problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given $\tan \alpha=2$, then $\sin \alpha\sin \left( \frac {\pi}{2}-\alpha\right)=($ $)$
A: $\frac {2}{5}$
B: $\frac { \sqrt {2}}{5}$
C: $\frac {2}{3}$
D: $\frac { \sqrt {2}}{3}$ | A |
14. There are three interest groups $A$, $B$, and $C$, with 5, 4, and 7 members respectively. Among them, 3 members are in both $A$ and $B$ interest groups, and 2 members are in $A$, $B$, and $C$ interest groups. Let $(A \cup B) \cap C$ consist of $x$ members, and $A \cup B \cup C$ consist of $y$ members. Find the rang... | 2\leqslantx\leqslant5,8\leqslanty\leqslant11 |
Given the function $f(x) = p\ln x + (p-1)x^2 + 1$.
(I) Discuss the monotonicity of the function $f(x)$;
(II) When $p = 1$, $f(x) \leq kx$ always holds, find the range of values for the real number $k$;
(III) Prove that: $\ln(n + 1) < 1 + 2 + \cdots + n$ ($n \in \mathbb{N}^*$). | k \geq 1 |
Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 121 |
The absolute value of $-\frac{3}{2}$ is ( ).
A: $-\frac{2}{3}$
B: $-\frac{3}{2}$
C: $\frac{2}{3}$
D: $\frac{3}{2}$ | D |
1. Find all pairs of positive integers $(k, n)$ such that $\left(7^{k}-3^{n}\right) \mid\left(k^{4}+n^{2}\right)$ | (2,4) |
2. The equation $x^{2}-2 a \sin (\cos x)+a^{2}=0$ has only one solution. Then the real number $a=$ | 2 \sin 1 |
Given that the line $x-2y+2=0$ intersects with the circle $C: x^2+y^2-4y+m=0$, and the length of the chord cut by the line is $\frac{2\sqrt{5}}{5}$.
(1) Find the equation of circle $C$;
(2) Given $P(2, 4)$, two tangents are drawn from $P$ to circle $C$ and intersect with the parabola $y=x^2$ at points $Q$ and $R$ (... | \text{tangent} |
Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$.
[i]Proposed by A. Antropov[/i] | 160 |
9th CanMO 1977 Problem 5 A right circular cone has base radius 1. The vertex is K. P is a point on the circumference of the base. The distance KP is 3. A particle travels from P around the cone and back by the shortest route. What is its minimum distance from K? | \frac{3}{2} |
Prove that in every triangle with sides $a, b, c$ and opposite angles $A, B, C$, is fulfilled (measuring the angles in radians) $$\frac{a A+bB+cC}{a+b+c} \ge \frac{\pi}{3}$$
Hint: Use $a \ge b \ge c \Rightarrow A \ge B \ge C$. | \frac{aA + bB + cC}{a + b + c} \geq \frac{\pi}{3} |
1. In the plane 2014 lines are arranged in three groups of parallel lines. What is the largest number of triangles formed by the lines (each side of the triangle lies on one of the lines). | 672\cdot671^{2} |
Given point $P(2,2)$, find the equation of the tangent line $l$ that touches the circle $(x-1)^{2}+y^{2}=5$. | x+2y-6=0 |
Example 9. Find the integral $\int \sin ^{2} \frac{x}{2} d x$. | \frac{1}{2}x-\frac{1}{2}\sinx+C |
G3.2 There are $R$ zeros at the end of $\underbrace{99 \ldots 9}_{2009 \text { of }} \times \underbrace{99 \ldots 9}_{2009 \text { of } 9^{\prime} s}+1 \underbrace{99 \ldots 9}_{2009 \text { of } 9^{\prime} s}$, find the value of $R$. | 4018 |
# Problem 10. (4 points)
In all cells of a $6 \times 8$ table, except for the corner cells, non-negative numbers are placed such that the sum of the numbers in each cross of five cells does not exceed 10. What is the maximum possible value of the sum of the numbers in the entire table? | 160 |
In square $ABCD$, the coordinates of $A$ and $C$ are $\left(1,-2\right)$ and $\left(4,1\right)$, respectively. $AB$ is parallel to the $x$-axis. What are the coordinates of point $B$?
A: $(4,-2)$
B: $(1,-1)$
C: $(-1,1)$
D: $(1,1)$ | A |
10. (10 points) A school's entrance exam has set a passing score. Among the students who took the exam, only $\frac{1}{4}$ were admitted. The average score of admitted students is 10 points higher than the passing score, while the average score of those not admitted is 26 points lower than the passing score. The averag... | 87 |
At a supermarket promotional event, customers who spend over 100 yuan are entitled to 3 opportunities to participate in a blindfolded ball drawing contest. In the draw box, there are red, yellow, and black balls that are identical except for their color, with one of each color. Each customer will draw one ball and then... | \frac{2}{9} |
In a convex quadrilateral $A B C D$, the length of side $A D$ is 4, the length of side $C D$ is 7, the cosine of angle $A D C$ is $\frac{1}{2}$, and the sine of angle $B C A$ is $\frac{1}{3}$. Find the side $B C$, given that the circle circumscribed around triangle $A B C$ also passes through point $D$. | \frac{\sqrt{37}}{3\sqrt{3}}(\sqrt{24}-1) |
1. Solve the equation
$$
5^{x} \sqrt[x]{8^{x-1}}=500
$$ | x_{1}=3,x_{2}=-\log_{5}2 |
Given that the sum of the first $n$ terms of a sequence $\{a_{n}\}$ is ${S}_{n},{S}_{n}={n}^{2}+n$, where $n\in N$, $n\geqslant 1$.
$(1)$ Find the general formula for $\{a_{n}\}$;
$(2)$ Find the sum of the first $n$ terms of the sequence $\{\frac{1}{{a}_{n}{a}_{n+1}}\}$, denoted as $H_{n}$. | H_{n} = \frac{n}{4(n+1)} |
17) The census employee on the Island of Knights and Knaves must determine the type (Knight or Knave) and the educational level of the residents (Knights always tell the truth, while Knaves always lie). In an apartment inhabited by a married couple, he only gets these responses:
Husband: We are both graduates.
Wife: ... | 3 |
If $\sin \theta = 1 - \log_{2}x$, then the range of $x$ is ( )
A: $[1,4]$
B: $[\frac{1}{4},1]$
C: $[2,4]$
D: $[\frac{1}{4},4]$ | A |
The equation of the circle symmetric to the circle $(x-1)^2+(y-2)^2=1$ with respect to the line $x-y-2=0$ is ( )
A: $(x-4)^2+(y+1)^2=1$
B: $(x+4)^2+(y+1)^2=1$
C: $(x+2)^2+(y+4)^2=1$
D: $(x-2)^2+(y+1)^2=1$ | \text{A} |
When $x=1$, $ax^{4}+bx^{2}+2=-3$; when $x=-1$, $ax^{4}+bx^{2}-2=\left(\ \ \right)$
A: $3$
B: $-3$
C: $-5$
D: $-7$ | D |
Given two lines $a$ and $b$, and two planes $\alpha$ and $\beta$, which of the following statements is correct? ( )
A: If $a \subset \beta$, and $\alpha \parallel \beta$, then $a \parallel \alpha$
B: If $b \subset \alpha$, $a \parallel b$, then $a \parallel \alpha$
C: If $a \parallel \beta$, $\alpha \parallel \beta$, t... | \text{A} |
10. Given that $f(x)$ is a function defined on
$\mathbf{R}$, $f(1)=1$ and for any $x \in \mathbf{R}$ we have
$$
f(x+5) \geqslant f(x)+5, f(x+1) \leqslant f(x)+1 \text {. }
$$
If $g(x)=f(x)+1-x$, then $g(2002)=$ | 1 |
N4. Find all integers $x, y$ such that
$$
x^{3}(y+1)+y^{3}(x+1)=19
$$
| (2,1),(1,2),(-1,-20),(-20,-1) |
[Theorem on the lengths of a tangent and a secant; the product of the entire secant and its external part. Pythagorean Theorem (direct and inverse).
From a point $A$ outside a circle, a tangent and a secant are drawn to the circle. The distance from point $A$ to the point of tangency is 16, and the distance from point... | 13 |
The number halfway between $\frac{1}{12}$ and $\frac{1}{10}$ is
(A) $\frac{1}{11}$
(B) $\frac{1}{120}$
(C) $\frac{11}{60}$
(D) $\frac{11}{120}$
(E) $\frac{1}{22}$ | \frac{11}{120} |
5.51 Proof: If the roots of the polynomial $x^{2}+p x+1$ are $\alpha$ and $\beta$, and the roots of the polynomial $x^{2}+q x+1$ are $\gamma$ and $\delta$, then
$$
(\alpha-\gamma)(\beta-\gamma)(\alpha+\delta)(\beta+\delta)=q^{2}-p^{2} .
$$ | q^{2}-p^{2} |
The students of the second grade of a certain school took a bus to visit the "Five Years of Struggle" large-scale achievement exhibition at the Beijing Exhibition Hall. The Beijing Exhibition Hall is 12 kilometers away from the school. The first bus departed 3 minutes before the second bus, and both buses arrived at th... | A |
Given the curves \\(C_1:\\begin{cases}x=-4+\cos t \\ y=3+\sin t\end{cases} (t \\) is the parameter\\()), \\(C_2:\\begin{cases}x=8\cos \theta \\ y=3\sin \theta\end{cases} (\theta \\) is the parameter\\()).
\\((1)\\) Convert the equations of \\(C_1,C_2 \\) into standard form;
\\((2)\\) If the point \\(P \\) on \\(C_1 ... | \dfrac{8 \sqrt{5}}{5} |
Given that the positive arithmetic sequence $\\{ {a_{n}} \\}$ satisfies $({a_{1}}+{{a}_{6}}={{a}_{2}}({{a}_{3}}-1))$, and the geometric sequence $\\{ {b_{n}} \\}$ with common ratio $q$ has its sum of the first $n$ terms ${{S}_{n}}$ satisfying $(2{{S}_{1}}+{{S}_{3}}=3{{S}_{2}})$, with $({{a}_{1}}={{b}_{1}}=1)$.
(1) Fin... | 3 |
18. (3 points) If $a b c+c b a=1069$, then there are $\qquad$ such $\underline{a b c}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 8 |
8. As shown in Figure 2, given that $A B$ is the diameter of circle $O$, $P Q$ is a chord of circle $O$, $P Q$ is not parallel to $A B$, and $R$ is the midpoint of $P Q$. Construct $P S \perp A B, Q T \perp A B$, with the feet of the perpendiculars being $S$ and $T (S \neq T)$, and $\angle S R T=60^{\circ}$. Then the v... | \frac{1}{2} |
The sequence $\{a_{n}\}$ satisfies: $a_{1}=1$, $a_{2}=2$, and the sequences $\lambda a_{2n-1}$, $a_{2n}$, $a_{2n+1}$ form an arithmetic progression, while $\lambda a_{2n}$, $a_{2n+1}$, $a_{2n+2}$ form a geometric progression. The following propositions are given:① If $\lambda =1$, then $a_{3}=3$;② If $\lambda =-1$, the... | C |
If the major axis of the ellipse $\frac{{x}^{2}}{{a}^{2}}+{y}^{2}=1(a＞1)$ is $4$, then its eccentricity is ______. | \frac{\sqrt{3}}{2} |
The contrapositive of the proposition "If $abc=0$, then $a=0$ or $b=0$ or $c=0$" is ( )
A: If $a=0$ or $b=0$ or $c=0$, then $abc \neq 0$
B: If $a=0$ or $b=0$ or $c=0$, then $abc = 0$
C: If $abc \neq 0$, then $a \neq 0$ and $b \neq 0$ and $c \neq 0$
D: If $a \neq 0$ and $b \neq 0$ and $c \neq 0$, then $abc \neq 0$ | D |
The moving line $2ax+(a+c)y+2c=0$ ($a\in\mathbb{R}, c\in\mathbb{R}$) passes through a fixed point $(m,n)$, and $x_1+x_2+m+n=15$ with $x_1>x_2$. The minimum value of $$\frac{x_1^2+x_2^2}{x_1-x_2}$$ is \_\_\_\_\_\_. | 16 |
In recent years, China's Beidou chip has achieved a breakthrough in the $22$-nanometer process, surpassing the GPS chip. It is known that $22$ nanometers $=0.000000022$ meters. The data $0.000000022$ can be expressed in scientific notation as ( ).
A: $0.22\times 10^{-7}$
B: $2.2\times 10^{-8}$
C: $2.2\times 10^{-9}$... | B |
Given that $F$ is the right focus of the hyperbola $C: x^{2}- \frac {y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, with $A(0, 6 \sqrt {6})$, determine the y-coordinate of point $P$ when the perimeter of $\triangle APF$ is at its minimum. | 2\sqrt{6} |
Given a complex number $z$ satisfying $z-|z|=-8+12i$, then the real part of $z$ is ( )
A: $9$
B: $7$
C: $5$
D: $3$ | C |
20. Given a triangle $D E F$, obtuse at $E$, let $O$ be the center of the circumcircle of the triangle. Let $P$ be the point where the angle bisector from $E$ intersects the side $D F$. It is known that the convex quadrilateral $F E P O$ is cyclic. Given that $\widehat{E D F}=52^{\circ}$, what is the measure of the ang... | 24 |
4. In rectangle $A B C D$, $A B=2, A D=1$, point $P$ on side $D C$ (including points $D, C$) and point $Q$ on the extension of $C B$ (including point $B$) satisfy $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$. Then the dot product $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ of vectors $\overrightarrow{P A}$ and... | \frac{3}{4} |
3rd USAMO 1974 Problem 5 A point inside an equilateral triangle with side 1 is a distance a, b, c from the vertices. The triangle ABC has BC = a, CA = b, AB = c. The sides subtend equal angles at a point inside it. Show that sum of the distances of the point from the vertices is 1. | 1 |
From 7 cards labeled with numbers 1, 2, 3, 4, 5, 6, 7 respectively, one card is randomly drawn, replaced, and then another card is randomly drawn. The probability that the number on the first card drawn is greater than the number on the second card drawn is ______. | \frac{3}{7} |
Given propositions $p$: $\forall x \in \mathbb{R}_{+}$, $\lg x \geqslant 0$; $q$: $\exists x_{0} \in \mathbb{R}$, $\sin x_{0} = \cos x_{0}$, then among the following propositions, the true one is ( ).
A: $\neg q$
B: $p \land q$
C: $\neg p \land q$
D: $\neg q \lor p$ | C |
5. A. Given that $a, b$ are non-zero real numbers, on the parabola $y=a x^{2}+b$ there are three different points $A, B, C$, and it is known that $B C \parallel x$-axis, $\angle B A C=90^{\circ}$. Then the distance from point $A$ to the line $B C$ is ( ).
(A) $\frac{1}{|a|}$
(B) $\frac{1}{2|a|}$
(C) $|a|$
(D) $2|a|$ | \frac{1}{||} |
5.
In a 6-liter vessel, 4 liters of a 70% (by volume) sulfuric acid solution are poured, and in a second vessel of the same capacity, 3 liters of a 90% solution of sulfuric acid are poured. A certain amount of the solution is transferred from the second vessel to the first so that it becomes an $r-\%$ solution of sul... | 76 |
1. The functions $y=f(x)$ and $y=g(x)$ have domains and ranges of $\mathbf{R}$, and both have inverse functions. Then the inverse function of
$$
y=f^{-1}\left(g^{-1}(f(x))\right)
$$
is $(\quad)$.
(A) $y=f\left(g\left(f^{-1}(x)\right)\right)$
(B) $y=f\left(g^{-1}\left(f^{-1}(x)\right)\right)$
(C) $y=f^{-1}(g(f(x)))$
(D... | C |
Let $i$ be the imaginary unit. If $(x-i)i=y+2i$ where $x, y \in \mathbb{R}$, then the sum of the real numbers $x+y=$ \_\_\_\_\_\_. | 3 |
Given \\(\alpha, \beta \in \left( \frac{3\pi}{4}, \pi \right)\\), \\(\cos (\alpha + \beta) = \frac{4}{5}\\), \\(\cos \left(\beta - \frac{\pi}{4}\right) = -\frac{5}{13}\\), then \\(\sin \left(\alpha + \frac{\pi}{4}\right) = \) ()
A: \\(\frac{33}{65}\\)
B: \\(-\frac{33}{65}\\)
C: \\(-\frac{16}{65}\\)
D: \\(\frac{16}{65... | \text{B} |
In the reasoning "Since the square of any number is non-negative, $(2i)^{2} \geqslant 0$" the error occurs because of \(\)
A: The form of reasoning does not meet the requirements of a syllogism
B: The major premise is incorrect
C: The minor premise is incorrect
D: The conclusion of the reasoning is incorrect | B |
95 Let the function $f(x)$ be defined for $x \in \mathbf{R}$, $f(x) \neq 0 (x \in \mathbf{R})$, and there exists a positive number $a$ such that $f(x-a) = \frac{1}{f(x)} (x \in \mathbf{R})$, then $f(x)=$
A. is definitely not a periodic function
B. is definitely a periodic function
C. may be a periodic function, or may ... | f(x)=f(x+2a) |
The general form of the quadratic equation $2x^{2}-1=6x$ is ______, where the coefficient of the quadratic term is ______, the coefficient of the linear term is ______, and the constant term is ______. | 2x^{2}-6x-1=0, 2, -6, -1 |
In triangle $A B C$, side $A B=6, \angle B A C=30^{\circ}$, the radius of the circumscribed circle is 5. Find side $A C$.
# | AC=3\sqrt{3}\4 |
Problem 11. Find the maximum value of the function $y=3 x+4 \sqrt{1-x^{2}}$
$$ | 5 |
5. What is the maximum number of rooks that can be placed on the cells of a $300 \times 300$ board so that each rook attacks no more than one other rook? (A rook attacks all cells it can reach according to chess rules, without passing through other pieces.)
# | 400 |
For any set $A=\left\{a_{1}, a_{2}, a_{3}, a_{4}\right\}$ of four distinct positive integers with sum $s_{A}=a_{1}+a_{2}+a_{3}+a_{4}$, let $p_{A}$ denote the number of pairs $(i, j)$ with $1 \leq i<j \leq 4$ for which $a_{i}+a_{j}$ divides $s_{A}$. Among all sets of four distinct positive integers, determine those sets... | \{d, 5 d, 7 d, 11 d\} \text{ and } \{d, 11 d, 19 d, 29 d\} |
The equation of the tangent line to the curve $y=x^2 + \frac{1}{x}$ at the point (1, 2) is _________. | x - y + 1 = 0 |
Find the sum of the squares of the distances from the vertices of a regular $n$-gon inscribed in a circle of radius $R$ to an arbitrary line passing through the center of the polygon.
# | \frac{1}{2}nR^{2} |
The coefficient of $x^3$ in the expansion of $(1+x)^6(1-x)^4$ is ______. | -8 |
4. 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 face of each die is determined. What is the probability that this sum is a perfect square? | \frac{7}{36} |
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$ | 1, 4, 9, 16, 25 |
Calculate:
1. $(-1)^{2}+(\pi-3.14)^{0}+2\sin60°+|1-\sqrt{3}|-\sqrt{12}$;
2. Solve the quadratic equation $x(x-2)=x-2$;
3. Simplify and evaluate $(x+2+\frac{4}{x-2})÷\frac{x^{3}}{x^{2}-4x+4}$, where $x$ is a suitable non-negative integer that satisfies $x\leqslant 2$. | -1 |
In a triangle $\triangle ABC$ with a circumcircle diameter of $1$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given vectors $\overrightarrow{m}=(a,\cos B)$ and $\overrightarrow{n}=(b,\cos A)$, and $\overrightarrow{m}$ is parallel to but not equal to $\overrightarrow{n}$.
1. Find the ran... | [2 \sqrt {2},+\infty) |
4. In $\triangle A B C$, it is known that the lengths of the three sides are $A B=\sqrt{7}$,
$$
\begin{aligned}
B C= & \sqrt{3}, C A=\sqrt{2} \text {. Then } \\
& \overrightarrow{A B} \cdot \overrightarrow{B C}+\sqrt{2} \overrightarrow{B C} \cdot \overrightarrow{C A}+\sqrt{3} \overrightarrow{C A} \cdot \overrightarrow{... | \sqrt{2}-3 \sqrt{3}-4 |
21. The diagram shows two congruent equilateral triangles whose overlap is a hexagon. The areas of the smaller triangles, which are also equilateral, are $1,1,9,9,16$ and 16 , as shown.
What is the area of the inner hexagon?
A 68
B 58
C 48
D 38
E 28 | 38 |
12. As shown in the right figure, fill the numbers $1.2, 3.7, 6.5, 2.9, 4.6$ into five ○s respectively, then fill each $\square$ with the average of the three $\bigcirc$ numbers it is connected to, and fill the $\triangle$ with the average of the three $\square$ numbers. Find a way to fill the numbers so that the numbe... | 3.1 |
If the function $f(x)=\sin x-kx$ has an extremum, then the range of the real number $k$ is ( )
A: $(-1,1)$
B: $[0,1)$
C: $(1,+\infty)$
D: $(-\infty,-1)$ | A |
Given that $M(a,3)$ and $N(4,b)$ are symmetric with respect to the $y$-axis, the value of $a+b$ is ______. | -1 |
Given $l$ and $m$ are two different lines, and $\alpha$, $\beta$, $\gamma$ are three different planes, which of the following statements is correct?
A: If $l$ is parallel to $\alpha$ and $m$ is parallel to $\alpha$, then $l$ is parallel to $m$.
B: If $l$ is parallel to $\alpha$ and $l$ is parallel to $\beta$, then $\... | \text{C} |
Problem 1. A group of schoolchildren decided to buy pizza. Each pizza is cut into 12 pieces. They calculated that if each boy eats 7 pieces and each girl eats 2 pieces, then three pizzas would not be enough. If they buy four pizzas, then each boy can have 8 pieces and each girl can have 4 pieces, and there will still b... | 1 |
The solution set for the inequality $-x^{2}+3x-2 \geqslant 0$ is $\boxed{(\text{ } \text{ })}$.
A: $(-\infty,1] \cup [2,+\infty)$
B: $[1,2]$
C: $(-\infty,1) \cup (2,+\infty)$
D: $(1,2)$ | B |
35. The fish soup prepared by the tourists on the seashore after a successful fishing trip was good, but due to lack of culinary experience, they put too little salt in it, so they had to add salt at the table. The next time they made the same amount of fish soup, they put in twice as much salt as the first time, but e... | \frac{1}{3} |
21st Putnam 1960 Problem B1 Find all pairs of unequal integers m, n such that m n = n m . Solution | (2,4),(4,2),(-2,-4),(-4,-2) |
Problem 2. On a shelf, there are three books. The first has 90, the second 110, and the third 150 pages. The covers of the books are of equal thickness, and each of them is 2 mm thick. How many millimeters thick are the books together if it is known that 10 pages have a thickness of $1 \mathrm{~mm}$? | 47\mathrm{~} |
Zaytsev C.A.
A checkerboard paper square $8 \times 8$ was folded several times along the cell lines so that a $1 \times 1$ square was obtained. It was then cut along a segment connecting the midpoints of two opposite sides of the square. Into how many pieces could the square have split as a result? | 9 |
7.5. Lёsha colors cells inside a $6 \times 6$ square drawn on graph paper. Then he marks the nodes (intersections of the grid lines) to which the same number of colored and uncolored squares are adjacent. What is the maximum number of nodes that can be marked? | 45 |
10. (10 points) Arrange 11 identical rectangular bricks as shown in the figure to form a large rectangular prism. It is known that the volume of each brick is 288 cubic centimeters, and the surface area of the large rectangular prism is ( ) square centimeters.
A. 1944
B. 1974
C. 2014
D. 1368 | 1368 |
(AHSME 1990, P9)
Consider a cube whose edges are colored either red or black such that each face of the cube has at least one black edge. What is the minimum number of black edges? | 3 |
35. Let $f(x)=x^{2}+a x+b \cos x$, find all pairs of real numbers $(a, b)$, such that the equation $f(x)=0$ and $f(f(x))=0$ have the same and non-empty set of real solutions. | (,b)\mid0\leqslant<4,b=0 |
Let $a,b,c>0$ the sides of a right triangle. Find all real $x$ for which $a^x>b^x+c^x$, with $a$ is the longest side. | x > 2 |
Problem 3. The father gave 650 denars to his 5 sons and told them to divide them so that each of them receives 30 denars less than the immediately older one. How many denars will each of them receive | 70,100,130,160,190 |
13. In a trapezoid, the lengths of the diagonals are 6 and 8, and the length of the midline is 5. Find the height of the trapezoid. | 4.8 |
Let the function $f(x) = xe^x$, then ( )
A: $x=1$ is a local maximum point of $f(x)$
B: $x=1$ is a local minimum point of $f(x)$
C: $x=-1$ is a local maximum point of $f(x)$
D: $x=-1$ is a local minimum point of $f(x)$ | D |
3. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=B C=4, A A_{1}=2, P$ is a point on the plane $A_{1} B C$, and $\overrightarrow{D P} \cdot \overrightarrow{P B}=0$. Then, the area of the plane region enclosed by all points $P$ that satisfy the above conditions is $\qquad$. | \frac{36\pi}{5} |
1. Given $m^{2}+n^{2}+m n+m-n+1=0$. Then the value of $\frac{1}{m}+\frac{1}{n}$ is ( ).
(A) -1
(B) 0
(C) 1
(D) 2 | B |
8.4. From identical isosceles triangles, where the angle opposite the base is $45^{\circ}$ and the lateral side is 1, a figure was formed as shown in the diagram. Find the distance between points $A$ and $B$. | 2 |
In the Cartesian coordinate system, find the general equation for curve \( C \) described by the parametric equations:
$$
\begin{cases}
x=2+ \frac { \sqrt {2}}{2}t \\
y=1+ \frac { \sqrt {2}}{2}t
\end{cases}
$$
where \( t \) is the parameter. | x - y - 1 = 0 |
11. Let real numbers $x, y, z, w$ satisfy $x+y+z+w=1$. Then the maximum value of $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ is $\qquad$ . | \frac{3}{2} |
6. Let's call the distance between numbers the absolute value of their difference. It is known that the sum of the distances from eighteen consecutive natural numbers to some number $a$ is 1005, and the sum of the distances from these same eighteen numbers to the number $a^{2}$ is 865. Find all possible values of $a$. | -\frac{7}{3} |
Given a sequence $\{a_n\}$ where the sum of the first $n$ terms is $S_n$ ($n \in \mathbb{N}^*$), and it satisfies $S_n + S_{n+1} = 2n^2 + n$. If for every $n \in \mathbb{N}^*$, $a_n < a_{n+1}$ always holds, then the range of values for the first term $a_1$ is ______. | \left(- \frac{1}{4}, \frac{3}{4}\right) |
Let proposition $p$: real number $x$ satisfies $x^{2}-4mx+3m^{2} \lt 0$; proposition $q$: real number $x$ satisfies $|x-3|\leqslant 1$.
$(1)$ If $m=1$, and both $p$ and $q$ are true, find the range of real number $x$;
$(2)$ If $m \gt 0$, and $q$ is a sufficient but not necessary condition for $p$, find the range of... | \left(\frac{4}{3}, 2\right) |
Given the parabola $y^{2}=8x$ whose latus rectum passes through a focus of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$, and the eccentricity of the hyperbola is $2$, find the equation of the hyperbola. | x^{2}-\frac{y^{2}}{3}=1 |
Given that the line $x=t$ intersects the graphs of the functions $f\left(x\right)=x^{2}$ and $g\left(x\right)=\ln x$ at points $M$ and $N$ respectively, the value of $t$ when $|MN|$ is minimized is ____. | \frac{\sqrt{2}}{2} |
Which of the following sets of numbers can form the sides of a right triangle?
A: $5$, $12$, $14$
B: $6$, $8$, $9$
C: $7$, $24$, $25$
D: $8$, $13$, $15$ | C |
Given the function f(x) = xe^(kx), where k ≠ 0,
1. Find the equation of the tangent line to the curve y = f(x) at the point (0, f(0)).
2. Determine the range of k for which the function f(x) is monotonically increasing in the interval (-1, 1). | k \in [-1, 0) \cup (0, 1] |
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