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279. If the system of equations
$$
\begin{gathered}
x y=a \\
x+y=b
\end{gathered}
$$
is satisfied by $x_{1}=\alpha, y_{1}=\beta$, then the solution will also be $x_{2}=\beta$, $y_{2}=\alpha$, and there is no other solution. | 279. The equations are symmetric with respect to $x$ and $y$. There cannot be more than two systems of solutions, since substituting one unknown from the second equation into the first leads to a quadratic equation, which admits only two roots. | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,079 |
280. One pair of roots of the system
$$
\begin{gathered}
2 x^{2}+3 x y+y^{2}=70 \\
6 x^{2}+x y-y^{2}=50
\end{gathered}
$$
is $x_{1}=3, y_{1}=4$; write another pair. | 280. $x_{2}=-3, y_{2}=-4$. The left sides of the equations are even functions of both unknowns, i.e., expressions that do not change their values when $x$ is replaced by $-x$ and $y$ by $-y$. | x_{2}=-3,y_{2}=-4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,080 |
281. The system of equations:
$$
\begin{gathered}
x^{4}+y^{4}=272 \\
x+y=6
\end{gathered}
$$
has 4 pairs of roots. Why? If it is known that $x_{1}=4, y_{1}=2$; $x_{2}=3+\sqrt{-55}$, then how to write the value of $y_{2}$ and the other pairs of roots $x_{3}, y_{3} ; x_{4}, y_{4}$ without calculations? | 281. The coefficients of the equations are rational numbers, and therefore there must exist an irrational root conjugate to the value $x_{2}$, namely $y_{2}=3-\sqrt{55}$. The left sides of the equations are symmetric with respect to $x$ and $y$. Therefore, we can write:
$$
x_{3}=2, y_{3}=4 ; x_{4}=2-\sqrt{55}, y_{4}=3... | x_{3}=2,y_{3}=4;x_{4}=2-\sqrt{55},y_{4}=3+\sqrt{55} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,081 |
282. Given the system of equations:
$$
\begin{gathered}
x^{3}-y^{5}=2882 \\
x-y=2
\end{gathered}
$$
if it is known that $x_{1}=5, y_{1}=3 ; x_{3}=1+3 \sqrt{-2}$, then how to write $x_{2}, y_{2} ; y_{3} ; x_{4}, y_{4}$ without calculations? | 282. Reasoning as in the previous question, we have:
$$
x_{4}=1-3 \sqrt{2}
$$
the left sides of the equations will not change if we substitute $-y$ for $x$ and $-x$ for $y$; therefore:
$$
x_{2}=-3, y_{2}=-5 ; y_{3}=-1+3 \sqrt{2} ; y_{4}=-1-3 \sqrt{2}
$$ | x_{2}=-3,y_{2}=-5;y_{3}=-1+3\sqrt{2};y_{4}=-1-3\sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,082 |
283. System
\[
\begin{aligned}
& (x-y)\left(x^{2}-y^{2}\right)=160 \\
& (x+y)\left(x^{2}+y^{2}\right)=580
\end{aligned}
\]
has a pair of roots \(x_{1}=3, y_{1}=7\); what other pair of roots should this system have? | 283. The form of the equations will not change if $x$ is replaced by $y$, and $y$ is replaced by $x$; therefore, $x_{2}=7, y_{2}=3$. | x_{2}=7,y_{2}=3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,083 |
284. Show that the equation
$$
a x^{2}+2 b x y+c y^{2}=0
$$
allows determining the ratio of the unknowns $\frac{x}{y}$. | 284. The given equation can be written in the form
$$
a\left(\frac{x}{y}\right)^{2}+b\left(\frac{x}{y}\right)+c=0
$$
Dividing the equation by $y^{2}$ eliminates the root $y_{1}=0$, which in turn implies $x_{1}=0$. | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,084 |
285. Show that the equation
$$
a x^{3}+b x^{2} y+b x y^{2}+a y^{3}=0
$$
allows determining the ratio $\frac{x}{y}$ of the unknowns. | 285. By dividing by $\boldsymbol{y}^{3}$, we reduce the given equation to
$$
a\left(\frac{x}{y}\right)^{3}+b\left(\frac{x}{y}\right)^{2}+b\left(\frac{x}{y}\right)+a=0
$$
This is a symmetric equation that is easily solvable. | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,085 |
286. Show that a homogeneous equation with two unknowns of any degree leads to an equation defining the ratio of the unknowns. | 286. Dividing the general homogeneous equation with two unknowns
$$
a_{0} x^{n}+a_{1} x^{n-1} y+a_{2} x^{n-2} y^{2}+\ldots+a_{n-1} x y^{n-1}+a_{n} y^{n}=0
$$
by $y^{n}$, we obtain for determining $\frac{x}{y}$ :
$$
a_{0}\left(\frac{x}{y}\right)^{n}+a_{1}\left(\frac{x}{y}\right)^{n-1}+a_{2}\left(\frac{x}{y}\right)^{n... | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,086 |
287. How many solutions does the system of equations generally have
$$
\begin{gathered}
a x^{2}+b x y+c y^{2}=d \\
a_{1} x^{2}+b_{1} x y+c_{1} y^{2}=d_{1} ?
\end{gathered}
$$
In particular, how many solutions does the system of question 280 have? | 287. Generally four. Excluding $d$ and $d_{1}$, we obtain a homogeneous equation of the second degree, which, generally, will determine two values of the ratio $\frac{x}{y}$ (see question 284) $\frac{x}{y}=k_{1}$ and $\frac{x}{y}=k_{2}$, from which $x=k_{1} y, x=k_{2} y$. Substituting each of these expressions sequenti... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,087 |
288. Quickly write a system of two linear equations with two unknowns, the roots of which, after solving, turn out to be integers. | 288. We preselect two integer numbers that we want to be the roots of the system, and then write the equations, performing mental calculations to select suitable free terms to identically yield zero. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,088 |
289. What condition must the coefficients of two linear equations with two unknowns satisfy, a) for both unknowns to have the same value, b) for $x=0$, c) for $y=0$, d) for $x=y=0$? | 289. Let the system of equations be:
$$
\begin{gathered}
a_{1} x+b_{1} y=c_{1} \\
a_{2} x+b_{2} y=c_{2}
\end{gathered}
$$
In case a) $x=y$, and therefore
$$
\left(a_{1}+b_{1}\right) x=c_{1},\left(a_{2}+b_{2}\right) x=c_{2}
$$
hence,
$$
\frac{c_{1}}{a_{1}+b_{1}}=\frac{c_{2}}{a_{2}+b_{2}}
$$
b) the equations take t... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,089 |
290. Excluding the unknown $y$ from the system of equations
$$
\begin{aligned}
& a_{1} x+b_{1} y=c_{1} \\
& a_{2} x+b_{2} y=c_{2}
\end{aligned}
$$
we obtain
$$
x=\frac{c_{1} b_{2}-c_{2} b_{1}}{a_{1} b_{2}-a_{2} b_{1}}
$$
Write, without new calculations, the formula for $y$. | 290. Our equations will not change if we replace $x$ with $y$ and $y$ with $x$ and simultaneously replace the coefficients $a$ and $b$ with the coefficients $b$ and $a$ with the corresponding indices; if the same is done with the formula for $x$, we get
$$
y=\frac{c_{1} a_{2}-c_{2} a_{1}}{b_{1} a_{2}-b_{2} a_{1}}
$$
... | \frac{c_{1}a_{2}-c_{2}a_{1}}{b_{1}a_{2}-b_{2}a_{1}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,090 |
291. Are two linear equations with two unknowns always solvable? Examples. Geometric meaning. | 291. No; for example, $2 x+3 y=1 ; 6 x+9 y=10$. Give a general rule, i.e., the dependence between the coefficients of both equations. Geometric meaning. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,091 |
293. Given two equivalent systems of two linear equations with two unknowns; what can be said about the four lines graphically representing each of these equations? | 293. Compare the answer to the previous question.
Translate the text above into English, please retain the source text's line breaks and format, and output the translation result directly. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,093 |
294. Let $\boldsymbol{P}(\boldsymbol{x}, \boldsymbol{y})=0$ (1) and $\boldsymbol{P}^{\mathbf{1}}(\boldsymbol{x}, \boldsymbol{y})=0$ (2) be two linear equations; what can be said about the graphical representation of the equation $\boldsymbol{P}(x, \boldsymbol{y})+k \boldsymbol{P}^{1}(\boldsymbol{x}, \boldsymbol{y})$ (3... | 294. Equation (3) is also a linear equation, and therefore represents a straight line. The values satisfying equations (1) and (2) also satisfy equation (3), which means that the line (3) passes through the point of intersection of lines (1) and (2). If we assign different values to the arbitrary parameter sequentially... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,094 |
296. Why are the equations $\boldsymbol{A}^{2}=\boldsymbol{B}^{2}$ (1) and $\boldsymbol{A}=\boldsymbol{B}$ (2), where $\boldsymbol{A}$ and $\boldsymbol{B}$ are known functions of $x$, generally not equivalent? | 296. Let's write equation (1) in the form
$$
(\boldsymbol{A}-\boldsymbol{B})(\boldsymbol{A}+\boldsymbol{B})=0
$$
Therefore, its roots are not only the roots of equation (2), but also the roots of the equation $\boldsymbol{A}=-\boldsymbol{B}$, and thus equations (1) and (2) are not equivalent. (Compare the definition ... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,095 |
297. Show that for no values of the coefficients $a$ and $b$ can the equation $x^{3}+a x^{2}+b=0$ have both roots 1 and -1 simultaneously. | 297. If +1 and -1 satisfied the equation, then we would have $1+a+b=0$ and $-1+a+b=0$; but there are no finite values of $a$ and $b$ that satisfy these two equations. | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,096 |
301. What problem about a rectangular parallelepiped leads to the system of equations:
$$
\begin{gathered}
x+y+z=17 \\
2 x y+2 y z+2 z x=180 \\
x^{2}+y^{2}=100 ?
\end{gathered}
$$ | 301. Given the sum of the lengths of all edges, the total surface area, and the diagonal of one of the faces. Determine the edges. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,098 | |
306. In the problem: "For what period of time will a capital of 4351 rubles, given on compound interest at $5 \%$ per annum, with interest being added to the capital after each half-year, be doubled?" - is there no lack of data? | 306. The capital sum is superfluous, for if 1 ruble doubles in a known number of years, then in the same number of years $2,3, \ldots$ any number of rubles will also double. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,099 | |
311. Of two banks, one pays $p^{\%} \%$ annually, with interest compounded semi-annually; the second pays $q^{\%} \%$, with interest compounded monthly. What relationship must exist between $p$ and $q$ for placing capital in both banks to be equally profitable? | 311.
$$
\left(1+\frac{p}{2 \times 100}\right)^{2}=\left(1+\frac{q}{12 \times 100}\right)^{12}
$$ | (1+\frac{p}{2\times100})^{2}=(1+\frac{q}{12\times100})^{12} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,103 |
313. The given problem is solved in two ways: arithmetically and algebraically (using an equation); what, in general, are the advantages or disadvantages of each solution?
Given the problem is solved in two ways: arithmetically and algebraically (using an equation); what, in general, are the advantages or disadvantage... | 313. In algebraic solving, it is only at the stage of formulating the equation (translating the conditions of the problem into algebraic language) that one penetrates to the essence of the problem; all the rest of the equation solving consists of more or less mechanical calculation, the essence of the question being ob... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,104 |
320. Write the general form of first and second degree functions, as well as special forms of these functions. | $$
\begin{gathered}
y=a x+b ; y=a x \\
y=a x^{2}+b x+c \\
y=a x^{2}+c ; y=a x^{2}
\end{gathered}
$$ | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,106 |
321. What symmetry properties do the curves representing the functions have:
$$
y=a x ; y=a x^{3}+b x ; y=\frac{a x^{3}}{b x^{4}+c x^{2}+d} ?
$$
Generalize. | 321. All these curves are symmetric with respect to the origin of coordinates. This property is possessed in general by curves representing functions $y=\boldsymbol{P}(x)$, where $\boldsymbol{P}(x)$ denotes a polynomial containing only odd powers of $x$, or (more generally) by curves $y=\frac{Q(\boldsymbol{x})}{\boldsy... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,107 |
322. What symmetry properties do the curves representing the functions have:
$$
y=a x^{2} ; y=a x^{4}+b x^{2}+d ; y=\frac{a x^{4}+b x^{2}+c}{a_{1} x^{4}+b_{1} x^{2}+c_{1}} ?
$$ | 322. Curves are symmetric with respect to the $y$-axis. This property generally applies to curves $y=\boldsymbol{P}(x)$, where $\boldsymbol{P}(x)$ is a polynomial containing only even powers of the variable, or to curves of the type
$$
y=\frac{Q(x)}{\boldsymbol{R}(x)}
$$
where $Q(x)$ and $\boldsymbol{R}(x)$ denote po... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,108 |
323. Read the symmetry properties of the curves represented by the equations
$$
\begin{aligned}
& x^{2}+y^{2}=r^{2} \\
& b^{2} x^{2}+a^{2} y^{2}=b^{2} a^{2} \\
& b^{2} x^{2}-a^{2} y^{2}=b^{2} a^{2} \\
& y^{2}=2 p x \\
& y=\sin x \\
& y=\tan x \\
& y=\log x
\end{aligned}
$$
Note. Below we will use the expression “ $y$... | 323. Curves (1), (2), (3) are symmetric with respect to both axes and the origin 0; curve (4) with respect to the $x$-axis; curves (5) and (6) with respect to 0; curve (7) does not possess any symmetry properties. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,109 |
324. Two quantities $x$ and $y$ depend on each other, such that when one increases, the other also increases. Someone, not accustomed to precise mathematical formulations, expressed this as: « $y$ increases proportionally to $x$ ». What is wrong with this formulation? Does it express more, less, or something entirely d... | 324. It is clear that if $y$ increases together with $x$, it does not at all follow from this that the increments of $y$ are proportional to the increments of $x$. Thus, $x^{2}$ is not proportional to $x$; $\sqrt[3]{x}$ is not proportional to $x$; $\log x$ is not proportional to $x$, - although all these expressions in... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,110 |
327. The cost of a telegram sent abroad is calculated as follows: to a constant basic rate, amounting to $a$ kopecks per telegram, is added the charge for the number of words at $b$ kopecks per word. a) Is the cost of the telegram proportional to the number of words? b) Does the cost of the telegram increase proportion... | 327. a) No. b) Yes. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,113 |
332. How is interpolation performed in calculations using tables? What is its geometric meaning? What does the expression "PP, (partes proportionales)" mean? Why does interpolation generally yield approximate results, and why is this approximation practically sufficient? (Examples of different tables.) | 332. Interpolation is a method of calculating the value of a given function $f(x)$ (in logarithmic tables - the function $\log x$) for a value of the argument $x$ that does not appear directly in the table. Let $n$ and $n+1$ be two consecutive numbers (arguments) that appear in the table, such that $n < x < n+1$. Inter... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,117 |
333. Express in terms of proportionality the dependence $t=2 \pi \sqrt{\prime} \frac{\bar{l}}{g}$, where $t$ denotes the period of oscillation of a pendulum, $l$ its length, and $g$ the gravitational constant (acceleration due to gravity). | 333. The period of oscillation of a pendulum is proportional to the square root of its length ( $\pi$ and $g$ are constant numbers). | \propto\sqrt{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,118 |
334. Express in terms of proportionality the formula for the force of mutual attraction between two masses $m$ and $m^{\prime}$, which are at a distance $d$ from each other, stating: $f=k \frac{m m^{\prime}}{d^{2}}$. What is the meaning of the constant $k$ in this formula? | 334. The letter $k$ denotes the force of attraction between two masses, each equal to one unit of mass, separated by one unit of length (two masses of one gram each at a distance of one centimeter). | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,119 | |
336. Two variables $x$ and $y$ are directly proportional to each other; what can be said about their reciprocals? | 336. $\frac{1}{x}$ is directly proportional to $\frac{1}{y}$. | \frac{1}{x} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,121 |
337. Two variables $x$ and $y$ are inversely proportional; what can be said about their reciprocals? | 337. $\frac{1}{x}$ is inversely proportional to $\frac{1}{y}$. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,122 |
340. From the formulas $l=v t ; v=\frac{l}{t} ; t=\frac{l}{v}$ (where $l$ represents the distance traveled over time $t$ by a body moving uniformly at speed $v$), determine which of these quantities are directly proportional to each other and which are inversely proportional (assuming the third quantity remains unchang... | 340. Two quantities are directly proportional when their quotient is constant; they are inversely proportional when their product is constant. Thus: $l$ and $v, l$ and $t$ are directly proportional, $v$ and $t$ are inversely proportional. | l | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,125 |
346. Two circles are given. The length of the first one is 4 centimeters, the second one is 40,000 kilometers (approximately the length of the Earth's meridian); the periphery of each circle is extended by 1 meter (the length of the first became 104 centimeters, the second 40,000,001 meters). Which of these circles exp... | 346. The radii of both circles are increased by the same length, exactly by $\frac{1}{2 \pi}$ meters. | \frac{1}{2\pi} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,127 |
350. $y$ is a function of $x$, and it is known that equal increments of $x$ correspond to equal increments of $y$. Prove that the dependence between $y$ and $x$ can be expressed by the formula $y=a x+b$, where $a$ and $b$ are constant numbers. | 350. The increments of $y$ are proportional to the increments of $x$; let $b$ be the coefficient of proportionality; denote by $a$ the value taken by $y$ when $x=0$. We have $\frac{y-a}{x-0}=b$, hence $y=$ $=a+b x$. | +bx | Algebra | proof | Yes | Yes | olympiads | false | 40,129 |
351. Two variable quantities $x$ and $y$ are related by the equation $3 x-2 y=7$. Show that $y$ increases proportionally with $x$. Calculate the coefficient of proportionality. | 351. The dependence of $y$ on $x$ can be expressed as $y=$ $=\frac{3}{2} x-\frac{7}{2}$. | \frac{3}{2}x-\frac{7}{2} | Algebra | proof | Yes | Yes | olympiads | false | 40,130 |
352. Show that $x^{2}$ does not increase proportionally to $x$. | 352. If $x$ doubles, then $y$ quadruples, etc. Otherwise: we have proved (in question 350) that if $y$ increases proportionally to $x$, then $y$ is a linear function of $x$, but $x^{2}$ is a quadratic function. | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,131 |
354. Give an example of a function that, for $x$ equal to any of the numbers $a_{1}, a_{2}, a_{3} \ldots a_{n}$, would be equal to: 1) zero, 2) a given number $k$. | 354. a) $\left(x-\alpha_{1}\right)\left(x-\alpha_{2}\right)^{2}\left(x-\alpha_{3}\right)^{3} \ldots\left(x-\alpha_{n}\right)^{n}$
b) $\sin \left(x-\alpha_{1}\right) \sin \left(x-\alpha_{2}\right) \ldots \sin \left(x-\alpha_{n}\right)+$ $+k \cos \left(x-\alpha_{1}\right) \cos \left(x-\alpha_{2}\right) \ldots \cos \left(... | (x-\alpha_{1})(x-\alpha_{2})^{2}(x-\alpha_{3})^{3}\ldots(x-\alpha_{n} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,133 |
356. Provide an example of a function that, when $x$ is equal to a known number, takes the form $\frac{0}{0}$, but as $x$ approaches this number, it tends to a certain limit, | 356. Ex.: 1) $\frac{x^{2}-10 x+9}{x^{2}-9 x+8}$ as $x \rightarrow 1 ; \quad \frac{(x-1)(x-9)}{(x-1)(x-8)}=$ $=\frac{x-9}{x-8}$; as $x \rightarrow 1 \frac{x-9}{x-8} \rightarrow \frac{1-9}{1-8}=\frac{8}{7}$,
$$
\text { 2) } \frac{\tan x}{\cos x} \text { as } x \rightarrow 0
$$ | \frac{8}{7} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,135 |
358. Give an example of several functions continuous in the interval $-\infty<x<+\infty$, i.e., continuous for any value of the variable. | 358. $y=3x-7 ; y=a^x ; y=\sin x$. | 3x-7;^x;\sinx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,137 |
360. Let the function of a positive variable be defined by the formula $y=x-[x]$, where the symbol $[x]$ denotes the greatest integer contained in $x$. What values can $y$ take? For which values of $x$ is the function continuous, and for which is it discontinuous? | 360. 2) The function can take values between 0 (inclusive) and 1 (exclusive); 2) discontinuity occurs at all integer values of \( x \) ( \( y \) makes a jump from values close to 1 to the value 0 ). | 0\leqy<1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,139 |
368. Which of the three functions
\[
\begin{aligned}
& y=100 x \\
& y=1000+100 x \\
& y=10000+99 x
\end{aligned}
\]
increases faster than the others? | 368. $100(x+h)-100 x=100 h$
$$
[1000+100(x+h)]-(1000+100 x)=100 h \cdot .(2)
$$
$$
[10000+99(x+h)]-(10000+99 x)=99 h . .(3)
$$
The first two increase at the same rate and faster than the third. 369. $\quad[2(x+h)+3]-(2 x+3)=2 h$
$$
\begin{array}{r}
{\left[\frac{1}{2}(x+h)+18\right]-\left[\frac{1}{2} x+18\right]=\fr... | 100 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,147 |
371. Which of the two functions
$$
y=1000 x \text { and } y=\frac{x^{2}}{1000}
$$
increases faster as $x$ increases from 0 to $+\infty$? | 371. The increment of $y=1000 x$ will be $1000(x+h)-1000 h=$ $=1000 h$;
the increment of $y=\frac{x^{2}}{1000}$ will be $\frac{(x+h)^{2}}{1000}-\frac{x^{2}}{1000}=\frac{2 x h+h^{2}}{1000}$, or approximately $\frac{2 x h}{1000}$.
Let: $\frac{2 x h}{1000}1000 h$, from which $x>500$ 000. Therefore: for $0<x<5000001000 x... | for\0<x<500000,\1000x\\faster;\for\x>500000,\\frac{x^2}{1000}\\faster | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,149 |
372. What change does a positive fraction undergo when its numerator and denominator are increased (decreased) by the same number? | 372. $\frac{a+k}{b+k}-\frac{a}{b}=\frac{(b-a) k}{b(b+k)}$. Therefore, if the fraction was proper $(a<b)$, it approaches 1 from below, i.e., it is increasing; if improper $(a>b)$, it approaches 1 from above, i.e., it is decreasing. | \frac{(b-)k}{b(b+k)} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,150 |
374. What is the lower and what is the upper bound of the fraction $\frac{n}{n+1}$, where $n$ denotes any positive integer? | 374. Lower bound $\frac{1}{2}$, upper bound 1; the lower bound is reached (at $n=1$), the upper bound is never reached. | \frac{1}{2},1 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 40,152 |
376. The number 1, raised to any power, always gives 1; a number slightly greater than 1, when multiplied by itself, gives successively larger and larger numbers, its power increases without limit as the exponent increases; conversely, the power of a number slightly less than 1 decreases without limit and approaches ze... | 376. $(1+\alpha)^{n}$, where $\alpha>0$, increases with the increase of $n$, since the product of multiplication by a number greater than one is greater than the multiplicand. Let us now show that $(1+\alpha)^{n}$ increases beyond any bound when $n$ increases, i.e., that one can specify a value of $n$ for which $(1+\al... | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,154 |
377. It is known that the number $\boldsymbol{e}$ (the base of natural logarithms) is defined as the limit of the expression $\left(1+\frac{1}{n}\right)^{n}$ as $n$ increases without bound:
$$
e=\lim _{n \rightarrow \infty}\left(1+\frac{1}{n}\right)^{n}
$$
But one might reason as follows: if the exponent of a base sl... | 377. Of course, both conclusions drawn are unjustified, because the propositions about the limit of a power apply only to a power with a constant base (or $1+\alpha$, or 1), while the exponent increases, whereas here the base $1+\frac{1}{n}$ itself is variable. In more detailed algebra textbooks (or introductory differ... | 2.71828\ldots | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,155 |
378. a) Must a constantly increasing function increase beyond any limit? b) Must a constantly decreasing function become less than any limit? If not, then provide examples. | 378. No. a) $1-\frac{1}{x}$; b) $1+\frac{1}{x}$. | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,156 |
379. Give examples of a function which, as $x$ increases beyond all bounds, a) also increases beyond all bounds, b) tends to a finite limit $a$ (especially to zero); finally, functions which, as $x$ approaches a certain value $p$, c) increases beyond all bounds, d) tends to a certain limit $a$ (especially to zero). | 379. a) $x^{2}$, cot $\frac{1}{x} ;$ b) $a-\frac{1}{x^{2}} ; \log \left(1+\frac{1}{x}\right) ;$ c) $a+\operatorname{tg}(x-p)$; $\log (1+x-p)$. | \ | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,157 |
380. Give examples of functions that, as $x$ increases without bound, a) do not tend to any limit, oscillating between finite bounds $a$ and $b(a<b)$; b) do not tend to any limit, oscillating, with the oscillations increasing without bound. | 380. a) $a+(b-a) \sin ^{2} x$; b) $x \sin x$. | +(b-)\sin^{2}xx\sinx | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,158 |
381. Give examples of functions that, as $x$ approaches a certain value $p$, a) do not tend to any limit, oscillating between finite boundaries $a$ and $b(a<b)$;
b) do not tend to any limit, oscillating, with the oscillations growing without bound. | 381. a) $\sin \frac{1}{a-p}, a+(b-a) \sin ^{2} \frac{1}{x-p}$, b) $\frac{1}{x-p}, \sin \frac{1}{x-p}$. | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,159 |
386. What type of function is an algebraic function of several variables $x, y, z \ldots$, if the function does not change when $k x$ is substituted for $x$, $k y$ for $y$, and so on? | 386. Homogeneous, of zero degree; for example, $\frac{x^{3}+x^{2} y+y^{3}}{x^{3}-7 x y^{2}-5 y^{3}}$. | \frac{x^{3}+x^{2}y+y^{3}}{x^{3}-7xy^{2}-5y^{3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,164 |
400. Show that if in the quadratic equation $x^{2}+p x+q=0$ $q$ is constant, and $p$ is variable, then the roots vary inversely as each other. | 400. The product of the roots $x_{1} x_{2}=q$, and $q$ is constant. | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,174 |
414. Show that neither the angles of a quadrilateral inscribed in a circle, nor the sides of a circumscribed quadrilateral, taken in natural order, can generally form either an arithmetic or a geometric progression. (Why "generally"?) | 414. Let the angles (sides) of the inscribed (circumscribed) quadrilateral $\boldsymbol{A} \boldsymbol{B} C D$ form an arithmetic progression in natural order: $a, a+r, a+2 r, a+3 r$. Since in an inscribed (circumscribed) quadrilateral, the sums of opposite angles (sides) are equal, we have $a+a+2 r=a+r+a+3 r$, from wh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,185 |
417. What sequence is formed by the algebraic sums of the corresponding terms of several given arithmetic progressions? Generalize. | 417. The linear function of corresponding terms $a_{k}, b_{k}, c_{k}, \ldots$ of several arithmetic progressions $A a_{k}+B b_{k}+C c_{k} \ldots$ also represents an arithmetic progression. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,187 |
418. What sequence is formed by the products of the corresponding $a_{k}, b_{k}, c_{k} \ldots$ terms of several given geometric progressions? What more general function could be taken instead of the product without changing the answer to the question? | 418. a) Geometric progression; b) the product of different (but for each of the given progressions the same) powers of the corresponding terms $a_{k}^{p} b_{k}^{q} c_{k}^{r} \ldots$ | )Geometricprogression;b)a_{k}^{p}b_{k}^{q}c_{k}^{r}\ldots | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,188 |
419. If in this sequence of numbers each term, from the second to the penultimate, is the arithmetic mean of both adjacent terms, then the sequence is an arithmetic progression; if the geometric mean, then a geometric progression. Proof. | 419. Let the sequence of numbers $a_{1}, a_{2}, a_{3}, \ldots a_{n}$ have the property that $a_{k}=\frac{a_{k-1}+a_{k+1}}{2}$ (for $k=2,3 \ldots n-1$); from this it follows that $2 a_{k}-a_{k-1}+a_{k+1}$ or $a_{k+1}-a_{k}=a_{k}-a_{k-1}$ (for $k=$ $=2,3, \ldots n-1)$. This means that the differences between consecutive ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,189 |
420. What should two geometric progressions be so that the series formed by the sums of their corresponding terms is also a geometric progression? (Instead of the sum, what more general function can be taken here?)
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 420. Let $a, a p, a p^{2} \ldots a p^{n}$ and $b, b q, b q^{2} \ldots b q^{n}$ be two given geometric progressions. If the sums of corresponding terms $a+b, a p+b q, a p^{2}+b q^{2} \ldots a p^{n}+b q^{n}$ are to form a geometric progression, then in particular, it must be $\frac{a p+b q}{a+b}=\frac{a p^{2}+b q^{2}}{a ... | (p-q)^{2}=0 | Logic and Puzzles | other | Yes | Yes | olympiads | false | 40,190 |
422. Can the sum of a finite number of terms of an arithmetic progression be equal to zero? | 422. Perhaps, if some members are positive, some are negative; justify; give examples. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,191 |
423. Can the sum of a finite number of terms of a geometric progression be zero? | 423. If all members of the progression are real numbers, then in the trivial case only, when the progression coefficient \( q = -1 \) and the number of terms is even. | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,192 |
424. In both arithmetic and geometric progressions, there can be both positive and negative terms; what is the difference in their arrangement in one type of progression compared to the other? | 424. In an arithmetic progression, there are several terms with the same sign followed by a series of terms with the opposite sign (the sign can change only once throughout the progression). In a geometric progression, the terms alternate between positive and negative (if the common ratio of the progression is a negati... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,193 |
426. Can three numbers $a, b, c$ be consecutive terms of both an arithmetic and a geometric progression at the same time? | 426. Let these numbers be $a-r, a, a+r$. We express that they form a geometric progression $a^{2}=(a-r)(a+r)$ or $a^{2}=a^{2}-r^{2}$, from which $r=0$. Therefore, this is possible only in the trivial case when all 3 numbers are equal (geometric progression with common ratio $=1$, arithmetic progression with common diff... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,195 |
427. Can the sides of a right triangle form an arithmetic progression? a geometric progression? | 427.
$$
\begin{aligned}
& a) Only } 3,4,5 ; \text { b) progression with common ratio } \\
& \sqrt{\frac{1+\sqrt{5}}{2}}, \text { specifically: } 1, \sqrt{\frac{1+\sqrt{5}}{2}}, \frac{1+\sqrt{5}}{2}
\end{aligned}
$$ | 3,4,5\text | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,196 |
428. Can all existing rational numbers be enumerated, i.e., can a known rule be given that would allow writing them in a certain order, so that each rational number occupies a well-defined position in this sequence? In other words, can a one-to-one correspondence ${ }^{\mathbf{1}}$ be established between the set of rat... | 428. $\frac{1}{1},-\frac{1}{1}, \frac{1}{2},-\frac{1}{2}, \frac{2}{1},-\frac{2}{1}, \frac{1}{3},-\frac{1}{3}, \frac{3}{1},-\frac{3}{1}, \frac{2}{3},-\frac{2}{3}, \frac{3}{2},-\frac{3}{2}, \frac{1}{4}$, $-\frac{1}{4}, \frac{4}{1},-\frac{4}{1}, \frac{3}{4},-\frac{3}{4}, \frac{4}{3},-\frac{4}{3}, \frac{1}{5},-\frac{1}{5},... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 40,197 |
429. A set of numbers is called c o n t i n u o u s if it has the following properties: 1) between any two different numbers in the set, an infinite set of numbers belonging to the same set can always be inserted; 2) if all numbers in the set are divided into two classes ( $\boldsymbol{A}$ ) and ( $\boldsymbol{B}$ ) su... | 429. 1) The domain of integers does not satisfy any of the continuity conditions; 2) the domain of rational numbers satisfies the first condition, but does not satisfy the second: for example, all rational numbers can be divided into two classes: class $(\boldsymbol{A})$ contains all numbers whose square is $2 ;$ each ... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 40,198 |
430. Let the base of the logarithm system be a positive number greater than 1. Since then each positive number corresponds to a logarithm, there are as many logarithms as there are positive numbers; on the other hand, logarithms can take any value from $-\infty$ to $+\infty$, and therefore there are as many logarithms ... | 430. The formula $y=\log x$ assigns to each positive number $x$ (pairs with each positive number $x$) a real number $y$ (positive, zero, or negative) in a one-to-one manner. The apparent contradiction is based on a play on words “as many as.” Clearly, neither the number of all positive numbers nor the number of all rea... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,199 |
432. Provide a formula that assigns, in a one-to-one manner, an even number to each integer.
Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly. | 432. $m=2 n$, where $n$ is any integer. Each integer $\boldsymbol{n}$ corresponds to an even number $m=2 n$, and, conversely, each even number corresponds to an integer $n=\frac{m}{2}$. | 2n | Logic and Puzzles | other | Yes | Yes | olympiads | false | 40,200 |
433. Establish a one-to-one correspondence by formula between all positive real numbers \( x (0 < x < +\infty) \) and real numbers \( y \) confined within the interval between 0 and 1 \( (0 < y < 1) \).
## V. Functions of an integer variable. | 433. For example, 1) $y=\frac{1}{1+x}$ or 2) $y=2^{-x}$. 3) Establish a one-to-one correspondence between a segment of length 1, taken on the y-axis, and the positive half of the x-axis. Let $S$ be the point whose coordinates are ( $-\boldsymbol{a}$, 0), where $\boldsymbol{a}$ is an arbitrary number, $\boldsymbol{M}$ a... | \frac{y}{1-y} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,201 |
434. The well-known expression $\boldsymbol{F}(n)$, where $n$ denotes a variable positive integer, should be prefixed with a + sign when $n$ is even, and with a - sign when $n$ is odd. How can this be expressed concisely algebraically, without adding any verbal explanation to the formula? | 434. $(-1)^{n} F(n)$.
Translate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.
However, since the provided text is already in a form that is commonly used in both English and Chinese (a mathematical expression), it does not require transla... | (-1)^{n}F(n) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,202 |
437. What sequence do the differences between the differences («second differences») of the cubes of consecutive natural numbers form? | 437. The first differences of the cubes of consecutive natural numbers are given by the formula: $(n+1)^{3}-n^{3}=3 n^{2}+3 n+1$. The second differences are obtained by the formula: $\left[3(\boldsymbol{n}+1)^{2}+3(\boldsymbol{n}+1)+1\right]$ $\left[3 n^{2}+3 n+1\right]=6 n+6$; this expression is the $n$-th term of an ... | 6n+6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 40,203 |
439. In the table of square roots, the difference between the roots of two consecutive numbers decreases, approaching zero, if the table is extended indefinitely. Why? | 439. $\sqrt{n+1}-\sqrt{n}=\frac{1}{\sqrt{n+1}+\sqrt{n}}$; the right side of this equality decreases, tending to zero, as it grows infinitely.
$$
\sqrt[3]{n+1}=\sqrt[3]{n}=\frac{1}{\sqrt[3]{(n+1)^{2}}+\sqrt[3]{(n+1) n}+\sqrt[3]{n^{2}}}
$$
from which
$$
\begin{gathered}
\frac{\sqrt[3]{n+1}-\sqrt[3]{n}}{\sqrt{n+1}-\sqr... | \frac{4}{\sqrt[3]{n}} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 40,204 |
441. In the table of reciprocal values of natural numbers (school tables of Schlemilch, table VI), the difference between the reciprocal values of two consecutive numbers decreases and very quickly, as we move towards the reciprocal values of increasingly larger numbers. Verify this directly. Explain why this should be... | 441. $\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n(n+1)}<\frac{1}{n^{2}} ;$ for example, when $n=1000$ the corresponding difference is less than 0.000001. | \frac{1}{n(n+1)}<\frac{1}{n^2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,205 |
442. The difference of the logarithms of two consecutive natural numbers decreases as the numbers themselves increase. Verify this using a table and mathematically justify why this should be the case. Explain how it is possible, contrary to the general rule, that there are places in the tables where the difference does... | 442. $\log (n+1)-\log n=\log \frac{n+1}{n}=\log \left(1+\frac{1}{n}\right)$; as $n$ increases indefinitely, $1+\frac{1}{n}$ decreases, approaching 1; therefore, $\log \left(1+\frac{1}{n}\right)$ decreases, tending to $\log 1=0$. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,206 |
445. A table of values of a known function $f(n)$ for natural values of $n$ has been compiled; it turned out that the second difference is constantly equal to zero. By what formula can $f(n)$ be represented for any natural $n$? How does the situation stand for fractional values of $n$? | 445. From the fact that the second differences are zero, it follows that the first differences are constant; let us denote them by $a$, so $f(2)-f(1)=a, f(3)-f(2)=a, \ldots f(n)-f(n-1)=a$. Adding these $n-1$ equations side by side, we have $f(n)-f(1)=(n-1) a$, or, denoting $f(1)$ by the letter $b, f(n)=(n-1) a+b=n a+b-... | f(n)=na+ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,209 |
446. A table of values for two functions $f(\boldsymbol{n})$ and $\varphi(\boldsymbol{n})$ for natural values of $n$ has been compiled; it turned out that the second differences in both tables are identical. From this, it was concluded that the differences $f(1)-\varphi(1), f(2)-\varphi(2), f(3)-\varphi(3) \ldots$ and ... | 446. Consider the function $\boldsymbol{F}(\boldsymbol{n})=f(\boldsymbol{n})-\varphi(n)$; since the second differences of the functions $f(n)$ and $\varphi(n)$ are identical, the second differences of the function $\boldsymbol{F}(\boldsymbol{n})$ are zero; hence (see the previous question), $\boldsymbol{F}(\boldsymbol{... | proof | Algebra | proof | Yes | Yes | olympiads | false | 40,210 |
447. In the table of squares of natural numbers, the second difference is constantly equal to 2. Is $n^{2}$ the only function possessing this property? | 447. If to $n^{2}$ we add a function, the second differences of which are constantly zero, that is, the function $a n+b$, then the second differences of the expression $n^{2}+a n+b$ are also equal to 2. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,211 |
448. Give examples of functions whose tables for natural values of the variable have: a) the second difference constantly equal to 1; b) the second difference constantly equal to 7. | 448. a) $\frac{1}{2} n^{2}+a n+b$; b) $\frac{7}{2} n^{2}+a n+b$ ( $a$ and $b$ arbitrary numbers). | \frac{1}{2}n^{2}++b\frac{7}{2}n^{2}++b | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,212 |
449. The second difference of the function $f(n)$ is constantly equal to 1. Prove that for any natural $n$ it will be:
$$
f(n)=\frac{1}{2} n^{2}+a n+b
$$
where $\boldsymbol{a}$ and $\boldsymbol{b}$ denote constant numbers (independent of $\boldsymbol{n}$). | 449. It is known that the second differences of $\frac{1}{2} n^{2}$ are also equal to 1; therefore, the second differences of $f(n)-\frac{1}{2} n^{2}$ are constantly equal to 0 or (see question 445) $f(n)-\frac{1}{2} n^{2}$ is a linear function of $n$:
$f(n)-\frac{1}{2} n^{2}=a n+b$, from which $f(n)=\frac{1}{2} n^{2}... | f(n)=\frac{1}{2}n^{2}++b | Algebra | proof | Yes | Yes | olympiads | false | 40,213 |
451. How many values of the linear function $y=a x+b$ need to be given to fully define the function? Geometric meaning of the answer. | 451. Two. A straight line is defined by two points lying on it. | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,214 |
452. It is easy to compute the value of the function
$$
f(n)=1+2+3+\ldots+n
$$
for successive values of $n: f(1)=1, f(2)=3$, etc.; using this computation for several (how many?) values of $n$ and the answer to question № 450 (but not using the formula for the sum of an arithmetic progression), prove the identity
$$
... | 452. The second difference of $f(n)$ is constantly equal to 1, and therefore (question 448): $f(n)=\frac{1}{2} n^{2}+a n+b$. Substituting into this formula sequentially $n=1$ and $n=2$, we get two equations from which we can determine $a$ and $b: a=\frac{1}{2}, b=0$, hence
$$
f(n)=\frac{1}{2} n^{2}+\frac{1}{2} n=\frac... | f(n)=\frac{n(n+1)}{2} | Algebra | proof | Yes | Yes | olympiads | false | 40,215 |
453. The third difference (see the note to question 443) of the function $f(n)$ is constantly equal to zero. Prove that for integer values of $n$ the function $f(n)$ is expressed by a quadratic trinomial. | 453. If the third differences of $f(n)$ are constantly equal to zero, then the second differences are equal to a known constant number $a$. But the second differences of the function $\frac{n^{2}}{2}$ are also constantly equal to $a$; therefore, the function $f(n)-\frac{a n^{2}}{2}$, whose second differences are equal ... | f(n)=\frac{^{2}}{2}++ | Algebra | proof | Yes | Yes | olympiads | false | 40,216 |
454. Prove that the third difference of the function $n^{3}$ is constantly equal to 6. Give the general form of a function that possesses this property. | 454. The first differences of the function $n^{3}$ are expressed by the function $3 n^{2}+3 n+1$, the second differences by $6 n+6$; consequently, the third differences are constant and equal to 6. If we add to $n^{3}$ an arbitrary function $\varphi(n)$, the third differences of which are zero, then the third differenc... | 6 | Algebra | proof | Yes | Yes | olympiads | false | 40,217 |
456. Prove that the third difference of the function
$$
f(n)=1^{2}+2^{2}+\ldots+n^{2}
$$
is constantly equal to 2. | 456. $f(n+1)-f(n)=(n+1)^{2}$; therefore, the first differences of $f(n)$ are $(n+1)^{2}$; the second differences are $(n+1)^{2}-n^{2}=2 n+1$, from which it is clear that the third differences are constantly equal to 2. | 2 | Algebra | proof | Yes | Yes | olympiads | false | 40,218 |
457. How many values of the quadratic function
$$
y=a x^{2}+b x+c
$$
must be given to determine this function? How can this question be formulated geometrically? | 457. If for three different values of $x$ we know the corresponding values of $y$, then to determine $a, b, c$ we have a system of three equations. In other words: a parabola, whose axis is parallel to $0 y$, represented by the equation $y=a x^{2}+b x+c$, is defined by three points lying on it. | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 40,219 |
458. It is easy to compute the values of the function
$$
f(n)=1^{2}+2^{2}+\ldots+n^{2}
$$
for individual values of $n$:
$$
f(1)=1, f(2)=5, f(3)=14
$$
Using such computations performed for several (how many?) values of $n$, and the answers to questions 455 and 456, prove the identity
$$
\begin{aligned}
1^{2}+2^{2}+... | 458. The third difference of the function $f(n)$ is constantly equal to 2, and the third difference of the expression $\frac{1}{3} n^{3}$ is also 2; therefore, the third difference of $f(n)-\frac{1}{3} n^{3}$ is 0, and thus $f(n)-\frac{1}{3} n^{3}=a n^{2}+b n+c$, or $f(n)=\frac{1}{3} n^{3}+a n^{2}+b n+c$. By substituti... | f(n)=\frac{n(n+1)(2n+1)}{6} | Algebra | proof | Yes | Yes | olympiads | false | 40,220 |
463. In the very beginning of geometry, a series of axioms (or postulates) about points and lines are usually introduced. Why, then, is Euclid's postulate (the parallel postulate) introduced later, specifically after a whole series of theorems have been proven? | 463. Many geometric theorems do not depend on the Euclidean postulate, i.e., they can be proven without reference to this postulate. Therefore, the Euclidean postulate is only introduced when studying parallel lines, when it is no longer possible to do without it. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,224 |
464. At the beginning of geometry, a theorem is proved that the exterior angle of a triangle is greater than the interior angle not adjacent to it; much later, a corollary is derived from the theorem about the sum of the angles of a triangle, stating that the exterior angle is equal to the sum of the two interior angle... | 464. The theorem about the exterior angle in a triangle (which is very important for subsequent geometric positions) is valid independently of Euclid's postulate, and therefore there is no reason to postpone its proof until the introduction of this postulate and the theorem about the sum of the angles of a triangle. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,225 |
465. Does the theorem about the sum of the angles of a triangle, directly or indirectly, rely on the parallel postulate?
Questions on elem. math.
4 | 465. Yes, for in proving this theorem we refer to the equality of corresponding and alternate interior angles between parallel lines and a transversal, and this is equivalent to Euclid's postulate. | Yes | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,226 |
466. On what theorem is the construction of parallel lines by sliding a triangular ruler along a fixed one-sided ruler based? Can the triangular ruler be replaced with a tool of a different shape? | 466. On the theorem of the parallelism of two lines forming equal corresponding angles with a third line intersecting them. Of the three sides of a triangular ruler, the straight side should be only the one with which we draw the new straight line; the other sides can be arbitrarily modified as long as two fixed points... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,227 |
468. It is proved that the angle formed by two chords intersecting inside a circle is measured by the half-sum $\frac{\alpha+\beta}{2}$ of the arcs $\alpha$ and $\beta$ enclosed between its sides. Can it be deduced from this that the central angle, as a special case of that angle (when the chords become diameters), has... | 468. Such reasoning would be a vicious circle, for the theory about the angle formed by two chords was based precisely on the fact that the central angle is measured by the arc enclosed by its sides. | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,229 |
470. On what theorems does the immediate proof of the Pythagorean theorem rely? Does it depend on the parallel postulate? | 470. Directly on the theorems of similarity or on the theorems of equivalence of areas. Both are based on the theories of parallels, and therefore on Euclid's postulate. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,231 |
473. Suppose it has already been preliminarily proven that the length of a circle is proportional to its radius. What else needs to be done to derive the formula for the circumference of a circle? | 473. When it has been proven that the length of the circumference $C$ is proportional to the radius $r$, this establishes the formula $C=2 \pi r$, since this formula merely expresses the stated proportionality, with the coefficient of proportionality denoted by $2 \pi$. The numerical value of the symbol $\pi$ is determ... | 2\pir | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,234 |
475. Define a square as a) a special rectangle, b) a special rhombus. Provide several different definitions for both a) and b).
a) As a special rectangle:
1. A square is a rectangle where all four sides are of equal length.
2. A square is a rectangle with four equal sides and four right angles.
3. A square is a rectan... | 475. a) A rectangle, all sides of which are equal, or the diagonals of which are perpendicular, or the diagonals of which bisect the angles; b) a rhombus with right angles or with equal diagonals, or the largest of all rhombi with a constant side. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,235 |
477. What is the mutual relationship between the concepts of equality (equivalence) and similarity of two geometric figures? | 477. Equivalence and similarity are independent concepts and therefore compatible; thus: two figures can be equivalent and not similar, similar and not equivalent, and equivalent and similar. In the latter case, they are equal (congruent). The combination of the properties of equivalence and similarity is equivalent to... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,237 |
478. What lines on a spherical surface correspond to straight lines on a plane? | 478. There is a substantial analogy between straight lines on a plane and great circles on the surface of a sphere: In fact, the main properties of a straight line on a plane can be reduced to two: a) through any two different points on the plane, one and only one straight line can be drawn, and b) the straight line co... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,238 |
479. a) Is the definition of a tangent: "a tangent is a straight line that has one and only one common point with a curve" a correct definition? b) Should the curve in the neighborhood of the point of tangency lie on one side of the tangent? | 479. The condition for a line to have one and only one common point with a curve is neither necessary nor sufficient for it to be a tangent. Indeed: a tangent at one point of a curve can intersect it at another (sine curve); a line can have only one common point with a curve, for example, a line parallel to the axis of... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,239 |
481. What angles can be constructed exactly with a compass and a straightedge? | 481. All angles of the form $\pm k \frac{90^{\circ}}{2^{m}} \pm l \frac{30^{\circ}}{2^{n}}$, where $k, l, n$ denote integers. Calculate some of such angles. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,241 |
482. Draw an angle $\alpha$. What angles can be constructed with a compass and straightedge using the given angle $\alpha$? | 482. All angles of the form $\frac{k}{2^{n}} \alpha+\beta$, where $k$ and $n$ are integers ( $k$ can be negative), and $\beta$ denotes angles that can be constructed with a compass and straightedge (see question 481). | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,242 |
483. Show that with a compass and a straightedge, one can construct an angle arbitrarily close to a given angle. | 483. Let it be required to construct an angle that differs by no more than $\frac{90^{\circ}}{2^{m}}$ (where $m$ is a given large number) from a given angle $\alpha$. For each $m$, one can find an integer $l$ such that
$$
\frac{l}{2^{m}} 90^{\circ} \leqslant \alpha \leqslant \frac{l+1}{2^{m}} \cdot 90^{\circ}
$$
$\ma... | proof | Geometry | proof | Yes | Yes | olympiads | false | 40,243 |
485. What does Ptolemy's theorem on the product of the diagonals of a cyclic quadrilateral turn into if one of its sides is reduced to zero? What does this theorem turn into if the quadrilateral is a) a rectangle? b) a trapezoid? | 485. In the case of a triangle, one of the sides is a side of a quadrilateral, and therefore its product with the opposite side reduces to zero, the diagonals coincide with the two remaining sides; consequently, a trivial identity is obtained. For a rectangle, the Pythagorean theorem is obtained, for an isosceles trape... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 40,244 |
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