problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
# 3. Let the intersection point of segments
SK and AL be O. Note that CO is the median drawn to the hypotenuse of the right triangle ACL. Therefore, $\mathrm{AO}=\mathrm{OC}=\mathrm{OL}$ and
$\angle \mathrm{OCA}=\angle \mathrm{OAC}=\angle \mathrm{OAK}$
$. The eccentricity of the ellipse $C$ is $\dfrac {\sqrt {2}}{2}$. The distance from its right focus to an external point $P(2,1)$ is $\sqrt {2}$. A line $l$ passing through the origin $O$ intersects the ellipse $C$ at points $A$ and... | \dfrac {\sqrt {2}}{2} |
B4. Calculate exactly, without using a pocket calculator: $\frac{(-2)^{-3}}{(-0.2)^{3}}-\left(\frac{2}{5}\right)^{-3} \cdot(-3)^{-2} \cdot 0.1^{-1}$.
## PROBLEMS FOR THE THIRD YEAR
Before you are two sets of problems. Problems 1 to 6 of the first set are solved by selecting the correct answer from the five proposed a... | -\frac{125}{72} |
Teacher Wang has enough money to buy 72 pens at 5 yuan each. How many pens can he buy if each pen costs 6 yuan? | 60 |
If the coordinates of point $A(m,n)$ with respect to the origin are $\left(-3,2\right)$, then $m+n=$____. | -1 |
Given that the points (2, 1) and (-2, 3) are on opposite sides of the line $3x-2y+a=0$, the range of values for $a$ is ( )
A: $a 12$
B: $a = -4$ or $a = 12$
C: $-4 < a < 12$
D: $-12 < a < 4$ | C |
9.1. Given positive numbers $p$ and $r$. Let $f(x)$ and $g(x)$ be linear functions with roots $p$ and $r$. Find all roots of the equation $f(x) g(x)=f(0) g(0)$. | x_{1}=0,x_{2}=p+r |
2. Let positive integers $x \neq y$, and satisfy $-\frac{1}{x}+\frac{1}{y}=\frac{2}{5}$. Then the units digit of $x^{y}+$ $y^{x}$ is ( ).
(A) 2
(B) 3
(C) 5
(D) 7 | A |
In a non-isosceles triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$. Given that $a\cos^2 \frac{C}{2} + c\cos^2 \frac{A}{2} = \frac{3}{2}c$ and $2\sin (A-B) + b\sin B = a\sin A$, find the perimeter of triangle $ABC$. | a + b + c = 6 |
2. Vasya and Petya, participating in a school sports and entertainment game, need to cover a distance of 3 km as quickly as possible with only one pair of roller skates between them. They start simultaneously, one just running, the other running on roller skates. At any time, the one running on roller skates can leave ... | 0.5 |
I4.1 If $A$ is a positive integer such that $\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{(A+1)(A+3)}=\frac{12}{25}$, find the value of $A$. | 22 |
1. It is known that for three consecutive natural values of the argument, the quadratic function $f(x)$ takes the values 6, 5, and 5, respectively. Find the smallest possible value of $f(x)$. | \frac{39}{8} |
Example 2. Let $x_{0}=7, x_{1}=3, 3 x_{2}=2 x_{n-1}+x_{n-2}$ $(n \geqslant 2)$. And find $\lim _{n \rightarrow \infty} x_{n}$. | 4 |
Four. (20 points) Let $n$ be a positive integer greater than 1. If there exists a positive integer $i (i < n)$ such that
$$
\sum_{a=1}^{i} a^{2}-\sum_{b=i+1}^{n} b^{2}
$$
is a perfect square, then $n$ is called a "TI number." For example, by $\left(1^{2}+2^{2}+\cdots+9^{2}\right)-\left(10^{2}+11^{2}\right)=8^{2}$, we ... | 26 \text{ is not a TI number.} |
Example 16. Find the function that maps the upper half of the circle $|z|<1$, $\text{Im } z>0$, onto the upper half-plane $\text{Im } w>0$. | (\frac{1+z}{1-z})^2 |
Given the sets $M=\{x|x^2-4<0\}$, and $N=\{x|x=2n+1, n\in \mathbb{Z}\}$, then the set $M\cap N$ equals ( )
A: $\{-1, 1\}$
B: $\{-1, 0, 1\}$
C: $\{0, 1\}$
D: $\{-1, 0\}$ | A |
8. (5 points) The sum of two numbers is 363. When the larger number is divided by the smaller number, the quotient is 16 with a remainder of 6. The larger of the two numbers is $\qquad$.
| 342 |
Among $\frac{23}{7},\sqrt{16}, 0, \sqrt[3]{9}, -\frac{π}{2}, 2.2023010010001$, there are ____ rational numbers. | 4 |
41st Putnam 1980 Problem B3 Define a n by a 0 = α, a n+1 = 2a n - n 2 . For which α are all a n positive? Solution | \alpha\geq3 |
If the asymptote equation of a hyperbola passing through the point \\((4, \sqrt{3})\\) is \\(y= \frac{1}{2}x\\), then the standard equation of the hyperbola is \_\_\_\_\_\_. | \frac{x^2}{4}-y^2=1 |
If the function $f(x)$ satisfies $f(1-x) = f(3+x)$, and $y = f(x)$ has three zeros, then the sum of these three zeros is | 6 |
Given $n\in\mathbb{N}^{*}$, if $$C_{ n }^{ 1 }+2 C_{ n }^{ 2 }+2^{2} C_{ n }^{ 3 }+…+2^{n-2} C_{ n }^{ n-1 }+2^{n-1}=40$$, then $n= \_\_\_\_\_\_$. | 4 |
A new operation "☆" is defined for any two rational numbers $a$ and $b$ such that $a$☆$b=a-b+1$. Please calculate the value of $(2$☆$3)$☆$2$ according to the new operation. | -1 |
15. Find the number of positive integers $x$ that satisfy the inequality $\left|3+\log _{x} \frac{1}{3}\right|<\frac{8}{3}$. | 25 |
A cat broke into a wine store at night. It jumped onto a shelf where bottles of wine were lined up in a long row - the first third of the bottles at the edge cost 160 Kč each, the next third cost 130 Kč each, and the last third cost 100 Kč each. First, the cat knocked over a 160 Kč bottle at the very beginning of the r... | 36 |
## Task 4
On the Day of Liberation, Soviet Pioneers and Komsomol members come to the school. 15 Soviet Pioneers celebrate with the Young Pioneers and 17 with the Thälmann Pioneers. Eight Komsomol members celebrate with the FDJ members.
How many Soviet guests were at the school? | 40 |
Given \(X \sim B(8, \frac{1}{2})\), when \(P(X=k)\) (\(k \in \mathbb{N}, 0 \leqslant k \leqslant 8\)) takes the maximum value, the value of \(k\) is ( )
A: \(7\)
B: \(6\)
C: \(5\)
D: \(4\) | D |
If the distance from one focus of the hyperbola \\(x^{2}- \dfrac {y^{2}}{b^{2}}=1\\) to its asymptote is \\(2\\), then the eccentricity of this hyperbola is \_\_\_\_\_. | \sqrt {5} |
A sports lottery stipulates that 7 numbers are drawn from 01 to 36 as one bet, with each bet costing 2 yuan. Someone wants to select the lucky number 18 first, then choose 3 consecutive numbers from 01 to 17, 2 consecutive numbers from 19 to 29, and 1 number from 30 to 36 to form a bet. If this person wants to buy all ... | 2100 |
Given the function $f(x)=e^{x}-\ln(x+m)$, and $x=0$ is an extremum point of $f(x)$, then $m=$ ______, the increasing interval of the function is ______, and the decreasing interval is ______. | (-1,0) |
Ding 0. .
Detective Nero Wolfe is investigating a crime. There are 80 people involved in the case, one of whom is the criminal, and another is a witness to the crime (but it is unknown who they are). Each day, the detective can invite one or several of these 80 people to his office, and if the witness is among the inv... | Yes |
The sum $29+12+23$ is equal to
(A) $6^{2}$
(B) $4^{4}$
(C) $8^{8}$
(D) $64^{0}$
(E) $2^{6}$ | 2^{6} |
Triangle $ABC$ is inscribed in circle $\omega$ with $AB=5$, $BC=7$, and $AC=3$. The bisector of angle $A$ meets side $\overline{BC}$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $\overline{DE}$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = ... | 919 |
An aquarium on a level table has rectangular faces and is $10$ inches wide and $8$ inches high. When it was tilted, the water in it covered an $8"\times 10"$ end but only three-fourths of the rectangular room. The depth of the water when the bottom was again made level, was
$\textbf{(A) }2\textstyle{\frac{1}{2}}"\qqua... | 3.75 |
Form a five-digit number with the digits 1, 2, 3, 4, 5 without repetition and ensuring that the digits 1 and 2 are not adjacent. Calculate the number of such five-digit numbers. | 72 |
Given $x+x^{-1}=4 (x > 0)$, find the value of $x^{ \frac {1}{2}}+x^{- \frac {1}{2}}=$ ( )
A: $2$
B: $6$
C: $\sqrt {2}$
D: $\sqrt {6}$ | D |
Given that line $l$ passes through the point $(3,1)$ and is parallel to the line $x+y-1=0$.
$(1)$ Find the equation of line $l$;
$(2)$ If the plane figure enclosed by line $l$ and the $x$-axis, $y$-axis is revolved around the $y$-axis for one week to obtain a geometric solid, find the volume of this geometric solid. | \frac{64}{3}\pi |
1. Variant 1.
When multiplying a two-digit and a three-digit number, a four-digit number of the form $A=\overline{a b a b}$ is obtained. Find the largest $A$, given that $A$ is divisible by 14. | 9898 |
## Problem Statement
Calculate the indefinite integral:
$$
\int \frac{\operatorname{arctg} x + x}{1 + x^{2}} d x
$$ | \frac{1}{2}(\operatorname{arctg}^2x+\ln|1+x^2|)+C |
The domain of the function $f(x)= \frac {1}{ \sqrt {1-\log _{3}x}}$ is $(\quad)$
A: $(0,3]$
B: $(0,3)$
C: $(3,+\infty)$
D: $[3,+\infty)$ | B |
10. In the Cartesian coordinate system $x O y$, it is known that point $P(2,1)$ lies on the ellipse $C: \frac{x^{2}}{6}+\frac{y^{2}}{3}=1$. A line $l$ that does not pass through the origin $O$ intersects the ellipse $C$ at points $A$ and $B$, and the midpoint of segment $A B$ is $D$. The slope of line $O D$ is 1. Let t... | \frac{1}{2} |
## Task A-2.4.
Determine all triples of natural numbers ( $a, b, c$ ) for which
$$
2^{a} \cdot 5^{b}-1=11 \cdot 3^{c}
$$ | (2,2,2) |
Given the function $f(x)=\sin x+\cos x$, $f'(x)$ is the derivative of $f(x)$.
$(1)$ Find the minimum value of the function $g(x)=f(x)\cdot f'(x)$ and the set of corresponding $x$ values;
$(2)$ If $f(x)=2f'(x)$, find the value of $\tan \left(x+ \frac {\pi}{4}\right)$. | 2 |
9. How many pairs of integers solve the system $|x y|+|x-y|=2$ if $-10 \leq x, y \leq 10 ?$ | 4 |
Example 5 (1993 National High School League Question) Let the sequence of positive integers $a_{0}, a_{1}, a_{2}, \cdots$ satisfy $a_{0}=a_{1}=1$ and $\sqrt{a_{n} \cdot a_{n-2}}-\sqrt{a_{n-1} \cdot a_{n-2}}=2 a_{n-1}(n=2,3, \cdots)$, find the general term formula of this sequence. | a_{n}=(2^{n}-1)^{2}(2^{n-1}-1)^{2}\cdots(2^{1}-1)^{2} |
9. There are four numbers, their sum is 45, the first number plus 2, the second number minus 2, the third number multiplied by 2, and the fourth number divided by 2, the results obtained are all the same. Therefore, the original four numbers in sequence are $\qquad$ . | 8,12,5,20 |
The period of the function $y=2\sin ( \frac {x}{2}+ \frac {π}{5})$ is $\boxed{\text{answer}}$, and the amplitude is $\boxed{\text{answer}}$. | 2 |
3. Determine the smallest natural number $n$ for which none of the fractions
$$
\frac{7}{n+9}, \frac{8}{n+10}, \frac{9}{n+11}, \ldots, \frac{2015}{n+2017}
$$
can be simplified. | 2015 |
15. For the function $f(x)$, if $f(x)=x$, then $x$ is called a "fixed point" of $f(x)$. If $f(f(x))=x$, then $x$ is called a "stable point" of $f(x)$; the sets of "fixed points" and "stable points" of the function $f(x)$ are denoted as $A$ and $B$, respectively, i.e., $A=\{x \mid f(x)=x\}, B=\{x \mid f(f(x))=x\}$.
(1) ... | [-\frac{1}{4},\frac{3}{4}] |
12.338. The side of the base of a regular quadrilateral prism is equal to $a$, its volume is equal to $V$. Find the cosine of the angle between the diagonals of two adjacent lateral faces. | \frac{V^{2}}{V^{2}+^{6}} |
Let $A=\{1,2, \cdots, 2002\}, M=\{1001,2003,3005\}$. For any non-empty subset $B$ of $A$, if the sum of any two numbers in $B$ does not belong to $M$, then $B$ is called an $M$-free set. If $A=A_{1} \cup A_{2}, A_{1} \cap A_{2}=\varnothing$, and both $A_{1}$ and $A_{2}$ are $M$-free sets, then the ordered pair $\left(A... | 2^{501} |
"x≥3" is a ( ) condition for "x＞3" to be true.
A: Sufficient but not necessary
B: Necessary but not sufficient
C: Both necessary and sufficient
D: Neither sufficient nor necessary | \text{B} |
2. Friends Miha and Blaž are renting motorbikes for a weekend trip from different providers. Miha has to pay 100 euros at the beginning, and when he returns the motorbike, he has to pay an additional 4 euros for each kilometer driven. Blaž pays 200 euros at the beginning, and then 3 euros for each kilometer driven. How... | 101 |
5. In Rt $\triangle A B C$, $\angle A=30^{\circ}, \angle C=90^{\circ},$ the angle bisector $A D$ of $\angle A$ intersects $B C$ at $D$, and $A D=\sqrt{6}$. Then the opposite side of $\angle A$, $B C=$ $\qquad$ . | \frac{1}{2}(\sqrt{3} + 1) |
30th Putnam 1969 Problem A3 An n-gon (which is not self-intersecting) is triangulated using m interior vertices. In other words, there is a set of N triangles such that: (1) their union is the original n-gon; (2) the union of their vertices is the set consisting of the n vertices of the n-gon and the m interior vertice... | n+2m-2 |
What is the value of the following fraction:
$1234321234321 \cdot 2468642468641-1234321234320$
$\overline{1234321234320 \cdot 2468642468641+1234321234321}$. | 1 |
13.228. Two workers were assigned to manufacture a batch of identical parts. After the first worked for 2 hours and the second for 5 hours, it turned out that they had completed half of the entire work. Working together for another 3 hours, they found that they had 0.05 of the entire work left to complete. In what time... | 12 |
Let $S$ be the set $\{1,2,3,...,19\}$. For $a,b \in S$, define $a \succ b$ to mean that either $0 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \succ y$, $y \succ z$, and $z \succ x$?
$\textbf{(A)} \ 810 \qquad \textbf{(B)} \ 855 \qquad \textbf{(C)} \ 900 \qquad \textbf{(D)} \ ... | 855 |
Given that the graph of the quadratic function $y=ax^{2}+bx+3$ passes through the points $(-3,0)$ and $(2,-5)$.
$(1)$ Find the expression of this quadratic function.
$(2)$ Find the coordinates of the vertex of this quadratic function. | (-1, 4) |
1. (8 points) The calculation result of the expression $(9 \times 9-2 \times 2) \div\left(\frac{1}{12}-\frac{1}{19}\right)$ is | 2508 |
A function $f(x)$ is defined on $\mathbb{R}$ and is increasing on the interval $(1,+\infty)$. The graph of the function $F(x)=f(x+1)$ is symmetrical about the $y$-axis. Then $(\ \ \ \ )$
A: $f(-1) > f(2)$
B: $f(0) > f(2)$
C: $f(-2)=f(2)$
D: $f(-4)=f(2)$ | \text{A} |
The function $f(x)=x^{3}-3x^{2}+4$ attains its minimum value at $x=$ \_\_\_\_\_\_. | 2 |
Let set $M=\{1, 2, 4, 8\}$, $N=\{x|x \text{ is a multiple of } 2\}$, then $M \cap N=$ ( )
A: $\{2, 4\}$
B: $\{1, 2, 4\}$
C: $\{2, 4, 8\}$
D: $\{1, 2, 8\}$ | C |
The focal length of the hyperbola $3x^2 - y^2 = 3$ is ______. | 4 |
6.3. Oleg, Igor, and Anya are in the 6th grade. Among them, there is the best mathematician, the best chess player, and the best artist. It is known that:
a) the best artist did not paint their own portrait, but painted Igor's portrait
b) Anya has never lost to boys in chess.
Who in the class is the best mathematicia... | Oleg |
Given the complex number $z= \frac{1+2i}{i^{3}}$, find its complex conjugate $\overline{z}=$ \_\_\_\_\_\_. | -2-i |
Given that $f(x)$ is an even function defined on $\mathbb{R}$, and it is an increasing function on $(-\infty, 0]$, let $a=f\left( \log_{4}7 \right)$, $b=f\left( \log_{\frac{1}{2}}3 \right)$, $c=f\left( 0.2^{0.6} \right)$, then the relationship among $a$, $b$, $c$ is ( ).
A: $c < b < a$
B: $b < c < a$
C: $b < a < c$
D:... | \text{C} |
Given the circle $x^2+y^2+4x-2y-4=0$, the coordinates of the center and the length of the radius are respectively ( )
A: $(2, -1)$, 3
B: $(-2, 1)$, 3
C: $(-2, -1)$, 3
D: $(2, -1)$, 9 | \text{B} |
12.26 Find the function $F(x)$, the graph of which passes through the point $M_{0}(3 ;-2)$, given that $F^{\prime}(x)=4 x^{2}+9 x^{-2}$.
Calculate the integrals (12.27-12.30): | F(x)=\frac{4}{3}x^{3}-\frac{9}{x}-35 |
13.175. A motorboat left the pier simultaneously with a raft and traveled downstream $40 / 3$ km. Without stopping, it turned around and went upstream. After traveling $28 / 3$ km, it met the raft. If the speed of the river current is 4 km/h, what is the motorboat's own speed | \frac{68}{3} |
## Task 2 - 080622
During the summer holidays, Monika visited the capital of the USSR. For her mathematics study group, she brought the following problem:
In the "Gorky" ring of the Moscow underground, there are four escalators of different lengths.
The total length of the two middle-length escalators is $136 \mathr... | 84\mathrm{~},72\mathrm{~},64\mathrm{~},60\mathrm{~} |
Among the following groups of vectors, the one that can serve as a basis is ( )
A: $\overrightarrow{e_{1}} = (0, 0)$, $\overrightarrow{e_{2}} = (1, -2)$
B: $\overrightarrow{e_{1}} = (-1, 2)$, $\overrightarrow{e_{2}} = (5, 7)$
C: $\overrightarrow{e_{1}} = (2, -3)$, $\overrightarrow{e_{2}} = \left(\frac{1}{2}, -\f... | \text{B} |
10.248. To two externally tangent circles with radii $R$ and $r$, a secant is constructed such that the circles intercept three equal segments on it. Find the lengths of these segments. | \frac{1}{2}\sqrt{\frac{14Rr-R^2-r^2}{3}} |
## Problem Statement
Calculate the lengths of the arcs of the curves given by equations in a rectangular coordinate system.
$$
y=\ln \sin x, \frac{\pi}{3} \leq x \leq \frac{\pi}{2}
$$ | \frac{1}{2}\ln3 |
If the perimeter of the base of a right prism is $4$ and its height is $2$, then the surface area of the prism is ______. | 10 |
If the complex number $z = (m^2 - 5m + 6) + (m - 3)i$ is a real number, then the real number $m=$ . | 3 |
6. As shown in Figure 1, if
the three sides of $\triangle A B C$ are
$n+x, n+2 x, n$
$+3 x$, and the height $A D$ from $B C$ is $n$, where $n$
is a positive integer, and $0<x \leqslant$
1. Then the number of triangles that satisfy the above conditions is $\qquad$. | 12 |
2.119. $\frac{\sqrt[4]{7 \sqrt[3]{54}+15 \sqrt[3]{128}}}{\sqrt[3]{4 \sqrt[4]{32}}+\sqrt[3]{9 \sqrt[4]{162}}}$.
2.119. $\frac{\sqrt[4]{7 \sqrt[3]{54}+15 \sqrt[3]{128}}}{\sqrt[3]{4 \sqrt[4]{32}}+\sqrt[3]{9 \sqrt[4]{162}}}$.
The translation is the same as the original text because it is a mathematical expression, which ... | \frac{3}{5} |
3. If $x$ and $y$ are real numbers, find the minimum value of $\sqrt{4+y^{2}}+\sqrt{x^{2}+y^{2}-4 x-4 y+8}+\sqrt{x^{2}-8 x+17}$.
(1 mark)
若 $x 、 y$ 為實數, 求 $\sqrt{4+y^{2}}+\sqrt{x^{2}+y^{2}-4 x-4 y+8}+\sqrt{x^{2}-8 x+17}$ 的最小值。
(1 分) | 5 |
9. [6] Compute the side length of the largest cube contained in the region
$$
\left\{(x, y, z): x^{2}+y^{2}+z^{2} \leq 25 \text { and } x \geq 0\right\}
$$
of three-dimensional space. | \frac{5\sqrt{6}}{3} |
12. Given a moving point $P(x, y)$ satisfies the quadratic equation $10 x-2 x y-2 y+1=0$, then the eccentricity of this quadratic curve is $\qquad$. | \sqrt{2} |
G7.3 Given that $a+\frac{1}{a}=3$. Evaluate $a^{3}+\frac{1}{a^{3}}$. | 18 |
7.4. A biologist sequentially placed 150 beetles into ten jars. Moreover, in each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar? | 16 |
Simplify and find the value of: $a^{2}-\left(a-\dfrac{2a}{a+1}\right)\div \dfrac{a^{2}-2a+1}{a^{2}-1}$, where $a$ is a solution of the equation $x^{2}-x-\dfrac{7}{2}=0$. | \dfrac{7}{2} |
Given that $|\vec{a}|=2$, $|\vec{b}|=4$, and the angle between $\vec{a}$ and $\vec{b}$ is $\frac{π}{3}$, find the length of the shorter diagonal of the parallelogram formed by using $\vec{a}$ and $\vec{b}$ as adjacent sides. | 2\sqrt{3} |
Let $f(x)=\int_x^{x+\frac{\pi}{3}} |\sin \theta|\ d\theta$.
(1) Find $f'(x)$.
(2) Find the maximum and minimum value of $f(x)$ on $0\leq x\leq \pi.$ | f_{\text{min}} = 2 - \sqrt{3} |
Given the function $f\left(x\right)=4\cos x\sin (x+\frac{π}{6})-1$.
$(1)$ Find the value of $f(\frac{π}{6})$.
$(2)$ Find the maximum and minimum values of $f\left(x\right)$ on the interval $[-\frac{π}{6}$,$\frac{π}{4}]$. | -1 |
What is the value of $2 \times 9-\sqrt{36}+1$ ?
(A) 7
(B) 11
(C) 8
(D) 13
(E) 4 | 13 |
15. If $\sin 2 x \sin 3 x=\cos 2 x \cos 3 x$, then one value of $\mathrm{x}$ is
(A) $18^{\circ}$;
(B) $30^{\circ}$; (C) $36^{\circ}$;
(D) $45^{\circ}$;
(E) $60^{\circ}$. | A |
7. The minimum value of the function $y=\sqrt{x^{2}+2 x+2}+\sqrt{x^{2}-2 x+2}$ is $\qquad$ . | 2 \sqrt{2} |
Which of the following propositions is both a universal quantifier proposition and a true proposition?
A: The two diagonals of a rectangle are perpendicular.
B: For any $a$, $b\in R$, $a^{2}+b^{2}\geqslant 2\left(a-b-1\right)$.
C: $\exists x\in R$, $|x|+x=0$.
D: There is at least one $x\in Z$ such that $x^{2}\leqsl... | B |
Given that the complex number $z$ satisfies the equation $i(z+1)=-3+2i$, find the real part of $z$. | 1 |
Which of the following equations is a quadratic equation?
A: $x^2+\frac{1}{x}=0$
B: $x^{2}=1$
C: $3x+2=1$
D: $x-2y=0$ | B |
2. Given the hyperbola $\frac{x^{2}}{t}-\frac{y^{2}}{t}=1$ with the right focus $F$, any line passing through $F$ intersects the right branch of the hyperbola at $M, N$, and the perpendicular bisector of $M N$ intersects the $x$-axis at $P$. When $t$ takes any positive real number except 0, $\frac{|F P|}{|M N|}=$ $\qqu... | \frac{\sqrt{2}}{2} |
V OM - I - Task 2
Investigate when the sum of the cubes of three consecutive natural numbers is divisible by $18$. | 3 |
In the Cartesian coordinate plane, triangle $\triangle ABC$ is translated to triangle $\triangle DEF$, and point $A(-2,3)$ coincides with point $D(1,2)$ after the translation. If a point $M(3,-1)$ inside triangle $\triangle ABC$ is translated, the coordinates of the new position of point $M$ are ______. | (6,-2) |
If the domain of the function $y= \sqrt {(2-a)x^{2}+2(2-a)x+4}$ is $\mathbb{R}$, then the range of values for $a$ is \_\_\_\_\_\_. | -2 \leqslant a \leqslant 2 |
Among the following algebraic expressions, the number of polynomials is ( ).
$2a$; $\frac{m-n}{6}$; $\frac{3}{π}+a$; $\frac{5}{a-b}$; $2(x^{2}-4)$.
A: $4$
B: $3$
C: $2$
D: $5$ | A |
## Task B-3.1.
Solve the inequality $1-27^{x} \leqslant 6 \cdot 9^{3 x}$. | x\geqslant-\frac{1}{3} |
1. Given the equalities $\log _{a} b+\log _{b} c+\log _{c} a=\log _{b} a+\log _{c} b+\log _{a} c=3.5$. Find $\log _{a} b$.
If there are multiple answers, list them in ascending order separated by a semicolon. | 0.5;1;2 |
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