problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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Given the function $f(x)=\cos x\cdot \cos (x-\theta )-\frac{1}{2}\cos \theta$, where $\theta\in(0,\pi)$. It is known that $f(x)$ reaches its maximum value when $x=\frac{\pi }{3}$.
(1) Find the value of $\theta$;
(2) Given $g(x)=2f(\frac{3}{2}x)$, find the maximum value of the function $g(x)$ on the interval $[0,\frac... | 1 |
In the ancient Chinese mathematical masterpiece "Nine Chapters on the Mathematical Art," it is recorded: "There is a bamboo that is one zhang tall. When the top is broken and reaches the ground, it is three chi away from the base. How high is the broken part from the ground?" If we let the height of the broken part fro... | A |
The domain of the function $f\left(x\right)=\sqrt{x+2}+\dfrac{1}{x-1}$ is ( ).
A: $\left(-\infty ,1\right)\cup \left(1,+\infty \right)$
B: $ \left[-2,+\infty \right)$
C: $ \left[-2,1\right)\cup \left(1,+\infty \right)$
D: $ \left(1,+\infty \right)$ | C |
6.091. $\left\{\begin{array}{l}(x-y) \cdot x y=30 \\ (x+y) \cdot x y=120\end{array}\right.$ | (5;3) |
Line segment $P Q$ has endpoints at $P(6,-2)$ and $Q(-3,10)$. Point $R(a, b)$ is a point on $P Q$ such that the distance from $P$ to $R$ is one-third the distance from $P$ to $Q$. What is $b-a$ ? | -1 |
$22 \cdot 10$ A line $l$ intersects with the two half-planes of a right dihedral angle, and the angles formed with these two half-planes are $\alpha$ and $\beta$. Then the value that $\alpha+\beta$ must not take is
(A) $90^{\circ}$.
(B) $73^{\circ}$.
(C) $87^{\circ}$.
(D) $95^{\circ}$.
(High School Mathematics Competit... | 95 |
A box of strawberries, containing $12$ strawberries total, costs $\$ 2$. A box of blueberries, containing $48$ blueberries total, costs $ \$ 3$. Suppose that for $\$ 12$, Sareen can either buy $m$ strawberries total or $n$ blueberries total. Find $n - m$.
[i]Proposed by Andrew Wu[/i] | 120 |
6.073. $\left\{\begin{array}{l}y^{2}-x y=-12, \\ x^{2}-x y=28\end{array}\right.$ | (-7,-3),(7,3) |
Given the function $g(x)=a-x^{2}$ ($\frac{1}{e} \leqslant x \leqslant e$, where $e$ is the base of the natural logarithm) and $h(x)=2\ln x$, there exist points on the graph of $h(x)$ that are symmetric about the $x$-axis with points on the graph of $g(x)$. The range of the real number $a$ is ( ).
A: $\left[1, \frac{1}... | B |
If $f(x)$ is an odd function with a period of 5 on $\mathbb{R}$, and satisfies $f(1) = 1$ and $f(2) = 2$, then $f(3) - f(4) = \ $
A: 1
B: -1
C: -2
D: 2 | \text{B} |
1. $a, x, y$ are all natural numbers, and $\sqrt{a+2 \sqrt{7}}=\sqrt{x}+\sqrt{y}$, then $x+y=$ | 8 |
Let $P = \{x | x = 2k, k \in \mathbb{Z}\}$, $Q = \{x | x = 2k + 1, k \in \mathbb{Z}\}$, $R = \{x | x = 4k + 1, k \in \mathbb{Z}\}$, and $a \in P$, $b \in Q$. Then we have ( )
A: $a + b \in P$
B: $a + b \in Q$
C: $a + b \in R$
D: $a + b$ does not belong to any of the sets $P$, $Q$, or $R$ | B |
Given $\overrightarrow{a}=(-3,4)$, $\overrightarrow{b}=(-2,1)$, then the projection of $\overrightarrow{a}$ on $\overrightarrow{b}$ is ( )
A: $-2$
B: $2$
C: $-2 \sqrt {5}$
D: $2 \sqrt {5}$ | D |
10. (CZS 4) The square $A B C D$ is to be decomposed into $n$ triangles (nonoverlapping) all of whose angles are acute. Find the smallest integer $n$ for which there exists a solution to this problem and construct at least one decomposition for this $n$. Answer whether it is possible to ask additionally that (at least)... | 8 |
17. Which of these is equal to $0 . \dot{8}+0.07$ ?
A $0.8 \dot{7}$
B 0.88
C $0.9 \dot{5}$
D $0.9 \dot{6}$
E $0.9 \dot{8}$ | 0.9\dot{6} |
Example 1. The random variable $\xi$ is distributed according to the normal law with an unknown expected value $a$ and variance 1. There is a sample $x_{1}, \ldots, x_{100}$ of values of $\xi, \bar{x}=1.3$. Find the confidence interval for the expected value $a$ at the confidence level $p=0.98$. | \in(1.067,1.533)_{0.98} |
Given that the sum of the first $n$ terms of the positive arithmetic sequence ${a_n}$ is $S_n$ ($n\in\mathbb{N}^*$), and $a_{5}+a_{7}-a_{6}^{2}=0$, find the value of $S_{11}$ ( ).
A: 11
B: 12
C: 20
D: 22 | D |
Given points $A(3,2,3)$ and $B(1,1,4)$, the coordinates of the midpoint of $A$ and $B$ are:
A: $(1, \frac{1}{2}, -\frac{1}{2})$
B: $(2, \frac{3}{2}, \frac{7}{2})$
C: $(4,3,7)$
D: $(-1, -\frac{1}{2}, \frac{1}{2})$ | \text{B} |
2.1. Every month Ivan pays a fixed amount from his salary for a mortgage, and the remaining part of the salary is spent on current expenses. In December, Ivan paid $40 \%$ of his salary for the mortgage. In January, Ivan's salary increased by $9 \%$. By what percentage did the amount spent on current expenses increase ... | 15 |
1. If real numbers $x, y$ satisfy $(x+5)^{2}+(y-12)^{2}=14^{2}$, then the minimum value of $x^{2}+y^{2}$ is
A. 2
B. 1
C. $\sqrt{3}$
D. $\sqrt{2}$ | 1 |
B1. Calculate the quotient of numbers $a$ and $b$, if $a$ is the sum of the numbers $\frac{2}{3}$ and $\frac{3}{4}$, and the number $b$ is the difference between the number 1 and the quotient of the numbers $\frac{2}{3}$ and $\frac{3}{4}$.
(6 points) | \frac{51}{4} |
Calculate the value of $v_3$ using the Horner's Rule for the polynomial $f(x) = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64$ when $x = 2$.
A: $-10$
B: $-80$
C: $40$
D: $80$ | \text{B} |
7.1. The sum of the minuend, subtrahend, and difference is 25. Find the minuend. | 12.5 |
7. Xiaoming mistakenly calculated addition as subtraction, getting a result of 86, which is 186 less than the correct answer. The larger of the original addends is $(\quad)$. | 179 |
8. Solve the system $\left\{\begin{array}{l}3^{x} \cdot 2^{y}=972 ; \\ \log _{\sqrt{3}}(x-y)=2 .\end{array}\right.$ | {5;2} |
Given that the sequence $\{a_n\}$ is an arithmetic sequence with the first term being $1$, and the sequence $\{b_n\}$ is a geometric sequence with the common ratio $q (q \neq 1)$, it is also given that $a_2=b_1$, $a_3-a_1=b_2-b_1$, and $a_2b_2=b_3$.
$(1)$ Find the general term formulas for $\{a_n\}$ and $\{b_n\}$;
$(2)... | S_n = 2^{n+2} - 2n - 4 |
30. Observe the triangular number array in Figure 14. Then the last number in the 50th row is ( ).
(A) 125
(B) 1260
(C) 1270
(D) 1275 | D |
Given the function $f(x)=2\cos^2x+2\sin x\cos x$.
$(1)$ Calculate the value of $f\left( \dfrac {\pi}{8}\right)$;
$(2)$ Find the smallest positive period and the minimum value of the function $f(x)$. | 1- \sqrt {2} |
A1. Find all ordered triples $(x, y, z)$ of real numbers satisfying the following system of equations:
$$
\begin{aligned}
x^{3} & =\frac{z}{y}-2 \frac{y}{z} \\
y^{3} & =\frac{x}{z}-2 \frac{z}{x} \\
z^{3} & =\frac{y}{x}-2 \frac{x}{y}
\end{aligned}
$$
| (1,1,-1),(1,-1,1),(-1,1,1),(-1,-1,-1) |
In triangle $ABC$, $\angle A = 60^{\circ}$, and $c = \frac{3}{7}a$.
(Ⅰ) Find the value of $\sin C$;
(Ⅱ) If $a=7$, find the area of $\triangle ABC$. | 6\sqrt{3} |
Given positive numbers $x$, $y$, $z$ greater than 1, satisfying $$x+y+z=3 \sqrt {3}.$$
(1) Prove that: $$\frac {x^{2}}{x+2y+3z}+ \frac {y^{2}}{y+2z+3x}+ \frac {z^{2}}{z+2x+3y}\geq \frac { \sqrt {3}}{2}.$$
(2) Find the minimum value of $$\frac {1}{\log_{3}x+\log_{3}y}+ \frac {1}{\log_{3}y+\log_{3}z}+ \frac {1}{\log_... | 3 |
How many $6$-digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit. | 3003 |
Given that $\sin 2α = \frac{4}{5}$ and $α \in (0, \frac{π}{4})$, find the value of $\sin(\frac{π}{4} - α)$. | \frac{\sqrt{10}}{10} |
In a bag, there are one red ball and one white ball. A ball is drawn at random with replacement three times. Calculate the probabilities of the following events:
(1) Exactly two of the three colors drawn are the same;
(2) All three colors drawn are the same;
(3) The number of times a red ball is drawn is more tha... | \frac{1}{2} |
In an increasing arithmetic sequence $\{a_{n}\}$, it is known that $a_{4}$ and $a_{6}$ are two roots of the equation $x^{2}-10x+24=0$. Then $a_{20}=\left(\ \ \right)$
A: $19$
B: $20$
C: $21$
D: $22$ | B |
Problem 5.2. Several people were seated around a round table such that the distances between adjacent people are the same. One of them was given a sign with the number 1, and then everyone was given signs with numbers 2, 3, and so on, in a clockwise direction.
The person with the sign numbered 31 noticed that the dist... | 41 |
In the "Charity Day Donation" event, the Youth League Committee of a school in Cuiping District randomly selected the donation amounts of $10$ students for statistical analysis and presented the data in the following table.
| Amount (in yuan) | $5$ | $10$ | $15$ | $20$ |
|------------------|-----|------|------|------|... | 26400 \text{ yuan} |
In the Cartesian coordinate system $xOy$, the parametric equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=\frac{2+t}{6},\\ y=\sqrt{t}\end{array}\right.$ (where $t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\left\{\begin{array}{l}x=-\frac{2+s}{6},\\ y=-\sqrt{s}\end{array}\right.$ (whe... | (-1, -2) |
The maximum value of the function $f(x)= \frac {|x|}{ \sqrt {1+x^{2}} \sqrt {4+x^{2}}}$ is ( )
A: $ \frac {1}{4}$
B: $ \frac {1}{3}$
C: $ \frac {1}{2}$
D: $ \frac { \sqrt {2}}{2}$ | B |
$31 \cdot 12$ If $a, b$ are natural numbers not exceeding 10, then the number of pairs $(a, b)$ that make the solution of the equation $a x=b$ less than $\frac{1}{2}$ but greater than $\frac{1}{3}$ is
(A) five pairs.
(B) four pairs.
(C) three pairs.
(D) two pairs.
(China Shanghai Junior High School Mathematics Competit... | 5 |
2.81 The sum of the following 7 numbers is exactly 19:
$$\begin{array}{l}
a_{1}=2.56, a_{2}=2.61, a_{3}=2.65, a_{4}=2.71, a_{5}=2.79, a_{6}= \\
2.82, a_{7}=2.86 .
\end{array}$$
To approximate $a_{i}$ with integers $A_{i}$ $(1 \leqslant i \leqslant 7)$, such that the sum of $A_{i}$ is still 19, and the maximum value $M... | 61 |
The negation of the proposition "For all $a \in \mathbb{R}$, the equation $ax^2 - 3x - a = 0$ has a positive real root" is ( ).
A: For all $a \in \mathbb{R}$, the equation $ax^2 - 3x - a = 0$ does not have a positive real root
B: For all $a \in \mathbb{R}$, the equation $ax^2 - 3x - a = 0$ does not have a negative r... | \text{D} |
Example 7 Find a sequence $a_{0}, a_{1}, a_{2}, \cdots$, whose terms are all positive, $a_{0}=1$, and $a_{n}-a_{n+1}=a_{n+2}(n=$ $0,1,2, \cdots)$ Prove that such a sequence is unique. | a_{n}=(\frac{-1+\sqrt{5}}{2})^{n} |
20. (3 points) Xiao Qiang and Xiao Lin have more than 400 stamps in total. If Xiao Qiang gives some stamps to Xiao Lin, Xiao Qiang's stamps will be $\frac{6}{19}$ fewer than Xiao Lin's; if Xiao Lin gives the same number of stamps to Xiao Qiang, then Xiao Lin's stamps will be $\frac{6}{17}$ fewer than Xiao Qiang's. Ther... | 227,221 |
How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board? | 34 |
Among the following propositions for a plane $\alpha$ and skew lines $m$ and $n$, the true proposition is ( )
A: There exists a plane $\alpha$ such that $m \perp \alpha$ and $n \perp \alpha$.
B: There exists a plane $\alpha$ such that $m \subset \alpha$ and $n \subset \alpha$.
C: There exists a plane $\alpha$ such that... | \text{D} |
5.2. Masha eats a bowl of porridge in 12 minutes, and Bear eats it twice as fast. How long will it take them to eat six bowls of porridge? | 24 |
In the Cartesian coordinate system xOy, the parametric equations of the curve C₁ are $$\begin{cases} x=2t^{2} \\ y=2t \end{cases}$$ (where t is the parameter). Establish a polar coordinate system with the origin O as the pole and the positive semi-axis of the x-axis as the polar axis. The polar equation of the curve C₂... | 1 |
46. Calculate in the most rational way:
$$
333\left(\frac{71}{111111}+\frac{573}{222222}-\frac{2}{7 \cdot 37 \cdot 3}\right)
$$ | \frac{3}{14} |
4. If the inequality $\frac{4 x}{a}+\frac{1}{x} \geqslant 4$ holds for all $x$ in the interval $[1,2]$, then the range of the real number $a$ is ( ).
(A) $\left(0, \frac{4}{3}\right]$
(B) $\left(1, \frac{4}{3}\right)$
(C) $\left[1, \frac{4}{3}\right]$
(D) $\left[\frac{16}{7}, \frac{4}{3}\right]$ | A |
8. A rectangular garden is divided into 5 smaller rectangular areas of the same shape and size, each with an area of 24 square meters. The perimeter of this rectangular garden is $\qquad$ meters. | 44 |
5.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 105.75 |
2. Given $a \geqslant b>0$, and $\sin \alpha$ is a root of the quadratic equation $a x^{2}+b x-b=0$. Then the maximum value of $\sin \alpha$ is $\qquad$ . | \frac{\sqrt{5}-1}{2} |
## Task B-4.4.
For every natural number $n$, the sum of the first $n$ terms of a certain sequence is $S_{n}=2.5 n^{2}-4.5 n$.
Which terms of this sequence take values between 10085 and 10095? | a_{2019}=10088a_{2020}=10093 |
If $a \lt b \lt 0$, which of the following inequalities cannot hold?
A: $\frac{1}{a} \gt \frac{1}{b}$
B: $a-b \gt a$
C: $|a| \gt |b|$
D: $a^{3} \gt b^{3}$ | D |
Given the sequence $\sqrt{2}, \sqrt{5}, 2\sqrt{2}, \sqrt{11}, \ldots$, then $2\sqrt{17}$ is its $\boxed{\text{nth}}$ term. | 23 |
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six iden... | \sqrt{6}+1 |
Given the parabola $E$: $y^{2}=2px(p > 0)$, and the line $x=my+3$ intersects $E$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=6$, where $O$ is the origin.
$(1)$ Find the equation of the parabola $E$;
$(2)$ Given that the coordinates of point $C$ are $(-3,0)$, let the slopes of lines $CA$ an... | 24 |
Given that one asymptote of the hyperbola $\dfrac{x^{2}}{a^{2}}-y^{2}=1 (a > 0)$ has an angle of inclination of $30^{\circ}$, find the eccentricity of the hyperbola. | \dfrac{2\sqrt{3}}{3} |
In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$. | 90^\circ |
If $a$ is a real number randomly chosen from the interval $[0,10]$, the probability that the equation $x^{2}-ax+1=0$ has no real solutions is _______. | \dfrac{1}{5} |
2. If $x+y+z \neq 0, a=\frac{x}{y+z}, b=\frac{y}{x+z}$, $c=\frac{z}{x+y}$, then $\frac{a}{a+1}+\frac{b}{b+1}+\frac{c}{c+1}=(\quad)$.
(A) 0
(B) 1
(C) $a+b+c$
(D) Uncertain | B |
Given a parabola $y=x^2$ with a fixed point A(-1, 1) and two moving points P and Q. When $PA \perp PQ$, the range of the x-coordinate of point Q is ( )
A: $(-\infty, -3]$
B: $[1, +\infty)$
C: $[-3, 1]$
D: $(-\infty, -3] \cup [1, +\infty)$ | D |
Given that the length of vectors $\overset{→}{a}$ and $\overset{→}{b}$ are $|\overset{→}{a}|=4$ and $|\overset{→}{b}|=2$, respectively, and the angle between $\overset{→}{a}$ and $\overset{→}{b}$ is $120^{\circ}$. Find the value of $|3\overset{→}{a}-4\overset{→}{b}|=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$ | 4\sqrt{19} |
A company held a lottery during its annual year-end summary meeting. In an opaque box, there are $3$ red balls and $3$ white balls (the balls are of the same shape and size). The rules of the lottery are as follows: draw $3$ balls from the bag at once, turn the white balls into red balls, and put them all back into the... | 28 |
4. In an acute-angled triangle $A B C$, the altitudes $A A_{1}, B B_{1}$, and $C C_{1}$ are dropped. A point $T$ is chosen on the plane such that the lines $T A$ and $T B$ are tangents to the circumcircle of triangle $A B C$, and point $O$ is the center of this circle. The perpendicular dropped from point $T$ to the li... | 90 |
4. Given that the equation $2|x|-k=k x-3$ has no negative solutions, then the range of values for $k$ is ( ).
(A) $-2 \leqslant k \leqslant 3$
(B) $2<k \leqslant 3$
(C) $2 \leqslant k \leqslant 3$
(D) $k \geqslant 3$ or $k \leqslant-2$ | A |
2. Three points $A, B$, and $C$ are placed in a coordinate system in the plane. Their abscissas are the numbers $2, -3, 0$, and their ordinates are the numbers $-3, 2, 5$, not necessarily in that order. Determine the coordinates of these points if it is known:
- No point is in the III. quadrant.
- Point $A$ does not l... | A(-3,5),B(0,2),C(2 |
$7 \cdot 32$ Given that the base $BC$ and the height $AD$ of isosceles $\triangle ABC$ are both integers, then among $\sin A$ and $\cos A$,
(A) one is rational and the other is irrational.
(B) both are rational.
(C) both are irrational.
(D) whether they are rational or irrational depends on the values of $BC$ and $AD$.... | B |
In the Cartesian coordinate system, the coordinates of the point symmetric to point $P(-2,3)$ relative to the $x$-axis are ( ).
A: $(-2,-3)$
B: $(2,-3)$
C: $(-3,-2)$
D: $(3,-2)$ | A |
20. Given that $a$ is a positive real number, $b$ is a positive integer and $2 \leqslant b \leqslant 200$, then the number of pairs $(a, b)$ that satisfy $\left(\log _{b} a\right)^{2017}=\log _{b} a^{2017}$ is ( ).
(A) 198
(B) 199
(C) 398
(D) 399
(E) 597 | E |
3. In a Cartesian coordinate system, two circles have a common point $(9,6)$ and are both tangent to the $x$-axis. The product of their radii is 68. If their other external common tangent also passes through the origin, then its slope is $\qquad$ . | \frac{12 \sqrt{221}}{49} |
8.5. In Nastya's room, 16 people gathered, each pair of whom either are friends or enemies. Upon entering the room, each of them wrote down the number of friends who had already arrived, and upon leaving - the number of enemies remaining in the room. What can the sum of all the numbers written down be, after everyone h... | 120 |
Xiaochao is 1 meter and 36 centimeters tall, which, when written as a decimal, is ____ meters. | 1.36 |
In the expansion of $(x-1)(x-2)(x-3)(x-4)(x-5)$, the coefficient of the term containing $x^{4}$ is $\_\_\_\_\_\_$. | -15 |
Given a sequence $\{a_{n}\}$ with $a_{1}=1$ and $\frac{{a}_{n+1}}{{a}_{n}}=a_{n}+\frac{1}{{a}_{n}}$, then ()
A: $a_{n+1}\geqslant 2a_{n}$
B: $\{\frac{{a}_{n+1}}{{a}_{n}}\}$ is an increasing sequence
C: $\{a_{n+1}-4a_{n}\}$ is an increasing sequence
D: $a_{n}\geqslant n^{2}-2n+2$ | ABD |
The sequence $\{a_{n}\}$ satisfies ${a_{n+1}}=(2|\sin \frac{n\, !!\, \pi\, !!}{2}|-1){a_{n}}+2n$. Find the sum of the first 100 terms of the sequence $\{a_{n}\}$. | 5100 |
Given rational numbers $a$ and $b$ are opposite to each other and $a \neq 0$, $c$ and $d$ are reciprocals. Rational numbers $m$ and $-2$ are $3$ units apart on the number line. Find the value of $|m|-\frac{a}{b}+\frac{{a+b}}{{2020}}-cd$. | 1 \text{ or } 5 |
A volleyball costs 88 yuan each, and a soccer ball costs 72 yuan each. The school bought 11 volleyballs. If the money spent on volleyballs is used to buy soccer balls, the maximum number of soccer balls that can be bought is ______. | 13 |
The medicine is on sale everywhere. You can get it at \_\_\_ chemist’s.
A. each B. some C. certain D. any | D |
54. Continuing her research with fractions, Tanya added 4 to the numerator of a certain fraction and 10 to the denominator. After simplifying, to her surprise, she got the original fraction. She tried the same with another fraction, but the result did not repeat. What fraction did Tanya originally have? | \frac{2}{5} |
The 17th day of a month is Saturday. The first day of that month was
(A) Monday
(B) Tuesday
(C) Wednesday (D) Thursday
(E) Friday | D |
Given the function $f(x) = 4^x + m \cdot 2^x + 1$ has exactly one zero, find the range of values for $m$, and determine that zero. | 0 |
2. Let $x=\sqrt{5}, y$ be the greatest integer not exceeding the real number $x$. Then $\frac{1}{x-y}=(\quad$.
(A) $\sqrt{5}-2$
(B) $\sqrt{5}+2$
(C) $\sqrt{5}-1$
(D) $\sqrt{5}+1$ | B |
Example 2 Given
$$
\left(x+\sqrt{x^{2}+2000}\right)\left(y+\sqrt{y^{2}+2002}\right)=2002 \text {. }
$$
Then $x^{2}-3 x y-4 y^{2}-6 x-6 y+58=$ $\qquad$
(2002, Junior High School Mathematics Competition, Jiangsu Province) | 58 |
2. In the Cartesian coordinate system $x O y$, the graph of the function $f(x)=a \sin a x+\cos a x(a>0)$ over an interval of the smallest positive period length and the graph of the function $g(x)=\sqrt{a^{2}+1}$ enclose a closed figure whose area is $\qquad$ . | \frac{2\pi}{}\sqrt{^{2}+1} |
Forty-four interns have been kidnapped by the diabolical François, who, in his great kindness, gives them a chance to escape his trap. He has them enter one by one into a large room containing 44 drawers, each of which contains the name of an intern (the names are assumed to be distinct). If by opening fewer than 22 dr... | 0.32 |
Let's calculate the value of the following expressions, given that $\alpha+\beta+\gamma=180^{\circ}$.
$$
\sin \frac{\alpha+\beta}{2}-\cos \frac{\gamma}{2}
$$
$$
\begin{aligned}
& \tan \frac{\gamma}{2}+\tan \frac{\alpha+\beta}{2}-\left(\cot \frac{\alpha+\beta}{2}+\cot \frac{\gamma}{2}\right) \\
& \sin ^{2} \frac{\alph... | 0,0,1,2 |
[Text problems (other)]
Forty gold and forty silver coins were distributed among three empty chests, with both types of coins in each chest. In the first chest, there were 7 more gold coins than silver ones, and in the second chest, there were 15 fewer silver coins than gold ones. Which type of coin is more in the thi... | 22 |
Among the given functions, the one that has a period of $\pi$ and is a decreasing function in the interval $\left(0, \frac{\pi}{2}\right)$ is $(\quad)$.
A: $y=\sin \frac{x}{2}$
B: $y=\cos 2x$
C: $y=\tan \left(x- \frac{\pi}{4}\right)$
D: $y=\sin \left(2x+ \frac{\pi}{4}\right)$ | B |
47. As shown in the figure, $A B / / C D / / E F / / G H, A E / / D G$, point $C$ is on $A E$, point $F$ is on $D G$. Let the number of angles equal to $\angle \alpha$ be $m$ (excluding $\angle \alpha$ itself), and the number of angles supplementary to $\angle \beta$ be $n$. If $\alpha \neq \beta$, then the value of $m... | 11 |
Given the function $f(x)=\ln x-ax-3$ $(a\neq 0)$
$(1)$ Find the extreme values of the function $f(x)$;
$(2)$ If for any $a\in[1,2]$, the function $g(x)=x^{3}+ \frac{x^{2}}{2}[m-2f′(x)]$ has an extremum in the interval $(a,3)$, find the range of the real number $m$. | -\frac{32}{3} < m < -\frac{19}{2} |
2. The zero of the linear function is 2, and the initial value is $\frac{5}{2}$. Write the equation of the line that is parallel to the graph of the given function and intersects the $x$-axis at $\frac{4}{3}$, in implicit form. | 15x+12y-20=0 |
1. Let $x=\frac{1-\sqrt{5}}{2}$, then $\sqrt{\frac{1}{x}-\frac{1}{x^{3}}}-$ $\sqrt[3]{x^{4}-x^{2}}=$ $\qquad$ | \sqrt{5} |
7. Petya sequentially writes down integers, starting from 21, such that each subsequent number is 4 less than the previous one, and Vasya, looking at the current number, calculates the sum of all the numbers written down by this point. Which of the sums found by Vasya will be closest to $55?$
Answer: 56. | 56 |
[ Extremal Properties (other) ] [ Examples and Counterexamples. Constructions ]
The number of edges of a convex polyhedron is 99. What is the maximum number of edges that a plane, not passing through its vertices, can intersect? | 66 |
Given that a hyperbola and an ellipse \\( \frac {x^{2}}{25}+ \frac {y^{2}}{9}=1\\) share the same foci, and the sum of their eccentricities is \\( \frac {14}{5}\\), determine the equations of the hyperbola's asymptotes.
A: \\(y=± \frac { \sqrt {3}}{3}x\\)
B: \\(y=± \frac {5}{3}x\\)
C: \\(y=± \frac {3}{5}x\\)
D: \\(y=±... | D |
1. The average of Pažout's grades is exactly 3. If we did not include three of Pažout's fives in the average, the average of his grades would be exactly 2. Determine the maximum number of ones that Pažout could have received. (Possible grades are 1, 2, 3, 4, 5.) | 4 |
A sequence $(a_n)$ is defined recursively by $a_1=0, a_2=1$ and for $n\ge 3$,
\[a_n=\frac12na_{n-1}+\frac12n(n-1)a_{n-2}+(-1)^n\left(1-\frac{n}{2}\right).\]
Find a closed-form expression for $f_n=a_n+2\binom{n}{1}a_{n-1}+3\binom{n}{2}a_{n-2}+\ldots +(n-1)\binom{n}{n-2}a_2+n\binom{n}{n-1}a_1$. | 2n! - n - 1 |
$3+$ [Properties and characteristics of an isosceles triangle.]
On each side of a square, one point was taken. It turned out that these points are the vertices of a rectangle, the sides of which are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6.
# | 12 |
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