larger base \(AD=12\) and \(AB=6\). Find the distance from point \(O\) where the diagonals intersect to point \(K\) where the extensions of the lateral sides intersect, if the ext... | \frac{12(3-\sqrt{2})}{7} |
What is the smallest positive integer $n$ for which $\sqrt{2019-n}$ is an integer? | 83 |
The number of all sets $B$ that satisfy the condition $\{1, 3\} \cup B = \{1, 3, 5\}$ is ____. | 4 |
1. [5 points] Point $D$ lies on side $A C$ of triangle $A B C$. The circle with diameter $B D$ intersects sides $A B$ and $B C$ at points $P$ and $T$ respectively. Points $M$ and $N$ are the midpoints of segments $A D$ and $C D$ respectively. It is known that $P M \| T N$.
a) Find the angle $A B C$.
b) Suppose additi... | 5 |
7. How many zeros does the number
$$
(1 \underbrace{000 \ldots 001}_{2017 \text { zeros }})^{2017}-1 ?
$$
end with? | 2018 |
2. Find the value of the expression $\frac{p}{q}+\frac{q}{p}$, where $p$ and $q$ are the largest and smallest roots of the equation $x^{3}-8 x^{2}+8 x=1$, respectively. | 47 |
Example 6 Given three real numbers $x_{1}, x_{2}, x_{3}$, any one of these numbers plus five times the product of the other two always equals 6. The number of such triples $\left(x_{1}, x_{2}, x_{3}\right)$ is $\qquad$.
$(1995$, Dongfang Airlines Cup - Shanghai Junior High School Mathematics Competition) | 5 |
Question 5: In an $n \times n$ grid, 101 cells are colored blue. It is known that there is a unique way to cut the grid along the grid lines into some rectangles, such that each rectangle contains exactly one blue cell. Find the minimum possible value of $n$.
(2015, Bulgarian Mathematical Olympiad)
It is easy to give a... | 101 |
53. Find the particular solution of the differential equation $y^{\prime}=2+y$, if $y=3$ when $x=0$. | 5e^{x}-2 |
Assume $ a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{a_{k}} \ge M $ and $ b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{b_{k}} \le M $. Prove that for any $ m \in \{1,2, \dots, 107\} $, the arithmetic mean of the following numbers $$ \frac{a_{1}}{b_{1... | \frac{M}{N} |
13. Given $a_{1}, a_{2}, \cdots, a_{n}$ are positive real numbers, and $a_{1}+a_{2}+\cdots+a_{n}=1, k$ is a positive integer, find the minimum value of $\frac{1}{a_{1}^{k}\left(1+a_{1}\right)}+\frac{1}{a_{2}^{k}\left(1+a_{2}\right)}+\cdots+\frac{1}{a_{n}^{k}\left(1+a_{n}\right)}$. | \frac{n^{k+2}}{n+1} |
Determine the minimum value of the function $f(x) = \sin(2x - \frac{\pi}{4})$ within the interval $\left[ 0, \frac{\pi}{2} \right]$. | -\frac{\sqrt{2}}{2} |
Given the point $M(-1,0)$, find the coordinates of the point $M'$ that is symmetric to $M$ about the line $x+2y-1=0$. | (-\frac{1}{5},\frac{8}{5}) |
PROBLEM 3. Calculate $\lim _{n \rightarrow \infty}\left\{\sqrt{n^{2}+5 n+9}\right\}$, where $\{x\}$ represents the fractional part of the real number $x$. | \frac{1}{2} |
Given the line $l_{1}$: $x-y-1=0$ and the moving line $l_{2}$: $\left(k+1\right)x+ky+k=0\left(k\in R\right)$, which of the following conclusions is correct?
A: There exists a $k$ such that the slope of $l_{2}$ is $90^{\circ}$.
B: For any $k$, $l_{1}$ and $l_{2}$ have a common point.
C: For any $k$, $l_{1}$ and $l_{2... | ABD |
We built a cuboid from small cubes with an edge of $1 \mathrm{~cm}$. If we removed one column from the cuboid, the remaining structure would consist of 602 cubes. If we removed one row from the top layer instead, the remaining structure would consist of 605 cubes. What are the dimensions of the cuboid? | 14\, |
In $\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the
midpoints of $BF$. The point $E$ divides side $BC$ in the ratio
$\textbf{(A) }1:4\qquad \textbf{(B) }1:3\qquad \textbf{(C) }2:5\qquad \textbf{(D) }4:11\qquad \textbf{(E) }3:8... | 1:3 |
Given the set $A=\{x|x \text{ is a rhombus or a rectangle}\}$, and $B=\{x|x \text{ is a rectangle}\}$, then $C_{A}B=$ ( )
A: $\{x|x \text{ is a rhombus}\}$
B: $\{x|x \text{ is a rhombus with no right angles}\}$
C: $\{x|x \text{ is a square}\}$
D: $\{x|x \text{ is a rectangle with unequal adjacent sides}\}$ | B |
In the trapezoid $A B C D$, the length of the leg $B C$ is equal to the sum of the lengths of the bases $A B$ and $C D$. The midpoint of the leg $D A$ is $F$. What is the measure of angle $B F C$? | 90 |
27.15. How many digits does the number $2^{100}$ have?
## 27.3. Identities for logarithms | 31 |
## Task $9 / 66$
Calculate
$$
\sum_{k=1}^{n}\left(k x^{k-1}\right)=1+2 x+3 x^{2}+\ldots+n x^{n-1}
$$ | \frac{1-(n+1)x^{n}+nx^{n+1}}{(1-x)^{2}} |
## Problem Statement
Calculate the limit of the numerical sequence:
$\lim _{n \rightarrow \infty} \frac{\sqrt[3]{n^{2}+2}-5 n^{2}}{n-\sqrt{n^{4}-n+1}}$ | 5 |
The value of the algebraic expression $\frac{3}{{x+2}}$ is equal to the value of the algebraic expression $\frac{2}{{x-1}}$, then $x=$____. | x=7 |
(5) Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=0, a_{n+1}=a_{n}+1+2 \sqrt{1+a_{n}}(n=$ $1,2, \cdots)$, then $a_{2009}=(\quad)$.
(A) 4036080
(B) 4036078
(C) 4036082
(D) 4036099 | 4036080 |
Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and in... | 129 |
24. A train has five carriages, each containing at least one passenger. Two passengers are said to be 'neighbours' if either they are in the same carriage or they are in adjacent carriages. Each passenger has exactly five or exactly ten neighbours. How many passengers are there on the train?
A 13
B 15
C 17
D 20
E There... | 17 |
Let $AB$ be a diameter of a circle and let $C$ be a point on the segement $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $[ABD]/[CDE]$ to the nearest integer. | 13 |
Given $\cos \left( \frac{\pi}{4}+α\right)= \frac{ \sqrt{2}}{10}\left(0 < α < \frac{\pi}{2}\right)$, then $\sin α =$\_\_\_\_\_\_\_\_\_\_\_\_\_\_. | \frac{3}{5} |
Several teachers at a school decided to crowdfund a special New Year gift package. If each person contributes $18$ yuan, there will be a surplus of $3$ yuan. If each person contributes $17$ yuan, there will be a shortfall of $4$ yuan. Let $x$ be the number of teachers and $y$ be the price of the gift package. Which set... | B |
39. As shown in the figure, $P A=P B, \angle A P B=2 \angle A C B, A C$ intersects $P B$ at point $D$, and $P B=4, P D=3$, then $A D \cdot D C=$ . $\qquad$ | 7 |
5. In $\triangle A B C$, $\angle C=90^{\circ}, \angle B=30^{\circ}$, $A C=1$, $M$ is the midpoint of $A B$. Fold $\triangle A C M$ along $C M$ so that the distance between points $A$ and $B$ is $\sqrt{2}$. Then the distance from point $A$ to the plane $B C M$ is $\qquad$ . | \frac{\sqrt{6}}{3} |
Write the equation of the line that is symmetrical to the line $\sqrt{3}x + y + 1 = 0$ across the line $y = -x$. | x + \sqrt{3}y - 1 = 0 |
Given the complex number Z = $$\frac { \sqrt {3}+i}{(1- \sqrt {3}i)^{2}}$$, find the value of |z|.
A: $$\frac {1}{4}$$
B: $$\frac {1}{2}$$
C: 1
D: 2 | \text{B} |
22. A two-digit number, the sum of the unit digit and the tens digit plus their product equals this two-digit number. Then the unit digit of this two-digit number is ( ).
(A) 1
(B) 3
(C) 5
(D) 7
(E) 9 | 9 |
13. (2005 National High School Mathematics Competition) Let the set $T=\{0,1,2,3,4,5,6\}, M=$ $\left\{\left.\frac{a_{1}}{7}+\frac{a_{2}}{7^{2}}+\frac{a_{3}}{7^{3}}+\frac{a_{4}}{7^{4}} \right\rvert\, a_{i} \in T, i=1,2,3,4\right\}$, and arrange the elements of $M$ in descending order. The 2005th number is ( ).
A. $\frac... | \frac{1}{7}+\frac{1}{7^{2}}+\frac{0}{7^{3}}+\frac{4}{7^{4}} |
1. N1 (TWN) ${ }^{1 \mathrm{MO} 04}$ Find all pairs of positive integers $(x, p)$ such that $p$ is a prime, $x \leq 2 p$, and $x^{p-1}$ is a divisor of $(p-1)^{x}+1$. | (1,p),(2,2),(3,3) |
Given an ellipse $M$: $\frac{y^{2}}{a^{2}}+\frac{x^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity that is the reciprocal of the eccentricity of the hyperbola $x^{2}-y^{2}=1$, and is internally tangent to the circle $x^{2}+y^{2}=4$.
(1) Find the equation of the ellipse $M$;
(2) Given $A(-2, \sqrt{2}), F$ is the lower f... | 3 \sqrt {2} |
If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ form a basis for a space, then the following set of vectors that can form a basis for the space is:
A: $\overrightarrow{a}+\overrightarrow{b}, \overrightarrow{a}-\overrightarrow{c}, \overrightarrow{b}$
B: $\overrightarrow{c}, \overrightarrow{b}+\overrighta... | \text{A} |
97.1. Let A be a set of seven positive numbers. Determine the maximal number of triples $(x, y, z)$ of elements of A satisfying $x<y$ and $x+y=z$.
| 9 |
One, (40 points) Find all real-coefficient polynomials $p(x)$ satisfying
$$
\begin{array}{l}
\left(x^{3}+2 x^{2}+3 x+2\right) p(x-1) \\
=\left(x^{3}-4 x^{2}+5 x-6\right) p(x)
\end{array}
$$ | p(x)=C(x^{2}+x+2)x(x-1)(x+1)(x-2) |
If $f(x)$ is a decreasing function on $\mathbb{R}$, and the graph of $f(x)$ passes through points $A(0, 4)$ and $B(3, -2)$, then when the solution set of the inequality $|f(x+t)-1|<3$ is $(-1, 2)$, the value of $t$ is ( )
A: $-1$
B: $0$
C: $1$
D: $2$ | C |
If $|m-3|+(n+2)^2=0$, find the value of $m+2n$. | -1 |
Let $f(n) = (a+b)^n$ ($n \in \mathbb{N}^*$, $n \geq 2$). If in the expansion of $f(n)$, there exist three consecutive terms whose binomial coefficients form an arithmetic sequence, then $f(n)$ is said to possess property P.
(1) Prove that $f(7)$ has property P;
(2) If there exists $n \leq 2016$ such that $f(n)$ has pro... | 1934 |
Given in $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. The following four conditions are provided:
(1) $\tan(A-B) \cos C = 0$; (2) $\sin(B+C) \cos(B-C) = 1$; (3) $a\cos A = b\cos B$; (4) $\sin^2(A-B) + \cos^2 C = 0$.
If the condition is satisfied, then $\triangle AB... | (4) |
Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times... | 2^{n-m} |
Given that the lengths of two adjacent sides of a parallelogram are $3$ and $5$, then its perimeter is ( )
A: $8$
B: $15$
C: $16$
D: $20$ | C |
1. (16 points) After walking one-fifth of the way from home to school, Olya realized she had forgotten her notebook. If she does not go back for it, she will arrive at school 6 minutes before the bell, but if she returns, she will be 2 minutes late. How much time (in minutes) does the journey to school take? | 20 |
13. Find $\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}$. | \frac{3}{4} |
G1.3 In Figure $1, \triangle A B C$ is an isosceles triangle and $A B=A C$. Suppose the angle bisector of $\angle B$ meets $A C$ at $D$ and $B C=B D+A D$. Let $\angle A=c^{\circ}$, find the value of $c$. | 100 |
The left vertex of the hyperbola $C$: ${{x}^{2}}-\dfrac{{{y}^{2}}}{3}=1$ is $A$, and the right focus is $F$. A line passing through point $F$ intersects the right branch of the hyperbola $C$ at points $P$ and $Q$. Lines $PA$ and $QA$ intersect the line $l$: $x=\dfrac{1}{2}$ at points $M$ and $N$, respectively. Then, th... | C |
1. Given $P A 、 P B 、 P C$ are three non-coplanar rays emanating from point $P$, and the angle between any two rays is $60^{\circ}$. There is a sphere with a radius of 1 that is tangent to all three rays. Find the distance from the center of the sphere $O$ to point $P$. | \sqrt{3} |
Two subsets of the set $S=\lbrace a,b,c,d,e\rbrace$ are to be chosen so that their union is $S$ and their intersection contains exactly two elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?
$\mathrm{(A)}\ 20\qquad\mathrm{(B)}\ 40\qquad\mathrm{(C)}\ 60\... | 40 |
18. Gargamel is designing a trap to catch the Smurfs. He plans to use 56 small squares, each with an area of 1 square decimeter, to form a large rectangular trap. The minimum perimeter of the trap he can make is $\qquad$ decimeters. | 30 |
14. A line passing through the left focus $F_{1}$ of a hyperbola and perpendicular to the real axis of the hyperbola intersects the hyperbola at points $A$ and $B$. If there exists a point $C$ on the line of the imaginary axis of the hyperbola such that $\angle A C B=90^{\circ}$, then the range of the eccentricity $e$ ... | e \geqslant \frac{\sqrt{5}+1}{2} |
Given the set $A=\{x|ax^{2}+2x+a=0\}$, if the set $A$ has exactly two subsets, then the value of $a$ is ______. | 0 \text{ or } 1 \text{ or } -1 |
Which of the following four line segments are proportional? ( )
A: 1cm, 2cm, 4cm, 6cm
B: 3cm, 4cm, 7cm, 8cm
C: 2cm, 4cm, 8cm, 16cm
D: 1cm, 3cm, 5cm, 7cm | C |
Find the distance from the point $(2,0)$ to the line $y=x-1$. | \frac{\sqrt{2}}{2} |
Given the following conditions (where $l$ and $a$ are lines, and $\alpha$ is a plane):
1. $l$ is perpendicular to two sides of a convex pentagon within $\alpha$;
2. $l$ is perpendicular to three non-parallel lines within $\alpha$;
3. $l$ is perpendicular to countless lines within $\alpha$;
4. $l$ is perpendicular to th... | \text{D} |
5. $\frac{\sin 7^{\circ}+\sin 8^{\circ} \cos 15^{\circ}}{\cos 7^{\circ}-\sin 8^{\circ} \sin 15^{\circ}}=$ | 2-\sqrt{3} |
If the functions are defined as $f(x)= \sqrt{x}$ and $g(x)=x- \sqrt{x}$, then $f(x)+g(x)=$ ______. | x |
Given that $\frac{\sin \beta}{\sin \alpha} = \cos(\alpha + \beta)$, where $\alpha, \beta \in (0, \frac{\pi}{2})$,
(1) Prove that $\tan \beta = \frac{\sin 2\alpha}{3 - \cos 2\alpha}$;
(2) Find the maximum value of $\tan \beta$. | \frac{\sqrt{2}}{4} |
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