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LIX OM - I - Task 9
Determine the smallest real number a with the following property:
For any real numbers $ x, y, z \geqslant a $ satisfying the condition $ x + y + z = 3 $
the inequality holds | Answer: $ a = -5 $.
In the solution, we will use the following identity:
Suppose that the number $ a \leqslant 1 $ has the property given in the problem statement. The numbers
$ x = a, y = z = 2\cdot (\frac{3-a}{2}) $ satisfy the conditions $ x, y, z \geqslant a $ and $ x+y+z = 3 $, thus
by virtue of (1),... | -5 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 1,641 |
XLIV OM - I - Problem 11
In six different cells of an $ n \times n $ table, we place a cross; all arrangements of crosses are equally probable. Let $ p_n $ be the probability that in some row or column there will be at least two crosses. Calculate the limit of the sequence $ (np_n) $ as $ n \to \infty $. | Elementary events are determined by six-element subsets of the set of $n^2$ cells of the table; there are $\binom{n^2}{6}$ of them. Let $\mathcal{Z}$ be the complementary event to the event considered in the problem. The configurations favorable to event $\mathcal{Z}$ are obtained as follows: we place the first cross i... | 30 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,643 |
XVI OM - I - Problem 11
Prove the theorem: A closed broken line with five sides, none of whose three vertices lie on the same straight line, can intersect itself at one, two, three, or five points, but it cannot intersect itself at four points. | Let's call a point of a closed broken line an ordinary point if it belongs only to one side and each vertex belongs only to two sides, and a double point if it belongs only to two sides of the broken line but is not a vertex. If no three vertices of a closed broken line with five sides lie on the same straight line, th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,644 |
XIII OM - II - Problem 5
In the plane, a square $ Q $ and a point $ P $ are given. Point $ K $ runs along the perimeter of square $ Q $. Find the geometric locus of the vertex $ M $ of the equilateral triangle $ KPM $. | Let $ K $ be any point on the perimeter of a given square. On the plane of the square, there are two equilateral triangles with base $ KP $. The vertex of one of them is obtained by rotating point $ K $ around point $ P $ by an angle of $ 60^\circ $ in the direction of the clockwise; the vertex of the second one is fou... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,647 |
III OM - I - Task 1
Two common internal tangents and both common external tangents are drawn to two circles. Prove that the segment of the internal tangent contained between the external tangents is equal to the segment of the external tangent contained between its points of tangency. | We will obtain the proof of the theorem by applying the theorem that the segments of tangents drawn from a certain point to a circle are equal. For example, in Fig. 10, representing the considered figure, $ PK = PR $, $ PL = PS $, $ QM = QR $, $ QN = QS $.
Considering that due to the symmetry of the figure, the equalit... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,651 |
XXXII - III - Problem 1
In space, there are two intersecting lines $ a $ and $ b $. We consider all pairs of planes $ \alpha $ and $ \beta $ that are perpendicular and such that $ a\subset \alpha $, $ b\subset \beta $. Prove that there exists a circle such that through each of its points there passes a line $ \alpha \... | We will distinguish two cases.
1. If lines $a$ and $b$ are perpendicular, consider the plane $\alpha$ perpendicular to line $b$ and containing line $a$, and take any circle contained in this plane. For any point $A$ of this circle, draw the plane $\beta$ containing line $b$ and point $A$. Of course, the line $\alpha \c... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,652 |
XXXVIII OM - II - Problem 2
Prove that the sum of the planar angles at each vertex of a given tetrahedron is $180^{\circ}$ if and only if all its faces are congruent. | We unfold the surface of the considered tetrahedron $ABCD$ along the edges $AD$, $BD$, $CD$ and lay the faces $BCD$, $CAD$, $ABD$ on the plane of the triangle $ABC$. We obtain the net of this tetrahedron, that is, the hexagon $AMBKCL$, in which the triangles $BCK$, $CAL$, $ABM$ are respectively congruent to the mention... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,653 |
XXXIX OM - I - Problem 7
Given a transformation of the plane onto itself such that the image of every circle is a circle. Prove that the image of every line is a line. | Let the considered transformation be denoted by $ f $. Let $ l $ be any line, and $ f(l) $ - its image under the transformation $ f $.
The proof of the given theorem will be based on the following three statements:
[(1)] \textrm{Together with any two points, the set}\ f(l) \ \textrm{contains the line passing through t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,654 |
XXV OM - II - Task 1
Let $ Z $ be an $ n $-element set. Find the number of such pairs of sets $ (A, B) $, where $ A $ is a subset of $ B $ and $ B $ is a subset of $ Z $. We assume that each set includes itself and the empty set. | As is known, if $ 0 \leq r \leq s $, then the number of $ r $-element subsets of an $ s $-element set is equal to $ \binom{s}{r} $, and the number of all subsets of an $ s $-element set is equal to $ 2^s $.
A $ k $-element subset $ A $, where $ 0 \leq k \leq n $, can be chosen from a set $ Z $ in $ \binom{n}{k} $ ways.... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,655 | |
XXIV OM - III - Problem 3
The polyhedron $ W $ has the following properties:
a) it has a center of symmetry,
b) the intersection of the polyhedron $ W $ with any plane containing the center of symmetry and any edge of the polyhedron is a parallelogram,
c) there exists a vertex of the polyhedron $ W $ that belongs to e... | A polyhedron is a bounded set of points in space that is not contained in a single plane and is the intersection of a finite number of half-spaces. Since a half-space is a convex set, and the intersection of convex sets is a convex set, every polyhedron is a convex set. (Some other definitions of a polyhedron are also ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,657 |
XLIX OM - I - Problem 6
In triangle $ABC$, where $|AB| > |AC|$, point $D$ is the midpoint of side $BC$, and point $E$ lies on side $AC$. Points $P$ and $Q$ are the orthogonal projections of points $B$ and $E$ onto line $AD$, respectively. Prove that $|BE| = |AE| + |AC|$ if and only if $|AD| = |PQ|$. | Note 1: In the official print of the problem texts for the first-level competition of the 49th Mathematical Olympiad, problem 6 was formulated with an error: the assumption that $ |AB| > |AC| $ was missing. (The corrected version of the problem was sent to schools as an Errata a few weeks later.) Without this assumptio... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,658 |
L OM - II - Task 1
Given is a function $ f: \langle 0,1 \rangle \to \mathbb{R} $ such that $ f\left(\frac{1}{n}\right) = (-1)^n $ for $ n = 1, 2,\ldots $. Prove that there do not exist increasing functions $ g: \langle 0,1 \rangle \to \mathbb{R} $, $ h: \langle 0,1 \rangle \to \mathbb{R} $, such that $ f = g - h $. | Suppose there exist functions $ g $ and $ h $ satisfying the conditions of the problem. Since these functions are increasing, for every $ k \in \mathbb{N} $ we have
From the monotonicity of the function $ g $ and the above inequalities, we get
Thus, $ g(1)-g(0) > 2k $ for any $ k \in \mathbb{N} $, which i... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,659 |
XXXI - I - Problem 10
A plane is divided into congruent squares by two families of parallel lines. $ S $ is a set consisting of $ n $ such squares. Prove that there exists a subset of the set $ S $, in which the number of squares is no less than $ \frac{n}{4} $ and no two squares have a common vertex. | A set of squares formed on a plane can be divided into two subsets in such a way that two squares sharing a side belong to different subsets (similar to the division of a chessboard into black and white squares). Such a division can be described as follows. We introduce a coordinate system on the plane, whose axes are ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,662 |
XV OM - I - Problem 7
Given a circle and points $ A $ and $ B $ inside it. Find a point $ P $ on this circle such that the angle $ APB $ is subtended by a chord $ MN $ equal to $ AB $. Does the problem have a solution if the given points, or only one of them, lie outside the circle? | Suppose that point $ P $ of a given circle $ C $ with radius $ r $ is a solution to the problem (Fig. 7).
Since points $ A $ and $ B $ lie inside the circle $ C $, points $ M $ and $ N $ lie on the rays $ PA $ and $ PB $ respectively, and angle $ APB $ coincides with angle $ MPN $. Triangles $ APB $ and $ MPN $ have eq... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,666 |
XL OM - II - Task 2
For a randomly chosen permutation of the set $ \{1,\ldots, n\} $, denote by $ X(\mathbf{f}) $ the largest number $ k \leq n $ such that $ f_i < f_{i+1} $ for all indices $ i < k $. Prove that the expected value of the random variable $ X $ is $ \sum_{k=1}^n \frac{1}{k!} $. | The considered random variable of course depends on $ n $. Let us denote it by $ X_n $. We will find a recursive relationship between the expected values of the variables $ X_{n-1} $ and $ X_n $.
Take any permutation $ f = (f_1, \ldots , f_n) $ of the set $ \{1,\ldots, n\} $. By discarding the term $ f_n $, we obtain a... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,669 |
LIX OM - I -Zadanie 10
Dana jest liczba pierwsza p. Ciąg liczb całkowitych dodatnich $ a_1, a_2, a_3, \dots $ spełnia warunek
Wykazać, że pewien wyraz tego ciągu jest $ p $-tą potęgą liczby całkowitej.
(Uwaga: Symbol $ [x] $ oznacza największą liczbę całkowitą nie przekraczającą $ x $.)
|
Zdefiniujmy ciągi liczb całkowitych nieujemnych $ b_1, b_2, b_3, \dots $ oraz $ r_1, r_2, r_3, \dots $
w następujący sposób:
dla $ n =1,2,3,\dots $. Inaczej mówiąc, $ r_n $ jest różnicą między wyrazem an a największą nie
przekraczającą tego wyrazu $ p $-tą potęgą liczby całkowitej. Warunek (1) przepisujemy te... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,670 |
XV OM - I - Problem 12
The tetrahedron $ABCD$ is cut into two parts by a plane passing through vertex $D$, point $M$ on edge $AB$, and point $N$ on edge $BC$. Prove that the area of triangle $DMN$ is less than the area of one of the triangles $DAB$, $DBC$, or $DCA$. | The theorem we need to prove can be considered the spatial counterpart of the following theorem from plane geometry:
The segment $CM$ connecting vertex $C$ of triangle $ABC$ with an internal point $M$ on the base $AB$ of this triangle is smaller than one of the sides $AC$ and $BC$.
A short proof of this theorem can be ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,673 |
XXI OM - I - Problem 11
Prove that in any division of the plane into three sets, there exist two points in at least one of them that are 1 unit apart. | Suppose this theorem is false. Consider equilateral triangles $ABC$ and $ABD$ (Fig. 7) with a common base $\overline{AB}$ and a side length of $1$. From our assumption, it follows that each vertex of triangle $ABC$ belongs to a different one of the three distinct subsets into which we have divided the plane. Similarly,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,674 |
XLVIII OM - I - Problem 10
Points $ P $, $ Q $ lie inside an acute-angled triangle $ ABC $, such that $ |\measuredangle ACP| = |\measuredangle BCQ| $ and $ |\measuredangle CAP| = |\measuredangle BAQ| $. Points $ D $, $ E $, $ F $ are the orthogonal projections of point $ P $ onto the sides $ BC $, $ CA $, $ AB $, resp... | In the assumptions of equality
they are equivalent to the equalities
Angles $ ACP $ and $ BCQ $, considered as areas of the plane, can be disjoint (as in Figure 6) or they can overlap; this has no significance. A similar remark applies to each of the other three pairs of angles in relations (1) and (2).
Let $ U... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,675 |
XXIV OM - I - Problem 2
Prove that among 25 different positive numbers, one can choose two such that neither their sum nor their difference is equal to any of the remaining numbers. | Let us denote a given set of numbers by $A = \{a_1, a_2, \ldots, a_{25}\}$, where $0 < a_1 < a_2 < \ldots < a_{25}$. Suppose that for any $r$, $s$, where $1 \leq r < s \leq 25$, we have: $a_s + a_r \in A - \{a_r, a_s\}$ or $a_s - a_r \in A - \{a_r, a_s\}$. Of course, $a_r - a_s < 0$ and therefore the number $a_r - a_s$... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,677 |
VII OM - I - Problem 9
Prove that a triangle can be constructed from segments of lengths $ a $, $ b $, $ c $ if and only if | From segments of lengths $ a $, $ b $, $ c $, a triangle can be constructed if and only if
If the numbers $ a $, $ b $, $ c $ satisfy inequality (1) and are (as measures of segments) positive, then they also satisfy inequalities (2) and vice versa. To this end, we transform inequality (1). Writing this inequality in t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,680 |
VI OM - III - Task 6
Through points $ A $ and $ B $, two oblique lines $ m $ and $ n $ perpendicular to the line $ AB $ have been drawn. On line $ m $, a point $ C $ (different from $ A $) has been chosen, and on line $ n $, a point $ D $ (different from $ B $). Given the lengths of segments $ AB = d $ and $ CD = l $,... | Fig. 20 shows the considered figure in an orthogonal projection onto the plane determined by the lines $AB$ and $n$, hence on this drawing $AB \bot n$. For simplicity, we denote the projections of points and lines with the same letters as the points and lines themselves. Let us draw through point $A$ a line $p$ paralle... | \sqrt{(\frac{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,681 |
XI OM - III - Task 2
A plane passing through the height of a regular tetrahedron intersects the planes of the lateral faces along $ 3 $ lines forming angles $ \alpha $, $ \beta $, $ \gamma $ with the plane of the base of the tetrahedron. Prove that | We accept the notation given in Fig. 27, where $HD$ represents the height of the regular tetrahedron $ABCD$, and $MD$, $ND$, $PD$ are the lines of intersection of the plane through $HD$ with the lateral faces $BCD$, $CAD$, and $ABD$, respectively. The angles $\alpha$, $\beta$, $\gamma$ are the angles that the segments ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,684 |
XVII OM - III - Problem 5
Given a convex hexagon $ABCDEF$, in which each of the diagonals $AD, BE, CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals pass through one point. | \spos{1} The area of a polygon $ ABC\ldots $ will be denoted by the symbol $ (ABC\ldots) $. According to the assumption (Fig. 15)
From (1) and (2) we obtain
from which it follows that $ AE \parallel BD $, since the vertices $ A $ and $ E $ of triangles $ ABD $ and $ EDB $ with the common side $ BD $ lie on ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,686 |
XVIII OM - I - Problem 7
On a plane, there are $2n$ points. Prove that there exists a line that does not pass through any of these points and divides the plane into half-planes, each containing $n$ of the given points. Formulate and prove an analogous theorem for space. | Let $A_1, A_2, \ldots, A_{2n}$ be given points in the plane and let $a_{ik}$ ($i, k = 1, 2, \ldots, 2n$, $i \ne k$) denote the line passing through points $A_i$ and $A_k$. The set $Z$ of all lines $a_{ik}$ contains at most $\binom{2n}{2} = n(2n-1)$ distinct lines. Therefore, through any point $P$ in the plane, one can ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,687 |
XLV OM - II - Task 3
Prove that if a circle can be circumscribed around a hexagonal cross-section of a cube passing through its center, then the cross-section is a regular hexagon. | We place a cube in a coordinate system so that its vertices are points with all coordinates equal to $+1$ or $-1$. The center of the cube is then the point $O = (0,0,0)$, the origin of the coordinate system. The plane considered in the problem passes through $O$ and thus has an equation of the form
\[ ax + by + cz = 0... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,690 |
XXVII OM - I - Problem 9
In a circle of radius 1, a triangle with side lengths $a, b, c$ is inscribed. Prove that the triangle is acute if and only if $a^2 + b^2 + c^2 > 8$, right-angled if and only if $a^2 + b^2 + c^2 = 8$, and obtuse if and only if $a^2 + b^2 + c^2 < 8$. | From the Law of Sines, we have $ \displaystyle 2 = \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} $, where $ \alpha $, $ \beta $, $ \gamma $ are the angles of the considered triangle opposite to the sides of lengths $ a $, $ b $, $ c $, respectively. Therefore,
From the Law of Cosines, we ha... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,691 |
XII OM - II - Task 1
Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers. | Suppose the statement made in the text of the problem is not true and that the equality holds
where $ k $, $ r $, $ n $ denote natural numbers. According to the formula for the sum of terms of an arithmetic progression, we obtain from this
Each of the natural numbers $ 2k + r $ and $ r + 1 $ is greater than... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,692 |
XXX OM - II - Task 5
Prove that among any ten consecutive natural numbers there exists one that is relatively prime to each of the other nine. | Every common divisor of two natural numbers is also a divisor of their difference. Therefore, if among ten consecutive natural numbers, some two are not relatively prime, then they have a common divisor which is a prime number less than $10$, i.e., one of the numbers $2$, $3$, $5$, $7$.
Among ten consecutive natural nu... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,694 |
XLIII OM - III - Problem 5
The base of a regular pyramid is a $2n$-sided regular polygon $A_1, A_2, \ldots, A_n$. A sphere passing through the vertex $S$ of the pyramid intersects the lateral edges $SA_i$ at points $B_i$ ($i = 1,2,\ldots, 2n$). Prove that | om43_3r_img_12.jpg
Let's adopt the following notation:
(Figure 12). It is tacitly assumed in the problem that $ B_i $ is the point of intersection of the given sphere with the edge $ SA_i $, different from $ S $. Therefore, $ x_i > 0 $.
Let the center of the sphere be denoted by $ O $. The following vector equali... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,695 |
XXXVII OM - I - Problem 10
Prove that the sequence $ (\sin n) $ does not have a limit. | Let's consider sequences of open intervals
($ k = 1, 2, 3, \ldots $). The length of each of them (equal to $ \pi/2 $) is greater than $ 1 $. Therefore, in each interval $ I_k $ there is a natural number $ m_k $ (if there are two, let's choose one of them, for example the smaller one, and denote it by $ m_k $). Similar... | proof | Calculus | proof | Yes | Yes | olympiads | false | 1,699 |
XXI OM - II - Problem 3
Prove the theorem: There does not exist a natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $. | \spos{1} Suppose that $ n $ is the smallest natural number greater than one such that $ n \mid 2^n - 1 $, and $ k $ is the smallest natural number such that $ n \mid 2^k - 1 $. By the theorem given to prove in problem 9 of the first stage, we have $ k < n $.
Let $ n = km + r $, where $ 0 \leq r < k $. Since $ n \mid 2^... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,700 |
XX OM - III - Task 6
Given a set of $ n $ points in the plane not lying on a single line. Prove that there exists a circle passing through at least three of these points, inside which there are no other points of the set. | \spos{1} Let us choose from the given set of points $ Z = \{A_1, A_2, \ldots, A_n \} $ two points whose distance is the smallest; suppose these are points $ A_1 $ and $ A_2 $. In the circle $ K $ with diameter $ A_1A_2 $, there is then no other point of the set $ Z $ different from $ A_1 $ and $ A_2 $.
Consider the set... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,701 |
XLIX OM - III - Problem 3
The convex pentagon $ABCDE$ is the base of the pyramid $ABCDES$. A plane intersects the edges $SA$, $SB$, $SC$, $SD$, $SE$ at points $A'$, $B'$, $C'$, $D'$, $E'$ (different from the vertices of the pyramid). Prove that the points of intersection of the diagonals of the quadrilaterals $ABB'A'$... | Let $ \pi $ and $ \pi' $ be the planes containing the points $ A $, $ B $, $ C $, $ D $, $ E $ and $ A $, $ B $, $ C $, $ D' $, $ E $, respectively. Furthermore, let $ K $, $ L $, $ M $, $ N $, $ O $ be the points of intersection of the diagonals of the quadrilaterals $ ABB' $, $ BCC' $, $ CDD' $, $ DEE' $, $ EAA' $ (r... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,702 |
I OM - B - Task 20
The sides of a right-angled triangle are expressed by natural numbers. One of the legs is expressed by the number 10. Calculate the remaining sides of this triangle. | Natural numbers $ x $ and $ y $ representing the length of the hypotenuse and the length of the other leg of the considered triangle satisfy, according to the Pythagorean theorem, the equation
Since $ 10^2 $ is an even number, it follows from equation (1) that the numbers $ x^2 $ and $ y^2 $ are either both even or bo... | x=26,y=24 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,703 |
XL OM - II - Task 6
In triangle $ABC$, through an internal point $P$, lines $CP$, $AP$, $BP$ are drawn intersecting sides $AB$, $BC$, $CA$ at points $K$, $L$, $M$ respectively. Prove that if circles can be inscribed in quadrilaterals $AKPM$ and $KBLP$, then a circle can also be inscribed in quadrilateral $LCMP$. | For a circle inscribed in a quadrilateral (convex or concave), we understand a circle contained within this quadrilateral and tangent to the four lines containing its sides. A necessary and sufficient condition for the existence of such a circle is the equality of the sums of the lengths of opposite sides of the quadri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,705 |
XXVI - II - Problem 5
Prove that if a sphere can be inscribed in a convex polyhedron and each face of the polyhedron can be painted in one of two colors such that any two faces sharing an edge are of different colors, then the sum of the areas of the faces of one color is equal to the sum of the areas of the faces of ... | Connecting the point of tangency of the sphere with any wall with all vertices belonging to that wall, we obtain a decomposition of that wall into a sum of triangles with disjoint interiors, one of whose vertices is the point of tangency of the sphere, and the others are two vertices of the polyhedron belonging to one ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,707 |
I OM - B - Task 4
Find two natural numbers $ a $ and $ b $ given their greatest common divisor
$ D=12 $ and least common multiple $ M=432 $. Provide a method for finding
solutions in the general case. | If the greatest common divisor of numbers $ a $ and $ b $ is 12, then
where numbers $ x $ and $ y $ are coprime. In this case, the least common multiple of numbers $ 12x $ and $ 12y $ is $ 12xy $, hence
Numbers $ x $ and $ y $ can be found by factoring 36 into a product of two coprime factors.
There are two such fact... | (12, | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,712 |
XV OM - I - Problem 3
Prove that the points symmetric to the intersection point of the altitudes of triangle $ABC$ with respect to the lines $AB$, $BC$, and $CA$ lie on the circumcircle of triangle $ABC$. | Let $S$ be the intersection point of the altitudes $AH$ and $BK$ of triangle $ABC$, and let $S'$ be the point symmetric to $S$ with respect to the line $AB$. We will consider three cases:
a) Angles $A$ and $B$ of the triangle are acute (Fig. 1). The intersection point $S$ of the lines $AH$ and $BK$ then lies on the sa... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,714 |
XXVII OM - I - Problem 5
We consider the following two-player game: The first player draws a token with a natural number written on it (let's call it $ n $) and subtracts from it a proper divisor $ d $ of that number (i.e., a divisor satisfying the inequality $ 1 \leq d < n $). The second player proceeds similarly wit... | Every divisor of an odd number is an odd number, and the difference between odd numbers is an even number. Therefore, a player who receives an odd number will pass an even number to the partner. If a player passes an odd number to the partner, then he will receive an even number from him, which is different from $1$. T... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,716 |
XXXI - I - Problem 4
Prove that if a polynomial with integer coefficients takes the value 1 for four different integer arguments, then for no integer argument does it take the value -1. | Integers $ a $, $ b $, $ c $, $ d $, for which the given polynomial $ w(x) $ takes the value $ 1 $ are roots of the polynomial $ w(x) - 1 $. This latter polynomial is therefore divisible by $ x-a $, $ x-b $, $ x-c $, $ x-d $, and thus $ w(x) - 1 = (x - a) (x - b) (x - c) (x - d) \cdot v (x) $, where $ v (x) $ is some p... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,717 |
XLIII OM - III - Problem 1
Segments $ AC $ and $ BD $ intersect at point $ P $, such that $ |PA|=|PD| $, $ |PB|=|PC| $. Let $ O $ be the center of the circumcircle of triangle $ PAB $. Prove that the lines $ OP $ and $ CD $ are perpendicular. | om43_3r_img_11.jpg
Let line $ l $ be the bisector of angles $ APD $ and $ BPC $, and let line $ m $ be the bisector of angles $ APB $ and $ CPD $. Lines $ l $ and $ m $ are perpendicular. From the given equalities of segments, it follows that triangle $ PCD $ is the image of triangle $ PBA $ under axial symmetry with r... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,719 |
LV OM - III - Task 1
Point $ D $ lies on side $ AB $ of triangle $ ABC $. Circles tangent to lines $ AC $ and $ BC $ at points $ A $ and $ B $, respectively, pass through point $ D $ and intersect again at point $ E $. Let $ F $ be the point symmetric to point $ C $ with respect to the perpendicular bisector of segmen... | Suppose that point $ E $ lies inside triangle $ ABC $. Then
Adding these equalities side by side, we have $ 180^\circ > \measuredangle AEB = 180^\circ + \measuredangle ACB > 180^\circ $. The obtained contradiction proves that points $ C $ and $ E $ lie on opposite sides of line $ AB $ (Fig. 1).
From the equality $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,720 |
XVIII OM - I - Problem 11
Three circles of the same radius intersect pairwise and have one common point. Prove that the remaining three intersection points of the circles lie on a circle of the same radius. | We introduce the notation: $ O_1, O_2, O_3 $ - the centers of the given circles, $ M $ the common point of all three circles, $ A_1, A_2, A_3 $ different from $ M $ are the points of intersection of the circles with centers $ O_2 $ and $ O_3 $, $ O_3 $ and $ O_1 $, $ O_1 $ and $ O_2 $ (Fig. 4).
All sides of the quadril... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,722 |
XXX OM - II - Task 6
On the side $ \overline{DC} $ of rectangle $ ABCD $, where $ \frac{AB}{AD} = \sqrt{2} $, a semicircle is constructed externally. Any point $ M $ on the semicircle is connected to $ A $ and $ B $ with segments, intersecting $ \overline{DC} $ at points $ K $ and $ L $, respectively. Prove that $ DL^... | Let $ S $ and $ T $ be the points of intersection of the line $ AB $ with the lines $ MD $ and $ MC $, respectively (Fig. 14). Let us choose a point $ P \in AB $ such that $ DP || CT $. Then $ \measuredangle SDP = \measuredangle DMC = \frac{\pi}{2} $ as an inscribed angle in a circle subtended by a diameter. We obvious... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,724 |
XLV OM - III - Task 1
Determine all triples $ (x,y,z) $ of positive rational numbers for which the numbers $ x+y+z $, $ \frac{1}{x}+\frac{1}{y}+\frac{1}{z} $, $ xyz $ are natural. | Let $ (x,y,z) $ be a triple of rational numbers satisfying the given condition. The numbers
are natural; therefore, the product
is also a natural number. The numbers $ x $, $ y $, $ z $ are roots of the polynomial
whose all coefficients are integers, and the coefficient of the highest power of the variable $ t $ is ... | (1,1,1),(2,2,1),(2,1,2),(1,2,2) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,725 |
LII OM - III - Problem 5
Points $ K $ and $ L $ lie on sides $ BC $ and $ CD $ of parallelogram $ ABCD $, respectively, such that $ BK \cdot AD = DL \cdot AB $. Segments $ DK $ and $ BL $ intersect at point P. Prove that $ \measuredangle DAP = \measuredangle BAC $. | Let $ Y $ be the intersection point of lines $ AP $ and $ CD $, and let $ X $ be the intersection point of lines $ DK $ and $ AB $ (Fig. $ 1 $ and $ 2 $). Then
Hence, and from the equality given in the problem, we obtain:
om52_3r_img_1.jpg
om52_3r_img_2.jpg
This proves that triangles $ ABC $ and $ ADY $ are s... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,726 |
XI OM - III - Task 6
On the perimeter of a rectangle, a point $ M $ is chosen. Find the shortest path whose beginning and end is point $ M $ and which has some common point with each side of the rectangle. | Suppose that point $M$ lies on side $AB$ of rectangle $ABCD$ (Fig. 31).
On sides $BC$, $CD$, and $DA$, we select points $N$, $P$, $Q$ in the following manner. If point $M$ does not coincide with either point $A$ or $B$, we draw $MN \parallel AC$, $NP \parallel BD$, $PQ \parallel AC$; then $MQ \parallel BD$ and we obtai... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,727 |
XXXIV OM - I - Problem 10
Do there exist three points in the plane with coordinates of the form $ a+b\sqrt[3]{2} $, where $ a, b $ are rational numbers, such that at least one of the distances from any point in the plane to each of these points is an irrational number? | Such points do exist. We will show that, for example, the triplet of points $ A = (\sqrt[3]{2},0) $, $ B=(-\sqrt[3]{2},0) $, $ C = (0,0) $ has the desired property. Suppose, for instance, that the distance from some point $ P=(x,y) $ to each of the points $ A $, $ B $, $ C $ is a rational number. In this case, the numb... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,730 |
XLII OM - II - Problem 3
Given are positive integers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ such that $ a+b = c+d = e+f = 101 $. Prove that the number $ \frac{ace}{bdf} $ cannot be expressed in the form of a fraction $ \frac{m}{n} $ where $ m $, $ n $ are positive integers with a sum less than $ 101 $. | From the assumption of the task, the following congruences hold:
We multiply these three relations side by side:
Suppose, contrary to the thesis, that
where $ m $, $ n $ are positive integers such that $ m + n < 101 $. We then have the relationship
which means
This implies that the number $ 101 $ is a divisor of ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,734 |
XXVI - I - Task 1
At the ball, there were 42 people. Lady $ A_1 $ danced with 7 gentlemen, Lady $ A_2 $ danced with 8 gentlemen, ..., Lady $ A_n $ danced with all the gentlemen. How many gentlemen were at the ball? | The number of ladies at the ball is $ n $, so the number of gentlemen is $ 42-n $. The lady with number $ k $, where $ 1 \leq k \leq n $, danced with $ k+6 $ gentlemen. Therefore, the lady with number $ n $ danced with $ n+ 6 $ gentlemen. These were all the gentlemen present at the ball. Thus, $ 42-n = n + 6 $. Solving... | 24 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,735 |
XXVIII - II - Task 2
Let $ X $ be an internal point of triangle $ ABC $. Prove that the product of the distances from point $ X $ to the vertices $ A, B, C $ is at least eight times greater than the product of the distances from this point to the lines $ AB, BC, CA $. | Let $ h_a $, $ h_b $, $ h_c $ be the lengths of the altitudes of triangle $ ABC $ drawn from vertices $ A $, $ B $, $ C $ respectively, $ x_a $, $ x_b $, $ x_c $ - the distances from point $ X $ to the lines $ BC $, $ CA $, $ AB $ respectively, and $ S_a $, $ S_b $, $ S_c $ - the areas of triangles $ XBC $, $ XCA $, $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,736 |
I OM - B - Task 5
A motorboat set off upstream at 9:00, and at the same moment, a ball was thrown from the motorboat into the river. At 9:15, the motorboat turned around and started moving downstream. At what time did the motorboat catch up with the ball? | On standing water, the motorboat would have returned to the ball within the next 15 minutes, i.e., at 9:30. The same would happen on the river, as the current equally carries the motorboat and the ball. The motorboat will catch up with the ball at 9:30.
If someone did not come up with the above simple reasoning, they ... | 9:30 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,737 |
VIII OM - II - Task 5
Given a segment $ AB $ and a line $ m $ parallel to this segment. Find the midpoint of segment $ AB $ using only a ruler, i.e., by drawing only straight lines. | We arbitrarily choose a point $ C $ lying on the opposite side of line $ m $ from segment $ AB $ (Fig. 16). Segments $ AC $ and $ BC $ intersect line $ m $ at points, which we will denote by the letters $ D $ and $ E $, respectively. We will denote the point of intersection of the diagonals of trapezoid $ ABED $ by the... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,738 |
VI OM - I - Task 4
Prove that if there exists a sphere tangent to all edges of a tetrahedron, then the sums of the opposite edges of the tetrahedron are equal, and that the converse theorem is also true. | a) We will show that if there exists a sphere tangent to all the edges of the tetrahedron $ABCD$, then $AB + CD = AC + BD = AD + BC$. Let $O$ denote the center, and $r$ the radius of such a sphere. The points of tangency of the sphere with the edges of the tetrahedron divide each edge into two segments, with each of th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,739 |
XLVII OM - I - Problem 11
In a ski jumping competition, 65 participants are involved. They jump one by one according to a predetermined order. Each participant performs one jump. We assume that the scores obtained are all non-zero and that every final ranking is equally likely. At any point during the competition, the... | We will consider an analogous problem for any number of participating athletes. Let us denote this number by $ n $. The task is to determine the probability of the event:
For each number $ k \in \{0,1, \ldots, n-1 \} $, we define the following event:
According to the assumption, each final order is equally ... | p>\frac{1}{16} | Combinatorics | proof | Yes | Yes | olympiads | false | 1,741 |
XXXV OM - III - Task 4
We toss a coin $ n $ times and record the result as a sequence $ (a_1, a_2, \ldots, a_n) $, where $ a_i = 1 $ or $ a_i = 2 $ depending on whether an eagle or a tail appeared in the $ i $-th toss. We assume $ b_j = a_1 + a_2 + \ldots + a_j $ for $ j = 1, 2, \ldots, n $, $ p(n) $ is the probabilit... | We directly observe that $ p(1) = \frac{1}{2} $, $ p(2) = \frac{1}{4} $. Suppose that $ n \geq 3 $. Notice that $ b_j \geq j $ for every $ j $. The number $ n $ can appear in the sequence $ (b_1, b_2, \ldots, b_n) $ in the following two cases:
1. Some term of this sequence is equal to $ n-1 $, for example, $ b_k = n-1 ... | p(n)=\frac{1}{2}p(n-1)+\frac{1}{2}p(n-2) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,743 |
XXII OM - I - Problem 8
Given a cube with edge length 1; let $ S $ denote the surface of this cube. Prove that there exists a point $ A \in S $ such that
1) for every point $ B \in S $, there exists a broken line connecting $ A $ and $ B $ with a length not greater than 2 contained in $ S $;
2) there exists a point $ ... | Let $ A $ be the midpoint of one of the faces of a cube, and $ B $ - the midpoint of the opposite face (Fig. 9). We will prove that point $ A $ satisfies condition 1) of the problem, and points $ A $ and $ B $ satisfy condition 2).
1) Let us cut the surface $ S $ of the cube along the broken lines $ BPC $, $ BQD $, $ B... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,744 |
LIV OM - I - Task 10
We have a deck of 52 cards. Shuffling will be called the execution of the following actions: an arbitrary division of the deck into an upper and a lower part, and then an arbitrary mixing of the cards while maintaining the order within each part. Formally, shuffling is any mixing of the cards wher... | We will prove that after five shuffles, it is impossible to obtain the reverse order of the initial order.
Notice that cards that end up in the same part during a certain shuffle do not change their relative order after that shuffle.
Among the 52 cards, during the first shuffle, at least 26 cards will be in one of the ... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,748 |
XLIX OM - I - Zadanie 2
Proste zawierające wysokości trójkąta $ ABC $, wpisanego w okrąg o środku $ O $, przecinają się w punkcie $ H $, przy czym $ |AO| =|AH| $. Obliczyć miarę kąta $ CAB $.
|
Oznaczmy przez $ M $ środek boku $ BC $. Rozwiązanie zadania będzie oparte na równości
która zachodzi w każdym trójkącie, niezależnie od założenia, że $ |AO| =|AH| $. Oto jej dowód:
Role punktów $ B $ i $ C $ są symetryczne; można przyjąć, że $ |AB|\geq|AC| $ i wobec tego kąt ABC jest ostry. Każdy z dwóch pozost... | 60or120 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,749 |
L OM - II - Task 5
Let $ S = \{1, 2,3,4, 5\} $. Determine the number of functions $ f: S \to S $ satisfying the equation $ f^{50} (x) = x $ for all $ x \in S $.
Note: $ f^{50}(x) = \underbrace{f \circ f \circ \ldots \circ f}_{50} (x) $. | Let $ f $ be a function satisfying the conditions of the problem. For numbers $ x \neq y $, we get $ f^{49}(f(x)) = x \neq y = f^{49}(f(y)) $, hence $ f(x) \neq f(y) $. Therefore, $ f $ is a permutation of the set $ S $. Denote by $ r(x) $ ($ x \in S $) the smallest positive integer such that $ f^{r(x)}(x) = x $. Then ... | 50 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,750 |
XXII OM - III - Task 6
Given a regular tetrahedron with edge length 1. Prove that:
1) On the surface $ S $ of the tetrahedron, there exist four points such that the distance from any point on the surface $ S $ to one of these four points does not exceed $ \frac{1}{2} $.
2) On the surface $ S $, there do not exist thre... | 1) First, let us note that in an equilateral triangle with a side length of $1$, the midpoints of two sides have the property that any point belonging to the triangle is no more than $\frac{1}{2}$ away from one of them. Indeed, by drawing circles with centers at these points (Fig. 15) and radii of length $\frac{1}{2}$,... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,752 |
XXXVIII OM - I - Problem 10
Let $ n = 4^k \cdot 5^m $ ($ k $, $ m $ — natural numbers). Prove that every natural number less than $ n $ is a divisor of $ n $ or a sum of different divisors of $ n $. | Note. By a natural number, we mean a positive integer in this task. For $ k = 0 $, the thesis is - as easily noticed - false.
Let $ k \geq 1 $ and $ m \geq 1 $ be fixed and let $ s < 4^k5^m $ be a natural number. We divide $ s $ by $ 5^m $ to get the quotient $ a $ and the remainder $ r $:
Let's write the numbers... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,755 |
XXXIII OM - I - Problem 1
A regular 25-gon $ A_1, A_2, \ldots, A_{25} $ is inscribed in a circle with center $ O $ and radius of length $ r $. What is the maximum length of the vector that is the sum of some of the vectors $ \overrightarrow{OA_1}, \overrightarrow{OA_2}, \ldots, \overrightarrow{OA_{25}} $? | Consider a certain subset $X$ of the set of data vectors, for which the vector $\overrightarrow{a}$ will be equal to the sum of all vectors in the subset $X$. If the vector $\overrightarrow{OA_i}$ does not belong to $X$ and forms an acute angle with the vector $\overrightarrow{a}$, then the sum of all vectors in the se... | 2r\cos\frac{6\pi}{25} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,757 |
XXX OM - I - Task 2
In the plane, there are $ n $ points ($ n > 4 $), no three of which are collinear. Prove that there are at least $ \binom{n-3}{2} $ convex quadrilaterals whose vertices are some of the given points. | If a quadrilateral is not convex, then in each pair of its opposite sides, there is a side such that the line containing it intersects the other side of the pair. Therefore, if in a certain quadrilateral there exists a pair of opposite sides such that no line containing one of these sides intersects the other, then the... | \binom{n-3}{2} | Combinatorics | proof | Yes | Yes | olympiads | false | 1,761 |
XXXII - II - Task 3
Prove that there does not exist a continuous function $ f: \mathbb{R} \to \mathbb{R} $ satisfying the Condition $ f(f(x)) = - x $ for every $ x $. | The function $ f $ satisfying the given condition must be injective, because if for some $ x_1 $, $ x_2 $ it were $ f(x_1) = f (x_2) $, then $ f(f(x_1)) = f(f(x_2)) $, which implies $ -x_1 = -x_2 $, so $ x_1 = x_2 $. A continuous and injective function must be monotonic. If $ f $ were an increasing function, then $ f(f... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,762 |
LX OM - I - Task 8
The diagonals of the base $ABCD$ of the pyramid $ABCDS$ intersect at a right angle at point $H$, which is the foot of the pyramid's height. Let $K$, $L$, $M$, $N$ be the orthogonal projections of point $H$ onto the faces $ABS$, $BCS$, $CDS$, $DAS$, respectively. Prove that the lines $KL$, $MN$, and ... | At the beginning, we will show that the lines $KL$ and $AC$ lie in the same plane.
From the assumptions of the problem, it follows that the lines $AC$, $BD$, and $HS$ are mutually perpendicular. This means that any line contained in the plane determined by two of these lines is perpendicular to the third. In particular... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,763 |
XIII OM - III - Task 3
What condition should the angles of triangle $ ABC $ satisfy so that the angle bisector of $ A $, the median drawn from vertex $ B $, and the altitude drawn from vertex $ C $ intersect at one point? | The common point of the bisector $AD$, the median $BS$, and the altitude $CH$ of triangle $ABC$ can only be an internal point of the triangle, since $AD$ and $BS$ intersect inside the triangle. Such a point can therefore exist only if the line $CH$ contains a segment lying inside the triangle, i.e., if angles $A$ and $... | \tanA\cdot\cosB=\sinC | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,764 |
XXIX OM - I - Problem 9
Prove that from any sequence of natural numbers, one can select a subsequence in which any two terms are coprime, or a subsequence in which all terms have a common divisor greater than 1. | Let $ p $ be a fixed prime number. If infinitely many terms of the sequence are divisible by $ p $, then the terms divisible by $ p $ form a subsequence, all of whose terms have a common divisor greater than $ 1 $.
Consider, then, the case where for each prime number $ p $ only a finite number of terms of the sequence ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,765 |
XIV OM - I - Problem 4
In the great circle of a sphere with radius $ r $, a regular polygon with $ n $ sides is inscribed. Prove that the sum of the squares of the distances from any point $ P $ on the surface of the sphere to the vertices of the polygon equals $ 2nr^2 $. | Let the symbol $ \overline{AB} $ denote the vector with initial point $ A $ and terminal point $ B $, and $ AB $ denote the length of this vector. Then
from which
it follows that
Therefore
Thus, the sum of vectors $ \overline{OA_1} + \overline{OA_2} + \ldots + \overline{OA_n} $ equals zero. For ... | 2nr^2 | Geometry | proof | Yes | Yes | olympiads | false | 1,766 |
XXV - I - Problem 8
Given a convex polyhedron and a point $ A $ in its interior. Prove that there exists a face $ S $ of this polyhedron such that the orthogonal projection of point $ A $ onto its plane belongs to $ S $. | Let $ S $ be such a face of a given polyhedron that the distance $ d $ from point $ A $ to the plane of face $ S $ is not greater than the distance from this point to the plane of any other face of the polyhedron. Let $ k $ be the ray with origin at point $ A $ perpendicular to the plane of face $ S $. If the ray $ k $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,769 |
XXIV OM - III - Task 5
Prove that every positive proper fraction $ \frac{m}{n} $ can be represented as a finite sum of the reciprocals of different natural numbers. | \spos{1} Let us choose a natural number $ k $ such that $ n \leq 2^k $. Let $ q $ and $ r $ be the quotient and remainder, respectively, of the division of $ 2^km $ by $ n $, i.e., let $ 2^km = qn + r $, where $ 0 \leq r < n $.
Then
We have $ qn \leq gn + r = 2^km < 2^kn $, since $ \displaystyle \frac{m}{n} < 1 $.... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,770 |
XXXVIII OM - I - Problem 8
The sequence of points $ A_1, A_2, \ldots $ in the plane satisfies for every $ n > 1 $ the following conditions: $ |A_{n+1}A_n| = n $ and the directed angle $ A_{n+1}A_nA_1 $ is $ 90^{\circ} $. Prove that every ray starting at $ A_1 $ intersects infinitely many segments $ A_nA_{n+1} $ ($ n =... | Let's denote: $ |A_1A_n| = r_n $, $ |\measuredangle A_nA_1A_{n+1}| = a_n $ for $ n = 2, 3, 4, \ldots $, and moreover $ a = r_2^2 - 1 $ (the problem statement does not specify the length of the segment $ A_1A_2 $, so $ a $ can be any number greater than $ -1 $). From the relationships in the right triangle $ A_1A_nA_{n+... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,772 |
XXXIX OM - I - Problem 9
From the vertices of a regular $ n $-gon ($ n \geq 4 $), four different ones are chosen. Each selection of four vertices is equally probable. Calculate the probability that all the chosen vertices lie on some semicircle (we assume that the endpoints of the semicircle belong to it). | Let $ W $ be the set of all $ n $ vertices of the considered regular $ n $-gon.
Denote the sought probability by $ P(n) $. It is expressed by the fraction
whose denominator is the number of all four-element subsets of the set $ W $, and the numerator is the number of four-point subsets of $ W $ contained in semicirc... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,773 |
LIII OM - III - Task 1
Determine all such triples of natural numbers $ a $, $ b $, $ c $, such that the numbers $ a^2 +1 $ and $ b^2 +1 $ are prime and | Without loss of generality, we can assume that $ a \leq b $. Let $ p = b^2 +1 $. Then the number $ (c^2 +1) - (b^2 + 1) = (c - b)(c + b) $ is divisible by $ p $. From the given equation in the problem, it follows that $ b < c $. Furthermore,
Thus $ c - b < p $ and $ c + b < 2p $. Since $ p | (c - b)(c + b) $, it must ... | (1,2,3)(2,1,3) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,774 |
V OM - II - Task 2
Prove that among ten consecutive natural numbers there is always at least one, and at most four numbers that are not divisible by any of the numbers $ 2 $, $ 3 $, $ 5 $, $ 7 $. | In a sequence consisting of ten consecutive natural numbers, there are five even numbers and five odd numbers. Therefore, the problem can be reduced to the following:
Prove that among five consecutive odd numbers, there is at least one, and at most four numbers not divisible by any of the numbers $3$, $5$, $7$.
Notice ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,776 |
XLVIII OM - III - Problem 6
On a circle of radius $ 1 $, there are $ n $ different points ($ n \geq 2 $). Let $ q $ be the number of segments with endpoints at these points and length greater than $ \sqrt{2} $. Prove that $ 3q \leq n^2 $. | We will call a segment with endpoints in the considered set long if its length is greater than $ \sqrt{2} $, and short otherwise. Thus, $ q $ is the number of long segments. We inscribe a square $ ABCD $ in a given circle so that none of the given $ n $ points is a vertex of the square. Let us assume that on the arcs $... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,778 |
IX OM - III - Task 2
Each side of the convex quadrilateral $ABCD$ is divided into three equal parts; a line is drawn through the division points on sides $AB$ and $AD$ that are closer to vertex $A$, and similarly for vertices $B$, $C$, and $D$. Prove that the centroid of the quadrilateral formed by the drawn lines coi... | We will adopt the notations shown in Fig. 27. We need to prove that the center of gravity of quadrilateral $ABCD$ coincides with the center of gravity of quadrilateral $MNPQ$, which is a parallelogram, since lines $MN$ and $QP$ (as well as lines $14$ and $85$) are parallel to line $AC$, and lines $MQ$ and $NP$ are para... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,779 |
L OM - II - Task 3
Convex quadrilateral $ABCD$ is inscribed in a circle. Points $E$ and $F$ lie on sides $AB$ and $CD$ respectively, such that $AE: EB = CF: FD$. Point $P$ lies on segment $EF$ and satisfies the condition $EP: PF = AB: CD$. Prove that the ratio of the areas of triangles $APD$ and $BPC$ does not depend ... | First, assume that lines $AD$ and $BC$ are not parallel and intersect at point $S$. Then triangles $ASB$ and $CSD$ are similar. Since $AE : EB = CF : FD$, triangles $ASE$ and $CSF$ are also similar. Therefore, $\measuredangle DSE = \measuredangle CSF$. Moreover,
from which we get $\measuredangle ESP = \measuredangle F... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,780 |
LVI OM - III - Task 3
In a square table of dimensions $ 2n \times 2n $, where $ n $ is a natural number, there are $ 4n^2 $ real numbers with a total sum of 0 (one number on each cell of the table). The absolute value of each of these numbers is no greater than 1. Prove that the absolute value of the sum of all number... | Let's number the rows and columns of the given table with the numbers $1, 2, \ldots, 2n$.
Let $r_1, r_2, \ldots, r_{2n}$ be the sums of the numbers written in the rows numbered $1, 2, \ldots, 2n$, and $c_1, c_2, \ldots, c_{2n}$ be the sums of the numbers in the columns numbered $1, 2, \ldots, 2n$. By changing the numbe... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,784 |
LX OM - I - Task 10
Point $ P $ is the midpoint of the shorter arc $ BC $ of the circumcircle of
triangle $ ABC $, where $ \measuredangle BAC =60^{\circ} $. Point $ M $ is the midpoint
of the segment connecting the centers of two excircles of the given triangle,
tangent to sides $ AB $ and $ AC $, respectively. Prove ... | Let $O_1$ and $O_2$ denote the centers of the excircles of triangle $ABC$ tangent to sides $AC$ and $AB$, respectively. Let $I$ be the incenter of triangle $ABC$, and let $D$ and $E$ be the points where lines $BI$ and $CI$ intersect the circumcircle of triangle $ABC$ for the second time (see Fig. 6).
Then the equality ... | PM=2\cdotBP | Geometry | proof | Yes | Yes | olympiads | false | 1,785 |
XII OM - III - Task 6
Someone wrote six letters to six people and addressed six envelopes to them. In how many ways can the letters be placed into the envelopes so that no letter ends up in the correct envelope? | Let $ F (n) $ denote the number of all ways of placing $ n $ letters $ L_1, L_2, \ldots, L_n $ into $ n $ envelopes $ K_1, K_2, \ldots, K_n $ so that no letter $ L_i $ ends up in the correct envelope $ K_i $, or more simply, so that there is no "hit". We need to calculate $ F (6) $.
Suppose we incorrectly place all let... | 265 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,789 |
XXXII - II - Problem 5
In the plane, there are two disjoint sets $ A $ and $ B $, each consisting of $ n $ points, such that no three points of the set $ A \cup B $ lie on the same line. Prove that there exists a set of $ n $ disjoint closed segments, each of which has one endpoint in set $ A $ and the other in set $ ... | For each one-to-one assignment of points $B_j$ from set $B$ to points $A_i$ from set $A$, we can calculate the sum of distances determined by this assignment for pairs of points $(A_i, B_j)$. Suppose that by assigning point $A_i$ to point $B_i$ for $i = 1,2,\ldots,n$, we obtain the minimum value of this sum. Then the s... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,790 |
IV OM - III - Task 5
A car departs from point $ O $ and drives along a straight road at a constant speed $ v $. A cyclist, who is at a distance $ a $ from point $ O $ and at a distance $ b $ from the road, wishes to deliver a letter to the car. What is the minimum speed at which the cyclist should ride to achieve his ... | We assume that $ b > 0 $; let the Reader formulate the answer to the question in the case when $ b = 0 $, i.e., when the cyclist is on the road.
Let $ M $ be the point where the cyclist is located, $ S $ - the meeting point, $ \alpha $ - the angle $ MOS $, $ t $ - the time that will elapse from the initial moment to th... | v\sin\alpha | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,794 |
XXVIII - I - Task 12
Determine in which of the polynomials $ (1 - 9x + 7x^2 - 6x^3)^{1976} $, $ (1 + 9x - 7x^2 + 6x^3)^{1976} $ the coefficient of $ x^{1976} $ is greater. | A monomial of degree 1976 in the first polynomial is the sum of a certain number of products of the form
\[
x^a y^b z^c
\]
where $ a, b, c \geq 0 $ and $ a + 2b + 3c = 1976 $. From the last equation, it follows that $ a + c = 1976 - 2b - 2c $ is an even number. Therefore, each monomial (1) has a positive coefficient.... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,795 |
LVIII OM - III - Problem 3
The plane has been divided by horizontal and vertical lines into unit squares.
A positive integer must be written in each square so that each
positive integer appears on the plane exactly once.
Determine whether this can be done in such a way that each written number
is a divisor of the sum ... | We will prove that the postulated method of writing numbers exists.
Let us shade some unit squares and draw a path passing through each unit square exactly once as shown in Fig. 15. Then, in the marked squares of the path, we write numbers according to the following rules:
We start from the square marked with a thick... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,796 |
XLII OM - II - Problem 1
The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2,\ldots,n $. Prove that | Induction. For $ n = 1 $, the inequality to be proven holds (with an equality sign; since $ a_1 + b_1 = c_1 + d_1 $, in accordance with the assumption).
Let us fix a natural number $ n \geq 1 $ and assume the validity of the given theorem for this very number $ n $. We need to prove its validity for $ n + 1 $.
Let then... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,798 |
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 | 1,801 |
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 | 1,803 |
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 | 1,804 |
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 | 1,805 |
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 | 1,807 |
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 | 1,808 |
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 | 1,809 |
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 | 1,810 |
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 | 1,813 |
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 | 1,814 |
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 | 1,816 |
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 | 1,817 |
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 | 1,818 |
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