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XVI OM - II - Task 1
Prove that the system of conditions
where $ a, b, c $ denote real numbers, is equivalent to the system of conditions
Note: The original text appears to have incomplete or missing mathematical conditions. If you can provide the complete conditions, I can translate them a... | 1. We will prove that from the system of conditions (1) and (2) follows the system of conditions (I) and (II). From equality (1) it follows that there exist values of the roots $ \sqrt{a+c} $ and $ \sqrt{b+c} $, hence
We square both sides of equality (1):
from which
From (2) and (4) it follows that
From (3) and (5)... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
LVII OM - I - Problem 6
In an acute triangle $ABC$, the altitudes intersect at point $H$. A line passing through $H$ intersects segments $AC$ and $BC$ at points $D$ and $E$, respectively. A line passing through $H$ and perpendicular to line $DE$ intersects line $AB$ at point $F$. Prove that | Let $ K $ be the intersection point of the line $ AH $ with the line passing through point $ B $ and parallel to the line
$ FH $ (Fig. 2). Denote by $ X $ the orthogonal projection of point $ A $ onto the line $ BC $, and by $ Y $ the intersection point of the
perpendiculars $ DE $ and $ BK $.
om57_1r_img_2.jpg
The se... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLVI OM - III - Problem 6
Given are three non-coplanar half-lines $ k $, $ l $, $ m $ with a common origin $ P $ and a point $ A $ different from $ P $ belonging to $ k $. Prove that there exists exactly one pair of points $ B $, $ C $, belonging to $ l $ and $ m $ respectively, such that | We will start with the analysis of the postulated conditions. Assume that points $ B \in l $, $ C \in m $ satisfy these conditions:
In the triangle $ ABC $, we inscribe a circle. The points of tangency of this circle with the sides $ BC $, $ CA $, $ AB $ are denoted by $ Q $, $ R $, $ S $ (Figure 15). Therefore, the f... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XLIV OM - I - Problem 5
Given is a half-plane and points $ A $ and $ C $ on its edge. For each point $ B $ of this half-plane, we consider squares $ ABKL $ and $ BCMN $ lying outside the triangle $ ABC $. They determine a line $ LM $ corresponding to the point $ B $. Prove that all lines corresponding to different pos... | Let's choose point $ B $ in the given half-plane. Due to the symmetry of the roles of the considered squares, we can assume, without loss of generality, that $ |AB| \geq |BC| $. Angle $ BAC $ is therefore acute; angle $ ACB $ can be acute or not (Figure 2). Let $ D $, $ P $, $ Q $ be the orthogonal projections of point... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XIII OM - I - Problem 12
From point $ A $ of circle $ k $, a segment $ AB $ perpendicular to the plane of the circle is drawn. Find the geometric locus of the orthogonal projection of point $ A $ onto the line passing through point $ B $ and point $ M $ moving along the circle $ k $. | Let $M$ be a point on the circle $k$ (Fig. 18). Draw the diameter $AC$ and the chords $CM$ and $AM$ of the circle $k$, and consider the projections $P$ and $D$ of the point $A$ onto the lines $BM$ and $BC$, respectively. Since $CM$ is perpendicular to the projection of $AM$ on the segment $BM$, according to the so-call... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXVI - II - Problem 2
In a convex quadrilateral $ABCD$, points $M$ and $N$ are chosen on the adjacent sides $\overline{AB}$ and $\overline{BC}$, respectively, and point $O$ is the intersection of segments $AN$ and $GM$. Prove that if circles can be inscribed in quadrilaterals $AOCD$ and $BMON$, then a circle can also ... | Let $ P $, $ Q $, $ R $, $ S $ be the points of tangency of the inscribed circle in quadrilateral $ AOCD $ with sides $ \overline{AO} $, $ \overline{OC} $, $ \overline{CD} $, $ \overline{DA} $ respectively, and similarly $ E $, $ F $, $ G $, $ H $ - the points of tangency of the inscribed circle in quadrilateral $ BMON... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XX OM - III - Task 1
Prove that if real numbers $ a, b, c $ satisfy the condition
where $ m $ denotes a positive number, then the equation
has a root contained between 0 and 1. | It is sufficient to conduct the proof when $ a \geq 0 $. The case of negative $ a $ reduces to the case $ a > 0 $ by changing the signs in equations (1) and (2) to their opposites.
I. When $ a = 0 $, the proof is short:
If $ b \ne 0 $, then equation (2) has a root
and since from condition (1) it follows that when $ a... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
LIII OM - III - Task 3
Three non-negative integers are written on the board. We choose two of these numbers $ k $, $ m $ and replace them with the numbers $ k + m $ and $ |k - m| $, while the third number remains unchanged. We proceed in the same way with the resulting triplet. Determine whether, from any initial trip... | Every triple of non-negative integers can be represented in the form
\[
(2^pa, 2^qb, 2^rc)
\]
where \( p \), \( q \), \( r \) are non-negative integers, and each of the numbers \( a \), \( b \), \( c \) is odd or equal to \( 0 \). The weight of a triple represented in form (1) will be the quantity \( a + b + c \).
W... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXVII OM - III - Problem 6
In triangle $ABC$, points $K$ and $L$ are the orthogonal projections of vertices $B$ and $C$ onto the angle bisector of $\angle BAC$, point $M$ is the foot of the altitude from vertex $A$ in triangle $ABC$, and point $N$ is the midpoint of side $BC$. Prove that points $K$, $L$, $M$, and $N$... | When $ |AB| = |AC| $, points $ K $, $ L $, $ M $, $ N $ coincide. We will exclude this trivial case from further considerations. We will present three methods of solving.
\spos{I} Without loss of generality, we can assume that $ |AB| < |AC| $. Let us first assume that angle $ ABC $ is not a right angle - it is therefor... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LIV OM - I - Problem 8
In the tetrahedron $ABCD$, points $M$ and $N$ are the midpoints of edges $AB$ and $CD$, respectively. Point $P$ lies on segment $MN$, such that $MP = CN$ and $NP = AM$. Point $O$ is the center of the sphere circumscribed around the tetrahedron $ABCD$. Prove that if $O \neq P$, then $OP \perp MN$... | Let's introduce the following notations:
Then, by the Law of Cosines, we have
Furthermore, using the Pythagorean theorem, we obtain
By subtracting equation (1) and utilizing the relationship (2), we get $ 2(a + b)x \cos \alpha = 0 $. Therefore, $ \alpha = 90^\circ $, which means $ OP \perp MN $. | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIX OM - II - Problem 6
Given is a tetrahedron $ABCD$. Prove that the edges $AB$ and $CD$ are perpendicular if and only if there exists a parallelogram $CDPQ$ in space such that $PA = PB = PD$ and $QA = QB = QC$. | Let $ k $ be the line perpendicular to the plane of triangle $ ABC $ and passing through the center of the circle circumscribed around triangle $ ABC $. Similarly, let $ \ell $ be the line perpendicular to the plane of triangle $ ABD $ and passing through the center of the circle circumscribed around triangle $ ABD $. ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXII OM - I - Problem 6
Let $ f(x,y,z) = \max(x^2 - yz, y^2 - xz, z^2 - xy) $. Find the set of values of the function $ f(x,y,z) $ considered for numbers $ x,y,z $ satisfying the conditions:
Note: $ \max (a, b, c) $ is the greatest of the numbers $ a, b, c $. | From (1) it follows that
thus $ x + y + z = 3 $. It follows from this that the conditions (1) added in the problem are equivalent to the conditions:
If we perform any permutation of the letters $ x $, $ y $, $ z $, the conditions of the problem will not change. Therefore, without loss of generality, we can l... | 4 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
X OM - II - Task 3
Prove that if $ 0 \leq \alpha < \frac{\pi}{2} $ and $ 0 \leq \beta < \frac{\pi}{2} $, then
| Transforming the left side of inequality (1), we obtain
Since $ 1 \geq \cos (\alpha - \beta) $, therefore
From relations (2) and (3), inequality (1) follows.
In addition to the above calculation, it also follows that equality
holds only when $ \cos (\alpha - \beta) = 1 $, i.e., when $ \alpha = \beta $... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XLVII OM - I - Problem 9
A polynomial with integer coefficients, when divided by the polynomial $ x^2 - 12x + 11 $, gives a remainder of $ 990x - 889 $. Prove that this polynomial does not have integer roots. | According to the assumption, the considered polynomial has the form
where $ Q(x) $ is some polynomial.
Further reasoning is based on the following observation: for every pair of different integers $ k, m $
[Justification: if $ P(x) = a_0 + a_1x + \ldots +a_nx^n $ ($ a_i $ integers), then the difference $ P(k) - P... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXIII OM - I - Problem 9
How many natural numbers less than $ 10^n $ have a decimal representation whose digits form a non-decreasing sequence? | Let $ A $ be the set of natural numbers less than $ 10^n $, having a decimal representation whose digits form a non-decreasing sequence. Let's agree to place as many zeros at the beginning of the representation of each number in the set $ A $ so that the representation of this number has $ n $ digits in total.
Let $ B ... | \binom{n+9}{9}-1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXXVI OM - II - Problem 1
Inside triangle $ ABC $, a point $ P $ is chosen. Let $ a, b, c $ be the lengths of sides $ BC $, $ CA $, $ AB $, respectively, and let $ x, y, z $ be the distances from point $ P $ to vertices $ B, C, A $. Prove that if
then
(Note: The original text had blank spaces where th... | From the first two equalities given in the problem, due to the positivity of the numbers $a$, $b$, $x$, $y$, $z$, it follows that $a = \sqrt{x^2 + xy + y^2} > \sqrt{x^2} = x$ and similarly $b > z$. Therefore, by the third equality, $c^2 = x^2 + xz + z^2 < a^2 + ab + b^2$, which was to be proven. | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXXV OM - III - Task 6
The towns $ P_1, \ldots, P_{1025} $ are served by the airlines $ A_1, \ldots, A_{10} $, such that for any towns $ P_k $ and $ P_m $ ($ k \neq m $) there exists an airline whose planes fly directly from $ P_k $ to $ P_m $ and directly from $ P_m $ to $ P_k $. Prove that one of these airlines can ... | Suppose that none of the lines $ A_1, \ldots, A_{10} $ can offer a journey with an odd number of landings that starts and ends in the same town. It follows that if there is a connection from town $ P_k $ to town $ P_m $ by planes of a certain line, with an even number of landings, then in any other connection from $ P_... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXXIX OM - III - Problem 4
Let $ d $ be a positive integer, and $ f \: \langle 0; d\rangle \to \mathbb{R} $ a continuous function such that $ f(0) = f(d) $. Prove that there exists $ x \in \langle 0; d-1\rangle $ such that $ f(x) = -f(x + 1) $. | We consider the function $ g(x) = f(x+1)-f(x) $ defined and continuous in the interval $ \langle 0; d-1 \rangle $. The sum
of the terms, by assumption, is zero, which means either all its terms are zero, or some two terms are of different signs. In any case, there exist integers $ a, b \in \langle 0; d-1 \rangle $ suc... | proof | Calculus | proof | Yes | Yes | olympiads | false |
VIII OM - II - Task 3
Given a cube with edge $ AB = a $ cm. Point $ M $ on segment $ AB $ is $ k $ cm away from the diagonal of the cube, which is skew to $ AB $. Find the distance from point $ M $ to the midpoint $ S $ of segment $ AB $. | If the vertices of the cube are marked as in Fig. 15, the diagonals of the cube skew to the edge $ AB $ will be $ A_1C $ and $ B_1D $. They are symmetrical to each other with respect to the plane bisector of the segment $ AB $, so it does not matter which of them is referred to in the problem. Let, for example, $ MN = ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
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 |
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 |
XIV OM - III - Task 3
Cut a rectangle of the largest area from the given triangle. | Suppose that the rectangle $MNPQ$ lies within a given triangle $ABC$. We will first prove that if all vertices $M$, $N$, $P$, $Q$ of the rectangle lie on the perimeter of triangle $ABC$, then the area $P$ of the rectangle is at most half the area $T$ of triangle $ABC$, and the equality $P = \frac{1}{2}T$ holds if and o... | proof | Geometry | math-word-problem | Incomplete | Incomplete | olympiads | false |
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 |
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 |
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 |
XVII OM - II - Problem 5
Each of the sides $ BC, CA, AB $ of triangle $ ABC $ is divided into three equal parts, and on the middle segments of these sides, as bases, equilateral triangles are constructed outside the triangle $ ABC $, with their third vertices marked by the letters $ A$ respectively. In addition, point... | We adopt the usual notation for the sides and angles of triangle $ABC$. Let $S$ denote the centroid of this triangle, and $K$ and $L$ the points dividing the sides $AB$ and $AC$ that are closer to vertex $A$ (Fig. 10).
In triangle $AKC$, $AK=KC$, $\measuredangle C$, from which we easily calculate that $AC$. Similarly, ... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
XVIII OM - I - Problem 9
Prove that if the numbers $ \alpha^2 $, $ \beta^2 $, $ \alpha\beta $ are rational, then there exist rational numbers $ k $, $ a $ and $ b $, such that
(Note: The last part of the text is left blank as it was in the original text.) | When $ \alpha \ne 0 $, $ \beta \ne 0 $, the number $ \frac{\alpha}{\beta} $ is rational, as it is the quotient of the rational numbers $ \alpha^2 $ and $ \alpha \beta $. Then the numbers $ k = \beta^2 $, $ a = \frac{\alpha}{\beta} $, and $ b = 1 $ satisfy the condition of the problem.
When $ \alpha \ne 0 $, $ \beta ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
LV OM - II - Task 5
Points $ D $ and $ E $ lie on sides $ BC $ and $ CA $ of triangle $ ABC $, respectively, such that $ BD = AE $. Segments $ AD $ and $ BE $ intersect at point $ P $. The angle bisector of angle $ ACB $ intersects segments $ AD $ and $ BE $ at points $ Q $ and $ R $, respectively. Prove that | Let $ \ell $ be any line perpendicular to the bisector of angle $ ACB $ (Fig. 2). Denote by $ A', $ B', $ C', $ D', $ E', $ P' the orthogonal projections of points $ A $, $ B $, $ C $, $ D $, $ E $, $ P $ onto the line $ \ell $.
Lines $ AC $ and $ BC $ form equal angles with line $ \ell $, so by the equality $ AE = BD ... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
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 |
XXVII OM - I - Problem 3
Inside an equilateral triangle with side length $ a $, a point is chosen whose distances from the vertices of the triangle are $ x, y, z $. Prove that
| Let's choose a coordinate system such that two vertices of a given equilateral triangle $ABC$ with side length $a$ lie on the x-axis, and the third vertex lies on the y-axis (Fig. 7). Then the vertices will have the following coordinates: $ \displaystyle A = \left( -\frac{a}{2}, 0\right) $, $ \displaystyle B = \left( \... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
VI OM - II - Task 6
Inside the trihedral angle $ OABC $, whose dihedral angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ equidistant from the faces of the angle is chosen. Through the point $ S $, a plane is drawn intersecting the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove... | We will calculate the volume $ V $ of the tetrahedron $ OMNP $ (Fig. 15) in two ways.
1°. Let $ PQ $ be the perpendicular dropped from point $ P $ to the plane $ AOB $, and let $ \alpha $ denote the angle $ AOB $, and $ \beta $ the angle of inclination of the edge $ OP $ to the plane $ AOB $; then $ PQ = OP \cdot \sin ... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
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 |
XI OM - III - Task 4
Prove that if the equation
has two equal roots, then
Note: The equation and the condition to prove are not provided in the text. If you have the specific equation and condition, please provide them for a complete translation. | Suppose that equation (1) has two roots equal to $ \alpha $; then $ \alpha^4 + a\alpha + b = 0 $, hence
where
the number $ \alpha $, as a double root of the polynomial $ x^4 + ax + b $, is a root of the polynomial $ Q (x) $, hence
By eliminating $ \alpha $ from the equations
we obtain condition (2):
Note. Conditio... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
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 |
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 |
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 |
XLIX OM - III - Problem 1
Find all systems of integers $ (a, b, c, x, y, z) $ satisfying the system of equations
as well as the conditions $ a\geq b \geq c \geq 1 $, $ x\geq y\geq z \geq 1 $. | Adding the equations of the given system side by side and transforming, we obtain the equality $ A + B + X + Y = 4 $, where
The numbers $ A $, $ B $, $ X $, $ Y $ are non-negative integers. If $ c \geq 2 $, then $ A \geq 3 $, $ B \geq 1 $, and consequently $ A = 3 $, $ B = 1 $, $ X = 0 $, $ Y = 0 $, which gives the so... | notfound | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
XXI OM - II - Problem 1
Prove that
where $ n $ is a natural number. Investigate for which values of $ n $, $ \alpha $, $ \beta $ the equality holds. | We will prove by induction that for every natural number $ n > 1 $ and number $ x \ne k\pi $ ($ k \in \mathbb{C} $), the inequality holds:
For $ n = 2 $, we have $ | \sin 2x | = 2|\sin x| |\cos x| < 2 |\sin x| $, because $ |\cos x| < 1 $ for $ x \ne k\pi $.
Assume the validity of (2) for some natural number $ n > 1 $;... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
XX OM - III - Task 4
Prove that if the natural numbers $ a, b, p, q, r, s $ satisfy the conditions
then
Note: The original text had blank lines and incomplete conditions. The translation preserves these elements. If you have the complete conditions, please provide them for a more accurate translation. | According to (2)
Thus $ aq-bp > 0 $, $ br-as > 0 $, and since the left sides of these inequalities are integers, therefore
We multiply both sides of the first inequality by $ s $, and the second by $ q $ and add them side by side:
Hence, taking into account condition (1), we obtain | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXXVII OM - I - Problem 8
Let $ E $ be any internal point on the side $ AB $ of a convex quadrilateral $ ABCD $. Denote by $ S_1, S_2, T $ the areas of triangles $ ADE $, $ BCE $, and $ CDE $, respectively.
a) Prove that if $ |\measuredangle A| + |\measuredangle B| = 180^{\circ} $, then $ T \geq 2\sqrt{S_1S_2} $.
b) P... | a) When $ |\measuredangle DAE| + |\measuredangle EBC| = 180^\circ $, lines $ BC $ and $ DA $ are parallel, so quadrilateral $ ABCD $ is a trapezoid. Let the lengths of sides $ DA $ and $ BC $ be denoted by $ a_1 $ and $ a_2 $, and the lengths of the heights of triangles $ DAE $ and $ BCE $ dropped from vertex $ E $ be ... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
VII OM - I - Problem 4
Prove that the midpoints of the segments connecting the center of the inscribed circle of a triangle with the centers of the excircles lie on the circumcircle of this triangle. | Let $ O $ denote the center of the circle inscribed in triangle $ ABC $ (Fig. 2), $ O_1 $ - the center of the excircle tangent to side $ AB $, and $ S $ - the midpoint of segment $ OO_1 $.
Points $ O $, $ O_1 $, and $ S $ lie on the angle bisector of angle $ C $.
To prove that point $ S $ lies on the circle passing thr... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXIV OM - I - Problem 4
In a tetrahedron $ABCD$, on the line connecting the midpoints of edges $\overline{AB}$ and $\overline{CD}$, an arbitrary point $P$ is chosen that does not lie on these edges. Prove that the lines $PA$, $PB$, $PC$, $PD$ intersect a plane parallel to edges $AB$ and $CD$ at points that are the ver... | Let $ \pi $ be a plane parallel to the edges $ \overline{AB} $ and $ \overline{CD} $. If for some point $ X $ the line $ PX $ intersects the plane $ \pi $, let $ X $ denote the point of intersection. Let $ R $ be the midpoint of segment $ \overline{AB} $, and $ S $ be the midpoint of segment $ \overline{CD} $. Then $ R... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
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 |
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 |
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 |
XV OM - I - Task 1
Prove that for every natural number $ n $ the following equality holds
Note: The original text was in Polish, and the task is to translate it into English while preserving the original formatting and line breaks. | \spos{1} We apply the method of induction. When $ n = 1 $, equation (1) is true. Suppose it is true for some natural number $ n $. Then it is also true for the number $ n + 1 $. Indeed, from the inductive hypothesis, we infer that
and since
we get from (2) that
Therefore, equation (1) holds for every ... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
IV OM - I - Problem 7
Prove that if $ n $ is an integer greater than $ 2 $, then | We can directly apply complete induction to prove the inequality
for any integers $ n > 2 $.
1° When $ n = 3 $, inequality (1) is true, since in this case
2° If inequality (1) is true for some $ k \geq 3 $, i.e., if
then
thus
From 1° and 2°, the inequality (1) is true for every integer $ n \geq 3 $. | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
XXVIII - I - Problem 9
The function $ f $ defined on the set of all integers takes only positive values and satisfies the condition:
Prove that the function $ f $ is constant. | Let $ g(n) = f(n + 1) - f(n) $ for every integer $ n $. Then from the conditions of the problem, we have $ f(n+1) - f(n) \leq f(n) - f(n - 1) $, which means . It follows that the function $ g $ is non-increasing. Therefore, for every natural number $ k $ we have $ g(n + k) \leq g(n) \leq g(n - k) $, which means
Adding... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLIII OM - I - Problem 1
Determine, depending on the real constants $ a $, $ b $, the number of solutions of the system of equations
| On the plane with Cartesian coordinate system $Oxy$, we consider sets $A$ and $B$ determined (respectively) by the first and second equation of the system:
om43_1r_img_1.jpg
Let $|A \cap B|$ denote the number of elements in the set $A \cap B$, i.e., the number of solutions to the considered system of equations.
When $... | notfound | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXXII - I - Problem 1
Given are sequences $ (x_n) $, $ (y_n) $ of real numbers satisfying the conditions
Prove that $ x_n^2 = y_n + 2 $ for $ n = 1, 2,\ldots $. | We will apply the principle of mathematical induction.
1. For $ n = 1 $, we have $ x_1 = x_0^3 $, so
2. Assume that for some natural number $ n $, $ x^2 = y_n + 2 $. We will prove that $ x^2_{n+1} = y_{n+1} + 2 $. Indeed,
By the principle of induction, $ x_n^2 = y_n + 2 $ for every natural number $ n $.
Note. The... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
XIV OM - III - Zadanie 5
Dowieść, że wielomian stopnia piątego
nie jest iloczynem dwóch wielomianów stopnia niższego o współczynnikach całkowitych.
|
Zauważmy, że jeżeli wielomian, w którym współczynnik najwyższego wyrazu równa się $ 1 $, jest iloczynem dwóch wielomianów o współczynnikach całkowitych, to można przyjąć, że w każdym z tych czynników współczynnik najwyższego wyrazu też równa się $ 1 $. Gdyż jeżeli ich najwyższymi wyrazami są $ a_0x^k $ i $ b_0x^k $, t... | proof | Algebra | proof | Incomplete | Incomplete | olympiads | false |
XLIV OM - I - Problem 2
Given a natural number $ n \geq 3 $. Solve the system of equations
(Note: The original text did not contain any specific equations, so the translation preserves the format and the lack of specific equations.) | Assume that the numbers $ x_1,\ldots, x_n $ satisfy the given system. Let us adopt
The considered system of equations takes the form
It follows, in particular, that the numbers $ t_1,\ldots ,t_n $ have the same sign. In this case, system (1) implies the system
Let us rewrite these equations in equival... | Other | math-word-problem | Incomplete | Yes | olympiads | false | |
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 |
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 |
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 |
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 |
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. 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 |
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
$$
\left\{\begin{array}{l}
x=... | Answer: : $\left\{\begin{array}{l}x=\frac{k(35-k)}{2}, \\ y=\frac{k(k-21)}{2},\end{array} k \in Z\right.$. | {\begin{pmatrix}\frac{k(35-k)}{2},\\\frac{k(k-21)}{2},\end{pmatrix}k\inZ.} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Each experience of throwing the circle corresponds to a point $M-$ the position of the center of the circle on the vertical segment $[A ; B]$ of length 1.
Let's introduce the following n... | Answer: $P(A)=\frac{2-\sqrt{2}}{2} \approx 0.29$. | P(A)=\frac{2-\sqrt{2}}{2}\approx0.29 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. The solutions to the second equation of the system are $x=1$ and $y=2$.
Case 1. $x=1$. Substituting $x=1$ into the first equation of the system, we get:
$$
4 \cos ^{2} a+(y-2-2 \sin a)^{2}=3
$$
or
$$
(y-2)^{2}-2 \sin a(y-2)+1=0
$$
This is a quadratic equation in terms of $(y-2)$. It has two solutions if its dis... | Answer: $a \in\left(\frac{\pi}{6}+\frac{\pi k}{2} ; \frac{\pi}{3}+\frac{\pi k}{2}\right), k \in Z$. | \in(\frac{\pi}{6}+\frac{\pik}{2};\frac{\pi}{3}+\frac{\pik}{2}),k\inZ | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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}$.
=16$, and $\operatorname{LCM}\left(\log _{8} x, \log _{12} y\right)=18$. Find these numbers. | Answer: $x=8^{9}, y=144 ; x=8^{18}, y=144$. | 8^{9},144;8^{18},144 | Number Theory | math-word-problem | Yes | Incomplete | olympiads | false |
3. Find all pairs of integers $(x ; y)$ 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 |
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 |
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 |
6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in the ratio $C N: N A=3$. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $1: 2$. 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 {MvCN }}: S=13: 42$.
# | 13:42 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
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 |
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 |
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 |
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 |
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. Natural numbers $x$ and $y$ are such that their LCM $(x, y)=5^{4} \cdot 2^{6}$, and LCM $\left(\log _{10} x, \log _{40} y\right)=4$. Find these numbers. | Answer: $x=10^{4}, y=40^{2}$. | 10^{4},40^{2} | Number Theory | math-word-problem | Yes | Incomplete | olympiads | false |
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 |
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 |
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 |
6. Point $N$ divides the diagonal $A C$ of trapezoid $A B C D$ in half. The lengths of the bases $B C$ and $A D$ of the trapezoid are in the ratio $1: 4$. 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 the area of quadril... | Answer: $S_{\text {MVC }}: S=1: 7$. | 1:7 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
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 |
3. $n-4=7 k, n=7 k+4,-20 \leq 7 k+4 \leq 100$, $k=-3 ;-2 ; \ldots 13$. In total, there are 17 integers.
The set of integers $n$ that satisfy case 1 do not satisfy the conditions of cases 2 and 3.
Note that cases 2 and 3 highlight the same set of integers: $7 m-3=7 k+4, m=k+1$.
Thus, the set $M$ contains 34 distinct ... | Answer: 1) 34 numbers; 2) $n_{\min }=-17, n_{\max }=96$. | 34;n_{\}=-17,n_{\max}=96 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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. A team of three pirates docked at an island to divide a treasure of gold coins left there. It was late, so they decided to postpone the division of the coins until morning. The first pirate woke up in the middle of the night and decided to take his share. He couldn't divide the coins into three equal parts, so he to... | Answer: the first pirate gets 127 coins, the second pirate gets 94 coins, the third pirate gets 72 coins. | notfound | Logic and Puzzles | math-word-problem | Yes | Problem not solved | olympiads | false |
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