problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
## Task 2
Multiply the numbers 7 and 6. Subtract 5 from the product.
Write the equation and solve.
| 37 |
B4. The infinite sequence of numbers
$$
0,1,2,2,1,-1,-2,-1,1,3, \ldots
$$
satisfies the following rule. For each quadruple of consecutive numbers $\ldots, a, b, c, d, \ldots$ in the sequence, it always holds that $d$ is equal to $c$ minus the smallest of the two numbers $a$ and $b$. Thus, the ninth number in the sequ... | 2187 |
4. Workers digging a tunnel for an underground railway complete 5 metres of tunnel on a typical day. Working every day, how long will it take them to dig a tunnel of length 2 kilometres?
A three months
B six months
$\mathrm{C}$ just under a year
D just over a year
E nearly two years | D |
3. Let $x, y$ be real numbers. Then
$$
f(x, y)=x^{2}+x y+y^{2}-x-y
$$
the minimum value is $\qquad$ . | -\frac{1}{3} |
$\sin 30^{\circ}+\cos 60^{\circ}=$____. | 1 |
Given the set $A=\{a_1, a_2, a_3, \ldots, a_n\}$, where $a_i \in \mathbb{R}$ ($1 \leq i \leq n$, $n > 2$), let $l(A)$ denote the number of distinct values of $a_i + a_j$ ($1 \leq i < j \leq n$).
(1) Given the sets $P=\{2, 4, 6, 8\}$ and $Q=\{2, 4, 8, 16\}$, find $l(P)$ and $l(Q)$ respectively;
(2) Find the minimum ... | 2n-3 |
Given the algebraic expression $x^2 + 2x - 3$ equals $7$, find the value of the algebraic expression $2x^2 + 4x + 1$. | 21 |
5. Given the equation about $x$: $(2 a-1) \sin x+(2-a) \sin 2 x=\sin 3 x$, if all solutions of the equation in the range $x \in[0,2 \pi]$ form an arithmetic sequence in ascending order, then the range of the real number $a$ is $\qquad$ . | (-\infty,-2]\bigcup{0}\bigcup[2,+\infty) |
10. Given point $P$ is on the line $x+2 y-1=0$, point $Q$ is on the line $x+2 y+3=0$, the midpoint of $P Q$ is $M\left(x_{0}, y_{0}\right)$, and $y_{0}>x_{0}+2$. Then the range of $\frac{y_{0}}{x_{0}}$ is $\qquad$ | (-\frac{1}{2},-\frac{1}{5}) |
Given condition $p$: $x^{2}-3x-4 \leqslant 0$; condition $q$: $x^{2}-6x+9-m^{2} \leqslant 0$, if $p$ is a sufficient but not necessary condition for $q$, then the range of $m$ is __( )__.
A: $[-1,1]$
B: $[-4,4]$
C: $(-\infty,-1] \cup [1,+\infty)$
D: $(-\infty,-4] \cup [4,+\infty)$ | D |
11. Two trains depart from two cities at the same time, heading towards each other, and meet after 3.3 hours. If Train A departs 24 minutes earlier, then 3 hours after Train B departs, the two trains still need to travel 14 kilometers to meet; if Train B departs 36 minutes earlier, then 3 hours after Train A departs, t... | 660 |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{\arcsin 2 x}{\ln (e-x)-1}$ | -2e |
2. Find all possible values of the quotient
$$
\frac{r+\rho}{a+b}
$$
where $r$ and $\rho$ are respectively the radii of the circumcircle and incircle of the right triangle with legs $a$ and $b$.
| \frac{1}{2} |
Some unit-radius circles are drawn on the plane, and we color the center of each circle blue. We mark some points on the circumference of the circles in red such that exactly 2 red points lie on each circumference. What is the maximum number of blue points if there are a total of 25 colored points? | 20 |
Given the function $f(x) = \begin{cases} x^2 + 1, & x \leqslant 1 \\ \log_{10} x, & x > 1 \end{cases}$, find the value of $f(f(10))$. | 2 |
Jack had a bag of $128$ apples. He sold $25\%$ of them to Jill. Next he sold $25\%$ of those remaining to June. Of those apples still in his bag, he gave the shiniest one to his teacher. How many apples did Jack have then?
$\text{(A)}\ 7 \qquad \text{(B)}\ 63 \qquad \text{(C)}\ 65 \qquad \text{(D)}\ 71 \qquad \text... | 71 |
The function $f(x)= \frac{\ln x}{x}$, then ( )
A: $x= e$ is the point of maximum value for the function $f(x)$
B: $x= e$ is the point of minimum value for the function $f(x)$
C: $x= \frac{1}{e}$ is the point of maximum value for the function $f(x)$
D: $x= \frac{1}{e}$ is the point of minimum value for the function $... | A |
Teacher Tan awarded a stack of exercise books to the students who were named "Outstanding Students" in the math Olympiad class. If each student is awarded 3 books, there are 7 books left over; if each student is awarded 5 books, there are 9 books short. How many students received the award? How many exercise books are ... | 31 |
Let the complex number $z$ satisfy $(1+i)z=2i$, then $|z|=$ __(__)
A: $\dfrac{1}{2}$
B: $\dfrac{\sqrt{2}}{2}$
C: $\sqrt{2}$
D: $2$ | C |
B2. What is $12 \%$ of the number $\sqrt[3]{(\sqrt{5}+2)(\sqrt{5}-2)}+2^{-\log _{10} 0.01}$? Write down the answer. | \frac{3}{5} |
Example 6 Let real numbers $a$, $b$, $c$ satisfy
$$
\left\{\begin{array}{l}
a^{2}-b c-8 a+7=0, \\
b^{2}+c^{2}+b c-6 a+6=0 .
\end{array}\right.
$$
Find the range of values for $a$.
(1995, Jilin Province Junior High School Mathematics Competition) | 1 \leqslant a \leqslant 9 |
Problem 5. The numbers $p$ and $q$ are prime and satisfy
$$
\frac{p}{p+1}+\frac{q+1}{q}=\frac{2 n}{n+2}
$$
for some positive integer $n$. Find all possible values of $q-p$.
Origin. Luxembourg (Pierre Haas).
| 2,3,5 |
13. Gerard cuts a large rectangle into four smaller rectangles. The perimeters of three of these smaller rectangles are 16,18 and 24 , as shown in the diagram.
What is the perimeter of the fourth small rectangle?
A 8
B 10
C 12
D 14
E 16 | 10 |
In the geometric sequence $\{a_n\}$, $a_3$ and $a_9$ are the two roots of the equation $3x^2-11x+9=0$. Then, the value of $a_5a_6a_7$ is ( )
A: $3 \sqrt {3}$
B: $\frac {11}{2}$
C: $\pm3 \sqrt {3}$
D: None of the above | C |
40. Hogwarts School of Witchcraft and Wizardry has 110 students, who are divided into three groups: Magic, Combat, and Potions. It is known that 52 students are in the Magic group, with 16 students only in the Magic group; 63 students are in the Combat group, with 21 students only in the Combat group; 61 students are i... | 8 |
Given a circle $x^2+y^2=4$ and a point $P(-2, -3)$ outside the circle, find the equation of the tangent line to the circle that passes through point $P$. | x=-2 \text{ or } 5x-12y-26=0 |
4. On the plane $(x ; y)$, plot the set of points whose coordinates satisfy the system $\left\{\begin{array}{l}(|x|-x)^{2}+(|y|-y)^{2} \leq 16, \\ 2 y+x \leq 0\end{array}\right.$, and find the area of the resulting figure. | 5+\pi |
Given that $(1+x) + (1+x)^2 + (1+x)^3 + \ldots + (1+x)^8 = a_0 + a_1x + a_2x^2 + \ldots + a_8x^8$, find the value of $a_1 + a_2 + a_3 + \ldots + a_8$. | 502 |
Example 1 The license plate numbers issued by a city consist of 6 digits (from 0 to 9), but it is required that any two license plates differ in at least 2 positions (for example, license plates 038471 and 030471 cannot be used simultaneously). Try to find the maximum number of different license plates the city can iss... | 100000 |
# 6. Task 6.1*
In a bag, there are mittens: right and left. In total, there are 12 pairs: 10 red and 2 blue. How many mittens need to be pulled out to guarantee getting a pair of mittens of the same color? Points for the task: 8.
# | 13 |
Among the following relations, the correct one is (__).
A: $0 \in 0$
B: $0 = 0$
C: $0 \subseteq 0$
D: $\emptyset = 0$ | B |
7. [5] In $\triangle A B C, D$ is the midpoint of $B C, E$ is the foot of the perpendicular from $A$ to $B C$, and $F$ is the foot of the perpendicular from $D$ to $A C$. Given that $B E=5, E C=9$, and the area of triangle $A B C$ is 84 , compute $|E F|$. | \frac{6\sqrt{37}}{5} |
Two circles $x^2+y^2+4x-4y=0$ and $x^2+y^2+2x-12=0$ intersect at points A and B. The equation of line AB is __________. | x-2y+6=0 |
Task B-2.1. Draw in the complex plane the set of points $(x, y)$, associated with complex numbers $z=x+yi$, for which
$$
\operatorname{Re} z \cdot \operatorname{Im} z<0 \quad \mathrm{and} \quad|z|^{2} \leq \operatorname{Im}\left(z^{2}\right)+1
$$
Calculate the area of this set of points. | 1 |
[Example 2.2.3] In a $7 \times 7$ grid, select $k$ small squares such that the centers of any 4 points do not form the vertices of a rectangle.
Find the maximum value of $k$ that satisfies the above requirement. | 21 |
11. (5 points) The two ends of a bus route are station $A$ and station $B$. The bus company has the following rules:
(1) Each bus completes a one-way trip, including stops at intermediate stations, within 50 minutes, and stops for 10 minutes when it reaches either end.
(2) Both station $A$ and station $B$ dispatch a bu... | 20 |
6. In $\triangle A B C$, $\frac{\cos A+2 \cos C}{\cos A+2 \cos B}=\frac{\sin B}{\sin C}$, determine the shape of $\triangle A B C$. | A=90orB=C |
The negation of the proposition "For any $x$ in $[0, +\infty)$, $x^3+x \geq 0$ is ( )
A: For any $x$ in $(-\infty, 0)$, $x^3+x < 0$
B: For any $x$ in $(-\infty, 0)$, $x^3+x \geq 0$
C: There exists an $x_0$ in $[0, +\infty)$ such that $x_0^3+x_0 < 0$
D: There exists an $x_0$ in $[0, +\infty)$ such that $x_0^3+x_0 \geq ... | \text{C} |
1. Petrov and Vasechkin were repairing a fence. Each had to nail a certain number of boards (the same amount). Petrov nailed two nails into some boards and three nails into others. Vasechkin nailed three nails into some boards and five nails into the rest. Find out how many boards each of them nailed, given that Petrov... | 30 |
Given the points $\overrightarrow{a}=(2,m)$ and $\overrightarrow{b}=(1,1)$, if $\overrightarrow{a} \cdot \overrightarrow{b}=|\overrightarrow{a}-\overrightarrow{b}|$, then the real number $m$ equals ( ).
A: $\dfrac {1}{2}$
B: $- \dfrac {1}{2}$
C: $\dfrac {1}{3}$
D: $- \dfrac {1}{3}$ | D |
27. Doraemon told Nobita, who was using a calculator to compute the product $2000 \times 2001 \times 2002 \times \cdots \cdots \times 2019 \times 2020$, "The product is too large, we can use a simpler method to determine the number of trailing zeros." So, how many consecutive zeros are there at the end of this product?... | 7 |
Given the function f(x) = $$\begin{cases} 2^{x},x\leq 0 \\ f(x-2),x>0 \end{cases}$$, find the sum of f(1) + f(2) + f(3) + ... + f(2017). | \frac {3025}{2} |
8.1. (12 points) In how many ways can eight of the nine digits $1,2,3,4,5,6$, 7,8 and 9 be placed in a $4 \times 2$ table (4 rows, 2 columns) so that the sum of the digits in each row, starting from the second, is 1 more than in the previous one? | 64 |
90. There and Back. Colonel Crackham asserts that his friend, Mr. Wilkinson, walks from his country house to the nearest town at a speed of 5 km/h, and on the way back, a little tired, he covers the same distance at a speed of 3 km/h. The round trip takes him exactly 7 hours.
How far from the town is Mr. Wilkinson's h... | 13\frac{1}{8} |
If the parabola $y=5x^{2}$ is shifted $2$ units to the left and then $3$ units up, the resulting parabola is ( )
A: $y=5\left(x-2\right)^{2}+3$
B: $y=5\left(x+2\right)^{2}-3$
C: $y=5\left(x+2\right)^{2}+3$
D: $y=5\left(x-2\right)^{2}-3$ | C |
1. The system of equations in $x, y, z$
$$
\left\{\begin{array}{l}
x y+y z+z x=1, \\
5 x+8 y+9 z=12
\end{array}\right.
$$
all real solutions $(x, y, z)$ are | (1,\frac{1}{2},\frac{1}{3}) |
The sum of the numerical coefficients of all the terms in the expansion of $(x-2y)^{18}$ is:
$\textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 19\qquad \textbf{(D)}\ -1\qquad \textbf{(E)}\ -19$ | 1 |
The domain of the function $f(x+1) = x^2 - 2x + 1$ is $[-2, 0]$, then the interval of monotonic decrease for $f(x)$ is \_\_\_\_. | [-1, 1] |
甲、乙两名魔方爱好者在$30$秒内复原魔方的概率分别是$0.8$和$0.6$.如果在$30$秒内将魔方复原称为"复原成功",且每次复原成功与否相互之间没有影响,求:（1）甲复原三次,第三次才成功的概率；（2）甲、乙两人在第一次复原中至少有一人成功的概率. | 0.92 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a^{2}=b^{2}+bc+c^{2}$, then $\angle A=$ _____ . | 120^{\circ} |
7. Seedlings of two varieties of black currant, 6 of the Selyanchenskaya variety and 8 of the Vologda variety, are prepared for planting in a garden plot and are accidentally mixed. What is the probability that the first 3 currant seedlings planted will be of the Selyanchenskaya variety? | \frac{5}{91} |
Example 8. In $\triangle ABC$, $D$ is the midpoint of $AC$.
(1) If $BC$ is trisected, with division points from $C$ being $C_{1}, C_{2}$, and $C_{1}A, C_{2}A$ intersect $BD$ at $D_{1}$, $D_{2}$. Find $S \wedge N_{1} D_{2}: S \triangle A C_{1} C_{2}$;
(2) If $BC$ is divided into four equal parts, with division points fr... | \frac{n^2}{2(n-1)(2n-1)} |
The number of solutions $(P, Q)$ of the equation $\frac{P}{Q}-\frac{Q}{P}=\frac{P+Q}{P Q}$, where $P$ and $Q$ are integers from 1 to 9 inclusive, is
(A) 1
(B) 8
(C) 16
(D) 72
(E) 81 | 8 |
$-0.9$ does not belong to which of the following sets?
A: Negative numbers
B: Fractions
C: Integers
D: Rational numbers | C |
Given the function $f(x) = \frac{x^2}{e^x}$, where $x \neq 0$ and $e$ is the base of the natural logarithm, if the equation $\sqrt{f(x)} + \frac{2}{\sqrt{f(x)}} - \lambda = 0$ has four distinct real roots, then the range of the real number $\lambda$ is ( )
A: $(0, \frac{2}{e})$
B: $(2\sqrt{2}, +\infty)$
C: $(e+ \fr... | \text{C} |
Fomin S.B.
Two people toss a coin: one tossed it 10 times, the other - 11 times.
What is the probability that the second one got heads more times than the first one? | \frac{1}{2} |
In a coordinate plane, given that the coordinates of point $A$ are $(2,-3)$, line segment $AB$ is parallel to the $x$-axis, and $AB=4$, then the coordinates of point $B$ are ( ).
A: $(-2,3)$
B: $(6,3)$
C: $(-2,-3)$ or $(6,-3)$
D: $(2,7)$ or $(2,-1)$ | C |
Given a sequence $\{a_{n}\}$ that satisfies $a_{1}+3a_{2}+\cdots \cdots +\left(2n-1\right)a_{n}=2n$, where ${b_n}=\frac{{{a_n}}}{{({2n+1})}}$, $S_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$. Which of the following statements is correct?
A: $a_{1}=2$
B: The general formula for the sequence $\{a_... | ACD |
In the geometric sequence $\{a\_n\}$, the sum of the first 3 terms is equal to 3 times the first term. What is the common ratio of this geometric sequence? | -2 \text{ or } 1 |
10.201. Given a triangle $A B C$ such that $A B=15 \text{~cm}, B C=12$ cm and $A C=18 \text{~cm}$. Calculate in what ratio the incenter of the triangle divides the angle bisector of angle $C$. | 2:1 |
Given the function $f(x) = \begin{cases} (a- \frac {1}{2})x, & x\geq2 \\ a^{x}-4, & x 0$ holds. Then, the range of the real number $a$ is ( )
A: $(1,2]$
B: $(\frac {13}{4},2]$
C: $(1,3]$
D: $(\frac {13}{4},3]$ | C |
If $a\in R$, then "$a^{2} \gt a$" is ( ) of "$a \gt 1$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | \text{B} |
## Task B-2.5.
Point $E$ is the midpoint of side $\overline{A B}$ of square $A B C D$. A point $F$ is chosen on diagonal $\overline{A C}$ such that triangle $E F D$ is a right triangle with a right angle at $F$. In what ratio does point $F$ divide diagonal $\overline{A C}$? | 3:1 |
4A. Let $ABCD$ be a triangular pyramid where all three angles at vertex $D$ are right angles. Calculate the height of the pyramid dropped from vertex $D$, given the lengths of the edges $\overline{AB}=\overline{AC}=5$ and $\overline{BC}=4\sqrt{2}$. | \frac{6}{17}\sqrt{34} |
The fraction $\frac {5x - 11}{2x^2 + x - 6}$ was obtained by adding the two fractions $\frac {A}{x + 2}$ and
$\frac {B}{2x - 3}$. The values of $A$ and $B$ must be, respectively:
$\textbf{(A)}\ 5x,-11\qquad\textbf{(B)}\ -11,5x\qquad\textbf{(C)}\ -1,3\qquad\textbf{(D)}\ 3,-1\qquad\textbf{(E)}\ 5,-11$ | D |
16. (1988 AIME Problem 6) Find the smallest positive integer $n$ such that the last three digits of its cube are 888. | 192 |
Evaluate the definite integral $\int_{1}^{e}{(1+ \ln x)dx}=\_\_\_\_\_\_\_\_\_\_\_\_\_.$ | e |
Given the hyperbola $\dfrac{y^{2}}{a^{2}} - \dfrac{x^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, one of its asymptote equations is $2x + y = 0$, and the distance from the vertex to the asymptote is $\dfrac{2\sqrt{5}}{5}$.
$(1)$ Find the equation of this hyperbola;
$(2)$ Let $P$ be a point on the hyperbola, and points $A$, $B$... | 2 |
Calculate the greatest common divisor (GCD) of 60 and 48 using the Euclidean algorithm. The number of division operations required is (__).
A: 1
B: 2
C: 3
D: 4 | B |
Let $n$ be a positive integer. Ana and Banana are playing the following game:
First, Ana arranges $2n$ cups in a row on a table, each facing upside-down. She then places a ball under a cup
and makes a hole in the table under some other cup. Banana then gives a finite sequence of commands to Ana,
where each command cons... | 2n^2 - n |
6. Find all three-digit numbers n with three different non-zero digits, which are divisible by the sum of all three two-digit numbers obtained by erasing one digit in the original number.
(Jaromír Šimša) | 138,294,351,459 |
16. As shown in Figure 14,
$\odot \mathrm{O}_{1}$ and $\odot \mathrm{O}_{2}$ intersect
at points $A$ and $B$, $\bigcirc \mathrm{O}_{2}$
passes through $\odot O_{1}$, and $A C$ and $A D$
are diameters of $\odot \mathrm{O}_{1}$ and $\odot \mathrm{O}_{2}$
respectively. Then the following conclusions:
(1) $C, B, D$ are col... | E |
24. If $a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}=p, b_{1}^{2}+b_{2}^{2}+\cdots+b_{n}^{2}=q$, then $\left(a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\right)_{\text {max }}=$ $\sqrt{p q}$ (when and only when $\sqrt{\frac{q}{p}} a_{i}=b_{i}(i=1,2, \cdots, n)$, the maximum value is achieved). | \sqrt{pq} |
2. Simplify $\sqrt{\frac{3+\sqrt{5}}{2}}-\sqrt{\frac{3-\sqrt{5}}{2}}=(\quad)$.
(A) $2 \sqrt{5}$
(B) 2
(C) 1
(D) $\sqrt{5}$ | C |
8,9}
Point $M$ lies inside an equilateral triangle $ABC$. Calculate the area of this triangle, given that $AM = BM = 2$, and $CM = 1$. | \frac{9\sqrt{3}+3\sqrt{15}}{8} |
17. Let $x>0$. Then
$$
f(x)=\frac{\left(x+\frac{1}{x}\right)^{4}-\left(x^{4}+\frac{1}{x^{4}}\right)}{\left(x+\frac{1}{x}\right)^{3}-\left(x^{3}+\frac{1}{x^{3}}\right)}
$$
the minimum value of $f(x)$ is | \frac{7}{3} |
Three positive numbers $a$, $b$, $c$ form a geometric sequence, then $\lg a$, $\lg b$, $\lg c$ are ( )
A: Geometric sequence
B: Arithmetic sequence
C: Both arithmetic and geometric sequence
D: Neither arithmetic nor geometric sequence | \text{B} |
Given the sets $M=\{x|x\geqslant -1\}$ and $N=\{x|-2 < x < 2\}$, then $M\cap N=$ ()
A: $(-\infty,-1]$
B: $[-1,2)$
C: $(-1,2]$
D: $(2,+\infty)$ | B |
Let $f:\mathbb{N} \rightarrow \mathbb{N},$ $f(n)=n^2-69n+2250$ be a function. Find the prime number $p$, for which the sum of the digits of the number $f(p^2+32)$ is as small as possible. | 3 |
Given a triangle \\(ABC\\) with side lengths of \\(4\\) meters, \\(5\\) meters, and \\(6\\) meters, if \\(x\\) meters are cut off from all three sides, and the remaining parts form an obtuse triangle, then the range of \\(x\\) is | (1,3) |
1. A driver traveled from place $A$ to place $B$, which are $800 \mathrm{~km}$ apart. Instead of driving at the planned speed of $80 \mathrm{~km} / \mathrm{h}$, on the first half of his journey, the driver was forced to drive at a speed of $(80-\mathrm{m})$ km/h. If on the second half of his journey the driver drives a... | t_{1}+t_{2}>10\mathrm{~} |
Given a circle $C$ passes through points $A(1,3)$ and $B(-1,1)$, and the center of the circle is on the line $y=x$.
(Ⅰ) Find the equation of circle $C$;
(Ⅱ) Suppose a line $l$ passes through the point $(2,-2)$, and $l$ intersects circle $C$ with a chord length of $2\sqrt{3}$, find the equation of line $l$. | 4x+3y-2=0 |
Solve the following system of equations:
$$
\begin{aligned}
& x y=z(x+y+z) \\
& y z=4 x(x+y+z) \\
& z x=9 y(x+y+z)
\end{aligned}
$$ | -3t,-2t,6t |
6. (5 points) There is a natural number, the difference between its smallest two divisors is 4, and the difference between its largest two divisors is 308, then this natural number is $\qquad$ | 385 |
5. The clock shows 00:00, at which the hour and minute hands of the clock coincide. Counting this coincidence as number 0, determine how much time (in minutes) will pass before they coincide for the 21st time. Round your answer to the nearest hundredth. | 1374.55 |
13.6.35 In a Cartesian coordinate system, a circle centered at the point $(0,1)$ intersects the parabola $y=x^{2}$ at four points $A, B, C, D$. Prove that $S_{A B C D}<\sqrt{2}$, and find the maximum possible value of $S_{A B C D}$. | \frac{4}{3}\sqrt{\frac{2}{3}} |
4. The radii of the two bases of a frustum are $R$ and $r(R > r)$, respectively. A plane parallel to the bases divides the lateral surface of the frustum into two equal areas. Then, the radius of the cross-sectional circle is
(A) $\sqrt{R r}$
(B) $\sqrt{\frac{R^{2}+r^{2}}{2}}$ | B |
9. The Fort Worth Zoo has several two-legged birds and four-legged mammals. During a visit to the zoo, Maggie counted a total of 200 heads and 522 legs among these animals. How many birds are there in the zoo? ( )
(A) 61
(B) 122
(C) 139
(D) 150
(E) 161 | C |
Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number. | 3 |
10. [20] A positive real number $x$ is such that
$$
\sqrt[3]{1-x^{3}}+\sqrt[3]{1+x^{3}}=1
$$
Find $x^{2}$. | x^{2}=\frac{\sqrt[3]{28}}{3} |
8. Quadrilateral $A B C D$ has right angles at $A$ and $D$. Inside the quadrilateral, there is a circle with radius 10 that touches all sides of the quadrilateral. Side $B C$ has a length of 24. Point $E$ is the midpoint of $A D$ and point $F$ is the midpoint of $B C$.
What is the length of $E F$?
A) $\frac{43}{2}$
B)... | 22 |
In the Cartesian coordinate system, with the origin _\\(O\\)_ as the pole and the positive half-axis of _\\(x\\)_ as the polar axis, the polar equation of curve _\\(C\\)_\\({\\,\\!}\_{1}\\) is _\\(ρ\\)_\\({\\,\\!}^{2}(1+3\\sin ^{2} \\)_\\(θ\\)_\\()=4.\\) Curve _\\(C\\)_\\({\\,\\!}\_{2}\\) is a circle with its center on... | \\dfrac{5}{4} |
Joy has $30$ thin rods, one each of every integer length from $1 \text{ cm}$ through $30 \text{ cm}$. She places the rods with lengths $3 \text{ cm}$, $7 \text{ cm}$, and $15 \text{cm}$ on a table. She then wants to choose a fourth rod that she can put with these three to form a quadrilateral with positive area. How ma... | 17 |
4. The maximum value of the function $y=-x-\frac{9}{x}+18(x>0)$ is ( ).
(A) 24 .
(B) 18
(C) 12 .
(D) 2 . | C |
Transform $y=2x^{2}-12x-12$ into the form $y=a\left(x-m\right)^{2}+n$______. The coordinates of the vertex are______. | (3, -30) |
Calculate: $-5^2-(-3)^3\times\frac{2}{9}-9\times|-\frac{2}{3}|$. | -25 |
287. $y=x^{2}+\frac{2}{x^{4}}-\sqrt[3]{x}$
287. $y=x^{2}+\frac{2}{x^{4}}-\sqrt[3]{x}$ | 2x-\frac{8}{x^{5}}-\frac{1}{3\sqrt[3]{x^{2}}} |
2.075. $\frac{\left(x^{2 / m}-9 x^{2 / n}\right)\left(\sqrt[m]{x^{1-m}}-3 \sqrt[n]{x^{1-n}}\right)}{\left(x^{1 / m}+3 x^{1 / n}\right)^{2}-12 x^{(m+n) /(m n)}}$. | \frac{x^{1/}+3x^{1/n}}{x} |
13.3.6 ** Given $P(3,4)$ is a point inside the circle $x^{2}+y^{2}=64$, and two moving points $A$ and $B$ on the circumference of the circle satisfy $\angle A P B-\frac{\pi}{2}$. A rectangle $A P B Q$ is constructed with $A P$ and $B P$ as adjacent sides. Find the equation of the trajectory of point $Q$. | x^{2}+y^{2}=103 |
Given the function $f(x)=\cos(2x+\frac{\pi}{3})+1$, in triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.
$(I)$ If angles $A$, $B$, and $C$ form an arithmetic sequence, find the value of $f(B)$;
$(II)$ If $f\left(\frac{B}{2}-\frac{\pi}{6}\right)=\frac{7}{4}$, and sides $a$... | 3+\sqrt{2} |
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