problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
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$\underline{\text { Folklore }}$
Given a polynomial $P(x)$ with integer coefficients. It is known that $P(1)=2013, P(2013)=1, P(k)=k$, where $k$ is some integer. Find $k$.
# | 1007 |
3. On a line, seven points A, B, C, D, E, F, G are marked in the given order. It turned out that $A G=23 \mathrm{~cm}, B F=17 \mathrm{~cm}$ and $\mathrm{CE}=9 \mathrm{~cm}$. Find the sum of the lengths of all segments with endpoints at these points. | 224 |
(1) Given $x > \frac{5}{4}$, find the minimum value of the function $y=4x+\frac{1}{4x-5}$.
(2) Given $a > 0, b > 0,$ and $2^a \cdot 4^b = 2$, find the minimum value of $\frac{2}{a} + \frac{1}{b}$. | 8 |
17・187 Given $\triangle A B C$ is determined by the given side $a$ (opposite angle $A$), angle $B$, and the altitude $h_{c}$ from $C$, $N$ is the number of non-overlapping triangles thus determined. Then $N$ is equal to
(A) 1.
(B) 2.
(C) 0.
(D) Infinite.
(E) 0 or Infinite.
(18th American High School Mathematics Examina... | E |
2. In the given parallelogram $A B C D$, point $E$ is the midpoint of side $B C$ and point $F$ lies inside side $A B$. The area of triangle $A F D$ is $15 \mathrm{~cm}^{2}$ and the area of triangle $F B E$ is $14 \mathrm{~cm}^{2}$. Determine the area of quadrilateral $F E C D$. | 57\mathrm{~}^{2} |
Let $a= \frac{2\pi}{3}$, then $\sin \left(\pi- \frac{a}{2}\right)+\tan \left(a- \frac{5\pi}{12}\right)=$
A: $\frac{2- \sqrt{2}}{2}$
B: $\frac{2+ \sqrt{2}}{2}$
C: $\frac{2+ \sqrt{3}}{2}$
D: $\frac{2- \sqrt{3}}{2}$ | C |
1. On the board, five different positive numbers are written. Determine the maximum number of ways in which pairs can be formed from them, the sum of which equals one of the five numbers written on the board. | 4 |
If the complex number $z= \frac {1+3i}{1-2i}$ (where $i$ is the imaginary unit), then the modulus of $z$, $|z|=$ ______. | \sqrt {2} |
If the inverse function of \\(f(x) = \sqrt{2x+3}\\) is \\(g(x)\\), then the zero of the function \\(g(x)\\) is __________. | \sqrt{3} |
# 7.3. Condition:
On the Christmas tree, there is a garland of 100 bulbs. It is known that the first and fifth bulbs are yellow. Moreover, among any five consecutive bulbs, exactly two are yellow and exactly three are blue. Father Frost can see only part of the garland from behind the tree. Help him determine the colo... | 97;98;99-Bluecolor,100-Yellowcolor |
1. Let $f$ be the function such that
$$
f(x)=\left\{\begin{array}{ll}
2 x & \text { if } x \leq \frac{1}{2} \\
2-2 x & \text { if } x>\frac{1}{2}
\end{array}\right.
$$
What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012 f^{\prime} s}(x) \ldots))$ from $x=0$ to $x=1$ ? | \sqrt{4^{2012}+1} |
Given $a \gt 0$, $b \gt 0$, if $4a+b=ab-4$, then the maximum value of $\log _{2}(a-1)\cdot \log _{2}(b-4)$ is ______. | \frac{9}{4} |
Points $P_{1}(-1$,$y_{1})$, $P_{2}(2$,$y_{2})$, $P_{3}(5$,$y_{3})$ are all on the graph of the quadratic function $y=-x^{2}+2x+c$. The relationship between $y_{1}$, $y_{2}$, and $y_{3}$ is ______. | y_{2} > y_{1} > y_{3} |
2. Find the maximum value of the quantity $x^{2}+y^{2}+z^{2}$, given that
$$
x^{2}+y^{2}+z^{2}=3 x+8 y+z
$$ | 74 |
5. Given the family of curves
$$
y=(m-1) x^{2}+2 m x+4
$$
a) Prove that all the given curves pass through two fixed points $A$ and $B$. Determine these points.
b) Determine the curve from the given family that touches the $O x$ axis, and then determine the curve whose vertex is point B ( $B$ is the common point of t... | A(-2,0),B(0,4),(x+2)^2,-x^2+4 |
5. If the complex number $z$ satisfies $|z|=1, A(-1,0)$, and $B(0,-1)$ are two points on the complex plane, then the function $f(z) = |(z+1)(\bar{z}-\mathrm{i})|$ reaches its maximum value when the figure formed by points $Z, A, B$ on the complex plane is ( ).
(A) Equilateral triangle
(B) Isosceles triangle
(C) Right t... | B |
(4) $\sin 7.5^{\circ}+\cos 7.5^{\circ}=$ | \frac{\sqrt{4+\sqrt{6}-\sqrt{2}}}{2} |
10.266. Find the third side of an acute-angled triangle if two of its sides are equal to $a$ and $b$ and it is known that the medians of these sides intersect at a right angle. | \sqrt{\frac{^{2}+b^{2}}{5}} |
If the coefficient of $x^{3}$ in the expansion of $((ax-l)^{6})$ is $20$, find the value of $a$ ___. | -1 |
The point with coordinates $(a, 0)$ is on the line with equation $y=x+8$. What is the value of $a$ ? | -8 |
5. In a regular quadrilateral pyramid $P-ABCD$, it is known that the side faces are equilateral triangles with a side length of 1, and $M, N$ are the midpoints of edges $AB, BC$ respectively. Then the distance between the skew lines $MN$ and $PC$ is $\qquad$. | \frac{\sqrt{2}}{4} |
In order to get an output result of 16 after running the following program, the keyboard input for $x$ should be ______.
INPUT $x$
IF $x<0$ THEN
$y=(x+1)\cdot(x+1)$
ELSE
$y=(x-1)\cdot(x-1)$
END IF
PRINT $y$
END | 5 \text{ or } -5 |
Given that function f(x) is an increasing function defined on the interval (-3, 3) and f(2m) < f(m+1), determine the range of values for m. | (-\frac {3}{2}, 1) |
1. What is the smallest positive prime factor of the integer $2005^{2007}+2007^{20015}$ ?
(A) 5
(B) 7
(C) 2
(D) 11
(E) 3 | 2 |
Given that \\(x_{1}\\) and \\(x_{2}\\) are the two roots of the quadratic equation \\(x^{2}-4x+1=0\\), then \\(x_{1} \cdot x_{2}\\) equals ( )
A: \\(-4\\)
B: \\(-1\\)
C: \\(1\\)
D: \\(4\\) | \text{C} |
1. Given the sets
$$
A=\{1, b, a+b\}, B=\{a-b, a b\} \text {, }
$$
and $A \cap B=\{-1,0\}$.
Then the values of $a$ and $b$ are ( ).
(A) $-1,0$
(B) $0,-1$
(C) $-1,1$
(D) $1,-1$ | A |
16. Let $f(n)$ denote the number of positive integral solutions of the equation $4 x+3 y+2 z=n$. Find $f(2009)-f(2000)$.
(2 marks)
設 $f(n)$ 表示方程 $4 x+3 y+2 z=n$ 的正整數解數目。求 $f(2009)-f(2000)$ 。 (2 分) | 1000 |
Given line $l$, $m$ and plane $\alpha$, $\beta$, determine the correct propositions among the following:
1. If $l$ is parallel to a line in $\alpha$, then $l \parallel \alpha$;
2. If $l$ is perpendicular to two lines in $\alpha$, then $l \perp \alpha$;
3. If $l \parallel \alpha$, $l \subset \beta$ and $\alpha \ca... | 3, 6 |
Problem 6. Yulia thought of a number. Dasha added 1 to Yulia's number, and Anya added 13 to Yulia's number. It turned out that the number obtained by Anya is 4 times the number obtained by Dasha. What number did Yulia think of? | 3 |
Problem 4. On each side of a cube, a natural number is written. At each vertex (corner) of the cube, the product of the three numbers written on the sides that form the vertex (corner) is written. The sum of the eight thus obtained products is 385. Determine the sum of the numbers written on the sides of the cube. | 23 |
7.1. In quadrilateral $A B C D$, $A B=B C=C D$. Let $E$ be the intersection point of $A B$ and $C D$ (with $B$ between $A$ and $E$). It turns out that $A C=C E$ and $\angle D B C=15^{\circ}$. Find $\angle A E D$. | 50 |
3. In the Cartesian coordinate system $x O y$, points with both coordinates as integers are called integer points. It is known that $k$ is a real number. When the intersection point of the two different lines $y=4 k x-\frac{1}{k}$ and $y=\frac{1}{k} x+2$ is an integer point, the number of values that $k$ can take is ( ... | 3 |
Given the set $A=\left\{ x \mid x^2-x-2\leqslant 0 \right\}$, and set $B$ is the set of integers, then $A\cap B=$ ()
A: $\left\{-1,0,1,2\right\}$
B: $\left\{-2,-1,0,1\right\}$
C: $\left\{0,1\right\}$
D: $\left\{-1,0\right\}$ | \text{A} |
Given: $m^{2}-2mn+2n^{2}-4n+4=0$, find the values of $m$ and $n$.
Solution:
Since $m^{2}-2mn+2n^{2}-4n+4=0$,
Therefore, $(m^{2}-2mn+n^{2})+(n^{2}-4n+4)=0$,
Therefore, $(m-n)^{2}+(n-2)^{2}=0$,
Therefore, $(m-n)^{2}=0$, $(n-2)^{2}=0$,
Therefore, $n=2$, $m=2$.
Based on your observation, explore the following... | 9 |
Which of the following calculations is correct?
A: $4\sqrt{3}-3\sqrt{3}=1$
B: $\sqrt{3}+\sqrt{5}=\sqrt{8}$
C: $3\sqrt{\frac{1}{3}}=\sqrt{3}$
D: $3+2\sqrt{2}=5\sqrt{2}$ | C |
A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$? | 9901 |
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is... | \frac{17}{8} |
To stabilize housing prices, a local government decided to build a batch of affordable housing for the community. The plan is to purchase a piece of land for 16 million yuan and build a residential community with 10 buildings on it. Each building has the same number of floors, and each floor has a construction area of ... | 1225 |
"$a > b$" is the \_\_\_\_\_\_ condition for "$\ln a > \ln b$" (Choose from "sufficient but not necessary," "necessary but not sufficient," "necessary and sufficient," and "neither sufficient nor necessary"). | \text{necessary but not sufficient} |
Variant 8.1.4. Dima and Seryozha were picking berries from a raspberry bush, on which 450 berries grew. Seryozha alternated actions while picking berries: he put one berry in the basket and ate the next one. Dima also alternated: he put two berries in the basket and ate the next one. It is known that Seryozha picks ber... | 50 |
12. (5 points) Before New Year's Day, Xiaofang made greeting cards for her five classmates. When putting the cards into envelopes, she made a mistake, and none of the five classmates received the card Xiaofang made for them; instead, they received cards Xiaofang made for others. In total, there are $\qquad$ possible sc... | 44 |
Example 2: Person A and Person B start from points $A$ and $B$ respectively at the same time and walk towards each other. They meet at point $C$, which is 10 kilometers away from $A$. After meeting, both continue at the same speed, reach $B$ and $A$ respectively, and immediately return. They meet again at point $D$, wh... | 27 |
A certain school randomly selected 20 ninth-grade students to investigate the time they spend on physical exercise each day. The data obtained (in minutes) are as follows: $30, 60, 70, 10, 30, 115, 70, 60, 75, 90, 15, 70, 40, 75, 105, 80, 60, 30, 70, 45$. After analyzing the data, the following Table 1 and Table 2 were... | 130 \text{ students} |
3. In a transparent sealed cubic container, there is exactly half the volume of the container filled with water. When the cubic container is rotated arbitrarily, the shape of the water surface in the container can be:
(1) triangle, (2) rhombus, (3) rectangle, (4) square, (5) regular hexagon.
The correct conclusion is ... | C |
From a compact regular tetrahedron with unit edge length, we cut off a regular tetrahedron of edge length $\frac{1}{n}$ at each vertex, where $n>1$ is an integer. For what value of $n$ will the ratio of the surface area to the volume of the remaining solid be the smallest? | 3 |
15. One of the biggest mathematical breakthroughs in 2019 was progress on an 82 -year old problem by the renowned mathematician and Fields medalist Terence Tao.
Consider the function
$$
f(n)=\left\{\begin{array}{cl}
\frac{n}{2} & \text { if } n \text { is even } \\
3 n+1 & \text { if } n \text { is odd }
\end{array}\ri... | Collatz |
3.117. $\frac{\tan^{4} \alpha-\tan^{6} \alpha}{\cot^{4} \alpha-\cot^{2} \alpha}$. | \tan^{8}\alpha |
Regular octagon $ABCDEFGH$ has area $n$. Let $m$ be the area of quadrilateral $ACEG$. What is $\tfrac{m}{n}?$
$\textbf{(A) } \frac{\sqrt{2}}{4} \qquad \textbf{(B) } \frac{\sqrt{2}}{2} \qquad \textbf{(C) } \frac{3}{4} \qquad \textbf{(D) } \frac{3\sqrt{2}}{5} \qquad \textbf{(E) } \frac{2\sqrt{2}}{3}$ | \frac{\sqrt{2}}{2} |
Example 9 Given the sequence $\left\{a_{n}\right\}$ with the sum of the first $n$ terms $S_{n}=$ $n^{2}$. Then
$$
\begin{array}{l}
\frac{1}{\sqrt{a_{1}}+\sqrt{a_{2}}}+\frac{1}{\sqrt{a_{2}}+\sqrt{a_{3}}}+\cdots \\
+\frac{1}{\sqrt{a_{998}}+\sqrt{a_{999}}}=(\quad) .
\end{array}
$$
(A) $\frac{\sqrt{1996}-1}{2}$
(B) $\frac{... | B |
Given vectors $\overrightarrow{a} = (\sin(x + \frac{\pi}{6}), 1)$ and $\overrightarrow{b} = (4, 4\cos x - \sqrt{3})$:
(I) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $\sin(x + \frac{4\pi}{3})$;
(II) Let $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$. If $\alpha \in [0, \frac{\pi}{2}]$ an... | \frac{3 + \sqrt{21}}{8} |
Example 2 Color each vertex of a square pyramid with one color, and make the endpoints of the same edge have different colors. If only 5 colors are available, then the total number of different coloring methods is $\qquad$ | 420 |
Given the function $f(x)=\sin (ωx+φ) (ω > 0)$. If the graph of $f(x)$ is shifted to the left by $\frac{{ }\!\!\pi\!\!{ }}{3}$ units and overlaps with the graph of $f(x)$, what is the minimum value of $ω$? | ω=6 |
Given two arithmetic sequences \\(\{a_n\}\) and \\(\{b_n\}\) with the sum of the first \\(n\\) terms denoted as \\(A_n\\) and \\(B_n\\) respectively, and \\( \dfrac{A_n}{B_n} = \dfrac{7n+45}{n+3} \\), then \\( \dfrac{a_5}{b_5} \\) is \\((\\) \\()\\)
A: \\(13\\)
B: \\(11\\)
C: \\(10\\)
D: \\(9\\) | \text{D} |
If $$\sin\theta= \frac {3}{5}$$ and $$\frac {5\pi}{2}<\theta<3\pi$$, then $$\sin \frac {\theta}{2}$$ equals \_\_\_\_\_\_. | -\frac {3 \sqrt {10}}{10} |
5. As shown in the figure, in parallelogram $A B C D$, $E C=2 A E, F C=2 B F$. If the area of $\triangle B E F$ is 90, then the area of $\triangle A C D$ is $\qquad$ . | 405 |
$1 \cdot 86$ Simplify $\frac{\sqrt{6}+4 \sqrt{3}+3 \sqrt{2}}{(\sqrt{6}+\sqrt{3})(\sqrt{3}+\sqrt{2})}$, the result is
(A) $\frac{2}{3}$.
(B) $\frac{2 \sqrt{3}}{3}$.
(C) $\sqrt{6}+\sqrt{2}$.
(D) $\sqrt{6}-\sqrt{2}$.
(China Beijing Junior High School Mathematics Competition, 1986) | \sqrt{6}-\sqrt{2} |
Given the function $f\left(x\right)=\sqrt{x}+1$, then $\underset{lim}{x→0}\frac{f(1+△x)-f(1)}{△x}=\left(\ \ \right)$
A: $\frac{3}{2}$
B: $1$
C: $\frac{1}{2}$
D: $2$ | C |
Given vectors $\overrightarrow{a}=(1,x-1)$, $\overrightarrow{b}=(y,2)$, if $\overrightarrow{a} \bot \overrightarrow{b}$, then the maximum value of $xy$ is ( ).
A: $-\dfrac{1}{4}$
B: $\dfrac{1}{2}$
C: $1$
D: $2$ | \text{B} |
Among the following propositions:
1. The converse of "The areas of congruent triangles are equal";
2. The negation of "If $ab=0$, then $a=0$";
3. The contrapositive of "The three interior angles of an equilateral triangle are all $60^{\circ}$".
The numbers of the true propositions are __________ (Fill in all the numb... | 2, 3 |
## Task A-4.1.
Determine all complex numbers $z$ for which $z^{2}=(1+i) \cdot \bar{z}$. | 0,\quad-1+i,\quad\frac{1+\sqrt{3}}{2}+i\frac{-1+\sqrt{3}}{2},\quad\frac{1-\sqrt{3}}{2}+i\frac{-1-\sqrt{3}}{2} |
Evaluate
$$\frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6}$$ | 108 |
Find the value of the following expressions:
$(1)\ \sin 20160^{\circ}\sin (-20180^{\circ}) + \cos 20190^{\circ}\sin (-840^{\circ})$
$(2)\ \sin \frac{4\pi}{3} \cdot \cos \frac{19\pi}{6} \cdot \tan \frac{21\pi}{4}$ | \frac{3}{4} |
7・75 Let $f(x)$ be a function defined on the set of real numbers $R$, and satisfies the following relations:
$$
f(10+x)=f(10-x), f(20-x)=-f(20+x) .
$$
Then $f(x)$ is
(A) an even function, and also a periodic function.
(B) an even function, but not a periodic function.
(C) an odd function, and also a periodic function.... | C |
In the sequence $\left\{ a_n \right\}$, $a_1=2$, $a_{n+1}=2a_n$, and $S_n$ is the sum of the first $n$ terms of $\left\{ a_n \right\}$. If $S_n=126$, then $n=$ ? | 6 |
In a certain store, the cost price of a product is $30$ yuan per item, and the selling price is $40 yuan per item. The store can sell $48$ items per day. In order to reduce inventory quickly, the store decides to reduce the price for promotion.
$(1)$ If the selling price decreases to $32.4$ yuan per item after two cons... | 2 \text{ yuan per item per day} |
43. Given a sequence of numbers $1,3,4,7,11,18, \ldots$, what is the 10th number in this sequence? | 123 |
A certain flower bed cultivates a certain kind of flower seedlings in flower pots. Through experiments, it is found that the profit per pot of flowers is related to the number of plants per pot. When $3$ plants are planted in each pot, the average profit per plant is $10$ yuan. Under the same cultivation conditions, if... | B |
3-ча 1. The sum of the reciprocals of three positive integers is equal to 1. What are these numbers? Find all solutions. | (2,4,4),(2,3,6),(3,3,3) |
The number of different divisors of 1998 is ( )
A: 20
B: 16
C: 14
D: 12 | B |
Given the sets $M = \{0, 1, 2\}$ and $N = \left\{x \in \mathbb{Z} \mid 0 < \log_{2}(x + 1) < 2\right\}$, find the intersection $M \cap N$. | \{1, 2\} |
9.1. First-grader Masha, entering the school, each time climbs the school porch stairs, which have 10 steps. Being at the bottom of the stairs or on one of its steps, she can either go up to the next step or jump over one step up (Masha cannot yet jump over two or more steps). What is the minimum number of times Masha ... | 89 |
Given a sphere with a radius of $5$, the circumferences of two of its parallel cross-sections are $6\pi$ and $8\pi$ respectively. Calculate the distance between these two cross-sections. | 7 |
Example 4 Given real numbers $a, b$ satisfy
$$
a^{2}+a b+b^{2}=1 \text {, and } t=a b-a^{2}-b^{2} \text {. }
$$
Then the range of values for $t$ is | -3 \leqslant t \leqslant -\frac{1}{3} |
\section*{Problem 3B - 111043B}
Dirk explains to Jürgen the usefulness of differential calculus using the solution to the following problem: Let \(A B C D E\) be a planar convex pentagon such that \(A, B, C, E\) are the vertices of a rectangle and \(C, D, E\) are the vertices of an equilateral triangle. The area of th... | 1+\frac{\sqrt{3}}{3} |
6. For the set $S=\{1,2,3, \cdots, 18\}$, a five-element subset $S_{1}=\left\{a_{1}, a_{2}, a_{3}, a_{4}, a_{5}\right\}$ is such that the difference between any two elements is not 1. The number of such subsets $S_{1}$ is ( ).
A. $C_{17}^{4}$
B. $C_{15}^{4}$
C. $C_{13}^{5}$
D. $C_{14}^{5}$ | D |
Given that in parallelogram $ABCD$, $E$ and $F$ are the midpoints of sides $CD$ and $BC$ respectively, then $\overrightarrow{EF}=$
A: $\frac{1}{2}\overrightarrow{AB}-\overrightarrow{AD}$
B: $\frac{1}{2}\overrightarrow{AB}+\overrightarrow{BC}$
C: $\frac{1}{2}\overrightarrow{AB}+\frac{1}{2}\overrightarrow{AD}$
D: $\f... | \text{D} |
[ Product of lengths of chord segments and lengths of secant segments ] [ Auxiliary similar triangles Area of a triangle (using height and base). ]
Diameter $A B$ and chord $C D$ of a circle intersect at point $E$, and $C E=D E$. Tangents to the circle at points $B$ and $C$ intersect at point $K$. Segments $A K$ and $... | \frac{27}{4} |
6. (5 points) The area of a square with a side length of 20 is exactly equal to the sum of the areas of two squares with side lengths of $a$ and $b$, respectively. If $a$ and $b$ are both natural numbers, then $a+b=$ $\qquad$ . | 28 |
7.5. In a certain triangle, one segment was drawn. It turned out that on the resulting drawing, all known types of triangles can be seen: equilateral, isosceles, scalene, right, acute, and obtuse.
a) Provide an example of such a triangle and the segment drawn in it.
b) Determine what values the angles of such a trian... | 90,30,60 |
$(6x^{4}-4x^{3}+2x^{2})\div 2x^{2}=\_\_\_\_\_\_.$ | 3x^{2} - 2x + 1 |
50. Given a triangle $ABC$. The tangent to the circumcircle of this triangle at point $B$ intersects the line $AC$ at point $M$. Find the ratio $|AM|:|MC|$, if $|AB|:|BC|=k$. | k^2 |
Based on the following set of equations:
$S_1 = 1$;
$S_2 = 2 + 3 = 5$;
$S_3 = 4 + 5 + 6 = 15$;
$S_4 = 7 + 8 + 9 + 10 = 34$;
$S_5 = 11 + 12 + 13 + 14 + 15 = 65$;
$S_6 = 16 + 17 + 18 + 19 + 20 + 21 = 111$;
$S_7 = 22 + 23 + 24 + 25 + 26 + 27 + 28 = 175$;
...
We can find that $S_1 + S_3 + S_5 + \ldots + S_{2n-1} =... | n^4 |
9.1. At a round table, 2015 people are sitting, each of them is either a knight or a liar. Knights always tell the truth, and liars always lie. Each person was given a card with a number on it; all the numbers on the cards are different. After looking at their neighbors' cards, each person said: "My number is greater t... | 2013 |
Given that $f\_0(x)=\cos x$, $f\_1(x)= f\_0'(x)$, $f\_2(x)= f\_1'(x)…$, $f\_{n+1}(x)= f\_n'(x), n\in\mathbb{N}^*$, find $f\_{2015}(x)=$ _____ . | \sin x |
4. Let the set $A=\left\{\frac{1}{2}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{15}, \frac{1}{32}\right\}$ have non-empty subsets $A_{1}, A_{2}, \cdots, A_{63}$. Denote the product of all elements in the set $A_{i}$ as $p_{i}$ (the product of elements in a singleton set is the element itself), $i=1,2, \cdots, 6... | \frac{79}{65} |
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=2: 3$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=3$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$. | AE:EQ=4:9 |
20. Given the ellipse $\frac{x^{2}}{5^{2}}+\frac{y^{2}}{4^{2}}=1$, a line is drawn through its left focus $F_{1}$ intersecting the ellipse at points $\mathrm{A}, \mathrm{B}$. $\mathrm{D}(a, \theta)$ is a point to the right of $F_{1}$, and lines $\mathrm{AD}$ and $\mathrm{BD}$ intersect the left directrix of the ellipse... | 5 |
Given that the graph of the quadratic function $y=mx^{2}+\left(m-2\right)x+2$ is symmetric about the $y$-axis, which of the following conclusions is incorrect?
A: $m=2$
B: The parabola opens upwards
C: $y$ increases as $x$ increases when $x \gt 0$
D: The function has a minimum value of $2$ when $x=2$ | D |
Find all integers $m \geqslant 1$ and $n \geqslant 1$ such that $\frac{5^{m}+2^{n+1}}{5^{m}-2^{n+1}}$ is the square of an integer. | m=n=1 |
Right triangle $A B C$ is divided by the height $C D$, drawn to the hypotenuse, into two triangles: $B C D$ and $A C D$. The radii of the circles inscribed in these triangles are 4 and 3, respectively. Find the radius of the circle inscribed in triangle $A B C$. | 5 |
In the complex plane, the coordinates of the point corresponding to the complex number $\dfrac{2i}{1-i}$ are
A: $(1,1)$
B: $(\,-1,1\,)$
C: $(\,-1,-1\,)$
D: $(1,-1)$ | \text{B} |
3. Simplify $\frac{\sqrt{2}+\sqrt{6}}{\sqrt{2+\sqrt{3}}}$, the result is $(\quad)$.
(A) 1
(B) $\sqrt{3}$
(C) 2
(D) 4 | C |
# Problem 7. (4 points)
Natural numbers $a, b, c$ are such that $\operatorname{GCD}(\operatorname{LCM}(a, b), c) \cdot \operatorname{LCM}(\operatorname{GCD}(a, b), c)=200$.
What is the greatest value that $\operatorname{GCD}(\operatorname{LCM}(a, b), c)$ can take? | 10 |
7.090. $27^{\lg x}-7 \cdot 9^{\lg x}-21 \cdot 3^{\lg x}+27=0$. | 1;100 |
In triangle $ABC$, if the lengths of the sides opposite to angles $A$, $B$, and $C$ which are $a$, $b$, and $c$ respectively, form a geometric progression, then the range of possible values for angle $B$ is ____. | (0, \frac{\pi}{3}] |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{π}{3}$, and $|\overrightarrow{a}|=3$, $|2\overrightarrow{a}+\overrightarrow{b}|=2\sqrt{13}$, find $|\overrightarrow{b}|$. | 2 |
12.401 The sides of a parallelogram are equal to $a$ and $\boldsymbol{b}(\boldsymbol{a}<\boldsymbol{b}$). From the midpoint of the larger side, the parallel side is seen at an angle $\alpha$. Find the area of the parallelogram. | \frac{4^{2}-b^{2}}{4}\cdot\tan\alpha |
Let $x \in \mathbb{R}$. Then, "$x^3 > 8$" is a __________ of "$|x| > 2$".
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D: Neither sufficient nor necessary condition | A |
5. [5 points] A magician has a set of $20^{2}$ different cards. Each card has one side red and the other side blue; on each card, there is a natural number from 1 to 20 written on both sides. We will call a card a duplicate if the numbers on both sides of the card are the same. The magician wants to draw two cards such... | 7030 |
In space, let $m$ and $n$ be different lines, and $\alpha$ and $\beta$ be different planes, with $m \subset \alpha$ and $n \subset \beta$. Which of the following statements is correct? ( )
A: If $m \parallel n$, then $\alpha \parallel \beta$
B: If $m$ and $n$ are skew lines, then $\alpha$ and $\beta$ are skew planes... | \text{D} |
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