problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
4. Any function $f(x)$ defined on $(-\infty,+\infty)$ can be expressed as the sum of an odd function $g(x)$ and an even function $h(x)$, given $f(x)=\lg \left(10^{x}+1\right), x \in(-\infty$, $+\infty)$, then ( ).
A. $g(x)=x, h(x)=\lg \left(10^{x}+10^{-x}+2\right)$
B. $g(x)=\frac{1}{2}\left[\lg \left(10^{x}+1\right)+x\... | C.\,(x)=\frac{x}{2},\,(x)=\lg(10^{x}+1)- |
3. Find all natural numbers $n$ for which $2^{n}+n^{2016}$ is a prime number. | 1 |
12.140. In what ratio does the point $O$, from which all three sides of the isosceles triangle $ABC$ are seen at the same angle $(\angle AOB = \angle BOC = \angle COA)$, divide the height of the isosceles triangle $ABC$ if the angle at the base of the triangle is $\alpha\left(\alpha>\frac{\pi}{6}\right)$? | \frac{2\sin(\alpha-\frac{\pi}{6})}{\cos\alpha} |
A certain company sends $5$ employees, including $2$ female employees, to study in $3$ different places. Each person must go to one place, and each place must have at least one person. The number of different allocation schemes where the $2$ female employees must study together is:
A: $24$
B: $32$
C: $36$
D: $48$ | C |
A math test consists of 12 multiple-choice questions, each worth 5 points. It is known that a student is confident in correctly answering 6 of these questions. For another three questions, the student can eliminate one incorrect option. For two questions, the student can eliminate two incorrect options. For the last qu... | 41.25 |
From a batch of 60 newly developed missiles numbered from 1 to 60, 5 missiles are randomly selected for a launch test. Using the systematic sampling method, the possible sequence of missile numbers selected could be ( )
A: 1, 3, 4, 7, 9, 5
B: 10, 15, 25, 35, 45
C: 5, 17, 29, 41, 53
D: 3, 13, 23, 33, 43 | \text{C} |
Given non-zero vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, then "$\overrightarrow{a}\cdot \overrightarrow{c}=\overrightarrow{b}\cdot \overrightarrow{c}$" is ( ).
A: Sufficient but not necessary condition
B: Necessary but not sufficient condition
C: Necessary and sufficient condition
D:... | \text{B} |
## Task 34/66
How many space diagonals does a rhombic dodecahedron have? | 43 |
626. Find $f^{\prime}(0)$, if $f(x)=e^{x} \arcsin x+\operatorname{arctg} x$. | 2 |
10.079. Find the bisectors of the acute angles of a right triangle with legs of 24 and $18 \mathrm{~cm}$. | 9\sqrt{5} |
Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$. | (4, 229) |
5. On 900 cards, all natural numbers from 1 to 900 are written. Cards with squares of integers are removed, and the remaining cards are renumbered, starting from 1.
Then the operation of removing squares is repeated. How many times will this operation have to be repeated to remove all the cards | 59 |
[Order relation]
If for yesterday tomorrow was Thursday, then what day will be yesterday for the day after tomorrow?
# | Friday |
Given the following four propositions:
① If $x^2 \neq y^2$, then $x \neq y$ or $x \neq -y$;
② If $2 \leq x < 3$, then $(x-2)(x-3) \leq 0$;
③ If $a$ and $b$ are both zero, then $|a| + |b| = 0$;
④ If $x, y \in \mathbb{N}$ and $x+y$ is an odd number, then one of $x, y$ is odd and the other is even.
Which of the ... | A |
Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?
| 1995 |
Let's denote the sequence as $A$.
The numerical sequence is defined by the conditions: $a_{1}=1, a_{n+1}=a_{n}+\left[\sqrt{a_{n}}\right]$.
How many perfect squares are there among the first terms of this sequence, not exceeding $1000000$? | 10 |
A factory produces product A and product B, each kilogram of product A requires $a\_1$ kilograms of material A and $b\_1$ kilograms of material B, and each kilogram of product B requires $a\_2$ kilograms of material A and $b\_2$ kilograms of material B. The profit per kilogram of product A and product B is $d\_1$ and $... | C |
1. Given $a+3=3a, b+4=4b, c+5=5c, d+6=6d$, then $a \times b \times c \times d=$ | 3 |
Given a sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n= \frac {1}{2}n^{2}+ \frac {1}{2}n$,
(1) find the expression for the general term $a_n$;
(2) let $b_n=a_n\cdot2^{n-1}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$. | T_n=(n-1)\cdot2^{n}+1 |
Given $A=\{a, b, c\}$, $B=\{a, b\}$, then the incorrect relation among the following is ( )
A: $A \cap B = B$
B: $\complement_A B \subseteq B$
C: $A \cup B \subseteq A$
D: $B \subsetneq A$ | \text{B} |
15. Let $S=10 \times \frac{2020^{2021}+2021^{2022}}{2020^{2020}+2021^{2021}}$, then the integer part of $S$ is | 20209 |
In triangle $\triangle ABC$, point $D$ satisfies $\overrightarrow{BC}=2\overrightarrow{CD}$, then $\overrightarrow{AD}=$
A: $\frac{3}{2}\overrightarrow{AC}-\frac{1}{2}\overrightarrow{AB}$
B: $\frac{1}{2}\overrightarrow{AC}+\frac{1}{2}\overrightarrow{AB}$
C: $\frac{3}{2}\overrightarrow{AC}+\frac{1}{2}\overrightarrow{... | A |
4. Given prime numbers $p$ and $q$, such that $p^{3}-q^{5}=(p+q)^{2}$. Then $\frac{8\left(p^{2013}-p^{2010} q^{5}\right)}{p^{2011}-p^{2009} q^{2}}=$ $\qquad$. | 140 |
Exercise 2. Anne drove her car for an integer (and non-zero) number of hours, and traveled an integer number of kilometers, at a speed of $55 \mathrm{~km} / \mathrm{h}$. At the beginning of the trip, the odometer read abc kilometers, where abc is a 3-digit number such that $a \geqslant 1$ and $a+b+c \leqslant 7$. At th... | 106 |
Let $f(x) = \cos x$, $f_1(x) = f'(x)$, $f_2(x) = f_1'(x)$, ..., $f_{n+1}(x) = f_n'(x)$, where $n \in \mathbb{N}^*$. Then, find $f_{2008}(x)$. | \cos x |
6. The line segments $P Q R S$ and $W X Y S$ intersect circle $C_{1}$ at points $P, Q, W$ and $X$.
The line segments intersect circle $C_{2}$ at points $Q, R, X$ and $Y$. The lengths $Q R, R S$ and $X Y$ are 7, 9 and 18 respectively. The length $W X$ is six times the length $Y S$.
What is the sum of the lengths of $P ... | 150 |
To obtain the graph of the function $y=\sin \left(2x- \frac {\pi}{6}\right)$, you can shift the graph of the function $y=\sin \left(2x+ \frac {\pi}{3}\right)$ by
A: Shifting to the right by $\frac {\pi}{2}$ units,
B: Shifting to the right by $\frac {\pi}{4}$ units,
C: Shifting to the left by $\frac {\pi}{2}$ units,
D: ... | B |
## Task B-3.7.
The base of a right pyramid is a triangle with side lengths $25 \mathrm{~cm}, 29 \mathrm{~cm}, 36 \mathrm{~cm}$. All lateral faces of the pyramid form an angle of $75^{\circ}$ with the base plane. The volume of the pyramid is $k+l \sqrt{m} \mathrm{~cm}^{3}$. Determine the natural numbers $k, l$ and $m$. | k=1920,=960,=3 |
9.90 To color some of the squares in a $2 \times n$ strip of squares such that no $2 \times 2$ square is completely colored, let $P_{n}$ denote the number of different coloring methods that satisfy this condition. Prove that $P_{1989}$ is divisible by 3, and find the highest power of 3 that divides $P_{1989}$. | 3^{994} |
Let $B_n$ be the set of all sequences of length $n$, consisting of zeros and ones. For every two sequences $a,b \in B_n$ (not necessarily different) we define strings $\varepsilon_0\varepsilon_1\varepsilon_2 \dots \varepsilon_n$ and $\delta_0\delta_1\delta_2 \dots \delta_n$ such that $\varepsilon_0=\delta_0=0$ and $... | n \cdot 4^{n-1} |
In the plane rectangular coordinate system $xOy$, $P$ and $Q$ are two points on the parabola $C: x^{2}=y$ (distinct from point $O$). The line $l$ passing through point $P$ and tangent to $C$ intersects the $x$-axis at point $M$, and the product of the slopes of lines $OQ$ and $l$ is $-2$.
- $(1)$ Prove that the line $P... | \frac{3}{8} |
127 Elements are non-empty sets $S$ of natural numbers that satisfy “if $x \in S$, then $14-x \in S$”. The number of such sets $S$ is $\qquad$ . | 127 |
5.4. Find all pairs $a_{n}, a_{n+1}$ of consecutive members of the sequence $a_{1}, a_{2} \ldots$, defined by $a_{n}=2^{n}+49$, so that
$$
a_{n}=p \cdot q, \quad a_{n+1}=r \cdot s
$$
where $p, q, r, s$ are prime numbers such that
$$
p<q, \quad r<s, \quad q-p=s-r
$$
## 6th BALKANIAN MATHEMATICAL OLYMPIAD
SPLIT, for... | a_{7}=2^{7}+49,a_{8}=2^{8}+49 |
(1) In the arithmetic sequence $\{a_n\}$, $a_1=-2$, $d=4$, find $S_8$.
(2) In the geometric sequence $\{a_n\}$, $a_4=27$, $q=3$, find $a_7$. | 729 |
Given that the sum of the first $n$ terms of the arithmetic sequence $\{a\_n\}$ is $S\_n$, and the sequence $\{b\_n\}$ is a geometric sequence satisfying $a\_1=3$, $b\_1=1$, $b\_2+S\_2=10$, and $a\_5-2b\_2=a\_3$.
(1) Find the general terms of the sequences $\{a\_n\}$ and $\{b\_n\}$;
(2) Let $c=\begin{cases} \frac{2}{... | \frac{2n}{2n+1}+\frac{2}{3}(4^n-1) |
18. (3 points) If $a b c+c b a=1069$, then there are $\qquad$ such $\underline{a b c}$.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 8 |
8. Given $a_{k}$ as the number of integer terms in $\log _{2} k, \log _{3} k, \cdots, \log _{2018} k$. Then $\sum_{k=1}^{2018} a_{k}=$ $\qquad$ | 4102 |
3. (7 points) A traveler was riding in a bus and saw a two-digit number on a kilometer post. He fell asleep and woke up an hour later to see a three-digit number on the kilometer post, where the first digit was the same as the second digit an hour ago, the second digit was zero, and the third digit was the same as the ... | 45 |
2nd APMO 1990 Problem 1 Given θ in the range (0, π) how many (incongruent) triangles ABC have angle A = θ, BC = 1, and the following four points concyclic: A, the centroid, the midpoint of AB and the midpoint of AC? Solution | 1 |
Given the complex number $z= \frac{1+i}{1-i}$ (where $i$ is the imaginary unit), find the value of $C_{ 8 }^{ 1 }$+ $C_{ 8 }^{ 2 }$•z+ $C_{ 8 }^{ 3 }$•z2+ $C_{ 8 }^{ 4 }$•z3+ $C_{ 8 }^{ 5 }$•z4+ $C_{ 8 }^{ 6 }$•z5+ $C_{ 8 }^{ 7 }$•z6+ $C_{ 8 }^{ 8 }$•z7. | -15i |
Square $ABCD$ has side length $4$. Points $E$ and $F$ are the midpoints of sides $AB$ and $CD$, respectively. Eight $1$ by $2$ rectangles are placed inside the square so that no two of the eight rectangles overlap (see diagram). If the arrangement of eight rectangles is chosen randomly, then there are relatively prime ... | 61 |
A linear function $f(x)$ satisfies $f\left[ f\left( x \right) \right]=4x+6$, then $f(x)=$ . | 2x+2 \text{ or } -2x-6 |
Find all non-integer solutions of the equation
$$
x+\frac{13}{x}=[x]+\frac{13}{[x]}
$$
where $[x]$ denotes the greatest integer less than or equal to the real number $x$.
(Proposed by the Problem Committee) | x=-\frac{13}{4} |
7. Let $n=1990$, then $\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{n}^{4}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right)=$ $\qquad$ | -\frac{1}{2} |
1. Solve the equation $\left(\cos ^{2} x+\frac{1}{\cos ^{2} x}\right)^{3}+\left(\cos ^{2} y+\frac{1}{\cos ^{2} y}\right)^{3}=16 \sin z \cdot(5$ points $)$ | \pin,\pik,\frac{\pi}{2}+2\pi, |
Problem No. 5 (15 points)
With the same amount of monatomic ideal gas, two cyclic processes $1-2-3-1$ and $1-3-4-1$ are performed. Find the ratio of their efficiencies.
Answer: $\frac{13}{1... | \frac{13}{12}\approx1.08 |
Example $6^{[12.4]}$ Let $a_{1}, a_{2}, a_{3}>0$, and $a_{1}+a_{2}+a_{3}=1$, then it holds that $\frac{1+a_{1}}{1-a_{1}} \cdot \frac{1+a_{2}}{1-a_{2}} \cdot \frac{1+a_{3}}{1-a_{3}} \geqslant 2^{3}$. | 2^3 |
Example 3 Place $r(n)$ chess pieces on an $n \times n$ chessboard such that no four pieces form a rectangle with sides parallel to the grid lines. Prove: $r(n) \leqslant \frac{n+n \sqrt{4 n-3}}{2}$, and find the maximum value of $r(7)$.
Place $r(n)$ chess pieces on an $n \times n$ chessboard such that no four pieces f... | 21 |
21. In a right-angled triangle, the angle bisector of an acute angle divides the opposite side into segments of length 1 and 2 . What is the length of the bisector?
A $\sqrt{2}$
B $\sqrt{3}$
C $\sqrt{4}$
D $\sqrt{5}$
E $\sqrt{6}$ | 2 |
The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. Find the sum of $d$ and $e$.
$\begin{tabular}{ccccccc} & 7 & 4 & 2 & 5 & 8 & 6 \\ + & 8 & 2 & 9 & 4 & 3 & 0 \\ \hline 1 & 2 & 1 & 2 & 0 & 1 & 6 \end{tabular}$
$\mathrm{(A) \ 4 } \qquad \... | 8 |
4. Given a tetrahedron $S-ABC$ with the base being an isosceles right triangle with $AB$ as the hypotenuse, $SA=SB=SC=2, AB=$ 2, and suppose points $S, A, B, C$ all lie on a sphere with center $O$. Then the distance from point $O$ to the plane $ABC$ is $\qquad$ | \frac{\sqrt{3}}{3} |
Use the Horner's method to write out the process of calculating the value of $f(x) = 1 + x + 0.5x^2 + 0.16667x^3 + 0.04167x^4 + 0.00833x^5$ at $x = -0.2$. | 0.81873 |
7.143. $\left\{\begin{array}{l}2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6 \\ x^{2}+5 y^{2}=6 x y\end{array}\right.$
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
7.143. $\left\{\begin{array}{l}2^{\frac{x+y}{3}}+2^{\frac{x+y}{6}}=6 \\ ... | (3;3),(5;1) |
Example: Given the radii of the upper and lower bases of a frustum are 3 and 6, respectively, and the height is $3 \sqrt{3}$, the radii $O A$ and $O B$ of the lower base are perpendicular, and $C$ is a point on the generatrix $B B^{\prime}$ such that $B^{\prime} C: C B$ $=1: 2$. Find the shortest distance between point... | 4 \sqrt{13-6 \sqrt{2}} |
3. Given the function
$$
\begin{aligned}
f(x)= & a \tan ^{2017} x+b x^{2017}+ \\
& c \ln \left(x+\sqrt{x^{2}+1}\right)+20,
\end{aligned}
$$
where $a$, $b$, and $c$ are real numbers. If $f\left(\ln \log _{2} 21\right)=17$, then $f\left(\ln \log _{21} 5\right)=$ $\qquad$ . | 23 |
Given points $A(1,-2)$ and $B(m,2)$, the equation of the perpendicular bisector of segment $AB$ is $x+2y-2=0$. Find the value of the real number $m$. | 3 |
In the plane a point $O$ is and a sequence of points $P_1, P_2, P_3, \ldots$ are given. The distances $OP_1, OP_2, OP_3, \ldots$ are $r_1, r_2, r_3, \ldots$ Let $\alpha$ satisfies $0 < \alpha < 1.$ Suppose that for every $n$ the distance from the point $P_n$ to any other point of the sequence is $\geq r^{\alpha}_n.$ De... | \beta = \frac{1}{2(1 - \alpha)} |
14. Let there be a rectangular parallelepiped $A_{1} B_{1} C_{1} D_{1}-A B C D$, with three edges $A A_{1}=a, B_{1} A_{1}=b, A_{1} D_{1}=c, M, N, P, Q$ being the midpoints of $A_{1} B_{1}, A_{1} D_{1}, B C, C D$ respectively. Find the distance between the centroids of the two triangles $A M N$ and $C_{1} P Q$. | \frac{1}{3}\sqrt{^{2}+4b^{2}+4^{2}} |
Given that the proposition "There exists an $x \in \mathbb{R}$ such that $e^{|x-1|} - m \leq 0$" is a false statement, the range of values for $m$ is $(-\infty, a)$. Determine the value of the real number $a$. | a=1 |
5. Given the proportion: \(\frac{x-2}{x-1}=\frac{x+4}{x+7}\). Find \(x\). | 5 |
18.99 Connecting the midpoints of the sides of a quadrilateral, the resulting quadrilateral is a rhombus. Then the shape of the original quadrilateral is
(A) Isosceles trapezoid.
(B) Rectangle.
(C) Rhombus.
(D) None of the above conclusions is correct.
(China Guangzhou, Wuhan, Fuzhou and other five cities junior high s... | D |
Exercise 8. Let $x, y, z$ be three real numbers satisfying $x+y+z=2$ and $xy+yz+zx=1$. Determine the maximum value that $x-y$ can take. | \frac{2}{\sqrt{3}} |
Given a sequence $\{a_n\}$ that satisfies $a_{n+1}-a_n=5$, and $a_1$, $a_2$, $a_5$ form a geometric sequence, then the sum of the first six terms of this sequence $S_6=$ ()
A: $60$
B: $75$
C: $90$
D: $105$ | C |
9. (40 points) The numbers $s_{1}, s_{2}, \ldots, s_{1008}$ are such that their sum is $2016^{2}$. It is known that
$$
\frac{s_{1}}{s_{1}+1}=\frac{s_{2}}{s_{2}+3}=\frac{s_{3}}{s_{3}+5}=\ldots=\frac{s_{1008}}{s_{1008}+2015}
$$
Find $s_{17}$. | 132 |
471. $\cos x + \cos 3x + \cos 5x = 0$. | x_{1}=\\frac{\pi}{6}+\frac{2\pik}{3},k\in{Z},\quadx_{2}=\\frac{2\pi}{3}+2\pik,k\in{Z} |
3. If the solution set of the inequality $a x^{2}+b x+4>0$ is $\{x \mid-2<x<1\}$, then the maximum and minimum values of the quadratic function $y=b x^{2}+4 x+$ $a$ on the interval $[0,3]$ are ( ).
(A) $0,-8$
(B) $0,-4$
(C) 4,0
(D) 8,0 | A |
Given that $f(x)=x^{3}-ax$ is monotonically increasing on $[1,+\infty)$, the maximum value of $a$ is \_\_\_\_\_\_. | 3 |
Example 1 Given the equation in $x$
$$
x^{4}+2 x^{3}+(3+k) x^{2}+(2+k) x+2 k=0
$$
has real roots. If the product of all real roots is -2, then the sum of the squares of all real roots is $\qquad$ . | 5 |
1. Variant 1. Petya thought of two numbers and wrote down their product. After that, he decreased the first of the thought numbers by 3, and increased the other by 3. It turned out that the product increased by 900. By how much would the product have decreased if Petya had done the opposite: increased the first number ... | 918 |
## Task 6 - 020816
Given a rectangle $A B C D$, whose sides are all divided in the ratio $1: 2$ as shown in the figure. We call the division points $P, Q, R, S$ and connect them in sequen... | \frac{5}{9} |
3 [ [sphere, circumscribed around a pyramid]
A regular quadrilateral pyramid is inscribed in a sphere. The radius of the sphere is 1. The plane angle at the vertex of the pyramid is $45^{\circ}$. Find the lateral surface area of the pyramid.
# | 4 |
The negation of the statement "$\forall x\in R$, $x^{2}+x+1 < 0$" is ( )
A: $\forall x\in R$, $x^{2}+x+1\geqslant 0$
B: $\forall x\notin R$, $x^{2}+x+1\geqslant 0$
C: $\exists x_{0}\notin R$, $x_{0}^{2}+x_{0}+1 < 0$
D: $\exists x_{0}\in R$, $x_{0}^{2}+x_{0}+1\geqslant 0$ | \text{D} |
## Problem Statement
Find the derivative.
$$
y=e^{\sin x}\left(x-\frac{1}{\cos x}\right)
$$ | e^{\sinx}\cdot(x\cdot\cosx-\frac{\sinx}{\cos^2x}) |
Given sets $A=\{x\left|\right.4 \lt x\leqslant 8\}$ and $B=\{x|5-m^{2}\leqslant x\leqslant 5+m^{2}\}$.$(1)$ If $m=2$, find $\complement _{B}A$;$(2)$ Let $p:x\in A$, $q:x\in B$. If $p$ is a necessary but not sufficient condition for $q$, find the range of real number $m$. | (-1, 1) |
Example 3 Let $a$ and $b$ be positive real numbers, and
$$
\begin{array}{l}
\frac{1}{a}-\frac{1}{b}-\frac{1}{a+b}=0 . \\
\text { Then }\left(\frac{b}{a}\right)^{3}+\left(\frac{a}{b}\right)^{3}=
\end{array}
$$ | 2 \sqrt{5} |
In the plane $xOy$, it is known that ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \, (a > b > 0)$ passes through point $P(2, 1)$, and its eccentricity $e = \frac{\sqrt{3}}{2}$.
$(1)$ Find the equation of ellipse $C$;
$(2)$ If line $l$ is given by the equation $y = \frac{1}{2}x + m$, and this line intersects wit... | 2 |
2. Three bear cubs were dividing three pieces of cheese weighing 10 g, 12 g, and 15 g. A fox came to help them. She can simultaneously bite and eat 1 g of cheese from any two pieces. Can the fox leave the bear cubs equal pieces of cheese? | 7 |
2. [5] Jerry and Neil have a 3 -sided die that rolls the numbers 1,2 , and 3 , each with probability $\frac{1}{3}$. Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is 3 . | \frac{11}{18} |
The coefficient of the second term of the expansion of the binomial $$(ax- \frac { \sqrt {3}}{6})^{3}$$ (where $a>0$) is $- \frac { \sqrt {3}}{2}$. Find the value of $$∫_{ -2 }^{ a }x^{2}$$dx. The options are:
A: 3 or $$\frac {7}{3}$$
B: $$\frac {7}{3}$$
C: 3
D: 3 or $$- \frac {10}{3}$$ | C |
# Problem 5.
In 10 minutes, Zhenya eats 5 cottage cheese vatrushki, while Sasha eats only 3. On Saturday, all the baked vatrushki went to Zhenya, and on Sunday, they went to Sasha. In total, 35 vatrushki were eaten in one and a half hours of pure time. How many cottage cheese vatrushki did each of the children get? | Zhenyagot20pancakes,Sashagot15 |
Given a complex number $z$ satisfies $\frac{z}{1+i}=|2-i|$, then the point corresponding to the conjugate of $z$ is located in the complex plane's
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | D |
Let $A=\{x||x| \lt 2\}$, $B=\{x|x^{2}-3x \lt 0\}$. Then $A\cup B$ is
A: $(-2,3)$
B: $(-2,0)$
C: $(0,2)$
D: $(2,3)$ | A |
If $x\in \left(1,+\infty \right)$, then the minimum value of $y=3x+\frac{1}{x-1}$ is ______. | 2\sqrt{3} + 3 |
Given vectors $\overrightarrow{a}=(\cos 45^\circ, \sin 45^\circ)$ and $\overrightarrow{b}=(\cos 15^\circ, \sin 15^\circ)$, calculate the dot product $\overrightarrow{a} \cdot \overrightarrow{b}$.
A: $-\frac{\sqrt{3}}{2}$
B: $-\frac{1}{2}$
C: $\frac{1}{2}$
D: $\frac{\sqrt{3}}{2}$ | D |
12. Let the function $f(x)$ be a differentiable function defined on $(-\infty, 0)$, with its derivative being $f^{\prime}(x)$, and it satisfies $2 f(x)+x f^{\prime}(x)>x^{2}$. Then the solution set of the inequality $(x+2017)^{2} f(x+2017)-f(-1)>0$ is $\qquad$ . | (-\infty,-2018) |
The graph of the even function $y=f(x)$ is symmetric about the line $x=2$, and $f(3)=3$. Then, $f(-1)=$\_\_\_\_\_\_\_\_. | 3 |
For a geometric sequence $\{a_n\}$, let $S_n$ be the sum of the first $n$ terms. Given that $\frac{S_4}{S_2}=3$, calculate the value of $2a_2 - a_4$. | 0 |
Example 5 (15th "Hope Cup" Senior High School Competition Question) The smallest positive period of the function $f(x)=|\tan 2 x|$ is ( )
A. $2 \pi$
B. $\pi$
C. $\frac{\pi}{2}$
D. $\frac{\pi}{4}$ | \frac{\pi}{2} |
Given the function $f(x)=\dfrac{1}{3}x^{3}-ax^{2}+\left(a^{2}-1\right)x+b$, and the tangent line at the point $\left(1,f(x)\right)$ is $x+y-3=0$.
(1) Find the values of $a$ and $b$, and determine the intervals where the function $f(x)$ is monotonic.
(2) If the inequality $f(x) \lt c^{2}-c$ holds for $x\in \left[-2,4\... | (-\infty,\dfrac{1-\sqrt{33}}{2})\cup (\dfrac{1+\sqrt{33}}{2},+\infty) |
65 people are lined up in the central column of a 17-column grid. At each step, each person takes a random step either to the left or to the right (all steps are of the same magnitude).
Show that there is always at least one column that contains at least 8 people. | 8 |
Four. (50 points) A fly and $k$ spiders are placed at some intersections of a $2012 \times 2012$ grid. An operation consists of the following steps: first, the fly moves to an adjacent intersection or stays in place, then each spider moves to an adjacent intersection or stays in place (multiple spiders can occupy the s... | 2 |
Example 5 Given positive real numbers $x, y, z$ satisfy
$$
\left\{\begin{array}{l}
x^{3}-x y z=-5, \\
y^{3}-x y z=2, \\
z^{3}-x y z=21 .
\end{array}\right.
$$
Find the value of $x+y+z$. | 6 |
8. There are three segments of length $2^{n}(n=0,1, \cdots, 1009)$. Then the number of non-congruent triangles that can be formed by these 3030 segments is (answer in digits). | 510555 |
4. Petya wants to test the knowledge of his brother Kolya, the winner of the "Higher Test" mathematics Olympiad. For this, Petya thought of three natural numbers $a, b, c$, and calculated $x = \text{GCD}(a, b)$, $y = \text{GCD}(b, c)$, $z = \text{GCD}(c, a)$. Then he wrote three rows of five numbers each on the board:
... | 8,14,18 |
The diagram below shows nine points on a circle where $AB=BC=CD=DE=EF=FG=GH$. Given that $\angle GHJ=117^\circ$ and $\overline{BH}$ is perpendicular to $\overline{EJ}$, there are relatively prime positive integers $m$ and $n$ so that the degree measure of $\angle AJB$ is $\textstyle\frac mn$. Find $m+n$.
[asy]
size(... | 29 |
11) An angle $A \widehat{O} B$ is trisected by the rays $O P$, $O Q$; the angle $B \widehat{O} C$ (supplementary to $A \widehat{O} B$) is also trisected by the rays $O R, O S$. What is the measure of the angle $Q \widehat{O} R$?
$\begin{array}{llll}\text { (A) } 45^{\circ} & \text { (B) } 60^{\circ} & \text { (C) } 90... | 60 |
3. Let $p$ be an odd prime, and $s(M)$ denote the sum of all elements in set $M$. For the non-empty subsets $T$ of set $F=\{1,2, \cdots, p-1\}$, if $p \mid s(T)$, find the number of such $T$.
| \frac{2^{p-1}-1}{p} |
Problem 8. Solve the system
$$
\left\{\begin{aligned}
x+y+z & =13 \\
x^{2}+y^{2}+z^{2} & =61 \\
x y+x z & =2 y z
\end{aligned}\right.
$$ | (4;3;6),(4;6;3) |
Let point \\(A\\) be a point on the ellipse \\(E\\): \\( \dfrac {x^{2}}{a^{2}}+ \dfrac {y^{2}}{b^{2}}=1(a > b > 0)\\), and let \\(F_{1}\\) and \\(F_{2}\\) be the left and right foci of the ellipse \\(E\\), respectively. Given that \\(\angle AF_{1}F_{2}=15^{\circ}\\) and \\(\angle AF_{2}F_{1}=75^{\circ}\\), and \\(|F_{1... | 3 \sqrt {3} |
1.006. $\frac{\left(13.75+9 \frac{1}{6}\right) \cdot 1.2}{\left(10.3-8 \frac{1}{2}\right) \cdot \frac{5}{9}}+\frac{\left(6.8-3 \frac{3}{5}\right) \cdot 5 \frac{5}{6}}{\left(3 \frac{2}{3}-3 \frac{1}{6}\right) \cdot 56}-27 \frac{1}{6}$. | 1 |
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