problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
For a set with $n$ elements, how many subsets are there whose cardinality is respectively $\equiv 0$ (mod $3$), $\equiv 1$ (mod $3$), $ \equiv 2$ (mod $3$)? In other words, calculate
$$s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}$$
for $i=0,1,2$. Your result should be strong enough to permit direct evalua... | \frac{2^{1000} - 1}{3} |
In an opaque bag, there are several red balls and $4$ yellow balls, which are indistinguishable except for their colors. After shaking the bag evenly, a ball is randomly drawn from it, the color is noted, and then the ball is put back into the bag. Through a large number of repeated ball-drawing experiments, it is foun... | A |
(1) Among the following functions, the one that is an even function with the smallest positive period of $\frac{\pi}{2}$ is ( ).
(A) $y=\sin 2 x+\cos 2 x$
(B) $y=\sin 2 x \cos 2 x$
(C) $y=\sin ^{2} x+\cos 2 x$
(D) $y=\sin ^{2} 2 x-\cos ^{2} 2 x$ | D |
The phrase "A great teacher produces outstanding students" can be interpreted as the teaching level of the teacher is positively correlated with the level of the students. ( )
A: \\(\checkmark\\)
B: \\(\times\\)
C:
D: | A |
7. Exchange a 2-jiao RMB note into 1-fen, 2-fen, and 5-fen coins. How many ways are there to do this? | 29 |
14. A ball was added to an urn containing one white ball - either white or black (with equal probabilities of selection). After this, one ball was randomly drawn from the urn. It turned out to be white. What is the conditional probability that the remaining ball in the urn is also white? | \frac{2}{3} |
Calculate:$(1)1\frac{1}{3}-2\frac{2}{9}×\frac{6}{5}+27÷({-3})$;$(2)\sqrt{{{({-2})}^2}}-\sqrt[3]{8}-|{3-π}|+{({-1})^3}$. | 2-\pi |
Calculate: $$\frac {1+i^{2017}}{1-i}$$ (where $i$ is the imaginary unit) | i |
3. Find the sum of the first 10 elements that are found both in the arithmetic progression $\{4,7,10,13, \ldots\}$ and in the geometric progression $\{10,20,40,80, \ldots\} \cdot(10$ points $)$ | 3495250 |
$\left[\begin{array}{l}\text { Algebraic inequalities (miscellaneous) } \\ {[\quad \text { Case analysis }}}\end{array}\right]$
$x, y>0$. Let $S$ denote the smallest of the numbers $x, 1 / y, y+1 / x$. What is the maximum value that $S$ can take? | \sqrt{2} |
6. (1992 National High School Competition Question) Let $y=f(x)$ be a function defined on $\mathbf{R}$, and satisfies the following relations: $f(10+x)=f(10-x)$, $f(20-x)=-f(20+x)$, then the function $f(x)$ is
A. An even function and a periodic function
B. An even function but not a periodic function
C. An odd function... | C |
Example 6. Given the general solution $y=C_{1} \sin 2 x+C_{2} \cos 2 x$ of the differential equation $y^{\prime \prime}+4 y=0$. What particular solutions are obtained when $C_{1}=2, C_{2}=3$? For what values of the parameters $C_{1}$ and $C_{2}$ do the particular solutions $y=\sin 2 x, y=\cos 2 x$ result? | 2\sin2x+3\cos2x,\,\sin2x\,(C_{1}=1,C_{2}=0),\,\cos2x\,(C_{1}=0,C_{2}=1) |
1489. Out of 5000 randomly taken parts, 32 were defective. Find the frequency of defective parts in this batch. | 0.0064 |
The graph of the function $f(x)=\sin 2x$ is shifted to the left by $\frac{π}{4}$ units to obtain the graph of the function $y=g(x)$. Which of the following statements about $g(x)$ is correct?
A: Odd function
B: Monotonically increasing on $({0, \frac{π}{4}})$
C: Symmetric about the point $({\frac{3π}{8}, 0})$
D: Sy... | D |
Given that the graphs of the inverse proportion function $y=\frac{2}{x}$ and the linear function $y=x+3$ intersect at the point $\left(m,n\right)$, the value of $\frac{1}{m}-\frac{1}{n}$ is ____. | \frac{3}{2} |
Owen spends $\$ 1.20$ per litre on gasoline. He uses an average of $1 \mathrm{~L}$ of gasoline to drive $12.5 \mathrm{~km}$. How much will Owen spend on gasoline to drive $50 \mathrm{~km}$ ?
(A) $\$ 4.80$
(B) $\$ 1.50$
(C) $\$ 4.50$
(D) $\$ 6.00$
(E) $\$ 7.50$ | 4.80 |
8. Given the function $f(x)=\sqrt{1+x \sqrt{1+(x+1) \sqrt{1+(x+2) \sqrt{1+(x+3) \sqrt{\cdots}}}}}(x>1$, $\left.x \in \mathbf{N}^{*}\right)$, then $f(2008)=$ $\qquad$ | 2009 |
If $A = \{a, b, 1\}$, then ( )
A: $1 \in A$
B: $1 \notin A$
C: $a = 1$
D: $b = 1$ | A |
A plane parallel to the base of a cone cuts the cone, resulting in a frustum whose top and bottom surface areas have a ratio of 1:16. The slant height of the removed cone is 3 cm. Find the slant height of the frustum. | 9\text{ cm} |
(4) Let the line $l$ with a slope of $\frac{\sqrt{2}}{2}$ intersect the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ at two distinct points $P$ and $Q$. If the projections of points $P$ and $Q$ on the $x$-axis are exactly the two foci of the ellipse, then the eccentricity of the ellipse is $\qquad$. | \frac{\sqrt{2}}{2} |
For which real numbers does the following inequality hold?
$$
\sqrt[3]{5 x+2}-\sqrt[3]{x+3} \leq 1
$$ | x\leq5 |
Problem 3. For natural numbers $x$ and $y$ and an odd integer $z$, it is known that $x! + y! = 48z + 2017$. Find all possible such triples of numbers $(x, y, z)$. (Recall that $1! = 1, 2! = 1 \cdot 2, n! = 1 \cdot 2 \cdot \ldots \cdot n$) Answer: $(1; 6; -27), (6; 1; -27), (1; 7; 63), (7; 1; 63)$. | (1;6;-27),(6;1;-27),(1;7;63),(7;1;63) |
701. Calculate the integral $\int_{1}^{9} \sqrt{6 x-5} d x$ using the Newton-Leibniz formula and approximate formulas for rectangles, trapezoids, and Simpson's method, dividing the integration interval into 8 equal parts. Then estimate the percentage error of the results obtained using the approximate formulas. | 38 |
Given $a$ and $b$ are real numbers, then "a > 0 and b > 0" is a ( ) condition for "a + b > 0 and ab > 0".
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Sufficient and necessary
D: Neither sufficient nor necessary | C |
Example 2-37 Find the value of the following $n$-order determinant:
$$
d_{n}=\left|\begin{array}{llllll}
2 & 1 & 0 & 0 & \cdots & 0 \\
1 & 2 & 1 & 0 & \cdots & 0 \\
0 & 1 & 2 & 1 & \cdots & 0 \\
& & & \vdots & & \\
0 & 0 & 0 & 0 & \cdots & 2
\end{array}\right|
$$
According to the properties of determinants | d_{n}=n+1 |
Determine the coefficient of the $x^{2}$ term in the expansion of $(1+2x)^{5}$. | 40 |
8. Let $f(x)=\cos 2x-2a(1+\cos x)$ have a minimum value of $-\frac{1}{2}$. Then $a=$ $\qquad$ . | -2+\sqrt{3} |
3. In triangle $\mathrm{ABC}$, the bisector $\mathrm{AD}$ is drawn. Angle $\mathrm{ADC}$ measures 66 degrees. -It turns out that $\mathrm{AD}=\mathrm{BD}$. Find the degree measure of angle C.
Assume it is known that the sum of the angles in a triangle is 180 degrees. | 81 |
50*. What is the greatest number of circles of radius 1 that can be placed on a plane so that they all intersect a certain fixed unit circle $S$ and no one of them contains the center of $S$ or the center of another circle inside it? | 18 |
6. (The 13th "Hope Cup" Grade 1 High School 2nd Test) Given that the function $f(x)$ is an increasing function on $\mathbf{R}$, and its graph passes through the points $A(0,-2), B(3,2)$, then the solution set of the inequality $|f(x+1)| \geqslant 2$ is
A. $(-\infty,-1) \cup[2,+\infty)$
B. $[2,+\infty)$
C. $(-\infty,-1]... | A.(-\infty,-1)\cup[2,+\infty) |
Example 2 In an isosceles right $\triangle ABC$, it is known that $\angle ACB=$ $90^{\circ}, P$ is a point inside $\triangle ABC$, such that $PA=11, PB=7, PC=$ 6. Then the length of side $AC$ is $\qquad$ [2] | \sqrt{85+42 \sqrt{2}} |
The expression $(x-3)(x-4)(x-5)\ldots(x-12)(x-13)$ (where $x\in\mathbb{N}^+$ and $x>13$) can be represented as ( )
A: $A_{x-3}^{10}$
B: $A_{x-3}^{11}$
C: $A_{x-13}^{10}$
D: $A_{x-13}^{11}$ | \text{B} |
Let \\(f(x)=e^{t(x-1)}-t\ln x\\), where \\(t > 0\\).
(I) If \\(t=1\\), prove that \\(x=1\\) is a local minimum point of the function \\(f(x)\\);
(II) Prove that: \\(f(x) \geqslant 0\\). | f(x) \geqslant 0 |
13. Given the sets $A=\left\{(x, y) \left\lvert\, \frac{y-3}{x-2}=a+1\right.\right\}, B=$ $\left\{(x, y) \mid\left(a^{2}-1\right) x+(a-1) y=15\right\}$. Then the possible values of $a$ that make $A \cap B=$ $\varnothing$ are . $\qquad$ | -4, -1, 1, \frac{5}{2} |
Let's determine how many regions the space is divided into by a) the six planes of the cube's faces?
b) the four planes of the tetrahedron's faces? | 15 |
Let $\theta \in \left(\frac{\pi}{4}, \frac{\pi}{2}\right)$, and $\sin 2\theta = \frac{1}{16}$. The value of $\cos\theta - \sin\theta$ is $\left( \ \right)$
A: $\frac{\sqrt{15}}{4}$
B: $-\frac{\sqrt{15}}{4}$
C: $\frac{3}{4}$
D: $-\frac{3}{4}$ | \text{B} |
Calculate:$(1)(2023-\sqrt{3})^0+|(\frac{1}{5})^{-1}-\sqrt{75}|-\frac{\sqrt{45}}{\sqrt{5}}$;$(2)(\sqrt{3}-2)^2-(\sqrt{2}+\sqrt{3})(\sqrt{3}-\sqrt{2})$. | 6 - 4\sqrt{3} |
Let set $M = \{x | -2 < x < 3\}$ and $N = \{-2, -1, 0, 1\}$. Then, $M \cap N =$ ( )
A: $\{-2, -1, 0\}$
B: $\{0, 1, 2\}$
C: $\{-1, 0, 1\}$
D: $\{-2, -1, 0, 1\}$ | C |
Given that the inequality $x^{2}-2y^{2}\leqslant cx(y-x)$ holds for any real numbers $x$ and $y$ satisfying $x > y > 0$, find the maximum value of the real number $c$. | 2\sqrt{2}-4 |
Determine the coefficient of the fourth term in the binomial expansion of $\left(x+\frac{2}{x^2}\right)^6$. (Express the answer in numerical form.) | 160 |
1.3. A ballpoint pen costs 10 rubles, a fountain pen costs 60 rubles, and a gel pen costs 70 rubles. What is the maximum number of fountain pens that can be bought if you need to buy exactly 25 pens in total and among them there must be pens of all three types, and you need to spend exactly 1000 rubles on them? | 9 |
Among the following functions, the one where $y$ is directly proportional to $x$ is:
A: $y=\frac{1}{2}x$
B: $y=2x+1$
C: $y=\frac{2}{x}$
D: $y=x^{2}$ | \text{A} |
Given that the projection of vector $\overrightarrow {a}$ in the direction of vector $\overrightarrow {b}$ is -1, the projection of vector $\overrightarrow {b}$ in the direction of vector $\overrightarrow {a}$ is - $\frac {1}{2}$, and $| \overrightarrow {b}|=1$, find $| \overrightarrow {a}+2 \overrightarrow {b}|=$ ( ... | \text{C} |
Among the following options, the most suitable for conducting a comprehensive survey (census) is:
A: Investigating the service life of a batch of energy-saving lamps
B: Investigating the water quality of Dongfeng Canal
C: Investigating the sports activities of middle school students in Henan Province
D: Testing the... | \text{D} |
## Task 5 - 200835
Two rays $s_{1}$ and $s_{2}$, which originate from a point $S$ and form a right angle with each other, are intersected by two parallel lines $g$ and $h$. The line $g$ intersects $s_{1}$ at $A$ and $s_{2}$ at $C$, the line $h$ intersects $s_{1}$ at $B$ and $s_{2}$ at $D$. Furthermore, $\overline{S B}... | 3\mathrm{~} |
1. (6 points) Calculate $10.37 \times 3.4 + 1.7 \times 19.26=$ | 68 |
9. Two circles are said to be orthogonal if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_{1}$ and $\omega_{2}$ with radii 10 and 13 , respectively, are externally tangent at point $P$. Another circle $\omega_{3}$ with radius $2 \sqrt{2}$ passes ... | \frac{92}{61} |
## SUBIECTUL IV
Să se calculeze:
$$
\lim _{x \rightarrow \infty}(a \sqrt{x+1}+b \sqrt{4 x+1}+c \sqrt{9 x+1}), a, b, c \in R
$$
## Notă:
Toate subiectele sunt obligatorii. Fiecare subiect se noteaza cu 7 puncte.
$>$ Timp de lucru: 3 ore.
## BAREM DE CORECTARE
etapa locală - 16 februarie 2014
## Clasa a XI- a
| L=0if+2b+3c=0,otherwiseL=\\infty |
A3. Let $A=\diamond-\odot+\boldsymbol{\phi}, B=-\boldsymbol{\phi}-\odot+\diamond$ and $C=\ominus-\diamond+\boldsymbol{\phi}$. Then $A-B-C$ is equal to:
(A) 0
(B) $3 \diamond-3 \odot-$
(C) $\odot$
(D) $\diamond$
(E) $\diamond-\odot+$ | \diamond-\odot+\boldsymbol{\phi} |
Given a function $f(x)$ on $\mathbb{R}$ defined as
$$
f(x)=
\begin{cases}
\log_{2}^{(3-x)} & \text{if } x\leq 0\\
f(x-1)-f(x-2) & \text{if } x>0
\end{cases}
$$
find the value of $f(2017)$. | \log_{2}3 - 2 |
6. Among the first 999 positive integers, the numbers that are neither divisible by 5 nor by 7 are
$\qquad$ in number. | 686 |
14. In Lion Tiger Mountain, there is a group of lions and tigers, and the $\frac{3}{5}$ of the number of lions is equal to the $\frac{2}{3}$ of the number of tigers. Therefore, Lion Tiger Mountain has at least $\qquad$ lions. | 10 |
7. (10 points) A certain two-digit number is a multiple of 2, adding 1 makes it a multiple of 3, adding 2 makes it a multiple of 4, adding 3 makes it a multiple of 5, then this two-digit number is $\qquad$ . | 62 |
Task 1. On the faces of a six-faced dice, the numbers 1, 2, 3, 4, 5, and 6 are placed. The dice is thrown and falls on the table. After this, the numbers on all faces except one are visible. The numbers on the five visible faces are multiplied. Find the probability that this product is divisible by 16.
Answer: $0.5$. | 0.5 |
2. (16 points) Find the minimum value of the function
$$
f(x)=3 \sin ^{2} x+5 \cos ^{2} x+2 \cos x
$$ | 2.5 |
2. The maximum volume of a cone inscribed in a sphere compared to the volume of the sphere is
保留了源文本的换行和格式。 | 8:27 |
If $a$ is a rational number, then the minimum value of $|a-1|+2$ is ( )
A: $0$
B: $1$
C: $2$
D: $3$ | C |
If $m, n \in \{x | x = a_2 \times 10^2 + a_1 \times 10 + a\}$, where $a_i \in \{1, 2, 3, 4, 5\}$ ($i=0, 1, 2$), and $m+n=735$, then the number of distinct points represented by the real number pairs $(m, n)$ on the plane is ( )
A: 32
B: 40
C: 50
D: 75 | A |
In right triangle $ABC$, $\angle C=90^{\circ}$, and the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively, satisfying $a+b=cx$. The range of the real number $x$ is \_\_\_\_\_\_. | (1, \sqrt {2}] |
The domain of the function $y= \sqrt{2-3x} - (x+1)^0$ is ( )
A: $(-1, \frac{2}{3}]$
B: $(-1, \frac{2}{3})$
C: $(-\infty, -1) \cup (-1, \frac{2}{3}]$
D: $[\frac{2}{3}, +\infty)$ | C |
12. A store has 3 types of scarves, priced at 14 yuan, 12 yuan, and 10 yuan respectively. There are 5 types of hats, priced at 13 yuan, 11 yuan, 9 yuan, 7 yuan, and 5 yuan respectively. If a hat and a scarf are paired into a set, then there are $\qquad$ different prices for the sets. | 7 |
5. In a triangle with sides 3 and 5 and an angle of $120^{\circ}$, the third side is 7. | 7 |
2. If $x^{2}+y^{2}=1$, then
$$
\sqrt{x^{2}-2 x+1}+\sqrt{4 y^{2}+4 y+1}+\sqrt{x y-2 x+y-2}
$$
is ( ).
(A) 0
(B) 1
(C) 2
(D) 3 | D |
Solve the following system of equations:
$$
\begin{aligned}
& 3 x^{2}-x y=1 \\
& 9 x y+y^{2}=22
\end{aligned}
$$ | x_{1}=1,y_{1}=2;x_{3}=-1,y_{3}=-2;x_{2}=-\frac{1}{6},y_{2}=5.5;x_{4}=\frac{1}{6},y_{4}=-5.5 |
Given a set $M = \{x\in \mathbb{R} \mid \frac {x+2}{x-1} \leq 0\}$ and $N$ as the set of natural numbers, which of the following options is correct?
(A) $M \subseteq \{x \mid x \geq 1\}$
(B) $M \subseteq \{x \mid x > -2\}$
(C) $M \cap N = \{0\}$
(D) $M \cup N = N$ | C |
In the arithmetic sequence $\{a_n\}$, $a_1+a_2=4$, and $d=2$. Find the value of $a_7+a_8$. | 28 |
4. (1988 American High School Mathematics Examination) Toss an "unfair" coin, the probability of heads is $p$, let $w$ be the probability of getting exactly 3 heads in 5 independent tosses, if $w=\frac{144}{625}$, then ( ).
A. $p$ must be $\frac{2}{5}$
B. $p$ must be $\frac{3}{5}$
C. $p$ must be greater than $\frac{3}{... | D |
2. Solve the system $\left\{\begin{array}{c}x^{2}-y-z=8 \\ 4 x+y^{2}+3 z=-11 \\ 2 x-3 y+z^{2}=-11\end{array}\right.$ | -3,2,-1 |
10.2. The number $a$ is a root of the quadratic equation $x^{2}-x-50=0$. Find the value of $a^{4}-101 a$. | 2550 |
1. Given $x^{2}=y^{2}+4 y+20$. Then the number of all integer pairs satisfying the equation is ( ).
(A) 4
(B) 6
(C) 8
(D) infinitely many | B |
[u]Set 3[/u]
[b]3.1[/b] Annie has $24$ letter tiles in a bag; $8$ C’s, $8$ M’s, and $8$ W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
[b]3.2[/b] Let $T$ be the answer from the previous problem. Charlotte is initially standi... | \frac{1}{5} |
## Task 15/89
Determine all natural numbers $n$ that satisfy the equation $n!+1=(10 n+1)^{2}$! | 7 |
Consider infinite sequences $a_1,a_2,\dots$ of positive integers satisfying $a_1=1$ and $$a_n \mid a_k+a_{k+1}+\dots+a_{k+n-1}$$ for all positive integers $k$ and $n.$ For a given positive integer $m,$ find the maximum possible value of $a_{2m}.$
[i]Proposed by Krit Boonsiriseth[/i] | 2^m - 1 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $60^{\circ}$ between them, and $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|=2$, then $\overrightarrow{a} \cdot \overrightarrow{b}=$ ()
A: $\frac{1}{2}$
B: $\frac{\sqrt{3}}{2}$
C: $1$
D: $2$ | C |
Given the universal set $U=\{1,2,3,4,5\}$, $\complement _{U}A=\{2,4\}$, $\complement _{U}B=\{3,4\}$, then ()
A: $1\in A$, $1\notin B$
B: $2\in A$, $2\in B$
C: $3\in A$, $3\notin B$
D: $5\notin A$, $5\in B$ | C |
The equation of the line passing through the point (1, 2) with a slope of 3 is ( )
A: y=3x-3
B: y=3x-2
C: y=3x-1
D: y=x-1 | C |
Given a line from a point A outside circle O intersects the circle O at points B and C, and AB•AC=60, OA=8, then the radius of this circle is. | 2 |
4. In a convex quadrilateral $ABCD$, $AB=\sqrt{3}, BC=CD=DA=1$. Let $S$ and $T$ be the areas of $\triangle ABD$ and $\triangle BCD$, respectively. Then the maximum value of $S^2 + T^2$ is ( ).
(A) $\frac{8}{7}$
(B) 1
(C) $\frac{7}{8}$
(D) 2 | C |
Suppose $x$ and $y$ are real numbers satisfying $x+y=5$. What is the largest possible value of $x^{2}+2 x y$ ? | 25 |
6. If $|x|+|y|=1$, then the range of values for $S=y-2 x$ is ( ).
(A) $-4 \leqslant S \leqslant 4$
(B) $-3 \leqslant S \leqslant 3$
(C) $-2 \leqslant S \leqslant 2$
(D) $-1 \leqslant S \leqslant 1$ | C |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $A=\frac{π}{6}$ and $(1+\sqrt{3})\sin B=2\sin C$.
$(1)$ Prove that $b=\sqrt{2}a$;
$(2)$ If the area of $\triangle ABC$ is $2+2\sqrt{3}$, find $b$. | 4 |
3. Let a tangent line of the circle $x^{2}+y^{2}=1$ intersect the $x$-axis and $y$-axis at points $A$ and $B$, respectively. Then the minimum value of $|AB|$ is $\qquad$ . | 2 |
11. Consider an equilateral triangle with a side length of 1, denoted as $A_{1}$ (as shown in Figure 3).
Trisect each side of $A_{1}$, and construct an equilateral triangle outward on the middle segment, then remove the middle segment to obtain the figure denoted as $A_{2}$ (as shown in Figure 4); trisect each side of ... | \frac{64}{9} |
A point mass is in a state of equilibrium under the action of three forces $F\_1$, $F\_2$, and $F\_3$ (in units of Newton) on the same plane. It is known that $F\_1$ and $F\_2$ form a $120^{\circ}$ angle, and both $F\_1$ and $F\_2$ have a magnitude of $6$ Newton. Calculate the magnitude of $F\_3$. | 6 |
Given a complex number $z=m-i$ where $m\in \mathbb{R}$, and $\overline{z}•(1+3i)$ is an imaginary number ($\overline{z}$ is the conjugate of $z$).$(1)$ Find the value of the real number $m$;$(2)$ Let $z_{1}=\frac{m+4i}{1-i}$, find $|z_{1}|$;$(3)$ Let $z_{2}=\frac{a-{i}^{2023}}{z}$, and the point corresponding to $z_{2}... | \left(\frac{1}{3}, +\infty\right) |
Gage skated $1$ hr $15$ min each day for $5$ days and $1$ hr $30$ min each day for $3$ days. How long would he have to skate the ninth day in order to average $85$ minutes of skating each day for the entire time?
$\text{(A)}\ \text{1 hr}\qquad\text{(B)}\ \text{1 hr 10 min}\qquad\text{(C)}\ \text{1 hr 20 min}\qquad\text... | (E)\2\ |
Given a sector with a radius of $2$ and a central angle of $90^{\circ}$, the area of the sector is ______. | \pi |
(1) If $a$ and $b$ are both positive numbers, and $a \neq b$, prove that: $$\frac {a}{ \sqrt {b}}+ \frac {b}{ \sqrt {a}} > \sqrt {a}+ \sqrt {b}$$
(2) For the sequence $\{a_n\}$, it is known that $a_n > 0$ and $(a_1+a_2+\ldots+a_n)^2 = a_1^3+a_2^3+\ldots+a_n^3$, find $a_1$, $a_2$, $a_3$, and conjecture $a_n$. | a_n = n |
In the sequence $\{a_n\}$, $a_{n+1}=ca_n$ (where $c$ is a non-zero constant), and the sum of the first $n$ terms is $S_n=3^n+k$, then the value of the real number $k$ is $\boxed{\text{(C) } -1}$. | -1 |
\section*{Problem 7}
A tangent to the inscribed circle of a triangle drawn parallel to one of the sides meets the other two sides at \(\mathrm{X}\) and \(\mathrm{Y}\). What is the maximum length \(\mathrm{XY}\), if the triangle has perimeter \(\mathrm{p}\) ?
| \frac{p}{8} |
11. As shown in the figure, the graph of the linear function $y=k x+b$ passes through the point $P(1,3)$, and intersects the positive halves of the $x$-axis and $y$-axis at points $A, B$, respectively. Point $O$ is the origin. The minimum value of the area of $\triangle A O B$ is $\qquad$ | 6 |
3. As shown in Figure 2, quadrilateral $ABCD$ is a parallelogram, $\angle ADC=60^{\circ}, AD=2$, $AB=\sqrt{3}+1, M$ is the midpoint of side $AD$, and the angle bisector of $\angle BCD$ intersects segment $BM$ at point $K$. Find the value of $\angle CKB$. | 75 |
(4) There is a regular quadrilateral pyramid, with the side length of its base and the side length of its lateral edges both being $a$. Now, it is to be completely wrapped with a square piece of wrapping paper (the paper cannot be cut, but can be folded). Then, the minimum side length of the wrapping paper should be ( ... | \frac{\sqrt{2}+\sqrt{6}}{2} |
Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$. | 5 |
Arrange the polynomial $-a + 3a^5b^3 + 5a^3b^5 - 9 + 4a^2b^2$ in descending powers of $a$. | 3a^5b^3 + 5a^3b^5 + 4a^2b^2 - a - 9 |
Given an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is $S_n$, and it is known that $a_3=3$, $S_4=10$.
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ If $b_n=2^{a_n}+1$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n=2^{n+1}+n-2 |
In $\triangle ABC$, the length of the side opposite to angle $A$ is $2$, and the vectors $\overrightarrow{m} = (2, 2\cos^2\frac{B+C}{2} - 1)$ and $\overrightarrow{n} = (\sin\frac{A}{2}, -1)$.
1. Find the value of angle $A$ when the dot product $\overrightarrow{m} \cdot \overrightarrow{n}$ is at its maximum.
2. Under t... | \sqrt{3} |
Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$, find the sum of digits of $E$
| 13 |
Find the unknown number $x$.
- $263-x=108$
- $25 \times x=1950$
- $x \div 15=64$ | 960 |
If the complex number $z=(1+i)(1+ai)$ corresponds to a point on the imaginary axis in the complex plane, then the real number $a=$ ______. | 1 |
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