problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
55. What is the maximum number of bishops that can be placed on a chessboard so that they do not threaten each other? Prove that the number of ways to arrange the bishops in such a way is the square of some number.
To understand the condition, of course, you need to know how a bishop moves. A bishop moves diagonally. ... | 55. The instruction to the problem shows a specific example of placing 14 bishops. Therefore, it is possible to place 14 bishops. If we prove that it is impossible to place more than 14 bishops, the first part of the problem will be solved. Let's first consider the black-square bishops. How many can be placed on a ches... | 14 | Combinatorics | proof | Yes | Yes | olympiads | false | 28,455 |
56. Mom has two apples and three pears. Every day for five consecutive days, she gives out one fruit. In how many ways can this be done? | 56. Here is one way to distribute apples and pears: 000 'on the 1st and 3rd day - apples, on the 2nd, 4th, and 5th days - pears),
## 000
(on the first three days - pears, on the last two days - apples). Thus, we need to count all the tables with two light and three dark circles; but we have already done this when sol... | 10 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,456 |
62. A father has five pairwise distinct oranges, which he gives to his eight sons so that each receives either one orange or nothing. In how many ways can this be done? | 62. Let the number of ways to distribute 5 pairwise distinct oranges to eight sons be denoted by $A_{8}^{5}$. We will calculate $A_{8}^{5}$ using two methods.
First method. The number of ways to choose 5 out of 8 sons (those who will receive the oranges) is $C_{8}^{5}$. For any such choice, there are 5! ways to distri... | 6720 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,460 |
63. Let $C_{n}^{k}$ denote the number of ways to light $k$ bulbs out of $n$ (each bulb can be in two states). Prove that the number $C_{n}^{k}$ is located at the $(k+1)$-th position in the $(n+1)$-th row of Pascal's Triangle. | 63. We will prove that the $(k+1)$-th place in the $(n+1)$-th row of Pascal's triangle contains $C_{n}^{k}$. We will prove this by induction on $n$. For $n=1$, the theorem is true. Indeed, in the second row of Pascal's triangle, the first place contains $1=C_{1}^{0}$ and the second place contains $1=C_{1}^{1}$. Suppose... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,461 |
66. In how many ways can three items be chosen from $n$ items? | 66. First, let's calculate the number of ways to choose three items from $n$ items in a specific order. The first item can be any of the $n$ items, the second can be any of the $n-1$ remaining items, and the third can be any of the $n-2$ remaining items. In total, there are $n(n-1)(n-2)$ ways. However, with this method... | \frac{n(n-1)(n-2)}{6} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,462 |
67. In how many ways can $k$ items be chosen from $n$ items? | 67. By definition of $C_{n}^{k}$, $k$ items can be chosen from $n$ in $C_{n}^{k}$ ways. We can solve our problem (i.e., find an explicit formula for $C_{n}^{k}$) in the same way as the previous one. Alternatively, we can use induction. We proved in problem 63 that
$$
C_{n}^{k-1}+C_{n}^{k}=C_{n+1}^{k}
$$
We will prove... | C_{n}^{k}=\frac{n!}{k!(n-k)!} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,463 |
70. How many members will result after expanding all the brackets in the expression: $(a+1)(b+1)(c+1)(d+1)(e+1)(f+1) \times$ $\times(g+1)$ ? | 70. Each of the members that result from expanding the brackets in the expression
$$
\begin{aligned}
(a+1)(b+1)(c+1)(d+1) & (e+1) \times \\
& \times(f+1)(g+1)
\end{aligned}
$$
is a product of seven factors (since there are seven brackets). Any of these factors is either a letter or the digit 1. Thus, we need to find ... | 128 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,464 |
72. In the expression $(1+x+y)^{20}$, the brackets are expanded, but similar terms are not combined. How many terms will be obtained in this case? | 72. Let's expand the expression $(1+x+y)^{20}$, but we will not combine like terms. Each term that results from this is a product of twenty factors. Each of these factors is either 1, or $x$, or $y$. Therefore, our problem is equivalent to the following: in how many ways can 20 traffic lights be lit, each of which can ... | 3^{20} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,466 |
74. In the expression $(1+x)^{56}$, the parentheses are expanded and like terms are combined. Find the coefficients of $x^{8}$ and $x^{48}$, | 74. Re-read the remark at the end of the solution to problem 71. Let's compare two expressions:
$$
(1+x)^{56} \text { and }\left(1+a_{1}\right)\left(1+a_{2}\right) \ldots\left(1+a_{55}\right) .
$$
If we set $a_{1}=a_{2}=\ldots=a_{56}=x$, then the second expression will turn into the first.
After expanding the bracke... | C_{56}^{8}=C_{56}^{48} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,468 |
76. From problems 75 and 63, the following statement follows. Let $a_{0}, a_{1}$, $a_{2}, \ldots, a_{n}$ be the numbers of the $(n+1)$-th row of Pascal's triangle. Then
$$
(1+x)^{n}=a_{0}+a_{1} x+a_{2} x^{2}+\ldots+a_{n} x^{n}
$$
Prove this fact directly based on the definition of Pascal's triangle (without using the... | 76. Let's conduct the proof by induction. For $n=1$, the statement is true: $(1+x)^{1}=1+1 \cdot x$, the binomial coefficients coincide with the numbers in the 2nd row of Pascal's triangle.
Assume that the statement is true for $n-1$:
$$
\begin{aligned}
& (1+x)^{n-1}=a_{0}+a_{1} x+\ldots+ \\
& \quad+a_{k-1} x^{k-1}+a... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 28,470 |
80. Find: a) $C_{n}^{0}+C_{n}^{2}+C_{n}^{4}+C_{n}^{6}+\ldots ;$ b) $C_{n}^{1}+C_{n}^{3}+C_{n}^{5}+\ldots$ | 80. If in the identity
$$
\begin{aligned}
& (1+x)^{n}=C_{n}^{0}+C_{n}^{1} x+C_{n}^{2} x^{2}+\ldots+ \\
& \quad+C_{n}^{n-1} x^{n-1}+C_{n}^{n} x^{n}
\end{aligned}
$$
we set $x=1$, we get
$$
2^{n}=C_{n}^{0}+C_{n}^{1}+C_{n}^{2}+\ldots+C_{n}^{n-1}+C_{n}^{n}
$$
For $x=-1$ we get
$$
0=C_{n}^{0}-C_{n}^{1}+C_{n}^{2}+\ldots... | C_{n}^{0}+C_{n}^{2}+C_{n}^{4}+\ldots=C_{n}^{1}+C_{n}^{3}+C_{n}^{5}+\ldots=2^{n-1} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,472 |
81. Determine the sum of the coefficients of the polynomial that results from expanding and combining like terms in the expression $\left(1+x-3 x^{2}\right)^{1965}$. | 81. If we expand the expression $\left(1+x-3 x^{2}\right)^{1965}$ and combine like terms, we get a polynomial $a_{0}+a_{1} \cdot x+$ $+a_{2} \cdot x^{2}+a_{3} \cdot x^{3}+\ldots$. Note that the sum of its coefficients is equal to the value of the polynomial at $x=1$
$$
\begin{aligned}
& a_{0}+a_{1} \cdot 1+a_{2} \cdot... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,473 |
86. How many diagonals can be drawn in a convex $n$-sided polygon? | 86. Answer: $C_{n}^{2}-n$. Solution: let's take all pairs of different vertices of the polygon (there are $C_{n}^{2}$ such pairs) and connect the points of each pair with segments. We will get $C_{n}^{2}$ segments. Among them, there will be $n$ sides, and the rest are diagonals, i.e., there will be $C_{n}^{2}-n$ diagon... | C_{n}^{2}-n | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,474 |
92. In the library, there is a certain number of readers (i.e., people who have read at least one book from this library). For any $k$ books ( $1 \leqslant k \leqslant \boldsymbol{n}$ ), it is known how many readers have read all of them. How can we find out how many readers are in the library? (There are a total of $n... | 92. Let's sum the number of readers who have read a given set of $k$ books over all sets of $k$ books. Denote this sum by $S_{k}$. We will prove that
$$
S=S_{1}-S_{2}+S_{3}-S_{4}+\ldots+(-1)^{n-1} S_{n}
$$
is the number of readers in the library.
Take a reader who has read exactly $k$ books, and let's see what contr... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,476 |
93. How many phone numbers contain the combination 12? (The number consists of six digits.) | 93. Let's consider the following five sets of phone numbers:
... | 49401 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 28,477 |
95. Two schoolchildren were assigned to keep a weather calendar. They were to mark the day with a "+" sign if the weather was good, and a "-" sign if the weather was bad. The first schoolchild proceeded as follows: He made observations three times a day - in the morning, at noon, and in the evening. If it rained at lea... | 95. If there was no rain at all during the day, both students will give a +. If it rained all the time, both will give a - . If it rained in the morning but was dry in the afternoon and evening, the first student will give a -, while the second will give a +.
The case of + - is impossible because the first student giv... | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 28,478 |
96. $\mathrm{K}$ Two schoolchildren from problem 95 were joined by a third, who makes observations at the same time as the first two and puts a «-» if it rained at least during two of the observations, and «+» in all other cases. Which of the eight evaluations $+++;++-$; +一+; - + + ; - +一; - + + + + +一; - can actually ... | 96. If the first student puts $t$, it means it did not rain at all. Then the other two will also put +. The overall assessment will be +++. If the first puts -, and the third puts +, it means it rained exactly once out of three times. In this case, the second student should put +. The overall assessment will be -++. If... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 28,479 |
97. a) Three hundred people are arranged in 30 rows and 10 columns. From each row, the tallest person was chosen, and from these 30 people, the shortest was chosen. Then, from each column, the shortest person was chosen, and from these 10 people, the tallest was chosen. Who will be taller: the tallest of the shortest, ... | 97. a) Let $A$ be the shortest among the tall, $B$ be the tallest among the short. Compare them with $B$ - a person who stands in the same row as $A$ and in the same column as $B$. Since $A$ is the tallest in his row, he is taller than $B$, and since $B$ is the shortest in his column, he is shorter than $B$. Therefore,... | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 28,480 |
99. Let's consider two definitions of an easy test:
a) In each version, every problem was solved by at least one student.
b) In each version, at least one student solved all the problems.
Can a test be easy in the sense of definition a) and difficult in the sense of definition b)? | 99. Suppose that each student solved only one problem, but in such a way that every problem was solved by someone. In this case, the test will be difficult in the sense of a) and easy in the sense of b). | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 28,481 |
105. Is it true that there exists a number $C$ such that for all integers $k$ the inequality
$$
\left|\frac{k^{8}-2 k+1}{k^{4}-3}\right|<C ?
$$ | 105. Let's see how the expression $\left|\frac{k^{3}-2 k+1}{k^{4}-3}\right|$ behaves for large (in absolute value) values of $k$. Clearly, in the numerator, the term $k^{3}$ plays the main role, and in the denominator, $k^{4}$. Therefore, we can expect that for large values of $k$, our expression is approximately equal... | 2 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 28,485 |
107. a) Prove that every trap is a feeder.
b) Invent such a sequence and such an interval that would be a feeder for this sequence, but not a trap for it. | 107. a) Let the segment $[a, b]$ be a trap. This means that outside this segment, there can only be a finite number of terms of the sequence. If this segment were not a feeder, then inside it there would also be only a finite number of terms of the sequence. But the sequence as a whole has an infinite number of terms. ... | proof | Logic and Puzzles | proof | Yes | Yes | olympiads | false | 28,487 |
109. Does there exist a sequence for which each of the intervals $[0,1]$ and $[2,3]$ is:
a) a feeder?
b) a trap? | 109. a) Sequence
$$
1 ; 3 ; \frac{1}{2} ; 2 \frac{1}{2} ; \ldots ; \frac{1}{n} ; 2 \frac{1}{n} ; \ldots
$$
b) Such a sequence does not exist. Indeed, suppose that for some sequence, the interval $[0,1]$ is a trap. Then there can only be a finite number of terms of the sequence outside this interval. This means that o... | proof | Other | math-word-problem | Yes | Yes | olympiads | false | 28,488 |
111. a) Does there exist a sequence that has no feeder?
b) $*$ Does there exist a sequence for which any segment is a feeder?
## § 2. Problems related to the definition of a limit
A number $a$ is called the limit of a sequence $\left\{x_{n}\right\}$ if for any positive number $\varepsilon$ (the Greek letter "epsilon"... | 111. a) The sequence $1 ; 2 ; 3 ; \ldots$; $n ; \ldots$ does not have any feeders, since in a segment of length $l$ there are no more than $l+1$ terms of the sequence.
b) We will now construct a sequence that contains all rational numbers. Since any segment contains an infinite set of rational numbers, any segment wil... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 28,490 |
113. a) Prove that if $x_{n} \rightarrow a$ as $n \rightarrow \infty$, then any interval centered at the point $a$ is a trap for the sequence $\left\{x_{n}\right\}$.
b) Is the converse statement true? | 113. a) Consider a segment centered at the point $\alpha$. The length of this segment is denoted by $2 \varepsilon$. By the definition of a limit, there exists a number $k$ such that for all $n>k$ the inequality $\left|x_{n}-a\right|<\varepsilon$ holds. This means that for all $n>k$ the term $x_{n}$ lies within our seg... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,491 |
114*. a) Prove that if $x_{n} \rightarrow a$ as $n \rightarrow \infty$, then every interval centered at point $a$ is a feeder, and no interval that does not contain point $a$ is a feeder for the sequence $\left\{x^{n}\right\}$.
b) It is known that for some sequence $\left\{x_{n}\right\}$, any interval centered at poin... | 114. a) Let a segment be given with its center at point $a$. Denote the length of the segment by $2 \varepsilon$. By the definition of a limit, there exists a number $k$ such that for $n > k$ the inequality $\left|x_{n} - a\right| < \varepsilon$ holds. This means that for $n > k$ the term $x_{n}$ is closer to point $a$... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,492 |
118. A number $a$ is called a limit point of the sequence $\left\{x_{n}\right\}$, if for any positive number $\varepsilon$ and any number $k$ there exists a number
$$
n>k
$$
such that the inequality
$$
\left|x_{n}-a\right|<\varepsilon
$$
holds.
a) Prove that if $a$ is a limit point of the sequence $\left\{x_{n}\ri... | 118. a) Let's show that any segment centered at point $a$ is a feeding ground. Suppose the length of the segment is $2 \varepsilon$. We need to show that infinitely many terms of the sequence fall within this segment, i.e., satisfy the inequality $\left|x_{n}-a\right| \leqslant \varepsilon$. Assume this is not the case... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,494 |
120. For each of the following sequences, specify all its limit points:
a) $x_{n}=\frac{n+1}{n}$,
b) $x_{n}=(-1)^{n}$
c) $x_{n}=\sin n^{\circ}$,
d) $x_{n}=n^{(-1)^{n}}$,
e) $x_{n}=n$,
f) $\frac{1}{2} ; \frac{1}{3} ; \frac{2}{3} ; \frac{1}{4} ; \frac{2}{4} ; \frac{3}{4} ; \frac{1}{5}$; $\frac{2}{5} ; \frac{3}{5} ; \... | 120. a) The sequence $x_{n}=\frac{n+1}{n}$ has the number 1 as its limit. Therefore (see problem 119), the number 1 is a limit point for this sequence. The sequence has no other limit points (see problems 114a) and 118a)).
b) The points +1 and -1 are clearly limit points for the sequence $x_{n}=(-1)^{n}$. For any othe... | notfound | Calculus | math-word-problem | Yes | Yes | olympiads | false | 28,496 |
122. a) Prove that if a sequence has a limit, then it is bounded.
b) Is the converse statement true? | 122. a) Let the sequence $\left\{x_{n}\right\}$ have a limit $x_{0}$. Take any interval $[a, b]$ centered at the point $\boldsymbol{x}_{0}$. Since this interval is a trap for the sequence $\left\{x_{n}\right\}$ (see problem 113a)), only a finite number of terms of the sequence lie outside this interval. Let $x_{k}$ be ... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,497 |
123. It is said that the sequence $\left\{x_{n}\right\}$ tends to infinity (this is written as: $x_{n} \rightarrow \infty$ as $n \rightarrow \infty$), if for any number $C$ there exists a number $k$ such that for all indices $n>k$ the inequality holds (Fig. 20):
$$
\left|x_{n}\right|>C .
$$
 Let $N$ be any positive number. For all $n>N$, the inequality $\left|x_{n}\right|>N$ holds. Therefore, the sequence $\left\{x_{n}\right\}$ tends to infinity.
b) The same solution as in case a).
c) The sequence is not bounded, since for any positive number $C$ there exists a number $n$ such that $\left|x_{n}\ri... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 28,498 |
128. for $\varepsilon>0$ there exists $k$ such that for $n>k \quad\left|x_{n}-a\right|<\varepsilon$. | 128. If the sequence $\left\{x_{n}\right\}$ or some of its subsequences do not have a greatest element, then as shown in the solution to problem 127, it is possible to extract an infinite increasing subsequence from $\left\{x_{n}\right\}$. Now consider the case where every subsequence has a greatest element. Let $x_{n_... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,502 |
130. for $\varepsilon>0$ there exists $k$ such that for $n>k\left|x_{n}-a\right|<\varepsilon$. | 130. Let a bounded sequence be given. We select from it an infinite monotonic subsequence (see the solution to problem 128). This subsequence is bounded and, by the Bolzano-Weierstrass axiom, has a limit. We will show that this limit is a limit point of the original sequence.
Indeed, any interval centered at this poin... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,504 |
132. for $\varepsilon>0$ there exists $k$ such that for $n>k \quad\left|x_{n}-a\right|<\varepsilon$. | 132. a) We need to prove that for any positive number $\varepsilon$, there exists a number $k$ such that $\left|x_{n}+y_{n}-a-b\right| < \varepsilon$ for all $n > k$. Since by the condition $\lim _{n \rightarrow \infty} x_{n}=a$, there exists a number $k_{1}$ such that $\left|x_{n}-a\right| < \frac{\varepsilon}{2}$ for... | proof | Calculus | proof | Yes | Yes | olympiads | false | 28,506 |
134. for $\varepsilon>0$ there exists $k$ such that for $n>k \quad\left|x_{n}-a\right|<\varepsilon$. | 134. Using the results of problem 133, we can write:
a) $\begin{aligned} & \lim _{n \rightarrow \infty} \frac{2 n+1}{3 n-5}=\lim _{n \rightarrow \infty} \frac{2+\frac{1}{n}}{3-\frac{5}{n}}= \\ &=\frac{\lim _{n \rightarrow \infty}\left(2+\frac{1}{n}\right)}{\lim _{n \rightarrow \infty}\left(3-\frac{5}{n}\right)}=\frac{2... | Calculus | proof | Yes | Yes | olympiads | false | 28,507 | |
147. Prove that the sequence
$$
2 ; 2+\frac{1}{2} ; 2+\frac{1}{2+\frac{1}{2}} ; 2+\frac{1}{2+\frac{1}{2+\frac{1}{2}}} ; \ldots
$$
has a limit, and find it. | 147. The sequence $\left\{x_{n}\right\}$ satisfies, obviously, the relation $x_{n+1}=$ $=2+\frac{1}{x_{n}}$. Suppose the sequence $\left\{x_{n}\right\}$ has a limit $a$. Then the left side of the equality tends to the number $a$, and the right side to the number $2+\frac{1}{a}$ (see problem 132). We thus obtain the equ... | 1+\sqrt{2} | Algebra | proof | Yes | Yes | olympiads | false | 28,515 |
148. To compute the square root of a positive number $a$, one can use the following method of successive approximations. Take any number $x_{0}$ and construct a sequence according to the following rule:
$$
x_{n+1}=\frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)
$$
Prove that if $x_{0}>0$, then $\lim _{n \rightarrow \in... | 148. First, let's prove that if the limit of $\{x_{n}\}$ exists, then it equals $\pm \sqrt{a}$. Indeed, let $\lim _{n \rightarrow \infty} x_{n}=b$. Then $\lim _{n \rightarrow \infty} \frac{1}{2}\left(x_{n}+\frac{a}{x_{n}}\right)=\frac{1}{2}\left(b+\frac{a}{b}\right)$. We obtain the equation $b=\frac{1}{2}\left(b+\frac{... | 2 | Algebra | proof | Yes | Yes | olympiads | false | 28,516 |
3. Prove that
a) if $A$ and $B$ are interior points of a convex figure $\Phi$, then all points of the segment $A B$ are interior points of $\Phi$;
b) if $A$ is an interior point and $B$ is a boundary point of a convex figure $\Phi$, then all points of the segment $A B$, except $B$, are interior points of $\Phi$;
c) ... | 3. а) Let $A$ and $B$ be two interior points of figure Ф. According to the definition of interior points, there exist two circles $C$ and $C'$ with centers at points $A$ and $B$, respectively, all points of which belong to the figure (Fig. 123, a). Let $MN$ and $PQ$ be the external common tangents of circles $C$ and $C... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,519 |
4. Prove that any line passing through an interior point of a convex figure intersects its boundary in no more than two points. If the convex figure is bounded, then every line passing through any of its interior points intersects the boundary of the figure exactly in two points. | 4. Let $\Phi$ be a convex figure, $O$ its interior point, and $l$ a line passing through point $O$. Since the line $l$ itself is a convex figure, the intersection of it with the figure $\Phi$ (according to problem 1) will be a convex figure located on the line (a one-dimensional convex figure), i.e., a segment, a ray, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,520 |
6. Prove that each of the two parallel supporting lines, the distance between which has the maximum possible value, contains a unique point of the boundary of the figure and that the segment connecting these points is perpendicular to both supporting lines (Fig. 13).
Fig. 133. $B$ divide the curve $K$ into two ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,526 |
11. (Inverse problem 10.) Prove that if any $n$ points of a bounded curve $K$ are vertices of some convex polygon, then this curve is convex. | 11. Suppose that the figure $F$, bounded by the curve $K$, satisfying the conditions of the problem, is not convex. Let's repeat the construction carried out in solving problem 5. Through the point $C$ and an arbitrary internal point
Fig. 27.
For non-convex figures, this sta... | 13. Let's denote our convex figures by the letters $\Phi_{0}, \Phi_{1}$, $\Phi_{2}$, and $\Phi_{3}$. Let $A_{0}$ be the point common to figures $\Phi_{1}, \Phi_{2}$, and $\Phi_{3}$, $A_{1}$ be the point common to figures $\Phi_{0}, \Phi_{2}$, and $\Phi_{3}$, $A_{2}$ be the point common to figures $\Phi_{0}, \Phi_{1}$, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,529 |
14. Helly's Theorem. On a plane, there are $n$ convex figures, any three of which have a common point. Prove that all $n$ figures have a common point.
Now, let's consider an infinite number of convex figures on a plane, any three of which have a common point. According to Helly's theorem, any finite number of these fi... | 14. We will prove the statement by mathematical induction. We have established (see problem 13) that if the number of figures is 4, the theorem is true. We will prove that if the theorem is true for $n$ figures, where $n \geqslant 4$, then it is also true for $n+1$ figures. Let $\Phi_{1}, \Phi_{2}, \ldots, \Phi_{n}, \P... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,530 |
21. a) Prove that if every three of $n$ half-planes have a common point, then all $n$ half-planes have a common point (Helly's theorem for half-planes).
b) Derive from the statement of problem 21a) that if every three of $n$ convex polygons (not necessarily bounded) have a common point, then all $n$ polygons have a co... | 21. a) We will conduct the proof by the method of mathematical induction. Suppose we have already proven that $k$ of our half-planes have a common point; we will show in this case that the $(k+1)$-th half-plane II intersects the common part of the first $k$ half-planes, i.e., in other words, that $k+1$ half-planes also... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,533 |
22. Prove that if every two of the given $n$ segments on a line have a common point, then all $n$ segments have a common point. | 22. Let the left endpoints of the given segments be denoted by $A_{1}$, $A_{2}, \ldots, A_{n}$, and the right endpoints of the same segments by $B_{1}$,
Fig. 154. $B_{2}, \ldots, B_{n}$. F... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,534 |
23. (Auxiliary problem.) Prove that if two bounded convex figures $\Phi_{1}$ and $\Phi_{2}$ on the plane do not intersect, then there exists a line $l$ such that $\Phi_{1}$ and $\Phi_{2}$ lie on opposite sides of $l$. | 23. Let $\Phi_{1}$ and $\Phi_{2}$ be two bounded convex figures with no common points, and let $A$ and $B$ be two points of these figures such that the distance between them is the smallest (Fig. 155) ${ }^{1}$. Draw lines $l_{1}$ and $l_{2}$ through points $A$ and $B$, perpendicular to the segment $A B$, and we will p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,535 |
24. Using the results of problems 22 and 23, prove that if every three of four given bounded convex figures on a plane have a common point, then all four figures have a common point.
From the statement of problem 24, the Helly's theorem for bounded convex figures (see the solution of problem 14) easily follows. | 24. Let's denote the given figures by $\Phi_{1}, \Phi_{2}, \Phi_{3}, \Phi_{4}$, and the convex figure that is the intersection of $\Phi_{1}, \Phi_{2}$, and $\Phi_{3}$ by $\Phi$ (see problem 1 from § 1). We need to prove that the figures $\Phi$ and $\Phi_{4}$ have a common point.
Suppose that $\Phi$ and $\Phi_{4}$ do n... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,536 |
25. Helly's Theorem for Space. In space, given $n$ bounded convex bodies, any four of which have a common point. Prove that all $n$ bodies have a common point.
Moreover, if in space there are infinitely many bounded convex bodies, any four of which have a common point, then it can also be asserted that all these bodie... | 25. We will only outline the stages of the proof, similar to the proof of Helly's theorem for the plane, as presented in the solutions to problems 22-24.
First, we prove that any two non-intersecting bounded convex bodies can be separated by some plane. Then, we prove that if any four of five bounded convex bodies $\P... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,537 |
28. a) On the plane, two convex figures $\Phi_{1}$ and $\Phi_{2}$ are given. Prove that there exists a line $l$ which simultaneously divides each of the figures $\Phi_{1}$ and $\Phi_{2}$ into two equal-area parts (Fig. 38, a).
 Let us choose in the plane a direction $O X$, which will serve as the origin of angular measurements; each direction $O Y$ will be defined by an angle $\alpha$, which
Fig. 160. this ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,540 |
34. a) Prove that any convex figure $\Phi$ can be enclosed within a centrally symmetric convex figure, the area of which is no more than twice the area of the figure $\Phi$.
b) Prove that a triangle cannot be enclosed in any centrally symmetric convex figure, the area of which would be less than twice the area of the ... | 34. a) Let us draw an arbitrary pair of parallel supporting lines $l$ and $l^{\prime}$ of the figure $\Phi$, which touch it at points $A$ and $B$ (which may not be unique; see Fig. 176). Now, draw supporting lines $m$ and $m^{\prime}$ parallel to line $AB$, touching the figure $\Phi$ at points $C$ and $D$ (which may al... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,543 |
37. Prove that if $\Phi_{1}$ and $\Phi_{2}$ are non-parallel segments, then the sum $\Phi_{1}+\Phi_{2}$ is a parallelogram; if $\Phi_{1}$ and $\Phi_{2}$ are parallel segments, then the sum $\Phi_{1}+\Phi_{2}$ is a segment parallel to the given ones, the length of which is equal to the sum of the lengths of $\Phi_{1}$ a... | 37. Let $A_{0}$ be some fixed point of the segment $A_{1} A_{2}$. It is easy to see that arbitrary sums $A_{0}+B$, where $B$ runs through all points of the segment $B_{1} B_{2}$, fill a segment equal to and parallel to $B_{1} B_{2}$, with endpoints at points $A_{0}+B_{1}$ and $A_{0}+B_{2}$; from the fact that triangles... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,545 |
38. Prove that the sum of convex figures is a convex figure (or, equivalently, that the sum of convex curves is a convex curve). | 38. Let $\Phi_{1}$ and $\Phi_{2}$ be two convex figures, $\Phi=\Phi_{1}+\Phi_{2}$ - their Minkowski sum, $C$ and $D$ - two points of figure $\Phi$. By the definition of the sum of two figures, there exist points $A_{1}$ and $A_{2}$ of figure $\Phi_{1}$ (which may coincide) and points $B_{1}$ and $B_{2}$ of figure $\Phi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,546 |
40. Prove that when the origin of coordinates is changed and the summands are subjected to a parallel translation, the shape of the figure, which is the sum of $\Phi_{1}+\Phi_{2}$ (or the shape of the curve $K_{1}+K_{2}$), does not change; this sum is only subjected to a parallel translation as a result.
Thus, the sha... | 40. Let the curve $K_{1}$ be subjected to parallel translation first, while the curve $K_{2}$ and the origin $O$ remain unchanged. Let $OA$ be the segment laid off from the origin $O$, equal and parallel to the segment by which each point of the curve $K_{1}$ is moved during the parallel translation. Let $K_{1}^{\prime... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,548 |
43. Prove that the sum $K_{1}+K_{2}$ of two convex polygons $K_{1}$ and $K_{2}$ is itself a convex polygon. If the convex polygons $K_{1}$ and $K_{2}$ do not have parallel and identically directed sides (see above, p. 26), then the number of sides of the convex polygon $K_{1}+K_{2}$ is equal to the sum of the number of... | 43. Let $K_{1}$ and $K_{2}$ be two polygons without pairwise parallel and identically directed sides (Fig. 187, a). According to problem 41, the curve that is the sum of these polygons is the geometric locus of the sums of corresponding points. Let $l_{1}$ be a supporting line of the polygon $K_{1}$, which is an extens... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,550 |
44. Prove that if $K_{1}$ and $K_{2}$ are two convex curves, and $L_{1}$ and $L_{2}$ are polygons with pairwise parallel and identically oriented sides, circumscribed around these curves, then $L_{1}+L_{2}$ is a polygon circumscribed around the curve $K_{1}+K_{2}$ (Fig. 58). Does this statement remain true if the sides... | 44. Let \( P_{1} Q_{1} \) and \( P_{2} Q_{2} \) be parallel and equally directed sides of polygons \( L_{1} \) and \( L_{2} \), and \( A_{1} \) and \( A_{2} \) be the points of tangency with curves \( K_{1} \) and \( K_{2} \) (Fig. 188). According to the definition of the addition of convex curves given in problem 41, ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,551 |
45. Let the sum $K=K_{1}+K_{2}$ of two convex curves $K_{1}$ and $K_{2}$ contain a straight line segment $A B$ of length $a$. Prove that in this case, either one of the curves $K_{1}, K_{2}$ contains a straight line segment parallel to $A B$ of length $a$, or both curves $K_{1}$ and $K_{2}$ contain straight line segmen... | 45. From the definition of the addition of convex curves given in problem 41,
it follows that if the curve $K_{1}+K_{2}$ contains a straight line segment $AB$, then this segment represents t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,552 |
47. Prove that the width $h$ of the curve $K_{1}+K_{2}$ in some direction is equal to the sum of the widths $h_{1}$ and $h_{2}$ of the curves $K_{1}$ and $K_{2}$ in the same direction.
48*. Prove that any convex polygon can be represented as the sum of triangles and segments. | 47. Let $K_{1}$ be a convex curve, $l_{1}$ and $l_{1}^{\prime}$ its two parallel supporting lines, $A_{1}$ and $A_{1}^{\prime}$ their points of tangency with the curve; similarly, $K_{2}$ is another convex curve, $l_{2}$ and $l_{2}^{\prime}$ its two supporting lines parallel to the lines $l_{1}$ and $l_{1}^{\prime}$, a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,554 |
51. Let $A, B$ and $C$ be three convex curves. Denote the distance between curves $A$ and $B$ by $\overline{A B}$, the distance between curves $B$ and $C$ by $\overline{B C}$, and the distance between curves $A$ and $C$ by $\overline{A C}$. Prove that the following inequalities hold:
$$
\overline{A B}+\overline{B C} \... | 51. Let $P$ be an arbitrary point on the curve $C$ (Fig. 199). Since the distance from the curve $C$ to the curve $B$ does not exceed $\overline{B C}$
Fig. 199. (recall that $\overline{B C}... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 28,557 |
56. a) Prove that among all triangles with a given perimeter, the one with the greatest area is equilateral.
6) Prove that among all quadrilaterals with a given perimeter, the one with the greatest area is a square. | 56. a) Let \(ABC\) be a scalene triangle, with \(AB\) being its larger side (or one of the two larger sides).
Fig. 210.
The isosceles triangle \(ABC'\), which has the same perimeter as tria... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,560 |
62. Using the result of problem 59, prove that a circle has a greater area than any other figure of the same perimeter. | 62. Let $\Phi$ be an arbitrary convex figure, and $K$ be a circle. We need to prove that the ratio of the area of the circle $K$ to the square of its perimeter is greater than the ratio of the area of the figure to the square of its perimeter. The area and perimeter of $\Phi$ and $K$ are defined as the limits of the ar... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,564 |
63. Prove that a polygon that can be inscribed in a circle has a greater area than any other convex polygon with the same side lengths.
Problem 63 is a generalization of problem 58. It can be formulated in a form analogous to problem 59b): among all convex polygons,
 inscribed in a circle and some other convex polygon \( N \) with the same side lengths. Extend the polygon \( N \) to some figure \( \Phi \) by attaching segments of a circle to its sides, which are cut off by the sides of \( M \) from the circle \( K \) circumscribed around \( M \) (Fig.... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,565 |
65. a) Prove that among all convex figures Ф bounded by a line segment of length $a$ and an arc of length $l (l > a)$, the one with the largest area is a circular segment.
b) Prove that among all convex figures Ф bounded by a line segment of arbitrary length and an arc of length $l$, the one with the largest area is a... | 65. a) Consider a segment $S$ with a chord of length $a$ and an arc of length $l$, and some other figure $\Phi$ with a perimeter of $a+l$, the boundary of which contains a straight-line segment of length $a$. Complete the segment $S$ to form a circle $K$; attach another such segment, identical to the one that completes... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,566 |
78. Calculate the total length of the Reuleaux triangle and the area it encloses. Which is larger: the area enclosed by a circle or the area enclosed by a Reuleaux triangle of the same width? Also determine the measure of the internal angles at the vertices of the Reuleaux triangle. | 78. If the width of the Reuleaux triangle (Fig. 252) is $h$, then each of the three arcs has a length of $\frac{2 \pi h}{6}$, and the length of the entire curve is $\pi h$. The area of the Reuleaux triangle is equal to the area of three circular sectors of radius $h$ with a central angle of $60^{\circ}$ minus twice the... | 0.7048^{2},0.7854^{2},120 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,572 |
79. Construct a curve of constant width $h$, composed of five, seven, or any odd number of arcs of circles of radius $h$. What is the length of each of these curves?
Curves of constant width $h$, composed of arcs of circles of radius $h$, will play a significant role in the subsequent exposition (see pp. 98-102, the t... | 79. Let $A B C D E$ be a regular pentagon, the larger of whose diagonals is equal to $h$. From each vertex of the pentagon, we draw an arc of a circle with radius $h$, connecting two opposite vertices (Fig. 253). The resulting convex curve will be a curve of constant width, because of any two parallel supporting lines,... | \pi | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,573 |
80. Prove that the distance between two points on a curve of constant width $h$ cannot be greater than $h$. | 80. If the chord $AB$ of the curve $K$ of constant width $h$ had a length greater than $h$, then the distance between the supporting lines perpendicular to $AB$ would also be greater than $h$, which contradicts the definition of a curve of constant width. | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,574 |
83. Prove that any two diameters of a curve of constant width intersect inside or on the curve. If they intersect on the curve, then the point $A$ of their intersection is a corner point of the curve, and the external angle of the curve (see § 1, p. 24) at point $A$ is not less than the angle between the considered dia... | 83. Let two diameters $A D$ and $B C$ intersect outside the curve or be parallel (Fig. 255). Then they are sides of
Fig. 255. the inscribed quadrilateral $A B C D$. Since the sum of the ang... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,577 |
84. Prove that the only curve of constant width having a center of symmetry is a circle. | 84. Let $K-$ be a curve of constant width and $O$ - its center of symmetry. It is not difficult to see that all diameters of the curve $K$ pass through the point $O$. Indeed, if the diameter $A B$ of the curve $K$ does not pass through the point $O$, then the segment $A^{\prime} B^{\prime}$, symmetric to $A B$ with res... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,578 |
85. Prove that if a curve of constant width $h$ has a corner point, then some arc of this curve is an arc of a circle of radius $h$. Conversely, if some arc of a curve of constant width $h$ is an arc of a circle of radius $h$, then this curve has a corner point. | 85. Let $A$ be a corner point of a curve of constant width $h$, and $AM$ and $AN$ be the half-tangents to the curve at this point (Fig. 258). Draw segments $AB$ and $AC$ of length $h$, perpendicular to the rays $AM$ and $AN$, respectively. Finally, draw the arc $BC$ of a circle of radius $h$ with center at $A$. We will... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,579 |
86. Prove that the internal angle at a vertex $A$ of a curve of constant width cannot be less than $120^{\circ}$. The only curve of constant width that has a vertex with an internal angle of $120^{\circ}$ is the Reuleaux triangle.
 Let $ABCD$ be a rhombus, $MN$ and $PQ$ be two lines perpendicular to the diagonal $BD$ of the rhombus, the distance between which has a fixed value $h$ (Fig. 85).
a) Prove that the perimeter of the hexagon $AMNCQP$ does not depend on the position of the lines $MN$ and $PQ$.
b) Prove that the a... | 87. a) Let the perimeter of the rhombus $ABCD$ be denoted by $p$, the length of its diagonal $BD$ by $a$, and the heights of triangles $BMN$ and $DPQ$, dropped from vertices $B$ and $D$ respectively, by $h_1$ and $h_2$. Then the perimeter of the hexagon $AMNCQP$ (Fig. 260) is given by $p - (BM + BN - MN) - (DP + DQ - P... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,581 |
88. From considering $2^{n}$-gons with equal angles inscribed around an arbitrary curve $K$ of constant width $h$ and around a circle $O$ of diameter $h$, derive Barbier's theorem: All curves of constant width $h$ have the same length, equal to $\pi$.
$89^{*}$. By considering $3 \cdot 2^{n}$-gons with equal angles ins... | 88. Let $O$ be a circle of diameter $h$, and $K$ be an arbitrary curve of constant width $h$. We will prove that the perimeters of $2^n$-gons with equal angles inscribed around $O$ and $K$ are equal. We will use induction on $n$. First, it is clear that the squares inscribed around $O$ and $K$ have equal perimeters—the... | \pi | Geometry | proof | Yes | Yes | olympiads | false | 28,582 |
92. Prove that the sum of an arbitrary curve of constant width $h$ and the same curve rotated by $180^{\circ}$ is a circle of radius $h$. Derive from this statement a new proof of Barbier's theorem (see problem 88). | 92. Let $K$ be an arbitrary curve of constant width $h$, $K^{\prime}$ be the curve obtained by rotating $K$ by $180^{\circ}$ around the origin $O$ (the curve symmetric to $K$
relative to t... | \pi | Geometry | proof | Yes | Yes | olympiads | false | 28,584 |
93. (Inverse of problem 92.) Prove that if the sum of some curve $K$ and the curve $K^{\prime}$, obtained from $K$ by a $180^{\circ}$ rotation, is a circle, then $K$ is a curve of constant width.
From problems 92 and 93, it follows that curves of constant width can be defined as those curves which, when summed with th... | 93. Let $K$ be a convex curve such that the sum $K^{*}=$ $=K+K^{\prime}$ of the curve $K$ and the curve $K^{\prime}$, which is symmetric to $K$ with respect to some point $O$ (obtained by rotating $K$ by $180^{\circ}$ around $O$), is a circle of radius $h$. If the width of the curve $K$ in some direction is $l$, then t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,585 |
94. a) Prove that among all convex curves of diameter 1 (see above, p. 20) the curves of constant width 1 have the greatest length.
Fig. 87.
b) Let $AB$ and $PQ$ be two non-parallel chor... | 94. a) Let $K$ be some convex curve of diameter 1, $K^{\prime}$ be the curve obtained from $K$ by symmetry relative to some point, and $K^{*}=K+K^{\prime}$. Clearly, $K^{*}$ is a centrally symmetric convex curve, the diameter of which is 2, and the length is twice the length of the curve $K$ (see § 6, problems 75a), b)... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,586 |
99. a) From the four vertices of a square with side $h$, four circles are described, each with a radius equal to the side of the square. Prove that the curve bounding the figure obtained from the intersection of these four circles (see Fig. 100) is a $\Delta$-curve of height $h$. Calculate the length of this curve and ... | 99. a) Let $ABCD$ be a square, and $MNPQ$ be the curve specified in the problem (Fig. 282). Triangle $CDM$ is equilateral by construction, so $\angle MDC = 60^\circ$, $\angle MDA = 30^\circ$. Triangle $AMD$ is isosceles, therefore
\[
\angle DAM = \frac{180^\circ - 30^\circ}{2} = 75^\circ, \quad \angle BAM = 15^\circ;
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,590 |
104. Prove that no $\Delta$-curve can have a vertex with an internal angle less than $60^{\circ}$. The only $\triangle$-curve that has a vertex with an internal angle of $60^{\circ}$ is the 1-biangle. | 104. If the angle at point $A \perp$-curve $K$ does not exceed $60^{\circ}$, then there exists an equilateral triangle circumscribed around $K$, one vertex of which coincides with $A$ (Fig. 289, a). If $B$ is the point of tangency of the opposite side of triangle $T$ with $\Delta$-curve $K$, then the distance $A B$ can... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,593 |
105. Prove that the only $\rfloor$-curve that maps onto itself under a rotation about some point $O$ by $120^{\circ}$ is a circle. | 105. In proving this theorem, we will need the following statement. Let $T$ be an equilateral triangle, $h$ be its height, $O$ be an arbitrary internal point, and $h_{1}, h_{2}$, and $h_{3}$ be the lengths of the perpendiculars dropped from point $O$ to the sides of the triangle; then the sum $h_{1}+h_{2}+h_{3}$ is equ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,594 |
110. Prove that the sum of an arbitrary $\Delta$-curve $K$ and the curves $K^{\prime}$ and $K^{\prime \prime}$, obtained from it by rotations of $120^{\circ}$ and $240^{\circ}$ respectively, is a circle. Derive from this a new proof of Barbier's theorem for $\Delta$-curves (see problem 108 above). | 110. Let $K$ be an arbitrary $\Delta$-curve, $K^{\prime \prime}$ and $K^{\prime \prime}$ - curves obtained from $K$ by rotating around some point $O$ by angles of $120^{\circ}$ and $240^{\circ}$, respectively, and $T$ -
 Prove that if the sum of a curve $K$ and the curves $K^{\prime}$ and $K^{\prime \prime}$, obtained from $K$ by rotations of $120^{\circ}$ and $240^{\circ}$ respectively, is a circle, then $K$ is a $\Delta$-curve.
From problems 110 and 111, it follows that $\Delta$-curves can be defined as c... | 111. Let $K$ be a convex curve such that the sum $M=K+K^{\prime}+K^{\prime \prime}$, where $K^{\prime \prime}$ and $K^{\prime \prime}$ are obtained from the curve $K$ by rotating around some point $O$ by $120^{\circ}$ and $240^{\circ}$ respectively, is a circle of radius $h$. Let $L$ be an equilateral triangle circumsc... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,597 |
113. The heights of an equilateral triangle are extended beyond the corresponding vertices, and segments equal to the side of the triangle are laid off on their extensions. The vertices of the triangle are connected by arcs, the centers of which are the obtained points (see Fig. 107). Prove that all rectangles circumsc... | 113. Let $ABC$ be a given triangle, $P, Q, R$ be the centers of the arcs specified in the problem, and $K$ be the curve formed by these arcs (Fig. 304). We will determine the angles of this curve at the angular points $A, B, C$. Draw the altitude $CH$ of triangle $ABC$ from $C$ to the segment $AP$ and $BP$ (where $P$ i... | ^2(\pi+2\sqrt{3}-6) | Geometry | proof | Yes | Yes | olympiads | false | 28,599 |
107. Using the properties of parallel projection, prove that the three medians of a triangle intersect at one point. | 107. Let $A B C$ be an ironized triangle. Project parallel to the plane $\pi$, in which this triangle is congruent, onto another plane $\pi'$ so that the triangle $A B C$ transforms into an equilateral triangle $A^{\prime} B^{\prime} C^{\prime}$. By the properties of parallel projection, the midpoints of the sides of t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,602 |
108. Using the properties of parallel projection, prove that the line connecting the point of intersection of the extensions of the non-parallel sides of a trapezoid with the point of intersection of its diagonals bisects the bases of the trapezoid. | 108. Let $A B C D$ be an arbitrary trapezoid, $E$ the point of intersection of its lateral sides, and $F$ the point of intersection of its diagonals. Projecting parallel to the plane $\pi$, in which the trapezoid is located, onto another plane such that the triangle $A B E$ transforms into an isosceles triangle $A^{\pr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,603 |
109. On a plane, two parallel lines $l$ and $l_{1}$ are given.
a) Divide the segment $A B$ on line $l$ into two equal parts using only a ruler.
b) Through a given point $M$, draw a line parallel to lines $l$ and $l_{1}$ using only a ruler.
 We take an arbitrary point $E$ on the plane, not lying on the lines $l$ and $l_{1}$, and connect it with points $A$ and $B$ on line $l$. Let $D$ and $C$ be the points of intersection of $E A$ and $E B$ with line $l_{1}$, and $F$ be the point of intersection of lines $A C$ and $B D$ (see Fig. 267, a). The line $... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,604 |
110. Let $M, N$, and $P$ be points located on the sides $AB, BC$, and $CA$ of triangle $ABC$ and dividing these sides in the same ratio (i.e., $\frac{AM}{MB} = \frac{BN}{NC} = \frac{CP}{PA}$). Prove that:
a) the point of intersection of the medians of triangle $MNP$ coincides with the point of intersection of the medi... | 110. a) Let's project the triangle $ABC$ onto some plane $\pi'$ in parallel so that it transforms into an equilateral triangle $A'B'C'$; by the property of parallel projection, points $M, N, P$ will transform into points $M', N', P'$, respectively, such that $\frac{A'M'}{M'B'} = \frac{B'N'}{N'C'} = \frac{C'P'}{P'A'}$ (... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,605 |
112. On the sides $AB$, $BC$, and $AC$ of triangle $ABC$, points $M$, $N$, and $P$ are given. Prove that:
а) if points $M_{1}$, $N_{1}$, and $P_{1}$ are symmetric to points $M$, $N$, and $P$ with respect to the midpoints of the corresponding sides of triangle $ABC$ (Fig. 12, a), then triangles $MNP$ and
 The properties of $B$, the parallel projection, and Theorem 1 on page 21 are sufficient to prove the statement of the problem for an equilateral triangle $A^{\prime} B^{\prime} C^{\prime}$ (Fig. 270, a; compare with the solutions of problems 107 and 110). Let us denote the common segment of the segments $D^{\pr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,606 |
114. On the sides of triangle $ABC$, three parallelograms are constructed with the sides of the triangle as their diagonals and with the same direction of sides. Prove that the second diagonals of these parallelograms intersect at one point (see Fig. 13). | 114. We will design a parallel projection of drawing 13 from plane $\pi$ to another plane $\pi^{\prime}$ such that the angles $A^{\prime} M^{\prime} N^{\prime}$ and $A^{\prime} R^{\prime} S^{\prime}$, into which the angles $A M N \| A R S$ of the original drawing are transformed, are equal. [Triangles $A^{\prime} M^{\p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,608 |
118. a) On a plane, there are two lines $l_{1} \parallel l_{2}$ and a point $P$, not lying on either of them. Through $P$, a pair of lines is drawn, intersecting $l_{1}$ and $l_{2}$ at points $A$ and $C$, respectively, and $B$ and $D$ (see Fig. 24, a). Prove that the points of intersection of the lines $AD$ and $BC$ fo... | 118. a) Let the lines $l_{1} \| l_{z}$ intersect at point $Q$. Project the plane $\pi$ from Fig. 24, a onto another plane $\pi^{\prime}$ so that the line $P Q$ becomes the distinguished line
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,612 |
121. On a plane, two points $A$ and $B$ are given. How can they be connected by a straight line if the only ruler available is shorter than the distance $AB$ (Fig. 27)?
In solving construction problems, it is always assumed that any two points on a plane can be connected by a straight line; this implies that the perso... | 121. Here is one of the possible constructions: draw lines $l_{1}$ and $l_{2}$ through point $A$, forming a small angle, inside which point $B^{2}$ lies (Fig. 284; by sliding our short ruler along the line, we can, of course, draw as long a segment of the line as we like). Through $B$, draw two lines intersecting $l_{1... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,615 |
123. Desargues' Theorem. Prove that if two triangles \(ABC\) and \(A_1B_1C_1\) are positioned on a plane such that the lines \(AA_1\), \(BB_1\), and \(CC_1\) intersect at a single point \(O\), then the points of intersection of the lines \(AB\) and \(A_1B_1\), \(AC\) and \(A_1C_1\), \(BC\) and \(B_1C_1\) lie on a singl... | 123. Let's first prove the first statement contained in the problem. Let $P, Q, R$ be the points of intersection of the lines $B C$ and $B_{1} C_{1}, C A$ and $C_{1} A_{1}, A B$ and $A_{1} B_{1}$. Project the plane $\pi$ of Fig. 29 onto a new plane $\pi^{\prime}$ such that the line $Q R$ is
 Given two lines $l$ and $l_{1}$, points $A, B, C$ located on line $l$, and points $A_{1}, B_{1}, C_{1}$ located on line $l_{1}$. Prove that the points of intersection of the lines $A B_{1}$ and $A_{1} B$, $A C_{1}$ and $A_{1} C$, $B C_{1}$ and $B_{1} C$ lie on the same line (see figure 33, a).
b) Given two poi... | 129. a) Let $P, Q$ be the points of intersection of the lines $A B_{1}$ and $A_{1} B, B C_{1}$ and $B_{1} C, C A_{1}$ and $C_{1} A$, and $O$ be the point of intersection of the lines $l$ and $l_{1}$. Project the plane $\pi$ of Fig. 33a onto a new plane $\pi^{\prime}$ such that the line $P Q$ becomes a line in the plane... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,623 |
130. a) Into which statement will the theorem of problem 129a) transform if the drawing 33, a is projected from the plane $\pi$ onto another plane $\pi^{\prime}$, so that the line $l_{1}$ becomes the distinguished line of the plane $\pi$?
b) On a plane, four lines are given, no two of which are parallel and no three in... | 130. a) Let $A, B, C$ be three points on a line $l$; through these points, lines $A C_{1} \| B C_{1}, \quad A B_{1} \| C B_{1}$ and $B A_{1} \| C A_{1}$ are drawn. In this case, the points of intersection of the lines $A B_{2}$ " $B A_{1}, A C_{1}$ " $C A_{1}, B C_{1}$ " $C B_{2}$ lie on one line (Fig. 292).
b) Let $l_... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,624 |
135. Let $A_{1} A_{2} A_{3} A_{4}$ be an arbitrary quadrilateral; $B_{1}, B_{2}, B_{3}, B_{4}$ - the points of intersection of its sides with the lines connecting point $N$ (the intersection of the diagonals of the quadrilateral) with points $P$ and $Q$ (the points of intersection of the opposite sides). The points of ... | 135. Let us project the quadrilateral \(A_{1} A_{2} A_{3} A_{4}\) into a square \(A_{1}^{\prime} A_{2}^{\prime} A_{3}^{\prime} A_{4}^{\prime}\) (this is possible according to Theorem 1, p. 60). In this case, points \(P\) and \(Q\) will pass to the "infinitely distant points" corresponding to the directions of the sides... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,629 |
139. Let $A B C D$ be a quadrilateral circumscribed around a circle $S$; $A_{1}, B_{1}, C_{1}$, and $D_{1}$ are the points of tangency of its sides with the circle $S$ (Fig. 57). Prove that:
a) the points of intersection of the diagonals of the quadrilaterals $A B C D$ and $A_{1} B_{1} C_{1} D_{1}$ coincide;
 there exist infinitely many triangles inscribed in the circle $S$ whose sides (or their extensions) pass through the point... | 140. Let's project the drawing of the problem onto a new plane $\pi^{\prime}$ such that the circle $S$ transforms into a new circle $S^{\prime}$, and the point $O$ transforms into the center $O^{\prime}$ of the circle $S_{2}$. In this case, the quadrilateral $A B C D$ will transform into a rectangle $A^{\prime} B^{\pri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,634 |
142. Given a circle $S$, a point $P$, and a line $I$ passing through $P$ and intersecting the circle at points $A$ and $B$; the intersection point of the tangents to $S$ at points $A$ and $B$ is denoted by $K$.
a) A variable line passes through $P$, intersecting $A K$ and $B K$ at points $M$ and $N$ (Fig. 58, a). Prov... | 142. a) Let's consider two possible cases.
$1^{\circ}$. Point $P$ is located outside $S$ (see Fig. $58, a$ in the text). We project the drawing of the problem onto another plane $\pi$
Fig.... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,636 |
144. Let $A A_{1}, B B_{1}$ and $C C_{1}$ be chords of a circle $S$, intersecting at a single point $O$, and let $X$ be an arbitrary point on the same circle. Prove that the points $P, Q$ and $R$, the intersections of the lines $X A_{1}, X B_{1}, X C_{1}$ with the sides $B C, C A, A B$ of triangle $A B C$, lie on a sin... | 144. The problem obviously makes no sense if point $O$ lies on circle $S$. Let's now consider two cases separately.
$1^{\circ}$. Point $O$ lies outside circle $S$ (see Fig. 59 in the text). Project Fig. 59 onto a new plane $\pi^{\prime}$ so that circle $S$ transforms into circle $S^{\prime}$,
.
Compare problem 147 with problem 153 of the following paragraph (p. 88). | 147. Consider a pentagon \(ABCDE\) inscribed in a circle (where points \(B, C, D, E\) are chosen arbitrarily). From Pascal's theorem, it follows that the point \(K\) of intersection of the tangent to the circle at point \(A\) and the side \(CD\) lies on the same circle with points \(L\) and \(M\) of intersection of the... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 28,640 |
150. Prove that if the distance from the center $O$ of the circle $S$ to the point $A$ is $d$, then the distance from $O$ to the polar $a$ of the point $A$ with respect to $S$ is $\frac{r}{d}$, where $r$ is the radius of $S$. | 150. Let point $A$ lie outside $S(d>r), AC$ and $AD$ be the tangents drawn from $A$ to $S, P$ be the point of intersection of $CD$ and $OA$ (Fig. 317, a). Clearly, $OA$ coincides with $CD$ (see above, p. 86). But from the similarity of triangles $OCA$ and $OPC$
. Consider the right trapezoids $O A P Q_{1}$ and $O B Q P_{1}$. Since $\angle A O Q_{1} = \angle B O P_{1}$, the sides are proportional: $\frac{O A}{O B} = \frac{O Q_{1}}{O P_{1}}$ (or from the result of the previ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 28,644 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.