problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
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VI OM - I - Task 2
A factory sends goods in packages of $ 3 $ kg and $ 5 $ kg. Prove that in this way, any whole number of kilograms greater than $ 7 $ can be sent. Can the numbers in this task be replaced by other numbers? | Every integer greater than $7$ can be represented in one of the forms $3k-1$, $3k$, $3k+1$, where $k$ is an integer greater than $2$. Since $3k - 1 = 3 (k - 2) + 5$, and $3k + 1 = 3 (k - 3) + 2 \cdot 5$, we conclude from this that every integer greater than $7$ can be represented in the form $3x + 5y$, where $x$ and $y... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,819 |
XIX OM - II - Problem 6
On a plane, $ n \geq 3 $ points not lying on a single straight line are chosen. By drawing lines through every pair of these points, $ k $ lines are obtained. Prove that $ k \geq n $. | We will first prove the theorem:
If $ n \geq 3 $ and $ A_1, A_2, \ldots, A_n $ are points in the plane not lying on a single line, then there exists a line that contains exactly two of these points.
Proof. From the assumption, it follows that some triples of points are non-collinear. If $ A_i $, $ A_k $, $ A_l $ is suc... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,820 |
XXX OM - III - Task 2
Prove that four lines connecting the vertices of a tetrahedron with the centers of the circles inscribed in the opposite faces have a common point if and only if the three products of the lengths of opposite edges of the tetrahedron are equal. | Let $ABCD$ be a given tetrahedron, and let points $P$ and $Q$ be the centers of the circles inscribed in the faces $ABC$ and $ABD$ (Fig. 15).
om30_3r_img_15.jpg
Let $R$ and $R'$ be the points of intersection of the line $AB$ with the lines $CP$ and $DQ$, respectively. Since the center of the circle inscribed in a trian... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,822 |
XLV OM - II - Problem 5
The circle inscribed in triangle $ABC$ is tangent to sides $AB$ and $BC$ of this triangle at points $P$ and $Q$, respectively. The line $PQ$ intersects the angle bisector of $\angle BAC$ at point $S$. Prove that this angle bisector is perpendicular to the line $SC$. | Let's denote the center of the inscribed circle in triangle $ABC$ by $I$, and the measures of angles $CAB$, $ABC$, and $BCA$ by $\alpha$, $\beta$, and $\gamma$ respectively. When $\beta = \gamma$, the theorem is obvious. Therefore, let's assume that $\beta \neq \gamma$. There are two possible cases: $\beta < \gamma$ or... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,823 |
LV OM - I - Task 10
Given is a convex polygon with an even number of sides. Each side of the polygon has a length of 2 or 3, and the number of sides of each of these lengths is even. Prove that there exist two vertices of the polygon that divide its perimeter into two parts of equal length. | Let the number of sides of the polygon be $2n$. Denote the vertices of the polygon in order as $A_1, A_2, A_3, \ldots, A_{2n}$. For $i = 1, 2, \ldots, 2n$ let
\[
f(i) = \text{the difference in the lengths of the parts into which the perimeter of the polygon is divided by the points } A_i \text{ and } A_{i+n}
\]
where... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,824 |
III OM - I - Problem 12
Prove that the area $ S $ of a quadrilateral inscribed in a circle with sides $ a $, $ b $, $ c $, $ d $ is given by the formula
where $ 2p = a + b + c + d $. | Marking the sides and angles of the quadrilateral as in Fig. 25, we have
and since in a cyclic quadrilateral $ A + C = 180^\circ $, therefore
The relationship between angle $ A $ and the sides of the quadrilateral can be found by calculating the length of $ BD $ from triangle $ ABD $ and from triangle $ BCD... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,827 |
XLIII OM - II - Problem 5
Determine the supremum of the volumes of spheres contained in tetrahedra with all altitudes no longer than $1$. | Suppose a sphere with center $ Q $ and radius $ r $ is contained within a tetrahedron with all altitudes no longer than $ 1 $. Let the volume of the tetrahedron be denoted by $ V $, and the areas of its faces by $ S_1 $, $ S_2 $, $ S_3 $, $ S_4 $; we arrange them such that $ S_1 \leq S_2 \leq S_3 \leq S_4 $.
The height... | \pi/48 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,828 |
XLI OM - I - Problem 4
From a chessboard with dimensions $8 \times 80$ squares, colored in the usual way, two squares of different colors have been removed. Prove that the remaining $62$ squares can be covered by 31 rectangles of dimensions $2 \times 1$. | Let $ R $ be the smallest rectangle containing both removed fields $ A $ and $ B $. Since these fields are of different colors and lie in opposite corners of the rectangle $ R $, the lengths of the sides of $ R $ are natural numbers of different parity. Without loss of generality, we can assume that the rectangle $ R $... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,830 |
XII OM - I - Problem 7
Given is a convex circular sector $ AOB $ (O - the center of the circle). Draw a tangent to the arc $ AB $ such that its segment contained in the angle $ AOB $ is divided by the point of tangency in the ratio $ 1 : 3 $. | Mech $ MN $ (Fig. 6) will be the sought segment, and $ P $ - its point of tangency with the given arc $ AB $. In triangle $ MON $, we are given the ratio of segments $ MP : PN =1:3 $, into which the height $ OP $ divides the base $ MN $, and the size of angle $ MON $. We will prove that these data determine the shape o... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,832 |
VII OM - I - Problem 11
In a triangle with sides $a$, $b$, $c$, segments $m$, $n$, $p$ tangent to the inscribed circle of the triangle have been drawn, with endpoints on the sides of the triangle and parallel to the sides $a$, $b$, $c$ respectively. Prove that | Let $2s$ denote the perimeter of the triangle, $P$ - its area, $r$ - the radius of the inscribed circle (Fig. 11).
The triangle cut off from the given triangle by the segment $m$ is similar to the given triangle, and its height relative to the side $m$ is equal to $h_a - 2r$, where $h_a$ denotes the height of the given... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,833 |
XXXVIII OM - I - Problem 7
Given are non-negative real numbers $ a_1, a_2, \ldots , a_n $, $ x_1, x_2, \ldots , x_n $.
We assume $ a = \sum_{i=1}^n a_i $, $ c = \sum_{i=1}^n a_ix_i $. Prove that | The inequality $ \sqrt{1+x^2} \leq 1+x $ is satisfied for all $ x > 0 $ (verification by squaring, which, when performed within the realm of non-negative numbers, yields an equivalent inequality to the original). When we substitute $ x = x_i $, multiply both sides by $ a_i $ ($ i = 1, \ldots , n $), and sum up, we obta... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,834 |
LI OM - III - Task 5
For a given natural number $ n \geq 2 $, find the smallest number $ k $ with the following property. From any $ k $-element set of fields of the $ n \times n $ chessboard, one can select a non-empty subset such that the number of fields of this subset in each row and each column of the chessboard ... | We will prove the following two statements:
From any set composed of $ m + n $ squares of a chessboard with dimensions $ m \times n $ ($ m,n \geq 2 $), one can select a non-empty subset having an even number of squares in each row and each column of the chessboard.
There exists a set of $ m+n-1 $ squares of a chessbo... | 2n | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,836 |
XXXV OM - I - Problem 5
Prove that there exists a multiple of the number $ 5^n $, whose decimal representation consists of $ n $ digits different from zero. | The number $5^n$ has in its decimal representation no more than $n$ digits: $5^n = c_0 + c_1 \cdot 10 + c_2 \cdot 10^2 + \ldots + c_{n-1} \cdot 10^{n-1}$, where some of the digits $c_i$ may be zeros. Suppose that $k$ is the smallest natural number for which $c_k = 0$. Consider the number $5^n + 10^k \cdot 5^{n-k}$. Thi... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,837 |
XXV - I - Problem 5
Prove that for every $ k $, the $ k $-th digit from the end in the decimal representation of the numbers $ 5, 5^2, 5^3, \ldots $ forms a periodic sequence from some point onwards. | Let $ a_n $ be the number formed by the last $ k $ digits of the number $ 5^n $. It suffices to prove that the sequence $ \{a_n\} $ is periodic. The number $ a_n $ is the remainder of the division of the number $ 5^n $ by $ 10^k $, i.e., $ 5^n = 10^k \cdot b_n + a_n $, where $ b_n \geq 0 $ and $ 0 \leq a_n < 10^k $. Th... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,838 |
XXXIV OM - I - Problem 12
A sphere is circumscribed around a tetrahedron $ABCD$. $\alpha, \beta, \gamma, \delta$ are planes tangent to this sphere at vertices $A, B, C, D$ respectively, where $\alpha \cap \beta = p$, $\gamma \cap \delta = q$, and $p \cap CD \neq \emptyset$. Prove that the lines $q$ and $AB$ lie in the... | Let $ S $ and $ E $ be the sphere circumscribed around the tetrahedron $ ABCD $ and the point of intersection of the lines $ p $ and $ CD $, respectively. Point $ E $ lies outside the segment $ CD $; it can be assumed that it belongs to the ray $ CD^\to $, as shown in Fig. 4. Since $ E \subset \alpha $, the line $ AE $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,839 |
IX OM - I - Problem 3
From triangle $ ABC $, three corner triangles have been cut off by lines parallel to the sides of the triangle and tangent to the inscribed circle of this triangle. Prove that the sum of the radii of the circles inscribed in the cut-off triangles is equal to the radius of the circle inscribed in ... | Let $ r $ denote the radius of the circle inscribed in triangle $ ABC $, $ S $ - its area, $ r_1 $, $ r_2 $, $ r_3 $ - the radii of the circles inscribed in the corner triangles, lying at vertices $ A $, $ B $, $ C $, respectively, and $ a $, $ b $, $ c $ - the sides of the triangle opposite these vertices, and $ h_a $... | r_1+r_2+r_3=r | Geometry | proof | Yes | Yes | olympiads | false | 1,841 |
XLVIII OM - III - Problem 3
The medians of the lateral faces $ABD$, $ACD$, $BCD$ of the pyramid $ABCD$ drawn from vertex $D$ form equal angles with the edges to which they are drawn. Prove that the area of each lateral face is less than the sum of the areas of the other two lateral faces. | We will prove the inequality
(the other two inequalities can be obtained analogously). Let $ K $, $ L $, $ M $ be the midpoints of the edges $ BC $, $ CA $, $ AB $, respectively. The segments $ DK $, $ DL $, $ DM $ form equal angles with the edges $ BC $, $ CA $, $ AB $, respectively; let this common angle be den... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,842 |
LII OM - III - Task 3
We consider the sequence $ (x_n) $ defined recursively by the formulas
where $ a $ and $ b $ are real numbers.
We will call the number $ c $ a multiple value of the sequence $ (x_n) $ if there exist at least two different positive integers $ k $ and $ l $ such that $ x_k = x_l = c $. Prove tha... | The sequence $ (x_n) $ satisfies the same recurrence relation as the Fibonacci sequence.
Let additionally $ F_0 = 0 $ and $ F_{-n} = (-1)^{n+1}F_n $ for $ n = 1,2,3,\ldots $. Then the equation $ F_{n+2} = F_{n+1} + F_n $ holds for any integer $ n $.
Let $ a = F_{-4001} $ and $ b = F_{-4000} $. Then $ x_n = F_{n-4002} ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,843 |
XXII OM - III - Task 3
How many locks at least need to be placed on the treasury so that with a certain distribution of keys among the 11-member committee authorized to open the treasury, any 6 members can open it, but no 5 can? Determine the distribution of keys among the committee members with the minimum number of ... | Suppose that for some natural number $ n $ there exists a key distribution to $ n $ locks among an 11-member committee such that the conditions of the problem are satisfied. Let $ A_i $ denote the set of locks that the $ i $-th member of the committee can open, where $ i = 1, 2, \ldots, 11 $, and let $ A $ denote the s... | 462 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,844 |
LIV OM - III - Task 6
Let n be a positive even integer. Prove that there exists a permutation $ (x_1,x_2, \ldots ,x_n) $ of the set $ \{1,2,\ldots,n\} $, satisfying for every $ i \in \{1,2,\ldots ,n\} $ the condition:
$ x_{i+1} $ is one of the numbers $ 2x_i $, $ 2x_i-1 $, $ 2x_i-n $, $ 2x_i-n-1 $,
where $ x_{n+1} =... | Let's adopt the notation:
We consider a directed graph with $ n $ vertices numbered $ 1,2,\ldots,n $, where from each vertex $ x $, two directed edges emerge: to vertex $ f(x) $ and to vertex $ g(x) $. The thesis reduces to the existence of a closed path that passes through each vertex exactly once (a Hamiltonia... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,845 |
XLVIII OM - I - Problem 12
A group of $ n $ people found that every day for a certain period, three of them could have lunch together at a restaurant, with each pair of them meeting for exactly one lunch. Prove that the number $ n $, when divided by $ 6 $, gives a remainder of $ 1 $ or $ 3 $. | Let $ m $ be the number of days during which the people in this group intend to go to lunch in threes. From $ n $ people, one can form $ \frac{1}{2}n(n-1) $ pairs. Each day, three pairs will meet for lunch, so $ \frac{1}{2}n(n- 1) = 3m $.
Each person will meet all the other $ n -1 $ people for lunch over the period. On... | n\equiv1orn\equiv3(\mod6) | Combinatorics | proof | Yes | Yes | olympiads | false | 1,846 |
XI OM - II - Problem 5
Given three distinct points $ A $, $ B $, $ C $ on a line and a point $ S $ outside this line; perpendiculars drawn from points $ A $, $ B $, $ C $ to the lines $ SA $, $ SB $, $ SC $ intersect at points $ M $, $ N $, $ P $. Prove that the points $ M $, $ N $, $ P $, $ S $ lie on a circle. | We will adopt the notation such that point $B$ lies between points $A$ and $C$, point $M$ lies on the perpendiculars to $SB$ and $SC$, point $N$ - on the perpendiculars to $SC$ and $SA$, and point $P$ - on the perpendiculars to $SA$ and $SB$.
If any of the lines $SA$, $SB$, $SC$ is perpendicular to a given line, then t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,848 |
XXXVII OM - III - Problem 5
In a chess tournament, $2n$ ($n > 1$) players participate, and any two of them play at most one game against each other. Prove that such a tournament, in which no three players play three games among themselves, is possible if and only if the total number of games played in the tournament d... | We need to prove the equivalence of the following two statements $A$ and $B$ (for given natural numbers $m$ and $n$, $n > 1$):
A. It is possible to organize a tournament with $2n$ participants such that
$A_1$) every player plays at most one game with each other player,
$A_2$) no trio of players plays three games am... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,850 |
LII OM - II - Problem 2
Points $ A $, $ B $, $ C $ lie on a straight line in that order, with $ AB < BC $. Points $ D $, $ E $ are vertices of the square $ ABDE $. The circle with diameter $ AC $ intersects the line $ DE $ at points $ P $ and $ Q $, where point $ P $ lies on the segment $ DE $. The lines $ AQ $ and $ ... | Angle $ APC $ is a right angle as an inscribed angle based on the diameter (Fig. 1). Hence
Furthermore, from the equality of arcs $ AP $ and $ CQ $, the equality of angles $ ACP $ and $ CAQ $ follows, hence
om52_2r_img_1.jpg
Triangles $ PAE $ and $ RBA $ are therefore congruent (angle-side-angle criterion), wh... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,851 |
XXIII OM - I - Problem 10
Given six points in space, not all in the same plane. Prove that among the lines determined by these points, there is a line that is not parallel to any of the others. | Let points $A_1, A_3, \ldots, A_6$ not lie in the same plane, and let $p_{ij}$ be the line containing points $A_i$ and $A_j$, where $\{i, j\} \ne \{s, t\}$. Suppose that each of the lines $p_{ij}$ is parallel to some line $p_{s,t}$. Since the number of lines $p_{ij}$ is $\binom{6}{2} = 15$, which is odd, there exists a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,856 |
XLVII OM - I - Zadanie 6
Dane są dwa ciągi liczb całkowitych dodatnich: ciąg arytmetyczny o różnicy $ r > 0 $ i ciąg geometryczny o ilorazie $ q > 1 $; liczby naturalne $ r $, $ q $ są względnie pierwsze. Udowodnić, że jeśli te ciągi mają jeden wspólny wyraz, to mają nieskończenie wiele wspólnych wyrazów.
|
Oznaczmy przez $ r_i $ resztę z dzielenia liczby $ q^i $ przez $ r\ (i = 1,2,3,\ldots) $. W nieskończonym ciągu $ (r_1,r_2,r_3,\ldots) $, którego wyrazy przybierają tylko skończenie wiele wartości, muszą wystąpić powtórzenia. Istnieją zatem różne numery $ i $, $ j $ takie, że $ r_i = r_j $. Różnica $ q^i-q^j $ jest wó... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,857 |
III OM - II - Task 2
Prove that if $ a $, $ b $, $ c $, $ d $ are the sides of a quadrilateral that can have a circle circumscribed around it and a circle inscribed in it, then the area $ S $ of the quadrilateral is given by the formula | If a quadrilateral with sides $a$, $b$, $c$, $d$ is inscribed in a circle, then the area $S$ is given by (see problem 27) the formula
in which
If such a quadrilateral is also circumscribed about a circle, then
From equation (2), it follows that in such a quadrilateral
Substituting these values into formula (1), we ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,858 |
XXXI - I - Problem 8
Find all subsets $ S $ of the set of rational numbers that satisfy the following conditions.
1. If $ a \in S $, $ b \in S $, then $ a + b \in S $.
2. If $ a $ is a rational number different from 0, then exactly one of the numbers $ a $ and $ -a $ belongs to $ S $. | From condition 2, it follows that the set $ S $ contains the number $ 1 $ or the number $ -1 $. Let's consider these two cases in turn.
Assume that $ 1 \in S $. Every natural number $ n $ is the sum of $ n $ terms, each equal to $ 1 $, so by condition 1, the number $ n $ belongs to $ S $. We will show that every positi... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,861 |
LVIII OM - I - Problem 4
For each natural number $ n\ge 3 $, determine the number of sequences $ (c_1,c_2,\ldots,c_n) $, where $ {c_i\in\{0,1,\ldots,9\}} $, with the following property:\break in every triple of consecutive terms, there are at least two terms that are equal. | Let $ z_n $ be the number of $ n $-term sequences having the desired properties (we will further call them {\it good} sequences). Let $ x_n $ be the number of good sequences $ (c_1,c_2,\ldots,c_n) $, whose last two terms $ c_{n-1} $ and $ c_n $ are equal, and let $ y_n $ denote the number of good sequences $ (c_1,c_2,\... | 5\cdot(4^n+(-2)^n) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,862 |
LIX OM - II - Task 4
In each field of a square table of size $ n \times n $, an integer is written. We can perform the following operation multiple times: We select any field of the table and decrease the number written in it by the number of adjacent fields (having a common side with the selected field), while increa... | Let $ P $ denote the diagonal of the table connecting its top-left corner with the bottom-right corner. We will prove that as a result of performing the described operation, the parity of the sum of the numbers located on the diagonal $ P $ (the fields lying on the diagonal $ P $ are shaded in Fig. 2) does not change.
... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,866 |
XXVI - II - Task 3
In a certain family, the husband and wife have made the following agreement: If the wife washes the dishes on a certain day, then the husband washes the dishes the next day. If, however, the husband washes the dishes on a certain day, then who washes the dishes the next day is decided by a coin toss... | If the husband washes the dishes on the $(n+1)$-th day, then either the husband washed the dishes on the $n$-th day (the probability of this event is $ \displaystyle p_n \cdot \frac{1}{2} $), or the wife washed the dishes on the $n$-th day (the probability of this event is $ 1- p_{n} $). Therefore, we have $ \displayst... | \frac{2}{3} | Algebra | proof | Yes | Yes | olympiads | false | 1,867 |
XV OM - II - Task 2
A circle is divided into four non-overlapping arcs $ AB $, $ BC $, $ CD $, and $ DA $. Prove that the segment connecting the midpoints of arcs $ AB $ and $ CD $ is perpendicular to the segment connecting the midpoints of arcs $ BC $ and $ DA $. | Let $ M $, $ N $, $ P $, $ Q $ denote the midpoints of arcs $ AB $, $ BC $, $ CD $, $ DA $ (Fig. 12).
The theorem will be proven when we show that the sum of the inscribed angles $ MNQ $ and $ PMN $ is $ 90^\circ $. The angle $ MNQ $ is subtended by the arc $ MAQ $ equal to the sum of the arcs $ MA $ and $ AQ $, and th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,870 |
XVIII OM - III - Problem 6
A sphere and a plane are given, with no points in common. Find the geometric locus of the centers of the circles of tangency with the sphere of those cones circumscribed around the sphere, whose vertices lie on the given plane. | We introduce the following notations: $ O $ - the center of a given sphere, $ r $ - its radius, $ M $ - the projection of point $ O $ onto a given plane $ \pi $, $ A $ - any point on the plane $ \pi $ different from $ M $, $ A_i $ and $ M $ - the centers of the circles of tangency of the cones circumscribed around the ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,871 |
I OM - B - Task 18
Provide a method for determining the least common multiple of the natural numbers $ 1, 2, 3, \dots , n $. | Let $ p $ denote the largest prime number contained in a given sequence of natural numbers $ 1, 2, 3, \dots, n $. If, for example, $ n=50 $, then $ p $ denotes the prime number 47.
To determine the least common multiple of the natural numbers $ 1, 2, 3, \dots, n $, one would need to factorize all these numbers into pri... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,872 |
LVI OM - I - Problem 5
Quadrilateral $ABCD$ is inscribed in a circle, and the circles inscribed in triangles $ABC$ and $BCD$ have equal radii. Determine whether it follows from these assumptions that the circles inscribed in triangles $CDA$ and $DAB$ also have equal radii. | The answer given to the question in the task is affirmative.
Let $ I $, $ J $ be the centers of the incircles of triangles $ ABC $ and $ BCD $ (Fig. 2).
om56_1r_img_2.jpg
om56_1r_img_3.jpg
From the equality of the radii of the incircles of triangles $ ABC $ and $ BCD $, it follows that lines $ BC $ and $ IJ $ are para... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,873 |
XL OM - II - Task 4
Given are integers $ a_1, a_2, \ldots , a_{11} $. Prove that there exists a non-zero sequence $ x_1, x_2, \ldots, x_{11} $ with terms from the set $ \{-1,0,1\} $, such that the number $ x_1a_1 + \ldots x_{11}a_{11} $ is divisible by 1989. | To each $11$-element sequence $(u_1, \ldots, u_{11})$ with terms equal to $0$ or $1$, we assign the number $u_1a_1 + \ldots + u_{11}a_{11}$. There are $2^{11}$ such sequences, which is more than $1989$. Therefore, there exists a pair of sequences $(u_1, \ldots, u_{11}) \neq (v_1, \ldots, v_{11})$, $u_i, v_i \in \{0,1\}... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,874 |
1. Let $\log _{2} x=m \in Z, m>0, \log _{6} y=n \in Z, n>0$.
Then $x=2^{m}, y=6^{n}$. As a result, we have
$$
\text { GCD }(x, y)=\text { GCD }\left(2^{m}, 6^{n}\right)=\text { GCD }\left(2^{m}, 2^{n} \cdot 3^{n}\right)=8=2^{3} .
$$
Case 1. $m \geq n$. Then $n=3, \quad y=6^{3}=216$,
GCD $\left(\log _{2} x, 3\right)... | Answer: 1) $x=8^{k}, k=1,2, \ldots ; y=216$;
2) $x=8, y=216^{s}, s=1,2, \ldots$ | 8^{k},k=1,2,\ldots;216\text | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,875 |
3. Since the right side of the equation $(x+y)^{2}=49(3 x+5 y)$ is divisible by 49, then $x+y=7 k$. Substituting this into the equation, we get $49 k^{2}=49(3 x+5 y)$ or $3 x+5 y=k^{2}$. Solving the system
$$
\left\{\begin{array}{c}
3 x+5 y=k^{2} \\
x+y=7 k
\end{array}\right.
$$
we find
$$
\left\{\begin{array}{l}
x=... | Answer: : $\left\{\begin{array}{l}x=\frac{k(35-k)}{2}, \\ y=\frac{k(k-21)}{2},\end{array} k \in Z\right.$. | {\begin{pmatrix}\frac{k(35-k)}{2},\\\frac{k(k-21)}{2},\end{pmatrix}k\inZ.} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,876 |
4. Each experience of throwing the circle corresponds to a point $M-$ the position of the center of the circle on the vertical segment $[A ; B]$ of length 1.
Let's introduce the following n... | Answer: $P(A)=\frac{2-\sqrt{2}}{2} \approx 0.29$. | P(A)=\frac{2-\sqrt{2}}{2}\approx0.29 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,877 |
6. Let's introduce the following notations: $S$ - the area of the trapezoid, $S_{1}$ - the area of triangle $ABC$, $S_{2}$ - the area of triangle $ACD$, $h$ - the height of the trapezoid, $\gamma=\frac{q}{p}$, $\mu=\frac{n}{m}$.
$ that satisfy the equation $(x-y)^{2}=25(2 x-3 y)$. | Answer: $\left\{\begin{array}{l}x=m(15-m), \\ y=m(10-m),\end{array} m \in Z\right.$. | (15-),(10-),\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,881 |
4. On a plane, an infinite number of parallel lines are drawn, each separated from the next by a distance of 1. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts such that the ratio of the areas (of the smaller to the larger p... | Answer: $P(A)=\frac{1}{2}$. | \frac{1}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,882 |
5. For what values of $a$ does the system of equations $\left\{\begin{array}{c}(x+2+2 \sqrt{2} \cos a)^{2}+(y-1-2 \sqrt{2} \sin a)^{2}=2 \\ (x-y+3)(x+y+1)=0\end{array}\right.$ have three
solutions? | Answer: $a_{1}=\frac{7 \pi}{12}+\pi k, a_{2}=\frac{11 \pi}{12}+\pi k, k \in Z$. | a_{1}=\frac{7\pi}{12}+\pik,a_{2}=\frac{11\pi}{12}+\pik,k\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,883 |
1. Natural numbers $x$ and $y$ are such that their LCM $(x, y)=3^{6} \cdot 2^{8}$, and GCD $\left(\log _{3} x, \log _{12} y\right)=2$. Find these numbers. | Answer: $x=3^{6}=729, y=12^{4}=20736$. | 3^{6}=729,12^{4}=20736 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,885 |
3. Find all pairs of integers $(x ; y)$ that satisfy the equation $(x+2 y)^{2}=9(x+y)$. | Answer: $\left\{\begin{array}{l}x=m(2 m-3), \\ y=m(3-m),\end{array} m \in Z\right.$. | (2-3),(3-),\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,886 |
4. On a plane, an infinite number of parallel lines are drawn, each separated from the next by a distance of 1. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts such that the ratio of the areas (of the smaller to the larger p... | Answer: $P(A)=\frac{2-\sqrt{3}}{2} \approx 0.13$. | \frac{2-\sqrt{3}}{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,887 |
5. For what values of $a$ does the system of equations $\left\{\begin{array}{c}(x-2-\sqrt{5} \cos a)^{2}+(y+1-\sqrt{5} \sin a)^{2}=\frac{5}{4} \text { have two solutions? } \\ (x-2)(x-y-3)=0\end{array}\right.$ | Answer: $a \in\left(\frac{\pi}{12}+\pi k ; \frac{\pi}{3}+\pi k\right) \cup\left(\frac{5 \pi}{12}+\pi k ; \frac{2 \pi}{3}+\pi k\right), k \in Z$. | \in(\frac{\pi}{12}+\pik;\frac{\pi}{3}+\pik)\cup(\frac{5\pi}{12}+\pik;\frac{2\pi}{3}+\pik),k\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,888 |
6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in the ratio $C N: N A=4$. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $2: 3$. A line is drawn through point $N$ and vertex $D$, intersecting the lateral side $A B$ at point $M$. What fraction of the area of the trapezoid is t... | Answer: $S_{\text {MBCN }}: S=124: 325$.
# | 124:325 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,889 |
3. Find all pairs of integers $(x ; y)$ that satisfy the equation $(2 x+3 y)^{2}=16(x-y)$. | Answer: 1$)\left\{\begin{array}{c}x=m(15 m \pm 4), \\ y=m(-10 m \pm 4)\end{array} ; 2\right)\left\{\begin{array}{c}x=(5 m+2)(3 m+2) \\ y=-2 m(5 m+2), m \in Z\end{array}\right.$. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,891 |
4. On a plane, an infinite number of parallel lines are drawn, each at a distance of 1 from each other. A circle with a diameter of 1 is randomly thrown onto the plane. Find the probability that a line intersecting the circle divides it into parts, the ratio of the areas of which (the smaller to the larger) does not ex... | Answer: $P(A)=\frac{2-\sqrt{2-\sqrt{3}}}{2} \approx 0.74$ | P(A)=\frac{2-\sqrt{2-\sqrt{3}}}{2}\approx0.74 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,892 |
5. For what values of $a$ does the system of equations $\left\{\begin{array}{c}(x-2-3 \cos a)^{2}+(y+2-3 \sin a)^{2}=1 \\ (y+2)(x+y)=0\end{array}\right.$ have a unique solution? | Answer: $a_{1}= \pm \arcsin \frac{1}{3}+\pi k, a_{2}=-\frac{\pi}{4} \pm \arcsin \frac{1}{3}+\pi k, k \in Z$. | a_{1}=\\arcsin\frac{1}{3}+\pik,a_{2}=-\frac{\pi}{4}\\arcsin\frac{1}{3}+\pik,k\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,893 |
1. Let $m$ be the number of coins in the treasure. Then the number of remaining coins $N$ in the morning is
$$
\begin{gathered}
N=\frac{2}{3}\left(\frac{2}{3}\left(\frac{2}{3}(m-1)-1\right)-1\right)=\frac{2^{3}}{3^{3}}(m-1)-\frac{2}{3}-\frac{2^{2}}{3^{2}} \\
N=\frac{2^{3}}{3^{3}}(m-1)-\frac{10}{9}=3 n \\
8(m-1)-30=81 ... | Answer: the first pirate 81 coins, the second pirate 60 coins, the third pirate 46 coins. | 81,60,46 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,895 |
3. For which natural numbers $n$ can the fraction $\frac{3}{n}$ be represented as a periodic decimal fraction of the form $0.1\left(a_{1} a_{2}\right)$ with a period containing two different digits? | Answer: $n=22 ; a_{1}=3, a_{2}=6$. | n=22;a_{1}=3,a_{2}=6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,897 |
4. Consider the set $M$ of integers $n \in[-100 ; 500]$, for which the expression $A=n^{3}+2 n^{2}-5 n-6$ is divisible by 11. How many integers are contained in $M$? Find the largest and smallest of them? | Answer: 1) 164 numbers; 2) $n_{\text {min }}=-100, n_{\text {max }}=497$. | 164 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,898 |
3. For which natural numbers $n$ can the fraction $\frac{4}{n}$ be represented as a periodic decimal fraction of the form $0.1\left(a_{1} a_{2} a_{3}\right)$ with a period containing at least two different digits? | Answer: $n=27 ; a_{1}=4, a_{2}=8, a_{3}=1$;
$
n=37 ; a_{1}=0, a_{2}=8, a_{3}=2
$ | n=27;a_{1}=4,a_{2}=8,a_{3}=1n=37;a_{1}=0,a_{2}=8,a_{3}=2 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,900 |
4. Consider the set $M$ of integers $n \in[-30 ; 100]$, for which the expression $A=n^{3}+4 n^{2}+n-6$ is divisible by 5. How many integers are contained in $M$? Find the largest and smallest of them. | Answer: 1) 78 numbers; 2) $n_{\min }=-29, n_{\max }=98$. | 78,n_{\}=-29,n_{\max}=98 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,901 |
3. For which natural numbers $n$ can the fraction $\frac{5}{n}$ be represented as a periodic decimal fraction of the form $0.1\left(a_{1} a_{2} a_{3}\right)$ with a period containing at least two different digits? | Answer: $n=27 ; a_{1}=8, a_{2}=5, a_{3}=1$;
$$
n=37 ; a_{1}=3, a_{2}=5, a_{3}=1
$$ | n=27;a_{1}=8,a_{2}=5,a_{3}=1orn=37;a_{1}=3,a_{2}=5,a_{3}=1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,903 |
4. Consider the set $M$ of integers $n \in[-50 ; 250]$, for which the expression $A=n^{3}-2 n^{2}-13 n-10$ is divisible by 13. How many integers are contained in $M$? Find the largest and smallest of them? | Answer: 1) 69 numbers; 2) $n_{\min }=-47, n_{\max }=246$. | 69,n_{\}=-47,n_{\max}=246 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,904 |
1. Solution. According to the problem, the sum of the original numbers is represented by the expression:
$$
\begin{aligned}
& \left(a_{1}+2\right)^{2}+\left(a_{2}+2\right)^{2}+\ldots+\left(a_{50}+2\right)^{2}=a_{1}^{2}+a_{2}^{2}+\ldots+a_{50}^{2} \rightarrow \\
& {\left[\left(a_{1}+2\right)^{2}-a_{1}^{2}\right]+\left[... | Answer: will increase by 150. | 150 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,906 |
1. A set of 60 numbers is such that adding 3 to each of them does not change the value of the sum of their squares. By how much will the sum of the squares of these numbers change if 4 is added to each number? | Answer: will increase by 240. | 240 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,908 |
1. A set of 70 numbers is such that adding 4 to each of them does not change the magnitude of the sum of their squares. By how much will the sum of the squares of these numbers change if 5 is added to each number? | Answer: will increase by 350. | 350 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,910 |
1. A set of 80 numbers is such that adding 5 to each of them does not change the magnitude of the sum of their squares. By how much will the sum of the squares of these numbers change if 6 is added to each number? | Answer: will increase by 480. | 480 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,912 |
5. Find the fraction $\frac{p}{q}$ with the smallest possible natural denominator, for which $\frac{1}{2014}<\frac{p}{q}<\frac{1}{2013}$. Enter the denominator of this fraction in the provided field | 5. Find the fraction $\frac{p}{q}$ with the smallest possible natural denominator, for which
$\frac{1}{2014}<\frac{p}{q}<\frac{1}{2013}$. Enter the denominator of this fraction in the provided field
Answer: 4027 | 4027 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,918 |
8. The residents of the village Razumevo, located 3 km away from the river, love to visit the village Vkusnotevo, situated 3.25 km downstream on the opposite bank of the river, 1 km away from the shore. The width of the river is 500 m, the speed of the current is 1 km/h, and the banks are parallel straight lines. The r... | 8. The residents of the village Razumevo, located 3 km away from the river, love to visit the village Vkusnotevo, situated 3.25 km downstream on the opposite bank of the river, 1 km away from the shore. The width of the river is 500 m, the speed of the current is 1 km/h, and the banks are parallel straight lines. The r... | 1.5 | Other | math-word-problem | Yes | Yes | olympiads | false | 1,921 |
6. Answer: $\Sigma_{\max }=\frac{\pi}{4}\left((\sqrt{2 a}-\sqrt{b})^{4}+6(3-2 \sqrt{2}) b^{2}\right)=4 \pi(19-12 \sqrt{2}) \approx 25.5$
Option 0
In rectangle $ABCD$ with sides $AD=a, AB=b(b<a<2b)$, three circles $K, K_{1}$, and $K_{2}$ are placed. Circle $K$ is externally tangent to circles $K_{1}$ and $K_{2}$, and ... | Answer: $\quad S_{\max }=\pi\left(R^{2}+\rho^{2}+r^{2}\right)_{\max }=\frac{\pi}{4}\left(6(3-2 \sqrt{2}) b^{2}+(\sqrt{2 a}-\sqrt{b})^{4}\right)$
$$
S_{\min }=\pi\left(R^{2}+\rho^{2}+r^{2}\right)=\pi\left(\frac{b^{2}}{4}+\frac{(\sqrt{a+b}-\sqrt{b})^{4}}{2}\right)
$$
Solution.
Let $\rho, r, R$ be the radii of circles ... | S_{\max}=\frac{\pi}{4}(6(3-2\sqrt{2})b^{2}+(\sqrt{2}-\sqrt{b})^{4}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,931 |
3. Let $c_{n}=11 \ldots 1$ be a number with $n$ ones in its decimal representation. Then $c_{n+1}=10 \cdot c_{n}+1$. Therefore,
$$
c_{n+1}^{2}=100 \cdot c_{n}^{2}+22 \ldots 2 \cdot 10+1
$$
For example,
$c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121$,
$c_{3}^{2}=111^{2}=100 \cdot 11^{2}+220+1=12100+220+1=... | Answer: 1) $a^{2}=121 ;$ 2) $b^{2}=12321 ;$ 3) $\sqrt{c}=11111111$. | \sqrt{}=11111111 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,949 |
4. If $\quad a=\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}, \quad$ then $\quad P(a)=\overline{a_{6} a_{1} a_{2} a_{3} a_{4} a_{5}}$, $P(P(a))=\overline{a_{5} a_{6} a_{1} a_{2} a_{3} a_{4}} \quad$ with $\quad a_{5} \neq 0, a_{6} \neq 0, a_{1} \neq 0 . \quad$ From the equality $P(P(a))=a$ it follows that $a_{1}=a_{5},... | Answer: 1) 81 is the number; 2) $a=\overline{t u t u t u}, t, u$, where $t, u$ - are any digits, not equal to zero. | 81 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,950 |
4. A cyclic permutation $P(a)$ of a natural number $a=\overline{a_{1} a_{2} \ldots a_{n}}, a_{n} \neq 0$ is the number $b=\overline{a_{n} a_{1} a_{2} \ldots a_{n-1}}$ written with the same digits but in a different order: the last digit becomes the first, and the rest are shifted one position to the right. How many eig... | Answer: 1) $9^{4} \quad$ number; 2) $a=\overline{t u \gamma w t u \gamma w}, t, u, v, w \quad-$ arbitrary digits, not equal to zero. | 9^4 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,952 |
3. Square the numbers $a=10001, b=100010001$. Extract the square root of the number $c=1000200030004000300020001$. | 1) $a^{2}=100020001$; 2) $b^{2}=10002000300020001$; 3) $\sqrt{c}=1000100010001$. | 1000100010001 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,955 |
3. Let $a > b$, denote $x, y, z$ as digits. Consider several cases for the decimal representation of the desired numbers.
Case 1. Let $a = x \cdot 10 + y, b = y \cdot 10 + z$.
a) Write the conditions of the problem for these notations
$$
\left\{\begin{array} { c }
{ 1 0 x + y = 3 ( 1 0 y + z ) , } \\
{ x + y = y + ... | Answer: $a=72, b=24 ; \quad a=45, b=15$. | =72,b=24;\quad=45,b=15 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,957 |
2. Solve the equation $2+2:(1+2:(1+2:(1+2:(3 x-8))))=x$.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | Answer: $x=\frac{14}{5}$. | \frac{14}{5} | Algebra | proof | Yes | Yes | olympiads | false | 1,959 |
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=2: 3$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=3$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$. | Answer: $A E: E Q=4: 9$.
# | AE:EQ=4:9 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,960 |
3. For two two-digit, integer, positive numbers $a$ and $b$, it is known that 1) one of them is 14 greater than the other; 2) in their decimal representation, one digit is the same; 3) the sum of the digits of one number is twice the sum of the digits of the other. Find these numbers. | Answer: $a_{1}=37, b_{1}=23 ; a_{2}=31, b_{2}=17$. | a_{1}=37,b_{1}=23;a_{2}=31,b_{2}=17 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,962 |
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=1: 2$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=2$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $PE: ED$. | Answer: $P E: E D=2: 9$.
# | PE:ED=2:9 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,963 |
3. For two two-digit, positive integers $a$ and $b$, it is known that 1) one of them is 12 greater than the other; 2) in their decimal representation, one digit is the same; 3) the sum of the digits of one number is 3 greater than the sum of the digits of the other. Find these numbers. | Answer: $a=11 t+10, b=11 t-2, t=2,3, \ldots, 8$;
$$
\tilde{a}=11 s+1, \quad \tilde{b}=11 s+13, \quad s=1,2,3, \ldots, 6
$$ | =11+10,b=11-2,=2,3,\ldots,8;\quad\tilde{}=11+1,\quad\tilde{b}=11+13,\quad=1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,965 |
5. Point $P$ is located on side $AB$ of square $ABCD$ such that $AP: PB=1: 4$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ: QC=5$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE: EQ$. | Answer: $A E: E Q=6: 29$. | AE:EQ=6:29 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,966 |
Problem 5. Answer: 1.
## Variant 0
Circles $K_{1}$ and $K_{2}$ have a common point $A$. Through point $A$, three lines are drawn: two pass through the centers of the circles and intersect them at points $B$ and $C$, the third is parallel to $BC$ and intersects the circles at points $M$ and $N$. Find the length of seg... | Answer: $a$
Solution.
Notations: $\alpha$ - angle $B A M, \beta-$ angle $C A N$
Angle $A B C=\alpha$, angle $A C B=\beta$ (vertical angles).
Angle $A M B=90^{\circ}$, angle $A N C=90^{\ci... | a | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,969 |
2. Let's find the greatest common divisor (GCD) of the numerator and the denominator. By the properties of GCD:
$$
\begin{aligned}
& \text { GCD }(5 n-7,3 n+2)=\text { GCD }(5 n-7-(3 n+2), 3 n+2)= \\
& =\text { GCD }(2 n-9,3 n+2)=\text { GCD }(2 n-9, n+11)= \\
& \quad=\text { GCD }(n-20, n+11)=\text { GCD }(31, n+11) ... | Answer: it can be reduced by 31 when $n=31k-11, k \in Z$. | 31k-11,k\in\mathbb{Z} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,970 |
4. Let $x_{1}, x_{2}$ be the integer solutions of the equation $x^{2} + p x + 3 q = 0$ with prime numbers $p$ and $q$. Then, by Vieta's theorem,
$$
x_{1} + x_{2} = -p
$$
From this, we conclude that both roots must be negative. Further, since the roots are integers and the numbers $q, 3$ are prime, there are only two ... | Answer: there are two pairs of solutions: $p=5, q=2 ; p=7, q=2$. | p=5,q=2;p=7,q=2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,971 |
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{5 n+3}{7 n+8}$ be reduced? For which integers $n$ can this occur? | Answer: it can be reduced by 19 when $n=19k+7, k \in Z$. | 19 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,973 |
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{4 n+3}{5 n+2}$ be reduced? For which integers $n$ can this occur? | Answer: can be reduced by 7 when $n=7 k+1, k \in Z$. | 7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,975 |
5. Based on the base $AC$ of the isosceles triangle $ABC$, a circle is constructed with $AC$ as its diameter, intersecting the side $BC$ at point $N$ such that $BN: NC = 7: 2$. Find the ratio of the lengths of the segments $AN$ and $BC$. | Answer: $\frac{A N}{B C}=\frac{4 \sqrt{2}}{9}$.
# | \frac{AN}{BC}=\frac{4\sqrt{2}}{9} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,977 |
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{3 n+2}{8 n+1}$ be reduced? For which integers $n$ can this occur? | Answer: it can be reduced by 13 when $n=13k-5, k \in Z$. | 13k-5,k\inZ | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,978 |
5. Based on the base $AC$ of the isosceles triangle $ABC$, a circle is constructed with $AC$ as its diameter, intersecting the side $BC$ at point $N$ such that $BN: NC = 5: 2$. Find the ratio of the lengths of the medians $NO$ and $BO$ of triangles $ANC$ and $ABC$. | Answer: $\frac{N O}{B O}=\frac{1}{\sqrt{6}}$. | \frac{NO}{BO}=\frac{1}{\sqrt{6}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,979 |
5. The angle at vertex $B$ of triangle $A B C$ is $130^{\circ}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $E$ and $D$. Find the acute angle between the diagonals of the quadrilateral with vertices at points $A, C, D$ and $E$.
P... | Solution. Let $a$ be the number of students in the first category, $c$ be the number of students in the third category, and $b$ be the part of students from the second category who will definitely lie in response to the first question (and say "YES" to all three questions), while the rest of the students from this cate... | 80 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,980 |
2. For what values of $a$ are the roots of the equation $x^{2}-\left(a+\frac{1}{a}\right) x+(a+\sqrt{35})\left(\frac{1}{a}-\sqrt{35}\right)=0$ integers? | Answer: $a= \pm 6-\sqrt{35}$. | \6-\sqrt{35} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,981 |
2. For what values of $a$ are the roots of the equation $x^{2}-\left(a+\frac{1}{a}\right) x+(a+4 \sqrt{3})\left(\frac{1}{a}-4 \sqrt{3}\right)=0$ integers? | Answer: $a= \pm 7-4 \sqrt{3}$. | \7-4\sqrt{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,982 |
2. For what values of $a$ are the roots of the equation $x^{2}-\left(a+\frac{1}{a}\right) x+(a+4 \sqrt{5})\left(\frac{1}{a}-4 \sqrt{5}\right)=0$ integers? | Answer: $a= \pm 9-4 \sqrt{5}$. | \9-4\sqrt{5} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,983 |
5. The angle at vertex $B$ of triangle $A B C$ is $58^{0}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $D$ and $E$. Find the angle between the diagonals of the quadrilateral with vertices at points $A, C, D$ and $E$. | Answer: $64^{0}$
## Grading Criteria for the Final Round of the Rosatom Olympiad 04.03.2023 8th Grade
In all problems, the correct answer without justification $\quad 0$ | 64^{0} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,984 |
5. For which $a$ does the set of solutions of the inequality $x^{2}+(|y|-a)^{2} \leq a^{2}$ contain all pairs of numbers $(x ; y)$ for which $(x-4)^{2}+(y-2)^{2} \leq 1$?
Problem 5 Answer: $a \in\left[\frac{19}{2} ;+\infty\right)$ | Solution.
Case 1. $a \leq 0$. The first inequality is satisfied by a single point (0, 0), so the situation where the solutions of the second inequality (points of the circle of radius 1) are... | \in[\frac{19}{2};+\infty) | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 1,991 |
6. Answer: 1) $S_{1}=\frac{3 \sqrt{3}}{2}$ 2) $S_{2}=6 \sqrt{6}$
Option 0
In a cube $A B C D A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ with edge $a$, a section is made by a plane parallel to the plane $B D A^{\prime}$ and at a distance $b$ from it. Find the area of the section. | Answer: 1) $S_{1}=\frac{\sqrt{3}}{2}(a-b \sqrt{3})^{2}$ for $a>b \sqrt{3}$ 2) $S_{2}=\frac{\sqrt{3}}{2}\left(a^{2}+2 a b \sqrt{6}-12 b^{2}\right)$ for $a>b \sqrt{6}$ Solution.
Fig. 1
^{2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,013 |
5. Answer: 1) $\alpha_{1}=\operatorname{arctg} \frac{24}{7}+2 \pi k, k \in Z \quad$ 2) $\alpha_{2}=\pi+2 \pi k, k \in Z$
$$
\text { 3) } \alpha_{3}=\operatorname{arctg} \frac{4+3 \sqrt{24}}{4 \sqrt{24}-3}+2 \pi k, k \in Z \quad \text { 4) } \alpha_{4}=\operatorname{arctg} \frac{3 \sqrt{24}-4}{4 \sqrt{24}+3}+2 \pi k, k... | Solution (Geometric)
The line with the equation $x \cos a + y \sin a - 2 = 0$ is tangent to a circle of radius 2 centered at the origin. The system has a unique solution if it is tangent (common tangent) to the circle with the equation $x^{2} + y^{2} + 6x - 8y + 24 = 0 \leftrightarrow (x + 3)^{2} + (y - 4)^{2} = 1$, r... | \alpha_{1}=\operatorname{arctg}\frac{24}{7}+2\pik,k\inZ\quad\alpha_{2}=\pi+2\pik,k\inZ\quad\alpha_{3}=\operatorname{arctg}\frac{4+3\sqrt{24}}{4\sqrt{24}-3}+2\pik,k\inZ\ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,018 |
2. The desired number is equal to
$$
(3+5+7+8) \cdot(3+5+7+8) \cdot(3+5+7+8)=23^{3}=12167
$$ | Answer: $23^{3}=12167$.
In a line, integers are written one after another, starting with 2, and each subsequent number, except the first, is the sum of the two adjacent numbers. The sum of the first 999 numbers is 6. What number stands in the 275th place in the line? | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,020 |
1. According to the problem, Sasha makes one step in 1 second, while Dan makes one step in $\frac{6}{7}$ seconds. Therefore, after 6 seconds, both Sasha and Dan will have made an integer number of steps, specifically, Sasha will have made 6 steps, and Dan will have made 7 steps. Consequently, we need to consider moment... | Answer: 1) $d_{\min }=29.6$ m; 2) Sasha made 432 steps, Dania made 504 steps. | 29.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,024 |
3. Rewrite the original equation in the form
$$
16 z^{2}+4 x y z+\left(y^{2}-3\right)=0
$$
This is a quadratic equation in terms of z. It has a solution if $D / 4=4 x^{2} y^{2}-16\left(y^{2}-3\right) \geq 0$. After transformations, we obtain the inequality
$$
y^{2}\left(x^{2}-4\right)+12 \geq 0
$$
If $x^{2}-4 \geq ... | Answer: $\quad x \in(-\infty ;-2] \cup[2 ;+\infty)$. | x\in(-\infty;-2]\cup[2;+\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,026 |
4. Let's write the natural number $a$ in its canonical form:
$$
a=p_{1}^{s_{1}} \cdot p_{2}^{s_{2}} \cdot \ldots \cdot p_{n}^{s_{n}}
$$
where $p_{1}, p_{2}, \ldots, p_{n}$ are distinct prime numbers, and $s_{1}, s_{2}, \ldots, s_{n}$ are natural numbers.
It is known that the number of natural divisors of $a$, includ... | Answer: 1) $\Sigma_{d}=\frac{a_{100}^{a}-1}{\sqrt[100]{a}-1}$; 2) $\Pi_{d}=\sqrt{a^{101}}$. | \Sigma_{}=\frac{\sqrt[100]{}-} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,027 |
1. At the intersection of roads $A$ and $B$ (straight lines) is a settlement $C$ (point). Sasha is walking along road $A$ towards point $C$, taking 50 steps per minute, with a step length of 50 cm. At the start of the movement, Sasha was 250 meters away from point $C$. Dan is walking towards $C$ along road $B$ at a spe... | Answer: 1) $d_{\text {min }}=15.8$ m; 2) the number of steps Sasha took - 470, the number of steps Dani took - 752. | 15.8 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,028 |
2. For what values of $a$ does the equation $\sin 5 a \cdot \cos x - \cos (x + 4 a) = 0$ have two solutions $x_{1}$ and $x_{2}$, such that $x_{1} - x_{2} \neq \pi k, k \in Z$? | Answer: $a=\frac{\pi(4 t+1)}{2}, t \in Z$. | \frac{\pi(4+1)}{2},\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,029 |
4. A natural number $a$ has 103 different divisors, including 1 and $a$. Find the sum and product of these divisors. | Answer: 1) $\Sigma_{d}=\frac{a \sqrt[102]{a}-1}{\sqrt[102]{a}-1}$; 2) $\Pi_{d}=\sqrt{a^{103}}$. | \Sigma_{}=\frac{\sqrt[102]{}-1}{\sqrt[102]{}-1};\Pi_{}=\sqrt{} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,030 |
5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 2 and $2 \sqrt{2}$ respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 1. Find the angle between the lines $B C$ and $A D$. | Answer: $\alpha=45^{\circ}$. | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,031 |
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