problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
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346. Prove that for any natural number $n$ and any real number $a$, the inequality
$$
|a||a-1||a-2| \ldots|a-n| \geqslant\langle a\rangle \frac{n!}{2^{n}}
$$
holds, where $\langle a\rangle$... is the distance from the number $a$ to the nearest integer to it, $n!=1,2, \ldots, n$. | 346. Let $a_{0}, a_{1}, \ldots, a_{n}$ be the numbers $0,1,2, \ldots, n$ rearranged such that $|a| \leqslant\left|a-a_{0}\right| \leqslant\left|a-a_{1}\right| \leqslant \ldots \leqslant\left|a-a_{n}\right|$. For all $1 \leqslant k \leqslant n$, the inequalities $\left|a-a_{k}\right| \geqslant \frac{k}{2}$ hold, and mul... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,537 |
347. a) Do there exist polynomials
$$
P=P(x, y, z), \quad Q=Q(x, y, z), \quad R=R(x, y, z)
$$
in variables $x, y, z$ such that the identity
$$
(x-y+1)^{3} P+(y-z-1)^{3} Q+(z-2 x+1)^{3} R=1
$$
holds?
b) The same question for the identity
$$
(x-y+1)^{3} P+(y-z-1)^{3} Q+(z-x+1)^{3} R=1
$$ | 347. a) no: at $x=1, y=2, z=1$ the left side equals 0.
b) yes. Let $u=x-y+1, v=y-z-1$, $w=z-x+1$. In the equality $(u+v+w)^{7}=1$ we expand the brackets. After grouping, we get $u^{3} P+v^{3} Q+w^{3} R=1$. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,538 |
348*. The vertices of the tetrahedron KLMN lie inside, on the faces, or on the edges of another tetrahedron $A B C D$. Prove that the sum of the lengths of all the edges of the tetrahedron $K I M N$ is less than $4 / 3$ of the sum of the lengths of all the edges of the tetrahedron $A B C D$.
17th All-Union Olympiad, 1... | 348. Let the face $K L M$ of the tetrahedron $K L M N$ have the largest perimeter. Let $A_{1}, B_{1}, C_{1}, D_{1}$ be the projections of points $A, B, C, D$ onto the plane $K L M$ and let $\Gamma$ be the broken line bounding the projection of the tetrahedron $K L M N$ onto this plane. Let $P_{R S T Q}$ be the sum of t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,539 |
349. In the grid depicted in Fig. 15, each cell has a size of $1 \times 1$. Can this grid be represented as the union of the following sets: a) eight broken lines, each of which has a length of 5; b) five broken lines, each of which has a length of 8?
 No two broken lines should have common segments. Therefore, 12 nodes of the grid, located on the boundary of the square and different from its vertices, should be the ends of the broken lines, and 5 broken lines have a total of 10 ends. | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,540 |
350. Three integers were written on the board. Then one of them was erased and replaced with the sum of the other two remaining numbers, decreased by one. This operation was repeated several times, and as a result, the numbers 17, 1967, 1983 were obtained. Could the numbers originally on the board have been: a) $2,2,2 ... | 350. Answer: a) no; b) yes.
It is useful to reason "from the end": the triplet ( $17,1967,1983$ ) could only have been obtained from the triplet ( $5,3,3$ ), and the triplet $(5,3,3)$ results from the triplet ( $3,3,3$ ) and does not result from the triplet $(2,2,2)$. | )no;b)yes | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,541 |
351. Three circles touched each other externally at points $X, Y, Z$. Then the radii of these circles were increased by more than $2 / \sqrt{3}$ times, keeping the centers. Prove that every point of triangle $X Y Z$ is covered by at least one of the enlarged circles. | 351. Let $O$ be a point inside triangle XYZ. Connect it with the centers $O_{1}, O_{2}, O_{3}$ of the given circles and the points $X, Y, Z$. One of the six resulting angles (acute!) is not less than $60^{\circ}$ (for example, angle $O_{1} X A$). Then $A O < O X / \sin 60^{\circ} = \frac{2}{\sqrt{3}} O X$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,542 |
353. Find all solutions of the system of equations
$$
\left\{\begin{array}{l}
y^{2}=x^{3}-3 x^{2}+2 x \\
x^{2}=y^{3}-3 y^{2}+2 y
\end{array}\right.
$$ | 353. Answer: $(0 ; 0),(2+\sqrt{2} ; 2+\sqrt{2}),(2-\sqrt{2} ; 2-\sqrt{2})$. Subtracting the second equation from the first, we get $f(x)=f(y)$, where the function $f(x)=x^{3}-2 x^{2}+2 x$ is increasing ( $f^{\prime}(x)>0$ ).
$$
\begin{aligned}
& \text { 354. If } k \geqslant a \underbrace{44 \ldots 45}_{n-2}, \quad(a ... | (0;0),(2+\sqrt{2};2+\sqrt{2}),(2-\sqrt{2};2-\sqrt{2}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,544 |
356. Will the sequences $\left(a_{n}\right)$ and ( $\left.\beta_{n}\right)$, consisting respectively of the last digits of the integers $\left[(\sqrt{10})^{n}\right]$ and $\left[(\sqrt{2})^{n}\right]$, be periodic? (Here $[x]$- the integer part of the number $x$.) | 356. Answer: a) no; b) no.
a) The last digit of the number $\alpha_{2 k+1}$ is equal to the $k$-th digit after the decimal point in the decimal expansion of the number $\sqrt{10}$.
b) Let $\gamma_{n}=0$ if $\beta_{n}$ is even and $\gamma_{n}=1$ if $\beta_{n}$ is odd. Since $\gamma_{2 n+1}$ coincides with the $n$-th d... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,545 |
357. The magnitudes $\alpha$ and $\beta$ of two acute angles satisfy the equation $\sin ^{2} \alpha+\sin ^{2} \beta=\sin (\alpha+\beta)$. Prove that $\alpha+\beta=\pi / 2$. | 357. From the condition, it follows that $\sin \alpha(\sin \alpha-\cos \beta)=$ $=\sin \beta(\cos \alpha-\sin \beta)$. If $\sin \alpha>\cos \beta$ and $\cos \alpha>\sin \beta$, then $\sin ^{2} \beta+\cos ^{2} \beta<1$. The inequalities $\sin \alpha<\cos \beta, \cos \alpha<\sin \beta$ are also impossible. Therefore, $\s... | \alpha+\beta=\pi/2 | Algebra | proof | Yes | Yes | olympiads | false | 25,546 |
358. The vertices of the tetrahedron $ABCD$ are orthogonally projected onto two planes. Points $A_{1}, B_{1}, C_{1}$, $D_{1}$ and $A_{2}, B_{2}, C_{2}, D_{2}$ are the projections of the corresponding vertices. Prove that one of the planes can be moved in space so that the lines $A_{1} A_{2}$, $B_{1} B_{2}, C_{1} C_{2},... | 358. Let $\alpha$ and $\beta$ be given planes and $l$ be their line of intersection. It is sufficient to rotate the plane $\alpha$ around the line $l$ until it coincides with the plane $\beta$ (points $A_{1}, B_{1}, C_{1}, D_{1}$ rotate together with the plane $\alpha$). | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,547 |
362. Is it possible to arrange integers in the cells of an infinite grid paper such that in every rectangle of size $4 \times 6$ cells, with sides along the grid lines, the sum of the numbers is a) 10; b) 1? | 362. Answer: a) it is possible; b) it is possible.
a) Let's take two infinite sheets of graph paper and on diagonals, separated from each other by 4 cells, on the first sheet place ones, and in all other cells place zeros, on the second sheet ones are placed on diagonals separated by 6 cells, after which we add the nu... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,550 |
365. The lengths of two parallel sides of a rectangle are 1 cm. In addition, it is known that it can be divided into four rectangles by two perpendicular lines, three of which have an area of no less than $1 \mathrm{~cm}^{2}$, and the fourth no less than $2 \mathrm{~cm}^{2}$. What is the minimum length of the other two... | 365. Answer: $3+2 \sqrt{2}$. If $S_{1}, S_{2}, S_{3}, S_{4}$ are the areas of rectangles numbered clockwise, then $S_{1} S_{8}=$ $=S_{2} S_{4} \geqslant 2$, and $S_{1}+S_{2}+S_{3}+S_{4} \geqslant 3+2 \sqrt{2}$. | 3+2\sqrt{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,553 |
366. Inside triangle $ABC$, an arbitrary point $O$ is chosen. Prove that the following equality holds:
$$
S_{A} \cdot \overrightarrow{O A} + S_{B} \cdot \overrightarrow{O B} + S_{C} \cdot \overrightarrow{O C} = 0
$$
where $S_{A}, S_{B}, S_{C}$ are the areas of triangles $BCO$, $CAO$, and $ABO$ respectively. | 366. Let $\boldsymbol{e}_{1}, \boldsymbol{e}_{2}, \boldsymbol{e}_{3}$ be unit vectors in the same direction as vectors $\overrightarrow{O A}, \overrightarrow{O B}, \overrightarrow{O C} ; \angle B O C=\alpha, \angle C O A=\beta, \angle A O B=\gamma$. It is sufficient to prove that $e_{1} \sin \alpha+e_{2} \sin \beta+e_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,554 |
367. Prove that among any $2 m+1$ different integers, not exceeding $2 m-1$ in absolute value, one can find three numbers whose sum is 0. | 367. Induction on $m$: for $m=1$ the statement is true. Let it be true for $m=k-1$ ($k \geqslant 2$). Consider any set $A$ consisting of $2 k+1$ numbers, not exceeding $2 k-1$ in absolute value. If among them there are $2 k-1$ numbers not exceeding $2 k-3$ in absolute value, then everything is clear. Otherwise, we can ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,555 |
369. The set $M$ consists of $k$ pairwise non-intersecting segments lying on the same line. It is known that any segment of length not greater than 1 can be placed on the line so that its ends belong to the set $M$. Prove that the sum of the lengths of the segments that make up $M$ is not less than $1 / k$. | 369. If on a line there are non-intersecting segments $\Delta_{1}$ and $\Delta_{2}$ with lengths $\alpha$ and $\beta$, then the set of lengths of segments with endpoints belonging to $\Delta_{1}$ and $\Delta_{2}$ fills some segment of length $\alpha+\beta$. Therefore, if $\delta_{1}, \delta_{2}, \ldots, \delta_{k}$ are... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,557 |
371. a) The product of some $n$ integers is equal to $n$, and their sum is zero. Prove that the number $n$ is divisible by 4.
b) Let $n$ be a natural number divisible by 4. Prove that there exist $n$ integers whose product is equal to $n$, and their sum is zero. | 371. a) If $n$ is not divisible by 4, then among the factors there is only one even number, and therefore their sum is not equal to 0.
b) Answer: for $n=4k$, $n=2 \cdot(-2k) \cdot 1^{3k-2} \cdot(-1)^{k}$, if $k$ is even, and $n=(-2)(-2k) \cdot 1^{3k} \cdot(-1)^{k-2}$, if $k$ is odd. | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,558 |
375. Prove that for all real $x>0, y>0$ and for all real $\alpha$ the inequality
$$
x^{\sin ^{2} \alpha} \cdot y^{\cos ^{2} \alpha}<x+y
$$
holds. | 375. $x^{\sin ^{2} \alpha} \cdot y^{\cos ^{2} \alpha} \leqslant \max (x, y)<x+y$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,560 |
376. There is a cube and two paints: red and green. Two people play such a game. The one who starts chooses 3 edges of the cube and paints them red. His partner chooses 3 edges from those that are not yet painted and paints them green. After that, the one who starts again paints 3 edges red, and then his partner paints... | 376. Answer: incorrect. The partner only needs to paint any three pairwise intersecting edges green, and he will always be able to do this. | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 25,561 |
379. For which integers $m$ and $n$ does the equality
$$
(5+3 \sqrt{2})^{m}=(3+5 \sqrt{2})^{n}
$$
hold? | 379. Answer: $m=n=0$. If $(5+3 \sqrt{2})^{m}=(3+5 \sqrt{2})^{n}$, then $(5-3 \sqrt{2})^{m}=(3-5 \sqrt{2})^{n}$, but this is impossible, since $0<5-3 \sqrt{2}<1$ and $3-5 \sqrt{2}<0$. | =n=0 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,564 |
380. In a row, n different real numbers were written in increasing order. Under them, in the second row, the same numbers were written, but perhaps in a different order. For each pair of numbers written one under the other, their sum was calculated. These sums formed the third row. It turned out that the numbers in the... | 380. If the first row contains numbers $a_{1}<a_{2}<\ldots$ $\ldots<a_{m}<a_{m+1}<\ldots<a_{n}$, and the second row consists of numbers $a_{k_{1}}, a_{k_{2}}, \ldots, a_{k_{m}}, a_{k_{m+1}}, \ldots, a_{k_{n}}$, then the third row contains numbers $a_{1}+a_{k_{1}}, a_{2}+a_{k_{2}}, \ldots, a_{m}+a_{k_{m}}$. Let $a_{1} \... | proof | Algebra | proof | Yes | Yes | olympiads | false | 25,565 |
381. Given a triangle $A B C$. Through point $P$, lines $P A, P B, P C$ are drawn, intersecting the circumcircle of this triangle at points $A_{1}, B_{1}, C_{1}$, distinct from the vertices of the triangle. It turns out that triangle $A_{1} B_{1} C_{1}$ is congruent to triangle $A B C$. Prove that there are no more tha... | 381. Here it is convenient to use the lemma: if the chord $A B$ of some circle is fixed, and the chord $A_{1} B_{1}$ slides with its ends along this circle, then the point of intersection of the lines $A A_{1}$ and $B B_{1}$ describes a circle passing through the points $A$ and $B$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,566 |
383. The teacher wrote the quadratic trinomial $x^{2}+10 x+20$ on the board, after which each student in turn either increased or decreased by one either the coefficient of $x$, or the constant term, but not both at the same time. As a result, the quadratic trinomial $x^{2}+20 x+10$ was written on the board. Is it true... | 383. Answer: correct. Since $f(-1)=11$, and $g(-1)=-9$, at some point the root of the obtained polynomial will be -1. 384. Answer: $\pi r^{2}+2 p r-\left(p r^{2} / 2 R\right)$. | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,568 |
388. On a straight line, four different points are taken, denoted in order by the letters $A, B$, $C, D$. Prove that for any point $E$, not lying on the line $A D$, the inequality
$$
A E+E D+|A B-C D|>B E+C E
$$
holds. | 388. If point $M$ lies inside or on the side of triangle $P Q R \quad(M \neq P)$, then $M Q+M R<P Q+P R$. The initial inequality is sufficient to prove for the case $A B=C D$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,572 |
389. The sequence $x_{n}$ is defined recursively: $\quad x_{1}=1 ; x_{2}=1 ; x_{n+2}=x_{n+1}^{2}-\frac{1}{2} x_{n}, \quad$ if $n \geqslant 1$. Prove that the sequence $x_{n}$ has a limit, and find it.
390 *. In the white cells of a chessboard of size $1983 \times 1984$, numbers 1 or -1 are written such that for any bl... | 389. Answer: $\lim _{n \rightarrow \infty} x_{n}=0\left(\left|x_{n+7}\right| \leqslant \frac{1}{4}\left(\frac{5}{6}\right)^{n}\right.$ for $\left.n \geqslant 1\right)$. | 0 | Algebra | proof | Yes | Yes | olympiads | false | 25,573 |
392. Which of the numbers is greater:
$$
\frac{2}{201} \text { or } \ln \frac{101}{100} ?
$$ | 392. Answer: $\ln 1.01 > 2 / 201$. The function $y=\ln (1+x)-\frac{2 x}{x+2}$ is increasing for $x>0$. | \ln1.01>2/201 | Calculus | math-word-problem | Yes | Yes | olympiads | false | 25,575 |
393. On a plane, there are three circles $c_{1}, c_{2}, c_{3}$ with centers $C_{1}, C_{2}, C_{3}$ and radii $r_{1}, r_{2}, r_{3}$ respectively, such that each lies outside the other two, $r_{1}>r_{2} ; r_{1}>r_{3}, A$ - the point of intersection of the external tangents to circles $c_{1}$ and $c_{2}$ - lies outside
cir... | 393. Answer: $r=r_{1} r_{2} r_{3} /\left(r_{1} r_{2}-r_{2} r_{3}+r_{1} r_{3}\right)$. | \frac{r_{1}r_{2}r_{3}}{r_{1}r_{2}-r_{2}r_{3}+r_{1}r_{3}} | Geometry | proof | Yes | Yes | olympiads | false | 25,576 |
395. In an acute-angled triangle, perpendiculars are dropped from the midpoint of each side to the other two sides. Prove that the area of the hexagon bounded by these perpendiculars is half the area of the triangle. | 395. Let $A_{1}, B_{1}, C_{1}$ be the midpoints of the sides of a given triangle $A B C$, and let $O$ be the center of its circumscribed circle. The segments $O A_{1}$, $O B_{1}, O C_{1}$ divide the hexagon into parallelograms. Therefore, the area of the hexagon is twice the area of triangle $A_{1} B_{1} C_{1}$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,578 |
398. In a regular $n$-gon, it is required to paint each side and each diagonal with some color so that any two of these segments, having a common point, are painted differently. What is the minimum number of colors needed for this? | 398. Answer: $n$ colors. Let $A, B$ and $C$ be three consecutive vertices. Any two of the $n$ segments: $B A, B C, A C$ and $n-3$ diagonals emanating from $B$, have a common point. Next, paint in one of $n$ colors all sides and diagonals forming with side $A B$ the same angle $\frac{k \cdot 180^{\circ}}{n}$ for $k=0,1,... | n | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,581 |
405. The sequence $a_{1}, a_{2}, a_{3}, \ldots$ is defined by the rules: $a_{2 n}=a_{n}$ for $n \geqslant 1$ and $a_{4 n+1}=1, a_{4 n+3}=0$ for $n \geqslant 0$. Prove that this sequence does not have a period. | 405. If $T=2^{r} \cdot q$ ( $q$ - odd) is the period of the given sequence, then for $q=4 m+3$ and $k \geqslant p+2: \quad 1=a_{2} k=$ $=a_{2^{k}+T}=0$. Contradiction. If $q=4 m+1: 1=a_{2^{k}}=a_{2^{k}}{ }_{3 T}=$ $=0$, which also leads to a contradiction. | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,584 |
406*. On a plane, $n$ lines ($n>2$) are drawn, dividing the plane into several regions. Some of these regions are colored, with the condition that no two colored regions can touch along a boundary. Prove that the number of colored regions does not exceed $\frac{1}{3}\left(n^{2}+n\right)$. | 406. If all the lines are pairwise parallel, the statement is obvious. Let $m_{2}, m_{\mathrm{s}}, \ldots, m_{k}$ be the numbers of colored regions having $k$ sides. Then $m_{2} \leqslant n$. Moreover, each line is divided by the others into no more than $n$ parts, so the total number of parts of the lines is no more t... | \frac{1}{3}(n^{2}+n) | Combinatorics | proof | Yes | Yes | olympiads | false | 25,585 |
407. There is a cube, a cubic box with a lid of the same size, and six paints. Each paint is used to color one face of the cube and one of the faces of the box. Prove that the cube can be placed in the box in such a way that each face of the cube touches a face of the box painted with a different color. | 407. Let $A$ and $B$ be the colors of the bottom and lid of the box. Two opposite faces of the cube are painted in other colors $C$ and $D$. Place the cube in the box so that the face of color $D$ is at the bottom of the box, and the face of the cube of color $E$ touches the face of the sixth color $F$ of the box. | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,586 |
409. Students of the school mathematics club have built a computing machine that transforms a quadruple of numbers $(a, b, c, d)$ into the quadruple $(a-b, b-c, c-d, d-a)$ with the press of a button. Prove that if not all numbers in the initial quadruple are equal, then after a certain number of button presses, a quadr... | 409. If the quadruple ( $a_{n}, b_{n}, c_{n}, d_{n}$ ) is obtained from the one at step $n-\mathrm{M}$ ( $n \geqslant 1$ ), then $a_{n}+b_{n}+c_{n}+d_{n}=0$ and $a_{n}^{2}+b_{n}^{2}+c_{n}^{2}+d_{n}^{2} \geqslant$ $\geqslant 2\left(a_{n-1}^{2}+b_{n-1}^{2}+c_{n-1}^{2}+d_{n-1}^{2}\right)$. | proof | Algebra | proof | Yes | Yes | olympiads | false | 25,588 |
410. The numbers $1,2,3, \ldots, 2 n-1,2 n$ are divided into two groups of $n$ numbers each. Let $a_{1}b_{2}>\ldots>b_{n}$ be the numbers of the second group in descending order. Prove that
$$
\left|a_{1}-b_{1}\right|+\left|a_{2}-b_{2}\right|+\ldots+\left|a_{n}-b_{n}\right|=n^{2}
$$ | 410. Each term $\left|a_{k}-b_{k}\right|(k=1,2, \ldots, n)$ is the difference between a number greater than $n$ and a number not exceeding $n$. Therefore,
$$
\begin{aligned}
|a_{1}-b_{1}|+\ldots+| a_{n}-b_{n} | & =(n+1)+(n+2)+\ldots \\
& \ldots+2 n-(1+2+\ldots+n)=n^{2}
\end{aligned}
$$ | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,589 |
411. A parallelepiped is composed of identical cubes. Three faces of the parallelepiped, having a common vertex, were painted. It turned out that at least one face of half of all the cubes is painted. How many cubes have painted faces? | 411. Answer: the number of painted cubes can take one of five values: $60, 72, 84, 90, 120$. Let the parallelepiped have dimensions $m \times n \times k(k \leqslant n \leqslant m)$. The total number of unpainted cubes will be $(m-1)(n-1)(k-1)$. According to the condition $m n k=$
$-2(m-1)(n-1)(k-1), \quad$ from which i... | 60,72,84,90,120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,590 |
415. From a regular pentagon with a side of 1 cm, all points that are at a distance less than 1 cm from all vertices of the pentagon have been removed. Find the area of the remaining part. | 415. Answer: $\frac{5 \sqrt{3}}{4}-\frac{\pi}{6}$. | \frac{5\sqrt{3}}{4}-\frac{\pi}{6} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,593 |
416. On an infinite grid sheet with a cell side of 1, cuts are only allowed along the grid lines. Prove that for any integer $m>12$, it is possible to cut out a rectangle with an area greater than $m$, from which it is impossible to cut out a rectangle with an area of $m$. | 416. A rectangle satisfies the conditions if for its sides $x$ and $y (x \leqslant y)$ the inequalities $x y > m$ and $x(y-1)12$ this system has solutions: $x=k-1, y=k+2$ for $m=k^{2} ; x=k, y=k+1$ for $k^{2}<$ $<m<k(k+1) ; x=k-1, y=k+3$ for $m=k(k+1)$ and $\alpha=y=k+1$ for $k(k+1)<m<(k+1)^{2}$. | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,594 |
417. The lengths of the edges of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ are 1 cm. Find the smallest distance between the points of the circles, one of which is inscribed in the base of the cube $A B C D$, and the other passes through the vertices $A, C$, and $B_{1}$.
20th All-Union Olympiad,
1986 (Ulyanovsk)
| Cl... | 417. Answer: $(\sqrt{3}-\sqrt{2}) / 2$. The considered circles lie on two concentric spheres: one circumscribed around the cube and the other touching all its edges. The smallest distance is equal to the difference in the radii of these spheres. | \frac{(\sqrt{3}-\sqrt{2})}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,595 |
418. The roots of the equation $x^{2}+a x+b+1=0$ are natural numbers. Prove that $a^{2}+$ $+b^{2}$ is a composite number. | 418. Let $x_{1}$ and $x_{2}$ be the roots of the equation. Then $a^{2}+b^{2}=\left(x_{1}^{2}+\right.$ $+1)\left(x_{2}^{2}+1\right)$ | proof | Algebra | proof | Yes | Yes | olympiads | false | 25,596 |
419. Two identical squares intersect to form an octagon. The sides of one square are blue, and the other is red. Prove that the sum of the lengths of the blue sides of the octagon is equal to the sum of the lengths of its red sides. | 419. When the blue square is translated parallel to one of its sides, the sum of the blue sides of the octagon does not change, while when the centers of the blue and red squares coincide, the sum of the blue sides is equal to the sum of the red ones. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,597 |
421. In one country, the king wants to build $n$ cities and $n-1$ roads between them so that it is possible to travel from any city to any other. (Each road connects two cities, roads do not intersect and do not pass through other cities.) The king wants the shortest distances along the network of roads between pairs o... | 421. a) It is possible (Fig. 122).
b) It is not possible.
If the required network of roads exists, then one of the numbers $n$ or $n-2$ is a perfect square of an integer.
Choose some city $A$ and call it "good." Any city is called "good" if the length of the path from $A$ to $B$ is an even number, and "bad" if it is... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,599 |
422. Prove that on the coordinate plane, it is impossible to draw a convex quadrilateral where one diagonal is twice as long as the other, the angle between the diagonals is $45^{\circ}$, and the coordinates of each vertex are integers. | 422. The equality $\overrightarrow{A C} \cdot \overrightarrow{B D}=B D^{2} \sqrt{2}$ is impossible (the numbers $\overrightarrow{A C} \overrightarrow{B D}$ and $B D^{2}$ are integers). | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,600 |
424. Two circles, the distance between the centers of which is equal to $d$, intersect at points $P$ and $Q$. Through points $P, Q$, and point $A$ of the first circle (different from $P$ and $Q$), lines are drawn intersecting the second circle at points $B$ and $C$ respectively. a) Prove that the radius of the circle c... | 424. a) Let $O_{1}$ and $O_{2}$ be the centers of the circles and point $O_{3}$ such that $\mathrm{AO}_{1} \mathrm{O}_{2} \mathrm{O}_{3}$ is a parallelogram. Then $\mathrm{O}_{3}$ is the center of the circle circumscribed around triangle $A B C$.
b) Answer: all points on the circle with center $\mathrm{O}_{2}$ of the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,602 |
425*. On a plane, a regular hexagon is given. Each of its sides is divided into 1000 equal parts, and the division points are connected by segments parallel to the sides of the hexagon. Choose any three nodes of the resulting grid that form the vertices of an equilateral triangle (of any size and orientation), and colo... | 425. Number the nodes of the grid with the numbers $0,1,2$ such that: a) in the vertices of any
Fig. 123 small triangle, all three numbers are present, b) in the vertices of the hexagon, t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,603 |
427. Prove that for any positive numbers $a_{1}, a_{2}, \ldots, a_{n}$, the inequality
$$
\begin{aligned}
\frac{1}{a_{1}}+\frac{2}{a_{1}+a_{2}}+\ldots+\frac{n}{a_{1}+\ldots+a_{n}} & < \\
& <4\left(\frac{1}{a_{1}}+\ldots+\frac{1}{a_{n}}\right) .
\end{aligned}
$$
holds. | 427. We can assume that $a_{1} \leqslant a_{2} \leqslant \ldots \leqslant a_{n}$. The inequality now follows from the estimates:
$$
\begin{aligned}
& \frac{2}{a_{1}+a_{2}} \leqslant \frac{1}{a_{1}} \\
& \frac{2 k-1}{a_{1}+a_{2}+\ldots+a_{2 k-1}} \leqslant \frac{2 k-1}{a_{k}+\ldots+a_{2 k-1}} \leqslant \\
& \leqslant \... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,605 |
429. A cube with edge length $n, n \geqslant 3$, consists of $n^{3}$ unit cubes. Prove that it is possible to write an integer in each of these unit cubes such that all $n^{3}$ numbers are distinct, and the sums of the numbers in any row parallel to any edge of the cube equal zero. | 429. The required arrangement is obtained from the arrangement of all zeros by repeating the following two operations: a) swapping the positions of two layers of identical cubes, parallel to one of the cube's faces; b) adding to the numbers standing in the vertices of the original cube marked with a "+" sign, and simul... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,607 |
431. Inside a convex dodecagon, there are two points located 10 cm apart from each other. For each of these points, the sum of the distances from it to the vertices of the dodecagon was found. Prove that the obtained sums differ by less than $1 \mathrm{m}$. | 431. Let $O$ and $O^{\prime}$ be given points, and for definiteness, point $O^{\prime}$ lies inside or on the sides of triangle $O A_{1} A_{2}$. Then $O^{\prime} A_{1} + O^{\prime} A_{2} < O A_{1} + O A_{2}$ and $O^{\prime} A_{1} - O A_{i} \leqslant 10$ cm $(i=3,4, \ldots, 12)$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,609 |
432. Milk is poured into 30 glasses. A boy is trying to make the amount of milk equal in all glasses. For this, he takes any two glasses and pours milk from one to the other until the amount of milk in them is equal. Is it possible to pour milk into the glasses in such a way that the boy cannot achieve his goal, no mat... | 432. Answer: it is possible. You need to pour 100 g of milk into all the glasses except one, and into the remaining one - 200 g
256 | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 25,610 |
433. A certain rectangle is divided by lines parallel to its sides into squares with a side of 1, which are painted in a checkerboard pattern in white and black. The diagonal of the rectangle has broken into white and black segments. Find the ratio of the sum of the lengths of the white segments to the sum of the lengt... | 433. Answer: a) 1; b) $5000 / 4999$.
a) under symmetry relative to the center of the rectangle, the coloring transitions to the opposite.
b) it is necessary to consider the projections of the white and black segments onto one of the sides of the rectangle. | )1;b)\frac{5000}{4999} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,611 |
434. On the plane, a regular $n$-gon $A_{1} A_{2} \ldots A_{n}$ is given. a) Prove that if $n$ is an even number, then for any point $M$ on the plane, in the expression $\pm \overrightarrow{M A}_{1} \pm \overrightarrow{M A}_{2} \pm \ldots \pm \overrightarrow{M A}_{n}$, the signs plus and minus can be chosen such that t... | 434. a) Let $o$ be the center of the circle circumscribed around the polygon
the sign “+” for terms with even indices, the sign “-” for the others.
b) Let $\overrightarrow{M A}_{i_{1}}, \ove... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,612 |
435. The cells of a square table of size $n \times n$ $(n \geqslant 3)$ are filled with numbers $\pm 1$ according to the following rules:
1) in all boundary cells of the table, the number -1 is placed;
2) the number placed in the next unfilled cell of the table - which can be chosen arbitrarily - is equal to the produc... | 435. Answer: a) 1 when $n=3 ;(n-1)^{2}-1$ when $n \geqslant 4$; b) $n-2$. | (n-1)^2-1whenn\geqslant4, | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,613 |
436. Prove that for each natural $n$ the following inequality holds:
$|\sin 1|+|\sin 2|+\ldots+|\sin (3 n-1)|+|\sin 3 n|>8 n / 5$. | 436. It is sufficient to prove for all $x$ the inequality $|\sin x| + |\sin (x+1)| + |\sin (x+2)| > 8 / 5$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,614 |
437. Prove that the sum of all numbers of the form $\frac{1}{m^{n}}$, where $m, n$ are natural numbers and $1 \leqslant m<n \leqslant 1986$, is not an integer. | 437. Among the numbers from 1 to 1986, only two, $729=3^{6}$ and $1458=2 \cdot 3^{6}$, are divisible by $3^{8}$. Bringing the sum of all fractions, except $\frac{1}{729 \cdot 1458}$, to a common denominator, we get the fraction $\frac{a}{3^{1!} \cdot b}$, where b does not divide by 3. | proof | Number Theory | proof | Yes | Yes | olympiads | false | 25,615 |
438. A square and a triangle are circumscribed around a circle of radius 1. Prove that the area of the common part of the square and the triangle is greater than 3.4. Can we assert that this area is greater than 3.5?
$439^{*}$. A polynomial $P(x)$ is called admissible if all its coefficients are equal to $0, 1, 2$ or ... | 438. Any tangent cuts off a right-angled triangle with an acute angle $\varphi$ and an area of $1-\frac{2}{\cos \varphi+\sin \varphi+1} \leqslant(\sqrt{2}-1)^{2}$. The area $S$ of the common part of the square and the triangle satisfies the inequality $S \geqslant 4-3(\sqrt{2}-1)^{2}>3.4$. | S>3.4 | Geometry | proof | Yes | Yes | olympiads | false | 25,616 |
449. Find a set of five different natural numbers in which any two numbers are coprime, and any several numbers give a composite number when summed. | 449. Answer: 121, 241, 361, 481, 601. The condition is satisfied by any set of $n$ numbers $a_{i}=i \cdot n!+1$ ( $i=1,2, \ldots, n$ ). The sum of any $k$ numbers from this set is divisible by $\boldsymbol{k}$. | 121,241,361,481,601 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 25,626 |
450. Prove that if in a convex pentagon $A B C D E$ the equality $\angle A B C=\angle A D E$ and $\angle A E C=\angle A D B$ holds, then $\angle B A C=\angle D A E$. | 450. Let $F$ be the point of intersection of the diagonals $B D$ and $C E$. Points $A, F, D, E$ lie on one circle. Points $A, B, C, F$ also lie on one circle. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,627 |
451. Find all values of $\alpha$, for each of which the sequence
$$
\cos \alpha, \cos 2 \alpha, \cos 4 \alpha, \cos 8 \alpha, \ldots, \cos 2^{n} \alpha, \ldots
$$ | 451. Answer: $\alpha= \pm \frac{2 \pi}{3}+2 k \pi, k \in \boldsymbol{Z}$. If $\alpha$ satisfies the condition, then $\cos \alpha \leqslant-\frac{1}{4}$ (otherwise $\cos 4 \alpha>0$). Similarly, for any natural $n: \cos 2^{n} \alpha \leqslant-\frac{1}{4}$, hence $\left|\cos 2^{n} \alpha-\frac{1}{2}\right| \geqslant \fra... | \alpha=\\frac{2\pi}{3}+2k\pi,k\in\boldsymbol{Z} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 25,628 |
452. Positive numbers $a, b, c, A, B, C$ satisfy the conditions $a+A=b+B=c+C=k$. Prove that
$$
a B+b C+c A \leqslant k^{2}
$$ | 452. $k^{3}=(a+A)(b+B)(c+C)=a b c+A B C+k(a B+$ $+b C+c A)$. | proof | Inequalities | proof | Yes | Yes | olympiads | false | 25,629 |
453. In each cell of a square table $1987 \times$ $\times 1987$, a number is written, not exceeding 1 in absolute value. In any $2 \times 2$ square of this table, the sum of the numbers is 0. Prove that the sum of all the numbers in the table does not exceed 1987. | 453. Let's highlight the top-left corner cell of the table. Then consider the areas: the corner square $3 \times 3$ without this cell, the square $5 \times 5$ without the square $3 \times 3$, the square $7 \times 7$ without the square $5 \times 5$, and so on. The sum of the numbers in the cells of any of the constructe... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 25,630 |
454. The vertex $B$ of angle $A B C$ lies outside the circle, and the rays $B A$ and $B C$ intersect it. From the point $K$ of intersection of the ray $B A$ and the circle, a line perpendicular to the bisector of the angle is drawn, intersecting the circle at points $K$ and $P$, and the ray $B C$ at point $M$. Prove th... | 454. Let the bisector of angle $ABC$ and the line $l$, symmetric to it with respect to the center of the circle, intersect the line at points $L$ and $N$ respectively. Then $NP=KL=LM, PM=LN$, | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,631 |
455. Two players take turns writing natural numbers on a board, not exceeding $p$. The rules of the game prohibit writing divisors of already written numbers on the board. The player who cannot make the next move loses.
a) Determine which player has a winning strategy for $p=10$, and indicate it.
b) Determine which p... | 455. Answer: a); b): the beginner wins. a) For example, the first move is to write the number 6, after which the second can write one of the numbers $(4,5),(7,8),(9,10)$, which we have grouped into pairs. The first should, in response to any move by the second, write the other number from the same pair.
b) Consider a ... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,632 |
456. Every evening, Uncle Chernomor appoints 9 or 10 out of 33 bogatyrs for duty, at his discretion. What is the smallest number of days after which it can happen that each of the bogatyrs has been on duty the same number of times? | 456. Answer: 7 days. Let $k$ be the number of days when 9 bogatyrs were on duty, and $l$ be the number of days when 10 bogatyrs were on duty, with each of them being on duty $m$ times. Then $9 k + 10 l = 33 m$. For $m=1$ there are no solutions, but for $m=2$ we have $k=4$ and $l=3$. | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,633 |
460. The graph of the function \( y = f(x) \), defined on the entire number line, maps to itself when rotated by an angle around the origin.
a) Prove that the equation \( f(x) = x \) has exactly one solution.
b) Provide an example of such a function. | 460. a) If $f(0)=a$, then $f(-a)=0$ and $f(0)=-a$, so $f(0)=0$.
The equality $f\left(x_{0}\right)=x_{0}$ for $x_{0} \neq 0$ is impossible.
b) Answer: $f(x)=\left\{\begin{array}{ccc}0 & \text { if } & x=0, \\ -x / 2 & \text { if } & 4^{k} \leqslant|x|<2 \cdot 4^{k}, \\ 2 x & \text { if } & 2 \cdot 4^{k-1} \leqslant|x|... | proof | Algebra | proof | Yes | Yes | olympiads | false | 25,637 |
461. All faces of a convex polyhedron are triangles. Prove that each edge of this polyhedron can be painted red or blue so that in the end, from any vertex to any other vertex, one can get by moving only along red edges, as well as only along blue edges. | 461. Let $A$ be a vertex of a polyhedron and $A A_{1}, A A_{2}, \ldots, A A_{n}$ be the edges emanating from it. The edge $A A_{1}$ is painted blue, the other edges - red. All edges of the broken line $A_{2} A_{3} \ldots A_{n}$ are painted blue. The edge $A_{1} A_{2}$ - red. We sequentially add faces adjacent to the al... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,638 |
1.2. In figure a, 7 lines intersect at 7 points, in figure b- at 9 points. Can 7 lines intersect at 8 points? How many points of intersection can 7 lines have in general? (List all possible values.) | 1.2. There may be no points of intersection, there may be one or any number from 6 to 21. We provide an example with 8 points of intersection: 6 lines running along the sides and diagonals of the trapezoid, and one more line parallel to the bases and passing through
the point of intersection of the diagonals. Find the ... | 6to21 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,641 |
1.3. Replace the ellipsis with such a natural number $p$ so that the answer to the following question is unique: how many lines are drawn on the plane, if it is known that they intersect at ... different points? | 1.3. Answer. $p=2$.
We will prove that if there are two intersection points, there can only be three lines. Suppose there are only two intersection points $A$ and $B$, and let $l_{1}$ and $l_{2}$ be the lines intersecting at point $A$, and $l_{3}$ be a line not passing through $A$ (such a line exists, otherwise all li... | 2 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,642 |
1.8. Arrange eleven identical squares on a plane so that they do not overlap and the following condition is met: no matter how these squares are colored with three colors, there will always be at least two squares of the same color that share a common segment of their boundary.
## VOLLEYBALL NET AND SCISSORS | 1.8. One of the possible arrangements is shown in the figure. Let's prove by "reductio ad absurdum" that it satisfies the condition.
Assume that there exists a coloring with three colors such that squares sharing a common border are painted in different colors. Then any two squares having two common neighbors of diffe... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,647 |
1.9. What is the maximum number of strings connecting adjacent nodes in a volleyball net with square cells that can be cut so that the net does not fall apart into separate pieces? The size of the net is $10 \times 100$ cells.
## PARALLEL SEGMENTS | 1.9. Answer. 1000.
Suppose that so many strings have been torn that the net has not yet split into pieces, but no more strings can be torn. This means that there are no closed loops of strings left in the net. Prove that in this case, the number of unbroken strings is one less than the total number of nodes (including... | 1000 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 25,648 |
2.1. Through a point $M$, taken on the extension of the diagonal of a trapezoid, and the midpoint of each base, two lines are drawn, intersecting the lateral sides of the trapezoid at points $H$ and $K$. Prove that the segment $H K$ is parallel to the bases of the trapezoid. | 2.1. Let $P$ and $Q$ be the midpoints of the bases $BC$ and $AD$ of trapezoid $ABCD$ (see figure), $M$ be a point on the extension of diagonal $AC$, and $R$ be the intersection point of lines $MP$ and $AD$. Since triangles $AHR$ and $BPH$ are similar and $PC \parallel AR$, we have $AH: HB = AR: BP = AR: PC = AM: CM$. S... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,649 |
2.2. The midpoints of sides $A B$ and $C D$, $B C$ and $E D$ of a convex pentagon $A B C D E$ are connected by segments. The midpoints $H$ and $K$ of the resulting segments are then connected. Prove that segment $H K$ is parallel to segment $A E$ and equal to $1 / 4 A E$.
## BISECTORS OF A TRAPEZOID | 2.2. Prove the following statement in advance. The segments connecting the midpoints of consecutive sides of an arbitrary quadrilateral
Fig. 2.2. form a parallelogram. Applying it to quadril... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,650 |
2.3. In trapezoid $ABCD (BC \| AD)$, the bisectors of the internal angles $A$ and $B$ intersect at point $M$, and the bisectors of angles $C$ and $D$ intersect at point $N$. Prove that the length of segment $MN$ is equal to half the difference between the sum of the bases and the sum of the legs of the trapezoid.
HEXA... | 2.3. First, prove that triangles $A B M$ and $C N D$ are right triangles. Then, by connecting points $M$ and $N$ to the midpoints $K$ and $L$ of the corresponding lateral sides, show that segments $L N$ and $K M$ are parallel to the base $A D$ and, thus, points $K, M, N$ and $L$ lie on the midline of the trapezoid. Fin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,651 |
2.7. Perpendiculars are erected from points $K, E, H$, lying on the sides $AB, BC, CA$ of a given triangle $ABC$, respectively, to these sides. Prove that these three perpendiculars intersect at one point if and only if the condition
$$
A K^{2}+B E^{2}+C H^{2}=K B^{2}+E C^{2}+H A^{2}
$$
is satisfied.
POLICE CHECK AN... | 2.7. Suppose that the perpendiculars intersect at one point 0. Then the validity of the equality stated in the condition can be easily proven by applying the Pythagorean theorem to the right triangles $A O K$, $K O B, \quad B O E, \quad E O C, \quad C O H$ and
,
Fig. 2.9.
Then $... | ^{2}+b^{2}=2^{2} | Geometry | proof | Yes | Yes | olympiads | false | 25,657 |
2.10. A point is taken on each side of a parallelogram. Prove that if the area of the quadrilateral with vertices at these points is half the area of the parallelogram, then one of the diagonals of the quadrilateral is parallel to one of the sides of the parallelogram.
## REFLECTED QUADRILATERAL | 2.10. Let points $K, L, M$ and $N$ be taken respectively on the sides $A B, B C, C D$ and $D A$ of a parallelogram. Suppose that $L N$ is not parallel to $A B$. Notice that if point $K$ is fixed and $M$ moves along side $C D$ from $C$ to $D$, then the area of $K L M N$ changes monotonically: it either always increases ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,658 |
3.1. In the plane of a convex quadrilateral $A B C D$, there is a point $M$. Consider the convex quadrilateral with vertices at the points symmetric to point $M$ with respect to the midpoints of the sides of the given quadrilateral. Prove that its area is twice the area of $A B C D$.
## EQUILATERAL AGAIN | 3.1. Prove that if the midpoints of the sides of a quadrilateral $A B C D$ are connected, a parallelogram $P Q R S$ is formed, the area of which is half the area of $A B C D$. Note that the "reflected quadrilateral" constructed in the condition can be obtained from the parallelogram $P Q R S$ by a similarity transforma... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,659 |
3.2. On two adjacent sides $A B$ and $B C$ of the parallelogram $A B C D$, equilateral triangles $A B E$ and $B C F$ are constructed outside it. Prove that triangle $D E F$ is equilateral.
## FAMILY OF TRIANGLES | 3.2. Prove that triangles $A D E, C F D$ and $B F E$ (see figure) are equal: triangle $C F D$ is obtained from $B F E$ by a $60^{\circ}$ rotation around point $F$, and $A D E$ from the same $B F E$ - by a $60^{\circ}$ rotation around point $E$. | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,660 |
3.3. Consider the set of isosceles triangles whose bases lie on a given line, one of the vertices is at a given point $A$ on this line, and the radii of the inscribed circles are equal to a given value $r$. Prove that all the lateral sides of these triangles, not passing through vertex $A$, touch one circle.
## FIXED ... | 3.3. Let $A B C$ be one of the triangles mentioned in the condition (see figure). We will perform a similarity transformation (homothety) with the center at point $C$ and the similarity coefficient $k=2$. Under this transformation, triangle $A B C$ will transform into triangle $A_{1} B_{1} C$, and the incircle of trian... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,661 |
3.4. a) A rectangular map of a city is completely covered by a second map of the same city, the scale of which is 5 times larger, and such that their corresponding sides are parallel. Prove that it is possible, without moving the maps, to pierce them with a needle simultaneously so that the point of the puncture indica... | 3.4. a) Let's draw both maps on one diagram (see figure). We will get two similar rectangles with a similarity coefficient of $1 / 5$, one of which is located inside the other. Prove that the four lines $A A_{1}, B B_{1}, C C_{1}$, and $D D_{1}$ intersect at one point $O$ (the center of similarity of the rectangles $A ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,662 |
3.5. From point $O$, 12 rays $l_{1}, l_{2}, \ldots, l_{12}$ are drawn such that each pair of adjacent rays forms an angle of $30^{\circ}$. An ant sitting on ray $l_{1}$, at some moment, started crawling along the perpendicular to ray $l_{2}$. Upon reaching $l_{2}$, it turned and crawled along the perpendicular to ray $... | 3.5. Suppose that the ant started its journey at point $A$ and after the first turn ended up at point $B$ (on the same ray $l_{1}$, see figure). Check that $O B / O A=(\sqrt{3} / 2)^{12}=729 / 4096$.
Prove that the length of each subsequent turn is $729/4096$ of the length of the previous one. From this, it follows th... | \frac{4096}{3367} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,663 |
4.1. A parallelogram $A B C D$ was drawn on the board, and point $E$ - the midpoint of side $B C$ and point $F$ - the midpoint of side $C D$ were marked. The duty student erased the parallelogram, leaving only points $A, E$ and $F$. How can the drawing be reconstructed from these data?
MIDPOINTS OF THREE EQUAL SIDES | 4.1. Prove that if on the extension of segment $A K$, where $K$ is the midpoint of segment $E F$, segment $K C = A K / 3$ is laid off, then point $C$ will be a vertex of the parallelogram. Then vertices $B$ and $D$ are found (see figure).
, and $O$ be the center of the inscribed circle. Prove that the angle $AOC$ (containing point $B$ inside it) is equal to $270^{\circ} - \angle ABC$ (for this, connect point $O$
, the circle constructed on the base \(AC\) as a diameter passes through points \(E\) and \(F\) - the feet of the altitudes dropped from the vertices to the sides (see the figure). Using the data from the problem, i.e., knowing points \(E, F\) and the line \(l\) on which \(AC\) li... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,670 |
4.7. Construct a triangle if the midpoints of two of its sides and the line on which the bisector, drawn to one of these sides, lies are known.
48. A line and two points are given on one side of it. Construct a triangle whose base lies on the given line, the point of intersection of the medians lies at one of the give... | 4.7. Let a line $l$ be given, on which the bisector of angle $B$ lies, point $E$ - the midpoint of side $B C$, and point $F$ - the midpoint of side $A C$ of triangle $A B C$. It is not difficult to indicate one point that definitely lies on line $A B$: this is point $D$, symmetric to point $E$ with respect to $l$. More... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,671 |
4.9. Given a segment. a) Divide it into 4 equal parts by drawing no more than 6 lines - straight lines and circles. b) Divide it into 6 equal parts by drawing no more than 8 lines.
SMALL INSTRUMENTS | 4.9. One of the possible solutions to task b) is shown in the figure (the numbers on the lines indicate the sequence in which they were constructed). Task a) is significantly easier. | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,672 |
4.11. On a plane, there are three points $A, B$ and $C$, which can be covered by one coin. Construct the fourth vertex $D$ of the parallelogram $A B C D$ using the coin. (It is allowed to apply the coin to two existing points and trace it with a pencil.) | 4.11. First, prove the following statement: if through point O three different circles of the same radius $r$ are drawn, then the three points of pairwise intersection of these circles, different from $O$, are the vertices of a triangle, the radius of the circumscribed circle of which is equal to $r$, and point O is th... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,674 |
4.12. Using three pennies, find the center of a circle whose diameter is equal to the diameter of the penny. (In addition to the constructions described in the previous problem, it is allowed to apply one penny to two others.)
## PROVIDE ANGLE | 4.12. Let's place one of the coins on the circle (why is this possible?). By successively applying the other two coins to this coin so that they touch each other pairwise, we will construct the vertices of a regular hexagon \(A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}\), inscribed in the circle. By applying a coin externally ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,675 |
5.1. The altitudes of triangle $ABC$ intersect at point $O$. It is known that $OC = AB$. Find the angle at vertex $C$. | ## 5.1. Answer. $45^{\circ}$.
Let $D$ be the foot of the perpendicular dropped from point $A$. Prove that triangles $C O D$ and $A B D$ are congruent. Then show that $\angle O B D=45^{\circ}$, and from this, deduce that the required angle is also $45^{\circ}$.
Another solution can be obtained by almost verbatim repea... | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,676 |
5.2. A circle constructed on the leg of a right triangle as a diameter divides the hypotenuse in the ratio $1: 3$. Determine the angles of the triangle. | 5.2. Answer. $30^{\circ}, 60^{\circ}$.
Notice that the segment connecting the vertex of the right angle to the point of division will be the height of the triangle. | 30,60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,677 |
5.3. Find the angle $C$ of triangle $ABC$, if the distance from vertex $C$ to the orthocenter of the triangle is equal to the radius of the circumscribed circle. | 5.3. Answer. $60^{\circ}$.
1st solution. Let $O$ be the center of the circumcircle of triangle $ABC$, $M$ be the orthocenter, and $CC'$ be the diameter of the circumcircle (see figure a). Inscribed angles $ABC$ and $AC'C$ are equal, so the acute angles $MCB$ and $ACO$, which complement them to $90^{\circ}$, are also e... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,678 |
5.4. Find the angle $C$ of triangle $A B C$, if vertex ' $A$ is equidistant from the centers of the excircles *) tangent to sides $A B$ and $B C$.
*) Excircle - a circle that is tangent to one side of the triangle and the extensions of the other two sides. | 5.4. Answer. $\angle C=90^{\circ}$.
Let $\mathrm{O}_{2}$ be the center of the excircle that touches side $A B$ and the extensions of sides $C B$ and
Fig. 5.4. $C A, O_{1}$ be the center of ... | \angleC=90 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,679 |
5.5. In triangle $A B C$ with an angle of $120^{\circ}$ at vertex $A$, the angle bisectors $A A_{1}, B B_{1}$ and $C C_{1}$ are drawn. Find the angle $C_{1} A_{1} B_{1}$. | 5.5. Answer. $90^{\circ}$.
Let the length of side $AB$ be denoted by $c$, $AC$ by $b$, and $BC$ by $a$. First, we will prove that $AA_1 = \frac{bc}{b+c}$.
Fig. 5.5.
Extend $AA_1$ and draw... | 90 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,680 |
5.7. In an isosceles triangle \(ABC\) with lateral sides \(AB = BC\), the angle \(ABC\) is \(80^\circ\). Inside the triangle, a point \(O\) is taken such that the angle \(OAC\) is \(10^\circ\) and the angle \(OCA\) is \(30^\circ\). Find the angle \(AOB\). | 5.7. Answer. $70^{\circ}$.
Let $K$ be the point of intersection of the altitude dropped from vertex $B$ and the bisector of angle $OAB$ (see figure). First, show that point $K$ lies on the extension of $OC$. Then prove that triangles
 and \( K_{2} \), where the circles inscribed in triangles \( A B C \) and \( A D C \) touch the diagonal \( A C \) (see the figure), is
\[
\frac{1}{2}|(A D+B C)-(A B+C D)|
\]
Then, use the theorem about a quadrilatera... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,683 |
6.2. a) Points $A_{0}, A_{1}$, and $A_{2}$ lie at the vertices of an equilateral triangle inscribed in circle $O$. An arbitrary point $M$, lying on circle $O$, is connected by segments to points $A_{0}, A_{1}$, and $A_{2}$. Prove that one of these segments is equal to the sum of the other two.
b) Points $A_{0}, A_{1}$... | 6.2. a) Let $M$ lie on the arc $A_{1} A_{2}$ of the circle $O$ (figure a). Rotate the triangle $A_{1} M A_{2}$ around the vertex $A_{2}$ by $60^{\circ}$ so that the segment $A_{2} A_{2}$ lies on the segment $A_{0} A_{2}$. Prove that then the vertex $M$ will fall on the point $K$ lying on the segment $A_{0} M$. The tria... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,684 |
6.3. Six pennies lie on the table, forming a closed chain (i.e., the first penny touches the second, the second touches the third, and so on, the sixth touches the first). A seventh penny, also lying on the table, rolls without slipping along the outer side of the chain, touching each of the six pennies in the chain in... | 6.3. Answer. 4 turns.
From figure $a$, it can be seen that during the time the moving coin, depicted with a dashed line, rolls along the arc $\alpha$ of the stationary coin with center $O$, it rotates by an angle of $2 \alpha$: in this figure, $M^{\prime} A^{\prime}$ is the new position of the radius $M A$, the radii ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 25,685 |
6.4. Given a convex quadrilateral $A B C D$ and four circles, each of which touches one side of the quadrilateral and the extensions of the two adjacent sides. Prove that the centers of these circles lie on a single circle. | 6.4. Connect the centers of the circles sequentially and prove that the vertices of the resulting quadrilateral are located on the sides of the constructed quadrilateral.
For this, it is sufficient to note that the centers of the circles lie on the bisectors of the external angles of the given quadrilateral. The same ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,686 |
6.5. a) In a decagon \(A_{1} A_{2} \ldots A_{10}\) inscribed in a circle, the sides \(A_{1} A_{2}, A_{2} A_{3}, A_{3} A_{4}, A_{4} A_{5}\) are parallel to their opposite sides. Prove that the side \(A_{5} A_{6}\) is also parallel to its opposite side.
b) In a decagon \(A_{1} A_{2} \ldots A_{10}\) circumscribed around ... | 6.5. a) Angles $A_{2}$ and $A_{7}$ of the decagon are equal as angles with respectively parallel sides.
Fig. 6.5.
 be the diameter of the circumscribed circle with center $O$, and let points $H$ and $K$ be the projections of points $C$ and $A$ onto $BD$. Draw $OQ \perp BD$. By the theorem of the diameter perpendicular to a chord, $DQ = QB$.
Since the radii $AO$ and $O... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,688 |
6.7. In a circle, three equal chords $A B, B C$ and $C D$ are drawn. Prove that the diameter $B E$ intersects the line $A D$ at a point $F$ such that segments $A B$ and $A F$ are equal, and the line $C E$ bisects the segment $F D$.
TRIANGLE AND THREE CIRCLES | 6.7. Triangle $A B F$ (see figure) is isosceles $(A B=A F)$, since $\angle A B F=\angle F B C=\angle A F B$ (the first equality is true because points $A$ and $C$ are symmetric with respect to the diameter $B E$, and the second because $B C \| A D$). In triangle $F E D$, the segment of the line $E C$ is a bisector (sin... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,689 |
6.8. Let points $P, Q$, and $R$ lie on the sides $AB, BC$, and $AC$ of triangle $ABC$, respectively. Prove that the circumcircles of triangles $APR$, $BQR$, and $CRQ$ intersect at one point.
CIRCUMCIRCLE AND PARALLELOGRAM | 6.8. Consider the intersection point of two of the circles mentioned in the condition and prove that it lies on the third. To do this, use the theorem about the sum of opposite angles of a cyclic quadrilateral and its converse.
For this, use the theorem about the sum of opposite angles of a cyclic quadrilateral and it... | proof | Geometry | proof | Yes | Yes | olympiads | false | 25,690 |
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