problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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28. In a square $A B C D$ with a side length of 1 cm, draw quarter circles with a radius of 1 cm, centered at $A, B, C, D$, intersecting at points $E, F, G, H$, as shown in the figure. The perimeter of the shaded area in the middle is $\qquad$ cm. | \frac{2}{3}\pi |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=1$ and the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, then $\overrightarrow{a}\cdot \overrightarrow{a}+ \overrightarrow{a}\cdot \overrightarrow{b}=$ ( )
A: $\frac{1}{2}$
B: $\... | B |
Here is a piece of deductive reasoning: "If a line is parallel to a plane, then the line is parallel to all the lines within the plane. Given that line $b \not\subset$ plane $\partial$, and line $a \subset$ plane $\partial$, and line $b \parallel$ plane $\partial$, then line $b \parallel$ line $a$." In this reasoning, ... | \text{A} |
Calculate: ${(-1)}^{2023}+\sqrt{16}-{(\pi-3)}^{0}-{2}^{-1}$. | \frac{3}{2} |
Given the function $f(x)=-x^{2}+4x-3$, if a real number $x_{0}$ is randomly chosen from the interval $[2,6]$, the probability that $f(x_{0}) \geqslant 0$ is ($\quad$).
A: $\frac {1}{4}$
B: $\frac {1}{3}$
C: $\frac {1}{2}$
D: $\frac {3}{4}$ | A |
Calculate the definite integral $\int_{1}^{2} -x \, dx$.
The options are:
A: $- \frac {3}{2}$
B: $\frac {3}{2}$
C: $-1$
D: 1 | A |
In the expansion of the binomial $(1+x)^{n} (n \in N^{*})$, there exists a pair of adjacent terms whose coefficients have a ratio of $5:7$. The minimum value of the exponent $n$ is ______. | 11 |
The range of the function f(x) = cos(x - π/4) + sin(x + π/4) is _______. | [-2, 2] |
2. In trapezoid $A B C D$, $D C / / A B$ and $A B=2 D C$, point $P$ is on $A B$. If the perimeters of $\triangle P B C$, $\triangle P C D$, and $\triangle P D A$ are all equal, then $A P: P B=$ $\qquad$ | 1:1 |
If the function $f(x) = x^2 + ax$ is an even function, then the real number $a=$ . | 0 |
Given the general term formula of the sequence ${a_n}$ as $a_n=$ $$\begin{cases} 2^{n}+4 & (n\leq 3) \\ -3+a_{n-1} & (n\geq 4) \end{cases}$$ (n∈ℕ*), if $|a_1|+|a_2|+…+|a_n|=80$, find the value of $n$. | 12 |
Solve the quadratic equation $x^{2}-8x+6=0$ using the completing the square method. The correct completion is:
A: $(x-4)^{2}=22$
B: $(x-4)^{2}=10$
C: $(x+4)^{2}=22$
D: $(x+4)^{2}=10$ | B |
G10.4 $A$ number $X$ consists of 2 digits whose product is 24 . By reversing the digits, the new number formed is 18 greater than the original one. What is $X$ ? (Reference: 1991 FG6.1-2) | 46 |
Problem 2. Calculate the value of the expression $\frac{x^{4}-2005 x^{3}+2005 x^{2}-x}{x-1}$ for $x=2005$. | 4022030 |
Determine the smallest positive period of the function $y=2\cos^2 \left( \frac{x}{2} \right) + 1$. | 2\pi |
Let there be a conic section $C$ centered at the origin, with the coordinate axes as its axes of symmetry, an eccentricity of $\sqrt{2}$, and passing through the point $(5,4)$. Then, its focal length is ( ).
A: $6\sqrt{2}$
B: $6$
C: $5\sqrt{2}$
D: $5$ | A |
Janne buys a camera which costs $\$ 200.00$ without tax. If she pays $15 \%$ tax on this purchase, how much tax does she pay?
(A) $\$ 30.00$
(B) $\$ 18.00$
(C) $\$ 20.00$
(D) $\$ 15.00$
(E) $\$ 45.00$ | A |
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list] | 1 |
Given a sequence $\{a_n\}$ that satisfies $a_{n+1}=a_n-2a_{n+1}a_n$, $a_n \neq 0$ and $a_1=1$
$(1)$ Prove that the sequence $\left\{ \frac{1}{a_n} \right\}$ is an arithmetic sequence, and find the general formula for $\{a_n\}$;
$(2)$ Let $b_n=a_na_{n+1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}... | T_n= \frac{n}{2n+1} |
10.38 In a regular 1976-gon, the midpoints of all sides and the midpoints of all diagonals are marked. How many of the marked points can lie on the same circle? | 1976 |
4. Find the number of pairs of integers $(x ; y)$ that satisfy the equation $y^{2}-x y=700000000$. | 324 |
Four boys, $A, B, C$, and $D$, made 3 statements each about the same number $x$. We know that each of them has at least one correct statement, but also that at least one of their statements is false. Can $x$ be determined? The statements are:
A: (1) The reciprocal of $x$ is not less than 1.
(2) The decimal representa... | 25 |
In triangle $\triangle ABC$, if $\angle A < \angle B$ then $a < b$. The major premise is: "In a triangle, a larger angle is opposite a longer side." | a < b |
2. (2 points) Point $M$ lies on the side of a regular hexagon with side length 12. Find the sum of the distances from point $M$ to the lines containing the other sides of the hexagon. | 36\sqrt{3} |
## Task 3 - 250623
Let $A B C D$ be a square with side length $14 \mathrm{~cm}$. The points $E, F, G$ and $H$ are the midpoints of the square's sides. Here, $E$ lies on $A B, F$ on $B C, G$ on $C D, H$ on $D A$.
a) Construct this square and connect the midpoints $E$ and $F, F$ and $G, G$ and $H$ as well as $H$ and $E... | 98 |
$(1)$ Prove: $rC_n^r=nC_{n-1}^{r-1}$;$(2)$ Find the sum: $C_n^1+2C_n^2+3C_n^3+…+nC_n^n$;$(3)$ Prove: For a random variable $X\sim B\left(n,p\right)$, $E\left(X\right)=np$. | E(X) = np |
Find the minimum value of the function $f(x) = \frac {3}{x} + \frac {1}{1-3x}$, where $x$ belongs to the interval $(0, \frac {1}{3})$. | 16 |
Example 2 Let $a, b, c \in \mathbf{R}_{+}$, and $abc=1$. Then $\frac{1}{a^{3}(b+c)}+\frac{1}{b^{3}(c+a)}+\frac{1}{c^{3}(a+b)} \geqslant \frac{3}{2}$.
(26th IMO) | \frac{3}{2} |
1.53 Let three real numbers $x, y, z$ satisfy the system of equations $\left\{\begin{array}{l}\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2, \\ \frac{1}{x^{2}}+\frac{1}{y^{2}}+\frac{1}{z^{2}}=1 .\end{array}\right.$ Then the value of $\frac{1}{x y}+\frac{1}{y z}+\frac{1}{z x}$ is
(A) 1 .
(B) $\sqrt{2}$.
(C) $\frac{3}{2}$.
(D) $... | \frac{3}{2} |
1. (40 points) Let real numbers $a_{1}, a_{2}, \cdots, a_{n} \in[0,2]$, and define $a_{n+1}=a_{1}$, find the maximum value of $\frac{\sum_{i=1}^{n} a_{1}^{2} a_{i+1}+8 n}{\sum_{i=1}^{n} a_{i}^{2}}$. | 4 |
Example 3 Given the point set $A=\left\{(x, y) \left\lvert\,(x-3)^{2}+(y-4)^{2} \leqslant\left(\frac{5}{2}\right)^{2}\right.\right\}, B=$ $\left\{(x, y) \left\lvert\,(x-4)^{2}+(y-5)^{2}>\left(\frac{5}{2}\right)^{2}\right.\right\}$, find the number of integer points in the point set $A \cap B$. | 7 |
1. Given a linear function $f(x)$. It is known that the distance between the points of intersection of the graphs $y=$ $x^{2}-1$ and $y=f(x)+1$ is $3 \sqrt{10}$, and the distance between the points of intersection of the graphs $y=x^{2}$ and $y=f(x)+3$ is $3 \sqrt{14}$. Find the distance between the points of intersect... | 3\sqrt{2} |
In the sequence $\{a_{n}\}$, $a_{1}=1$, $\sqrt{{a}_{n+1}}-\sqrt{{a}_{n}}=1$ ($n\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\{b_{n}\}$ is $S_{n}=2^{n}-m$. For $n\in N^{*}$, the smallest value of the real number $\lambda$ that satisfies $\lambda b_{n}\geqslant a_{n}$ for all $n$ is ______. | \frac{9}{4} |
1.21 Real numbers $x, y, z$ satisfy $x+y+z=x y z$, then the following statements I. If two of $x, y, z$ are opposite numbers, then the third number must be 0. II. If one of $x, y, z$ is 0, then the other two must be opposite numbers.
III. $z=\frac{x+y}{x y-1}$.
(A) Only I is true, II and III are not true.
(B) Only I is... | D |
Find all solutions to the following system of equations
$$
y=x+\sqrt{x+\sqrt{x+\sqrt{x+\ldots+\sqrt{x+\sqrt{y}}}}}, \quad x+y=6
$$
where there are 1974 square root signs in the equation. | 2,\quad4 |
In a certain knowledge competition, there are a total of 20 questions. Each correct answer earns 5 points, while each wrong answer or unanswered question deducts 1 point. If a player scores a total of 70 points in this competition, how many questions did he answer correctly? | 15 |
$\frac{(.2)^3}{(.02)^2} =$
$\text{(A)}\ .2 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 20$ | 20 |
Determine the coefficient of the fourth term in the expansion of $( 3x - \frac{1}{2 \cdot 3x} )^{8}$. | -63 |
From the numbers $2$, $3$, and $4$, two numbers are randomly selected, one to be the base and the other to be the true number of a logarithm. The probability that the value of the logarithm is greater than $1$ is ______. | \dfrac{1}{2} |
In the Cartesian coordinate system, given $A(1,0)$, $B(4,0)$, if there exists a point $P$ on the line $x+my-1=0$ such that $|PA|=2|PB|$, then the minimum value of the positive real number $m$ is ( ).
A: $\frac{1}{3}$
B: $3$
C: $\frac{{\sqrt{3}}}{3}$
D: $\sqrt{3}$ | D |
13. For a given positive integer $n(n \geqslant 6)$, the set $A$ consists of the sums of 5 consecutive positive integers not greater than $n$, and the set $B$ consists of the sums of 6 consecutive positive integers not greater than $n$. If the number of elements in the set $A \cap B$ is 2016, determine the maximum valu... | 12106 |
The distance between two parallel lines $2x+3y-3=0$ and $2x+3y+2=0$ is ______. | \frac{5\sqrt{13}}{13} |
I5.2 Let $f(x)=x^{3}-20 x^{2}+x-a$ and $g(x)=x^{4}+3 x^{2}+2$. If $h(x)$ is the highest common factor of $f(x)$ and $g(x)$, find $b=h(1)$. | 2 |
Define a function $f(x)$ on $\mathbb{R}$ that is an even function, and satisfies $f(1-x) = f(1+x)$. If $f(x) = x^2$ for $x \in [0, 1]$, then the value of $f(-3)$ is ( ).
A: -1
B: 3
C: 1
D: -3 | \text{C} |
Example 3 Find all integer solutions of the indeterminate equation
$$
\left(x^{2}-4\right)\left(y^{2}-4\right)=t^{2}+12
$$ | (0,0,\2),(0,\1,0),(\1,0,0) |
1. A robot starts from the origin on a number line, moving in the positive direction of the number line, with a program of advancing 4 steps and then retreating 3 steps. Suppose the robot moves forward or backward 1 step per second, and each step is one unit length, $x_{n}$ represents the number corresponding to the ro... | B |
## Task 2 - 281232
Given a point $A$ in a plane $\epsilon$ and a length $a$.
Determine the set of all points $C$ in $\epsilon$ for which there exist points $B$ and $D$ such that $A B C D$ is a parallelogram with $A B=a$ and $A C: A B=B D: A D$. | \sqrt{2} |
Given the function $f(x) = |x - 1| + |x + 2|$,
(Ⅰ) Solve the inequality $f(x) \geq 4$ with respect to $x$;
(Ⅱ) If the inequality $f(x) \geq c$ is always true, find the range of the real number $c$. | (-\infty, 3] |
Three, given real numbers $\alpha_{1}, \alpha_{2}, \cdots, \alpha_{n}$ and $\beta_{1}, \beta_{2}, \cdots, \beta_{n}$ satisfy $\alpha_{1}^{2}+\alpha_{2}^{2}+\cdots+\alpha_{n}^{2}=\beta_{1}^{2}+\beta_{2}^{2}+\cdots$ $+\beta_{n}^{2}=1$ and $\alpha_{1} \beta_{1}+\alpha_{2} \beta_{2}+\cdots+\alpha_{n} \beta_{n}=0$. Prove: $... | 1\geqslant\alpha_{1}^{2}+\beta_{1}^{2} |
Given vectors $\overrightarrow{a}=(-1,2)$, $\overrightarrow{b}=(-1,1)$, $\overrightarrow{c}=(-3,1)$, then $\overrightarrow{c}\cdot(\overrightarrow{a}+ \overrightarrow{b})=$ ()
A: $(6,3)$
B: $(-6,3)$
C: $-3$
D: $9$ | D |
16. Given $\vec{a}=\{1,2\}, \vec{b}=\{-3,2\}$, find the real number $k$ such that $k \vec{a}+\vec{b}$ is in the same direction or opposite direction to $\vec{a}-3 \vec{b}$. | -\frac{1}{3} |
Given that $\{a_n\}$ is a geometric sequence with a common ratio of $q$, and $a_m$, $a_{m+2}$, $a_{m+1}$ form an arithmetic sequence.
(Ⅰ) Find the value of $q$;
(Ⅱ) Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$. Determine whether $S_m$, $S_{m+2}$, $S_{m+1}$ form an arithmetic sequence and exp... | -\frac{1}{2} |
Calculate: $$(-3)^2 + 2017^0 - \sqrt{18} \times \sin 45^\circ.$$ | 7 |
[ Sum of angles in a triangle. Theorem about the exterior angle.]
The angles of a triangle are in the ratio $2: 3: 4$. Find the ratio of the exterior angles of the triangle. | 5:6:7 |
Provide the standard equation of a circle with center at point A (-3, 4) that is tangent to the x-axis. | (x + 3)^2 + (y - 4)^2 = 16 |
Given the set $M = {(x\_1, x\_2, x\_3, x\_4, x\_5) | x\_i \in {-1,0,1}, i = 1, 2, 3, 4, 5}$, the number of elements in set $M$ that satisfy the condition $1 \leqslant |x\_1| + |x\_2| + |x\_3| + |x\_4| + |x\_5| \leqslant 3$ is $(\ \ \ \ \ \ \ )$.
A: $60$
B: $90$
C: $130$
D: $140$ | C |
(IMO 2009 P2)
Let $ABC$ be a triangle and let $O$ be the center of its circumcircle. Points $P$ and $Q$ are located on the segments $[AC]$ and $[AB]$ respectively. Let $K, L,$ and $M$ be the midpoints of the segments $[BP], [CQ],$ and $[PQ]$ respectively. Let $k$ be the circle passing through the points $K, L,$ and $M... | OP=OQ |
## Task 1
Arrange the numbers in order of size. $\quad \begin{array}{lllllll}7 & 11 & 20 & 0 & 19 & 3\end{array}$ | 0\quad3\quad7\quad11\quad19\quad20 |
Given the function $f(x)=(x-a)\ln x-x+a-3$ $(a\in R)$, find:
1. When $a=0$, determine the minimum value of $f(x)$.
2. Discuss the monotonicity of $f'(x)$.
3. When $a=2$, if $\lambda \leq f(x)$ always holds, find the maximum integer value of $\lambda$. | -4 |
Given two real numbers $a$ and $b$, consider $ab = a^2 - ab + b^2$. What is the value of 1?
(a) 1
(b) 0
(c) 2
(d) -2
(e) -1 | 1 |
$28 \cdot 9$ If $a, b, c$ are all $n$-digit positive integers, then the number of digits in the positive integer $a b c$ is
(A) $3 n$.
(B) $3 n-1$.
(C) $3 n-2$.
(D) None of the above.
(2nd "Five Sheep Cup" Junior High School Mathematics Competition, 1990) | D |
Let $S_{n}$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_{n}\}$ $\left(n\geqslant 1, n\in \mathbb{N}\right)$. If $a_{2}+a_{4}=9-a_{6}$, then $S_{7}=\_\_\_\_\_\_$. | 21 |
Find all triples $(m,p,q)$ where $ m $ is a positive integer and $ p , q $ are primes.
\[ 2^m p^2 + 1 = q^5 \] | (1, 11, 3) |
Among the following functions, the one that is a decreasing function on the interval $(0,+\infty)$ is $(\ )$
A: $y=-|x-1|$
B: $y=x^{2}-2x+4$
C: $y=\ln(x+2)$
D: $y=\left(\frac{1}{2}\right)^{x}$ | D |
12. Given the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $\frac{1}{2}$, $A_{1}, A_{2}$ are the left and right vertices of the ellipse $C$, $F_{1}, F_{2}$ are the left and right foci of the ellipse $C$, $B$ is the upper vertex of the ellipse $C$, and the circumradius of $\trian... | (0,\frac{9\sqrt{5}}{2}) |
In the sequence $\{a_n\}$, $a_1=1$, $a_{n+1}-a_n=4$, the value of $a_{100}$ is. | 397 |
Find all positive integers such that they have $6$ divisors (without $1$ and the number itself) and the sum of the divisors is $14133$. | 16136 \text{ and } 26666 |
## Task B-3.2.
Determine two natural numbers that are divisible by four and for which the difference of their cubes is a four-digit number divisible by 91. | 24,20 |
Determine the relationship between $a$, $b$, and $c$ if $a=0.3^{0.2}$, $b=0.2^{0.3}$, and $c=0.3^{0.3}$.
A: $c>a>b$
B: $c>b>a$
C: $a>b>c$
D: $a>c>b$ | \text{D} |
Example 4 If the expansion of $\left(1+x+x^{2}\right)^{1000}$ is $a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2000} x^{2000}$,
then the value of $a_{0}+a_{3}+a_{6}+\cdots+a_{1998}$ is ( ).
(A) $3^{333}$
(B) $3^{666}$
(C) $3^{999}$
(D) $3^{2001}$
(2001, National High School Mathematics Competition) | C |
Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\geqslant 0$ holds for $x\in [-1,1]$, then the minimum value of $m$ is
A: $\dfrac{e^{2}-1}{e^{2}+1}$
B: $\dfrac{2}{e^{2}+1}$
C: $\dfrac{e^{2}+... | A |
For the function $f(x)=x^{3}+2\ln x$, the value of $f'(1)$ is \_\_\_\_\_\_. | 5 |
3. Given that the function $f(x)$ is an increasing function defined on $[-4,+\infty)$, to ensure that for all real numbers $x$ in the domain, the inequality $f\left(\cos x-b^{2}\right) \geqslant f\left(\sin ^{2} x-b-3\right)$ always holds, the range of the real number $b$ is $\qquad$. | [\frac{1}{2}-\sqrt{2},1] |
If the asymptotes of the hyperbola $x^{2}-\frac{y^{2}}{k^{2}}=1(k＞0)$ are tangent to the circle $x^{2}+\left(y-2\right)^{2}=1$, then $k=\left(\ \ \right)$
A: $2$
B: $\sqrt{3}$
C: $1$
D: $\frac{\sqrt{3}}{3}$ | B |
Point M is inside the plane of $\triangle ABC$, and $\overrightarrow {MA}+ \overrightarrow {MB}+ \overrightarrow {MC}= \overrightarrow {0}$. If D is the midpoint of BC, then the value of $\frac {S_{\triangle ABC}}{S_{\triangle MBC}}$ is ( )
A: $\frac {1}{2}$
B: 1
C: 2
D: 3 | D |
2. (mathematics) Segments $A B$ and $C D$, with lengths 3 and 4 respectively, are chords of a circle with radius 5. Points $M$ and $N$ are located on them such that $A M: M B=1: 2, \quad C N: N D=1: 3$. Find the smallest and largest possible lengths of segment $M N$. | d_{\}=\sqrt{23}-\sqrt{22}\quadd_{\max}=\sqrt{23}+\sqrt{22} |
## Task B-4.1.
Solve the equation
$$
\binom{x+1}{x-2}+2\binom{x-1}{3}=7(x-1)
$$ | 5 |
4. Given $\left\{\begin{array}{l}\frac{x+9 y}{9 x+7 y}=\frac{m}{n}, \\ \frac{x+9 y}{9 x+8 y}=\frac{m+a n}{b m+c n} .\end{array}\right.$
If all real numbers $x, y, m, n$ that satisfy (1) also satisfy (2), then the value of $a+b+c$ is $\qquad$. | \frac{41}{37} |
Point $P$ and line $\ell$ are such that the distance from $P$ to $\ell$ is $12$. Given that $T$ is a point on $\ell$ such that $PT = 13$, find the radius of the circle passing through $P$ and tangent to $\ell$ at $T$. | \frac{169}{24} |
The equation of the line that passes through the center of the circle $C: (x+1)^2+(y-2)^2=4$ and has a slope of 1 is ( )
A: $x-y+3=0$
B: $x-y-3=0$
C: $x+y-1=0$
D: $x+y+3=0$ | A |
In the school library, there are 140 comic books and 100 storybooks. If 4 of each type of book are borrowed every day, after how many days will the number of comic books be three times the number of storybooks? (Solve using an equation) | 20 |
1326 ** Find the area of the region enclosed by the graph of the equation $|x-60|+|y|-\left|\frac{x}{4}\right|$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 480 |
Example 11 In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the degree of the dihedral angle $A-B D_{1}-A_{1}$ is $\qquad$ .
(2004 National High School League Question) | 60 |
17th Chinese 2002 Problem A3 18 people play in a tournament of 17 rounds. There are 9 games in each round and each person plays in one game in each round. Each person plays every other person just once in the tournament. What is the largest n such that however the tournament is arranged we can find 4 people amongst who... | 7 |
2. If $1 \times 2 \times \cdots \times 100=12^{n} M$, where $M$ is a natural number, and $n$ is the largest natural number that makes the equation true, then $M$ ( ).
(A) is divisible by 2 but not by 3
(B) is divisible by 3 but not by 2
(C) is divisible by 4 but not by 3
(D) is not divisible by 3, nor by 2
(1991, Natio... | A |
2. Given the vertex of a cone is $P$, the radius of the base is 2, and the height is 1. Take a point $Q$ on the base of the cone such that the angle between the line $P Q$ and the base is no greater than $45^{\circ}$. Then the area of the region formed by the points $Q$ that satisfy the condition is $\qquad$. | 3\pi |
Let $(x+1)(2x+1)^{10}=a_0+a_1(x+2)+a_2(x+2)^2+\ldots+a_{11}(x+2)^{11}$, then the value of $a_1+a_2+a_3+\ldots+a_{11}$ is ( )
A: $-310$
B: $0$
C: $310$
D: $510$ | C |
Given the set $U=\{1,2,3,4,5\}$, $A=\{1,3\}$, $B=\{1,2,4\}$, then $(\complement _{U}B)\cup A=\left(\ \ \right)$
A: $\{1,3,5\}$
B: $\{1,3\}$
C: $\{1,2,4\}$
D: $\{1,2,4,5\}$ | A |
In $\triangle ABC$, we have $\angle C=3\angle A, a=27$ and $c=48$. What is $b$?
$\mathrm{(A)\ } 33 \qquad \mathrm{(B) \ }35 \qquad \mathrm{(C) \ } 37 \qquad \mathrm{(D) \ } 39 \qquad \mathrm{(E) \ }\text{not uniquely determined}$ | 35 |
Given $A=\{-1,2\}$, $B=\{x|mx+1=0\}$, if $A\cup B=A$, then the set of values of the real number $m$ is
A: $\left\{-1, \frac{1}{2}\right\}$
B: $\left\{- \frac{1}{2},1\right\}$
C: $\left\{-1,0, \frac{1}{2}\right\}$
D: $\left\{- \frac{1}{2},0,1\right\}$ | \text{D} |
Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$
(a) Prove that $\hat B=\hat C=80^\circ$.
(b) Find the measure of the angle $D \hat{A} C$. | 5^\circ |
2. Let $x_{i} \in \mathbf{R},\left|x_{i}\right| \leqslant 1(1 \leqslant i \leqslant n)$, find the minimum value of $F=\sum_{1 \leq i<j \leqslant n} x_{i} x_{j}$. | -\left[\frac{n}{2}\right] |
4. For which triples of numbers $(x ; y ; z)$, satisfying the system $\left\{\begin{array}{l}\sin 2 x-\sin 2 y=3(x-y) \\ \cos 3 x-\cos 3 z=4(x-z)\end{array}\right.$, does the expression $\frac{\sin x+\sin y}{2-\sin z}$ take its maximum possible value? | (\frac{\pi}{2}+2\pin;\frac{\pi}{2}+2\pin;\frac{\pi}{2}+2\pin),n\inZ |
1.20. Find the area of the part of the surface of the paraboloid $z=x^{2}-y^{2}$, bounded by the paraboloids $z=3 x^{2}+$ $+y^{2}-2$ and $z=3 x^{2}+y^{2}-4$. | \frac{\pi}{6}(27-5\sqrt{5}) |
41st Putnam 1980 Problem B2 S is the region of space defined by x, y, z ≥ 0, x + y + z ≤ 11, 2x + 4y + 3z ≤ 36, 2x + 3z ≤ 24. Find the number of vertices and edges of S. For which a, b is ax + by + z ≤ 2a + 5b + 4 for all points of S? Solution | 7vertices,11edges,2/3\leq\leq1,2- |
14.51. In a drawer, there are red and black socks. If two socks are randomly pulled out of the drawer, the probability that both are red is $1 / 2$.
a) What is the smallest number of socks that can be in the drawer?
b) What is the smallest number of socks that can be in the drawer if it is known that the number of bl... | 4 |
4. Find all integers $a \geqslant 2$ and $b \geqslant 2$ such that $a$ is even and all the digits in the decimal representation of $a^{b}+1$ are equal.
## Solutions | (2,2),(2,3),(2,5),(6,5) |
[Coordinate method in space] [equation of a plane]
Given points $A(1 ; 0 ; 1), B(-2 ; 2 ; 1), C(2 ; 0 ; 3)$. Form the equation of the plane $ABC$.
# | 2x+3y-z-1=0 |
2.078. $\left(\frac{1}{t^{2}+3 t+2}+\frac{2 t}{t^{2}+4 t+3}+\frac{1}{t^{2}+5 t+6}\right)^{2} \cdot \frac{(t-3)^{2}+12 t}{2}$. | 2 |
The proposition $p$: The real number $x$ satisfies $x^{2}-5ax+4a^{2} 0$), and the proposition $q$: The real number $x$ satisfies $(x-2)(x-5) < 0$.
(1) If $a=1$, and the proposition "$p∧q$" is true, find the range of values for the real number $x$;
(2) If $p$ is a necessary but not sufficient condition for $q$, find th... | [\frac{5}{4}, 2] |
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