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XXIX OM - I - Problem 7
For a fixed natural number $ n>2 $, we define: $ x_1 = n $, $ y_1=1 $, $ x_{i+1} = \left[\frac{1}{2}(x_i+y_i)\right] $, $ y_{i+1} = \left[ \frac{n}{x_{i+1}}\right] $. Prove that the smallest of the numbers $ x_1, x_2, \ldots, x_n $ is equal to $ [\sqrt{n}] $.
Note. $ [a] $ denotes the greatest ... | For any integer $ k $, the number $ \left[ \frac{k}{2} \right] $ is equal to $ \frac{k}{2} $ or $ \frac{k-1}{2} $ depending on whether $ k $ is even or odd. Therefore, $ \left[ \frac{k}{2} \right] \geq \frac{k-1}{2} $ for any integer $ k $. We also have $ a \geq [a] > a - 1 $ by the definition of the symbol $ [a] $. Fi... | [\sqrt{n}] | Number Theory | proof | Yes | Yes | olympiads | false | 573 |
XIX OM - III - Problem 6
Given a set of $ n > 3 $ points, no three of which are collinear, and a natural number $ k < n $. Prove the following statements:
1. If $ k \leq \frac{n}{2} $, then each point in the given set can be connected to at least $ k $ other points in the set in such a way that among the drawn segment... | $ 1^\circ $. Suppose $ k \leq \frac{n}{2} $. From the given set $ Z $, select a part $ Z_1 $ consisting of $ \left[ \frac{n}{2} \right] $ points*); the remaining part $ Z_2 $ contains $ \left[ \frac{n}{2} \right] $ points if $ n $ is even, and $ \left[ \frac{n}{2} \right] + 1 $ points if $ n $ is odd. Since $ k $ is an... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 574 |
LVII OM - II - Problem 3
Positive numbers $ a, b, c $ satisfy the condition $ ab+bc+ca = abc $. Prove that | Dividing the equality $ ab+bc+ca = abc $ by $ abc $ on both sides, we get
Let us substitute: $ x=1/a,\ y =1/b,\ z =1/c $. Then the numbers $ x, y, z $ are positive, and their sum is 1. Moreover,
Thus, the inequality to be proved takes the form
We will show that for any positive numbers $ x, y $, th... | proof | Algebra | proof | Yes | Yes | olympiads | false | 575 |
XXIX OM - I - Problem 10
Point $ O $ is an internal point of a convex quadrilateral $ ABCD $, $ A_1 $, $ B_1 $, $ C_1 $, $ D_1 $ are the orthogonal projections of point $ O $ onto the lines $ AB $, $ BC $, $ CD $, and $ DA $, respectively, $ A_{i+1} $, $ B_{i+1} $, $ C_{i+1} $, $ D_{i+1} $ are the orthogonal projectio... | The construction given in the task is not always feasible. For example, three of the points $A_{i+1}$, $B_{i+1}$, $C_{i+1}$, $D_{i+1}$ may lie on the same line. In this case, the task loses its meaning. Therefore, we will provide a solution to the task with the additional assumption that all considered quadrilaterals e... | proof | Geometry | proof | Yes | Yes | olympiads | false | 581 |
LI OM - I - Task 4
Each point of the circle is painted one of three colors. Prove that some three points of the same color are vertices of an isosceles triangle. | Let $A_1, A_2, \ldots, A_{13}$ be the vertices of an arbitrary regular 13-gon inscribed in a given circle. We will show that among any five vertices of this 13-gon, there will be three that form the vertices of an isosceles triangle. Since among any thirteen points on a given circle, there will always be five colored t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 584 |
IX OM - II - Problem 6
On a plane, there are two circles $ C_1 $ and $ C_2 $ and a line $ m $. Find a point on the line $ m $ from which tangents can be drawn to the circles $ C_1 $ and $ C_2 $ that are equally inclined to the line $ m $. | If the given circles have a common tangent $s$, intersecting the line $m$ at point $S$, then we can consider that point $S$ satisfies the conditions of the problem, as through this point pass two coinciding tangents to the given circles equally inclined to the line $m$.
Ignoring this trivial solution, we will look for ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 585 |
XXXV OM - I - Problem 8
Let $ n $ be an even natural number. Prove that a quadrilateral can be divided into $ n $ triangles, whose vertices lie at the vertices of the quadrilateral or inside the quadrilateral, and each side of the triangle is either a side of the quadrilateral or a side of another triangle. | At least one diagonal of the quadrilateral is contained within its interior. Let this be, for example, the diagonal $ \overline{AC} $. If $ n = 2 $, then the division of the quadrilateral $ ABCD $ into triangles $ ABC $ and $ ACD $ is the desired division. Suppose that $ n = 2k $ is an even number. Choose any points $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 588 |
V OM - I - Problem 7
In the plane, a line $ p $ and points $ A $ and $ B $ are given. Find a point $ M $ on the line $ p $ such that the sum of the squares $ AM^2 + BM^2 $ is minimized. | The geometric locus of points $ X $ in the plane for which $ AX^2 + BX^2 $ has a given value $ 2k^2 $ is a certain circle whose center lies at the midpoint $ S $ of segment $ AB $; the radius of this circle is larger the larger $ k $ is.
The smallest circle centered at $ S $ that intersects a line $ p $ is the circle t... | M | Geometry | math-word-problem | Yes | Yes | olympiads | false | 589 |
XXIX OM - I - Problem 12
Determine the least upper bound of such numbers $ \alpha \leq \frac{\pi}{2} $, that every acute angle $ MON $ of measure $ \alpha $ and every triangle $ T $ on the plane have the following property. There exists a triangle $ ABC $ isometric to $ T $ such that the side $ \overline{AB} $ is para... | First, note that if $Q$ is the midpoint of side $\widehat{AC}$, then $\measuredangle ABQ \leq \frac{\pi}{6}$. Indeed, denoting by $D$ the projection of point $A$ onto the line $BQ$, we get $AD \leq AQ = \frac{1}{2} AC \leq \frac{1}{2} AB$ (Fig. 9).
Therefore, $\sin \measuredangle ABQ = \frac{AD}{AB} \leq \frac{1}{2}$. ... | \frac{\pi}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 592 |
XII OM - III - Task 4
Prove that if each side of a triangle is less than $ 1 $, then its area is less than $ \frac{\sqrt{3}}{4} $. | In every triangle, at least one angle does not exceed $ 60^\circ $. Suppose that in triangle $ ABC $ with sides less than one unit, $ \measuredangle C \leq 60^\circ $. Then | proof | Geometry | proof | Yes | Yes | olympiads | false | 593 |
XLIII OM - I - Problem 12
On a plane, four lines are drawn such that no two of them are parallel and no three have a common point. These lines form four triangles. Prove that the orthocenters of these triangles lie on a single straight line.
Note: The orthocenter of a triangle is the point where its altitudes intersec... | Let's denote four given lines by $k$, $l$, $m$, $n$, and their points of intersection by $O$, $P$, $Q$, $R$, $S$, $T$, such that
None of these points coincide; this is guaranteed by the conditions of the problem.
Figure 5 shows one possible configuration; however, for the further course of reasoning, it does not matte... | proof | Geometry | proof | Yes | Yes | olympiads | false | 596 |
XVII OM - I - Problem 12
Prove the theorem: If the sum of the planar angles at the vertex of a regular pyramid is equal to $180^{\circ}$, then in this pyramid, the center of the circumscribed sphere coincides with the center of the inscribed sphere. | We introduce the following notations: $ AB $ - the edge of the base of the given regular pyramid, $ S $ - its apex, $ H $ - the projection of point $ S $ onto the base, $ O $ - the center of the sphere circumscribed around the pyramid, $ M $ - the center of the circle circumscribed around triangle $ ASB $ (Fig. 9).
To ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 597 |
XLV OM - I - Problem 4
Given a circle with center $ O $, a point $ A $ inside this circle, and a chord $ PQ $, which is not a diameter, passing through $ A $. Lines $ p $ and $ q $ are tangent to the considered circle at points $ P $ and $ Q $, respectively. Line $ l $ passing through point $ A $ and perpendicular to ... | om45_1r_img_1.jpg
We start by observing that points $P$ and $K$ lie on one side of the line $OA$, while points $Q$ and $L$ lie on the other side (regardless of whether point $A$ is closer to the endpoint $P$ or $Q$ on the chord $PQ$). The chord $PQ$ of the given circle is not a diameter, so the tangents $p$ and $q$ int... | proof | Geometry | proof | Yes | Yes | olympiads | false | 598 |
XL OM - I - Task 5
For a given natural number $ n $, determine the number of sequences $ (a_1, a_2, \ldots , a_n) $ whose terms $ a_i $ belong to the set $ \{0,1,2,3,4\} $ and satisfy the condition $ |a_i - a_{i+1}| = 1 $ for $ i = 1,2,\ldots,n-1 $. | We will divide all sequences satisfying the conditions given in the problem into three types:
sequences of type I — ending with the symbol $ 0 $ or $ 4 $;\\
sequences of type II — ending with the symbol $ 1 $ or $ 3 $;\\
sequences of type III — ending with the symbol $ 2 $.
Let $ x_n $, $ y_n $, $ z_n $ denote t... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 601 |
LVII OM - I - Problem 4
Participants in a mathematics competition solved six problems, each graded with one of the scores 6, 5, 2, 0. It turned out that
for every pair of participants $ A, B $, there are two problems such that in each of them $ A $ received a different score than $ B $.
Determine the maximum number ... | We will show that the largest number of participants for which such a situation is possible is 1024. We will continue to assume that the permissible ratings are the numbers 0, 1, 2, 3 (instead of 5 points, we give 4, and then divide each rating by 2).
Let $ P = \{0,1,2,3\} $ and consider the set
Set $ X $ obviou... | 1024 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 602 |
XXVI - I - Problem 12
In space, there is a cube with side $ a $, and spheres $ B_1, B_2, \ldots, B_n $ of arbitrary radii such that every point of the cube belongs to at least one of the spheres. Prove that among these spheres, one can select pairwise disjoint spheres such that the sum of their volumes is not less tha... | By induction on $ N $ we will prove
Theorem. If in a space there is a set $ F $ of volume $ V $ contained in the sum of $ N $ open balls $ B_1, B_2, \ldots, B_N $, then there exists a subset $ B_{i_1}, B_{i_2}, \ldots, B_{i_r} $ of these balls such that the balls belonging to this subset are pairwise disjoint and the s... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 604 |
XXXII - I - Problem 4
On the sides of an acute triangle $ ABC $, squares $ ABED $, $ BCGF $, $ ACHI $ are constructed outside the triangle. Prove that the medians of the triangle formed by the lines $ EF $, $ GH $, $ DI $ are perpendicular to the sides of triangle $ ABC $. | Let $K$, $L$, $M$ be the vertices of the formed triangle lying at the squares $ABED$, $BCGF$, $ACHI$, respectively. Let $KO$ and $LO$ be lines perpendicular to segments $\overline{AB}$ and $\overline{BC}$, respectively (Fig. 8).
om32_1r_img_8.jpg
Triangles $KLO$ and $BEF$ have corresponding sides that are parallel, so ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 608 |
LVIII OM - III - Problem 1
In an acute triangle $ABC$, point $O$ is the center of the circumscribed circle, segment $CD$ is the altitude, point $E$ lies on side $AB$, and point $M$ is the midpoint of segment $CE$. The line perpendicular to line $OM$ and passing through point $M$ intersects lines $AC$, $BC$ at points $... | Let's draw a line through point $ M $ parallel to side $ AB $, which intersects segments $ AC $ and $ BC $ at points $ P $ and $ Q $ respectively (Fig. 12). Then points $ P $ and $ Q $ are the midpoints of sides $ AC $ and $ BC $ respectively. Since point $ O $ is the center of the circumcircle of triangle $ ABC $, we ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 609 |
XXIII OM - I - Problem 7
A broken line contained in a square with a side length of 50 has the property that the distance from any point of this square to it is less than 1. Prove that the length of this broken line is greater than 1248. | Let the broken line $ A_1A_2 \ldots A_n $ have the property given in the problem. Denote by $ K_i $ ($ i= 1, 2, \ldots, n $) the circle with center at point $ A_i $ and radius of length $ 1 $, and by $ F_i $ ($ i= 1, 2, \ldots, n-1 $) the figure bounded by segments parallel to segment $ \overline{A_iA_{i+1}} $ and at a... | 1248 | Geometry | proof | Yes | Yes | olympiads | false | 613 |
XXV OM - II - Problem 6
Given is a sequence of integers $ a_1, a_2, \ldots, a_{2n+1} $ with the following property: after discarding any term, the remaining terms can be divided into two groups of $ n $ terms each, such that the sum of the terms in the first group is equal to the sum of the terms in the second. Prove ... | \spos{1} First, note that if a sequence of real numbers satisfies the conditions of the problem, then so does any sequence of the form
where $ k $ is any real number.
Suppose that a sequence (1) of integers satisfies the conditions of the problem and $ a_1 \ne a_2 $. Without loss of generality, we can assu... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 614 |
XXXVIII OM - III - Problem 3
Given is a polynomial $ W $ with non-negative integer coefficients. We define a sequence of numbers $ (p_n) $, where $ p_n $ is the sum of the digits of the number $ W(n) $. Prove that some number appears in the sequence $ (p_n) $ infinitely many times. | Let $ W(x) = a_mx^m + \ldots +a_1x+a_0 $ ($ a_i $ - non-negative integers) and let $ q $ be the number of digits in the decimal representation of the largest of the numbers $ a_0, \ldots, a_m $. Let $ \overline{a_i} $ denote the $ q $-digit decimal representation of $ a_i $; if $ a_i $ has fewer than $ q $ digits, we p... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 615 |
XXI OM - I - Problem 2
Given is the sequence $ \{c_n\} $ defined by the formulas $ c_1=\frac{a}{2} $, $ c_{n+1}=\frac{a+c_n^2}{2} $ where $ a $ is a given number satisfying the inequality $ 0 < a < 1 $. Prove that for every $ n $ the inequality $ c_n < 1 - \sqrt{1 - a} $ holds. Prove the convergence of the sequence $ ... | From the definition of the sequence $ \{c_n\} $, it follows that all its terms are positive. By applying the method of induction, we will show that for every natural number $ n $, the inequality
holds.
We first state that $ c_1 < 1 - \sqrt{1 - a} $, i.e.,
Specifically, we have
This implies (2).
If for some natural ... | 1-\sqrt{1-} | Algebra | proof | Yes | Yes | olympiads | false | 617 |
XLIX OM - I - Problem 11
In a tennis tournament, $ n $ players participated. Each played one match against each other; there were no draws. Prove that there exists a player $ A $ who has either directly or indirectly defeated every other player $ B $, i.e., player $ A $ won against $ B $ or player $ A $ defeated some ... | Let $ A $ be the participant in the tournament who defeated the largest number of opponents. (If several players have the same maximum number of victories, we choose any one of them.) We claim that the player $ A $ selected in this way has defeated, directly or indirectly, all other players.
Let $ B $ be any player who... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 619 |
II OM - III - Task 1
A beam of length $ a $ has been suspended horizontally by its ends on two parallel ropes of equal length $ b $. We rotate the beam by an angle $ \varphi $ around a vertical axis passing through the center of the beam. By how much will the beam be raised? | When solving geometric problems, a properly executed drawing is an important aid to our imagination. We represent spatial figures through mappings onto the drawing plane. There are various ways of such mapping. In elementary geometry, we most often draw an oblique parallel projection of the figure; in many cases, it is... | \sqrt{b^2-(\frac{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 621 |
XIV OM - III - Task 1
Prove that two natural numbers, whose digits are all ones, are relatively prime if and only if the numbers of their digits are relatively prime. | Let $ J_m $ denote the $ m $-digit number whose digits are all ones:
If $ m $ is divisible by $ d $ ($ m $, $ d $ - natural numbers), then $ J_m $ is divisible by $ J_d $. Indeed, if $ m = k \cdot d $ ($ k $ - a natural number), then
where the number $ M $ is an integer.
$ \beta $) If $ m > n $, then
Let the numbers... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 623 |
LX OM - III - Task 1
Each vertex of a convex hexagon is the center of a circle with a radius equal to the length of the longer of the two sides of the hexagon containing that vertex. Prove that if the intersection of all six circles (considered with their boundaries) is non-empty, then the hexagon is regular. | Let's denote the consecutive vertices of the given hexagon by the letters $A, B, C, D, E$, and $F$.
Assume that the intersection of the six circles mentioned in the problem is non-empty and that point
$P$ belongs to this intersection. Consider any side of the hexagon, for the sake of argument, let it be
side $AB$. From... | proof | Geometry | proof | Yes | Yes | olympiads | false | 624 |
LV OM - I - Task 8
Point $ P $ lies inside the tetrahedron $ ABCD $. Prove that | Let $ K $ be the point of intersection of the plane $ CDP $ with the edge $ AB $, and let $ L $ be the point of intersection of the plane $ ABP $ with the edge $ CD $ (Fig. 4).
Similarly, let $ M $ be the point of intersection of the plane $ ADP $ and the edge $ BC $, and let $ N $ denote the common point of the plane ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 625 |
LVII OM - III - Problem 2
Determine all positive integers $ k $ for which the number $ 3^k+5^k $ is a power of an integer with an exponent greater than 1. | If $ k $ is an even number, then the numbers $ 3^k $ and $ 5^k $ are squares of odd numbers, giving a remainder of 1 when divided by 4. Hence, the number $ 3^k + 5^k $ gives a remainder of 2 when divided by 4, and thus is divisible by 2 but not by $ 2^2 $. Such a number cannot be a power of an integer with an exponent ... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 626 |
L OM - I - Task 4
Given real numbers $ x $, $ y $, such that the numbers $ x + y $, $ x^2 + y^2 $, $ x^3 + y^3 $, and $ x^4 + y^4 $ are integers. Prove that for every positive integer $ n $, the number $ x^n + y^n $ is an integer. | Since $ 2xy= (x + y)^2 - (x^2 + y^2) $ and $ 2x^2y^2 = (x^2 + y^2)^2 - (x^4 + y^4) $, the numbers $ 2xy $ and $ 2x^2y^2 $ are integers. If the number $ xy $ were not an integer, then $ 2xy $ would be odd. But then the number $ 2x^2y^2 = (2xy)^2/2 $ would not be an integer. The conclusion is that the number $ xy $ is an... | proof | Algebra | proof | Yes | Yes | olympiads | false | 627 |
V OM - III - Task 5
Prove that if in a tetrahedron $ABCD$ the opposite edges are equal, i.e., $AB = CD$, $AC = BD$, $AD = BC$, then the lines passing through the midpoints of opposite edges are mutually perpendicular and are axes of symmetry of the tetrahedron. | Let $ K $, $ L $, $ M $, $ N $, $ P $, $ Q $ denote the midpoints of the edges of the tetrahedron $ ABCD $, as indicated in Fig. 42. It is sufficient to prove that any of the lines $ KL $, $ MN $, $ PQ $, for example, $ KL $, is an axis of symmetry of the tetrahedron and that it is perpendicular to one of the two remai... | proof | Geometry | proof | Yes | Yes | olympiads | false | 630 |
XXIV OM - I - Problem 12
In a class of n students, a Secret Santa event was organized. Each student draws the name of the person for whom they are to buy a gift, so student $ A_1 $ buys a gift for student $ A_2 $, $ A_2 $ buys a gift for $ A_3 $, ..., $ A_k $ buys a gift for $ A_1 $, where $ 1 \leq k \leq n $. Assumin... | In the result of any drawing, each student draws a certain student from the same class (possibly themselves), and different students draw different students. Therefore, the result of each drawing defines a certain one-to-one mapping (permutation) of the set of all students in the class onto itself. Conversely, each per... | \frac{1}{n} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 636 |
VIII OM - I - Task 6
Find a four-digit number, whose first two digits are the same, the last two digits are the same, and which is a square of an integer. | If $ x $ is the number sought, then
where $ a $ and $ b $ are integers satisfying the inequalities $ 0 < a \leq 9 $, $ 0 \leq b \leq 9 $. The number $ x $ is divisible by $ 11 $, since
Since $ x $ is a perfect square, being divisible by $ 11 $ it must be divisible by $ 11^2 $, so the number
is divisib... | 7744 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 638 |
XXV - I - Problem 3
Prove that the bisectors of the angles formed by the lines containing the opposite sides of a convex quadrilateral inscribed in a circle are respectively parallel to the bisectors of the angles formed by the lines containing the diagonals of this quadrilateral. | Let $ A $, $ B $, $ C $, $ D $ be consecutive vertices of a quadrilateral inscribed in a circle. Let $ O $ be the point of intersection of the lines $ AB $ and $ CD $, and let $ k $ be the angle bisector of $ \measuredangle AOD $ (Fig. 12). If $ A' $ and $ C' $ are the images of points $ A $ and $ C $, respectively, un... | proof | Geometry | proof | Yes | Yes | olympiads | false | 639 |
XIX OM - I - Problem 5
Given a set of $ 2n $ ($ n \geq 2 $) different numbers. How to divide this set into pairs so that the sum of the products of the numbers in each pair is a) the smallest, b) the largest? | By a partition of a set of numbers $ Z $ into pairs, we mean a set of pairs of numbers from $ Z $ such that each number in $ Z $ belongs to one and only one pair.
Let us denote the given numbers by the letters $ a_1, a_2, \ldots, a_{2n} $ in such a way that
The number of all partitions of the set of numbers (1) in... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 641 |
XXXIV OM - I - Problem 1
$ A $ tosses a coin $ n $ times, $ B $ tosses it $ n+1 $ times. What is the probability that $ B $ will get more heads than $ A $? | Consider the following events. $ Z_0 $ - the event that player $ B $ has flipped more heads than player $ A $, $ Z_r $ - the event that $ B $ has flipped more tails than player $ A $. These events are equally probable: if the coin is fair, it does not matter which side is called heads and which is called tails. Thus, $... | \frac{1}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 643 |
XXXVI OM - III - Problem 6
Prove that if in a convex polyhedron with $ k $ faces there are more than $ k/2 $ faces, no two of which have a common edge, then it is impossible to inscribe a sphere in this polyhedron. | Suppose that a sphere can be inscribed in a given polyhedron. Let $S_1, \ldots, S_k$ be the faces of the polyhedron, and $P_1, \ldots, P_k$ the points of tangency of the respective faces with the sphere. If $A$ and $B$ are common vertices of the faces $S_i$ and $S_j$, then the triangles $ABP_i$ and $ABP_j$ are congruen... | proof | Geometry | proof | Yes | Yes | olympiads | false | 645 |
XXIII OM - III - Problem 5
Prove that all subsets of a finite set can be arranged in a sequence, where consecutive terms differ by one element. | Assume that a given finite set $A$ has $n$ elements. By applying induction on $n$, we will prove that there exists such a sequence of all subsets of set $A$ that
(1)
all subsets of set $A$ are included, and
(2)
for any two consecutive subsets $A_i$ and $A_{i+1}$ in the sequence, one of the sets $A_{i+1} - A_i$ an... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 646 |
XI OM - II - Task 3
Given are two circles with a common center $ O $ and a point $ A $. Construct a circle with center $ A $ intersecting the given circles at points $ M $ and $ N $, so that the line $ MN $ passes through the point $ O $. | Suppose a circle with center $A$ intersects the given circles at points $M$ and $N$ lying on a straight line with point $O$. Let $S$ be the midpoint of segment $MN$. The problem will be solved when we can find point $S$. If points $M$ and $N$ lie on the same side of point $O$, then
if, however, $M$ and $N$ lie on oppo... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 648 |
LIV OM - II - Task 5
Point $ A $ lies outside the circle $ o $ with center $ O $. From point $ A $, two tangent lines to circle $ o $ are drawn at points $ B $ and $ C $. A certain tangent to circle $ o $ intersects segments $ AB $ and $ AC $ at points $ E $ and $ F $, respectively. Lines $ OE $ and $ OF $ intersect s... | Let $ X $ be the point of tangency of circle $ o $ with line $ EF $ (Fig. 3). From the equality
of
it follows that triangles $ BEP $ and $ XEP $ are congruent. Therefore, $ BP = XP $.
Similarly, we prove the equality $ CQ = XQ $. This means that triangle $ PQX $ is constructed from segments $ BP $, $ PQ $, and $ QC $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 649 |
XXXVII OM - III - Problem 3
Prove that if $ p $ is a prime number, and the integer $ m $ satisfies the inequality $ 0 \leq m < p—1 $, then $ p $ divides the number $ \sum_{j=1}^p j^m $. | Let's denote the sum $1^m + 2^m + \ldots + p^m$ by $A(p, m)$. We need to prove that under the given conditions
Proof is conducted by induction on $m$ (for a fixed $p$). When $m = 0$, we get the number $A(p, 0) = p$, which is clearly divisible by $p$.
Fix an integer $m$ satisfying $0 < m < p - 1$ and assume that
We wa... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 651 |
XLVI OM - II - Zadanie 1
Wielomian $ P(x) $ ma współczynniki całkowite. Udowodnić, że jeżeli liczba $ P(5) $ dzieli się przez 2, a liczba $ P(2) $ dzieli się przez 5, to liczba $ P(7) $ dzieli się przez 10.
|
Dowód opiera się na spostrzeżeniu, że jeśli $ P(x) = a_0 + a_1x + \ldots + a_nx^n $ jest wielomianem o współczynnikach całkowitych, to dla każdej pary różnych liczb całkowitych $ u $, $ v $ różnica $ P(u) - P(v) $ jest podzielna przez różnicę $ u-v $. Istotnie:
a każda z różnic $ u^k - v^k $ rozkłada się na czyn... | proof | Algebra | proof | Yes | Yes | olympiads | false | 652 |
LIX OM - II - Task 6
Given a positive integer $ n $ not divisible by 3. Prove that there exists a number $ m $ with the following property:
Every integer not less than $ m $ is the sum of the digits of some multiple of the number $ n $. | A good number will be understood as a positive integer that is the sum of the digits of some multiple of the number $ n $.
First of all, note that the sum of two good numbers is also a good number. Indeed, let $ k_1, k_2 $ be multiples of the number $ n $ with the sums of their digits being $ s_1, s_2 $, respectively.... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 653 |
XXVIII - I - Problem 7
Three circles with radii equal to 1 intersect such that their common part is bounded by three arcs, each belonging to a different circle. The endpoints of these arcs are denoted by $ K, L, M $. The first circle intersects the second at points $ C $ and $ M $, the second intersects the third at p... | If two circles of equal radii intersect at points $ P $ and $ Q $, then the lengths of the arcs determined by these points on one circle are equal to the lengths of the corresponding arcs determined by these points on the other circle. By performing a symmetry with respect to the line $ PQ $, the arcs with endpoints $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 656 |
XXXII - I - Problem 3
A cone $ S_1 $ is circumscribed around a sphere $ K $. Let $ A $ be the center of the circle of tangency of the cone with the sphere. A plane $ \Pi $ is drawn through the vertex of the cone $ S_1 $ perpendicular to the axis of the cone. A second cone $ S_2 $ is circumscribed around the sphere $ K... | Let $ O $ be the center of the sphere, $ r $ - the length of its radius. $ W_1 $, $ W_2 $ are the vertices of the cones $ S_1 $ and $ S_2 $ (Fig. 7).
om32_1r_img_7.jpg
The plane determined by the points $ O $, $ W_1 $, $ W_2 $ contains the point $ A $, which is the center of the circle of tangency of the cone $ S_1 $ w... | proof | Geometry | proof | Yes | Yes | olympiads | false | 657 |
XXVII OM - II - Problem 4
Inside the circle $ S $, place the circle $ T $ and the circles $ K_1, K_2, \ldots, K_n $ that are externally tangent to $ T $ and internally tangent to $ S $, with the circle $ K_1 $ tangent to $ K_2 $, $ K_2 $ tangent to $ K_3 $, and so on. Prove that the points of tangency between the circ... | Let circles $S$ and $T$ be concentric. We will show that the points of tangency $K_1$ with $K_2$, $K_2$ with $K_3$, etc., are equidistant from the common center $O$ of circles $S$ and $T$.
Let for $i = 1, 2, \ldots, A_i$ be the point of tangency of circles $K_i$ and $T$, $B_i$ - the point of tangency of circles $K_i$ a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 658 |
LI OM - II - Problem 4
Point $ I $ is the center of the circle inscribed in triangle $ ABC $, where $ AB \neq AC $. Lines $ BI $ and $ CI $ intersect sides $ AC $ and $ AB $ at points $ D $ and $ E $, respectively. Determine all possible measures of angle $ BAC $ for which the equality $ DI = EI $ can hold. | We will show that the only value taken by angle $ BAC $ is $ 60^\circ $.
By the Law of Sines applied to triangles $ ADI $ and $ AEI $, we obtain $ \sin \measuredangle AEI = \sin \measuredangle ADI $. Hence,
om51_2r_img_6.jpg
First, suppose that the equality $ \measuredangle AEI = \measuredangle ADI $ holds (Fig.... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 659 |
XXXIV OM - II - Problem 6
For a given number $ n $, let $ p_n $ denote the probability that when a pair of integers $ k, m $ satisfying the conditions $ 0 \leq k \leq m \leq 2^n $ is chosen at random (each pair is equally likely), the number $ \binom{m}{k} $ is even. Calculate $ \lim_{n\to \infty} p_n $. | om34_2r_img_10.jpg
The diagram in Figure 10 shows Pascal's triangle written modulo $2$, meaning it has zeros and ones in the places where the usual Pascal's triangle has even and odd numbers, respectively. Just like in the usual Pascal's triangle, each element here is the sum of the elements directly above it, accordin... | 1 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 663 |
L OM - I - Problem 8
Given a natural number $ n \geq 2 $ and an $ n $-element set $ S $. Determine the smallest natural number $ k $ for which there exist subsets $ A_1, A_2, \ldots, A_k $ of the set $ S $ with the following property:
for any two distinct elements $ a, b \in S $, there exists a number $ j \in \{1, 2, ... | We will show that $ k = [\log_2(n-1) +1] $, i.e., $ 2^{k-1} < n \leq 2^k $.
Given the number $ k $ defined as above, we construct sets $ A_j $ as follows: we number the elements of set $ S $ with $ k $-digit binary numbers (leading zeros are allowed). We thus have $ 2^k \geq n $ numbers available. Then, we take $ A_i $... | [\log_2(n-1)+1] | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 664 |
XLVIII OM - II - Problem 3
Given a set of $ n $ points ($ n \geq 2 $), no three of which are collinear. We color all segments with endpoints in this set such that any two segments sharing a common endpoint have different colors. Determine the smallest number of colors for which such a coloring exists. | Let's assume that the given points are vertices of some $n$-gon (if $n = 2$, of course, one color is enough). All segments of the same color determine disjoint pairs of vertices of the polygon. The maximum number of disjoint two-element subsets of an $n$-element set is $[n/2]$. Therefore, the number of colors is not le... | [n/2] | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 666 |
XXII OM - II - Problem 3
Given 6 lines in space, of which no 3 are parallel, no 3 pass through the same point, and no 3 are contained in the same plane. Prove that among these 6 lines, there are 3 lines that are mutually skew. | From the conditions of the problem, it follows that in every triple of the considered lines, there is a pair of skew lines. Let us associate the given lines with the vertices of a convex hexagon marked with the numbers $1, 2, \ldots, 6$ (Fig. 11). If two lines are skew, let us connect the corresponding vertices of the ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 667 |
XII OM - I - Problem 2
Prove that if $ a_1 \leq a_2 \leq a_3 $ and $ b_1 \leq b_2 \leq b_3 $, then | Inequality (1) will be replaced by an equivalent inequality
This, in turn, by the inequality
According to the assumption $ a_1 - a_2 \leq 0 $, $ b_1 -b_2 \leq 0 $, therefore
Hence
Similarly
And
Adding the above 3 inequalities side by side, we obtain inequality (2) equivalent to the desired inequality ... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 669 |
LX OM - II - Zadanie 3
Rozłączne okręgi $ o_1 $ i $ o_2 $ o środkach odpowiednio $ I_1 $ i $ I_2 $ są styczne do prostej
$ k $ odpowiednio w punktach $ A_1 $ i $ A_2 $ oraz leżą po tej samej jej stronie. Punkt $ C $ leży na odcinku
$ I_1I_2 $, przy czym $ \meqsuredangle A_1CA_2 =90^{\circ} $. Dla $ i=1,2 $ niech $ B_i... |
Niech prosta styczna do okręgu $ o_1 $ w punkcie $ B_1 $ przecina prostą $ A_1A_2 $ w punkcie $ D $.
(Punkt przecięcia istnieje, gdyż punkt $ C $ nie leży na prostej $ A_1I_1 $, a więc odcinek $ A_1B_1 $
nie jest średnicą okręgu $ o_1 $.) Trójkąt $ A_1B_1D $ jest wówczas równoramienny, gdyż dwa jego boki
są odcinkami ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 670 |
LVII OM - III - Problem 6
Determine all pairs of integers $ a $, $ b $, for which there exists a polynomial $ P(x) $ with integer coefficients, such that the product $ (x^2 + ax + b)\cdot P(x) $ is a polynomial of the form
where each of the numbers $ c_0,c_1,\dots ,c_{n-1} $ is equal to 1 or -1. | If the quadratic polynomial $x^2 + ax + b$ is a divisor of some polynomial $Q(x)$ with integer coefficients, then of course the constant term $b$ is a divisor of the constant term of the polynomial $Q(x)$. Therefore, in the considered situation, $b=\pm 1$.
No polynomial $Q(x)$ of the form $x^n \pm x^{n-1} \pm x^{n-2} \... | (-2,1),(-1,-1),(-1,1),(0,-1),(0,1),(1,-1),(1,1),(2,1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 672 |
II OM - III - Problem 5
In a circle, a quadrilateral $ABCD$ is inscribed. Lines $AB$ and $CD$ intersect at point $E$, and lines $AD$ and $BC$ intersect at point $F$. The angle bisector of $\angle AEC$ intersects side $BC$ at point $M$ and side $AD$ at point $N$; and the angle bisector of $\angle BFD$ intersects side $... | Ignoring for now the condition that points $A$, $B$, $C$, $D$ lie on a circle, consider any convex quadrilateral $ABCD$ where the extensions of sides $AB$ and $CD$ intersect at point $E$, and the extensions of sides $AD$ and $BC$ intersect at point $F$.
Draw the angle bisectors $EO$ and $FO$ of angles $E$ and $F$, and ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 674 |
XXX OM - II - Task 1
Given are points $ A $ and $ B $ on the edge of a circular pool. An athlete needs to get from point $ A $ to point $ B $ by walking along the edge of the pool or swimming in the pool; he can change his mode of movement multiple times. How should the athlete move to get from point $ A $ to $ B $ in... | Let the radius of the pool be $r$, and the arc $\widehat{AB}$ corresponds to the central angle $\alpha$, where $0 < \alpha \leq \pi$. Then the length of the arc $\widehat{AB}$ is $r\alpha$, and the length of the segment $\overline{AB}$ is (Fig. 10)
om30_2r_img_10.jpg
We will prove that $r\alpha < 2AB$. It will follow ... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 676 |
XXVIII - I - Task 3
Let $ a $ and $ b $ be natural numbers. A rectangle with sides of length $ a $ and $ b $ has been divided by lines parallel to the sides into unit squares. Through the interiors of how many squares does the diagonal of the rectangle pass? | Consider a rectangle $ABCD$ with base $\overline{AB}$ of length $a$ and height $\overline{BC}$ of length $b$. Let's consider its diagonal $\overline{AC}$ connecting the lower left vertex $A$ with the upper right vertex $C$ (Fig. 4).
First, assume that the numbers $a$ and $b$ are relatively prime. If a vertex $E$ of a u... | \gcd(,b) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 679 |
XLVI OM - II - Problem 5
The circles inscribed in the faces $ABC$ and $ABD$ of the tetrahedron $ABCD$ are tangent to the edge $AB$ at the same point. Prove that the points of tangency of these circles with the edges $AC$, $BC$ and $AD$, $BD$ lie on a single circle. | Let's assume that the circle inscribed in the face $ABC$ is tangent to the edges $AC$, $BC$, and $AB$ at points $P$, $Q$, and $T$, respectively, and the circle inscribed in the face $ABD$ is tangent to the edges $AD$, $BD$, and $AB$ at points $R$, $S$, and $T$ (the same point $T$, according to the assumption). We have ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 682 |
LX OM - II - Task 1
Real numbers $ a_1, a_2, \cdots , a_n $ $ (n \geqslant 2) $ satisfy the condition
$ a_1 \geqslant a_2 \geqslant \cdots \geqslant a_n > 0 $. Prove the inequality | We will apply mathematical induction with respect to the value of the number $ n $.
For $ n = 2 $, the thesis of the problem is true, as both sides of the inequality are equal to $ 2a_2 $.
Moving on to the inductive step, consider the real numbers
$ a_1, a_2, \cdots, a_{n+1} $, for which $ a_1 \geqslant a_2 \geqslant... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 683 |
XVIII OM - II - Task 6
Prove that points $ A_1, A_2, \ldots, A_n $ ($ n \geq 7 $) lying on the surface of a sphere lie on a circle if and only if the tangent planes to the sphere at these points have a common point or are parallel to a single line. | a) Suppose that points $ A_1, A_2, \ldots, A_n $ on the surface of a sphere with center $ O $ lie on a circle $ k $. Let $ \alpha_1 $ denote the tangent plane to the sphere at point $ A_i $ ($ i= 1,2, \ldots, n $), and $ \pi $ - the plane of the circle $ k $.
If the center of the circle $ k $ is point $ O $, then the p... | proof | Geometry | proof | Yes | Yes | olympiads | false | 684 |
XVI OM - I - Problem 4
The school organized three trips for its 300 students. The same number of students participated in each trip. Each student went on at least one trip, but half of the participants in the first trip, one-third of the participants in the second trip, and one-fourth of the participants in the third ... | Let $ x $ denote the number of participants in each trip, and $ y $, $ z $, $ u $, $ w $ denote, respectively, the numbers of students who went: a) only on the first and second trip, b) only on the first and third trip, c) only on the second and third trip, d) on all three trips.
The total number of students equals the... | x=120,y=14,z=27,u=53,w=37 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 685 |
XI OM - I - Problem 11
Prove that if a quadrilateral is circumscribed around a circle, then a circle can be circumscribed around it if and only if the chords connecting the points of tangency of the opposite sides of the quadrilateral with the circle are perpendicular. | Let $ABCD$ be a quadrilateral inscribed in a certain circle, and let $M$, $N$, $P$, $Q$ denote the points of tangency of the sides $AB$, $BC$, $CD$, $DA$ with the inscribed circle, and let $S$ be the point of intersection of segments $MP$ and $NQ$ (Fig. 16).
Since tangents to a circle drawn from the ends of a chord for... | proof | Geometry | proof | Yes | Yes | olympiads | false | 686 |
LIV OM - III - Task 3
Determine all polynomials $ W $ with integer coefficients that satisfy the following condition: for every natural number $ n $, the number $ 2^n-1 $ is divisible by $ W(n) $. | Let $ W(x) = a_0 + a_1x +\ldots + a_rx^r $ be a polynomial with integer coefficients, satisfying the given condition. Suppose that for some natural number $ k $, the value $ W(k) $ is different from $ 1 $ and $ -1 $; it therefore has a prime divisor $ p $. Let $ m = k + p $. The difference $ W(m) - W(k) $ is the sum of... | W(x)=1orW(x)=-1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 687 |
XXXVII OM - II - Problem 5
Prove that if a polynomial $ f $ not identically zero satisfies for every real $ x $ the equation $ f(x)f(x + 3) = f(x^2 + x + 3) $, then it has no real roots. | Suppose a polynomial $ f $ satisfying the given conditions has real roots. Let $ x_0 $ be the largest of them. The number $ y_0 = x_0^2 + x_0 + 3 > x_0 $ is also a root of the polynomial $ f $, because
- contrary to the fact that $ x_0 $ is the largest root. The obtained contradiction proves that $ f $ cannot have re... | proof | Algebra | proof | Yes | Yes | olympiads | false | 689 |
VI OM - II - Task 3
What should be the angle at the vertex of an isosceles triangle so that a triangle can be constructed with sides equal to the height, base, and one of the remaining sides of this isosceles triangle? | We will adopt the notations indicated in Fig. 9. A triangle with sides equal to $a$, $c$, $h$ can be constructed if and only if the following inequalities are satisfied:
Since in triangle $ADC$ we have $a > h$, $\frac{c}{2} + h > a$, the first two of the above inequalities always hold, so the necessary and suffic... | 106 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 690 |
XXXIV OM - I - Problem 5
Prove the existence of a number $ C_0 $ with the property that for any sequence $ x_1, x_2, \ldots, x_N $ of positive numbers and for any positive number $ K $, if the number of terms $ x_j $ not less than $ K $ is not greater than $ \frac{N}{K} $, then | Suppose the sequence $ x_1, x_2, \ldots, x_N $ satisfies the given condition. By rearranging the terms if necessary, we can assume that $ x_1 \geq x_2 \geq \ldots \geq x_N $. We will show that $ x_j \leq N/j $ for $ j = 1, \ldots, N $. Suppose this is not the case, i.e., for some index $ m $, the inequality $ x_m > N/m... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 692 |
XXXVI OM - III - Problem 2
Given a square with side length 1 and positive numbers $ a_1, b_1, a_2, b_2, \ldots, a_n, b_n $ not greater than 1 such that $ a_1b_1 + a_2b_2 + \ldots + a_nb_n > 100 $. Prove that the square can be covered by rectangles $ P_i $ ($ i = 1,2,\ldots,n $) with side lengths $ a_i, b_i $ parallel ... | Let $ AB $ be the base of a square. We can assume that $ 1 \geq a_x \geq a_2 \geq \ldots \geq a_n $. On the ray $ AB^\to $, we sequentially lay down segments of lengths $ b_1, b_2, \ldots, b_k $ until we exceed point $ B $, and over each subsequent segment of length $ b_i $, we construct a rectangle whose other side ha... | proof | Geometry | proof | Yes | Yes | olympiads | false | 693 |
XVII OM - I - Problem 8
In space, three lines are given. Find a line that intersects them at points $ A, B, C $ such that $ \frac{AB}{BC} $ equals a given positive number $ k $. | Let us first consider the case where the three given lines are pairwise skew, and at the same time do not lie in three mutually parallel planes.
Let the given lines be denoted by the letters $a$, $b$, $c$ and let $l$ be the line intersecting them at points $A$, $B$, $C$ respectively, such that $\frac{AB}{BC}=k$.
Let $\... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 695 |
XLV OM - I - Problem 8
Given natural numbers $ a $, $ b $, $ c $, such that $ a^3 $ is divisible by $ b $, $ b^3 $ is divisible by $ c $, and $ c^3 $ is divisible by $ a $. Prove that the number $ (a+b+c)^{13} $ is divisible by $ abc $. | The number $ (a + b + c)^{13} $ is the product of thirteen factors, each equal to $ a + b + c $. Multiplying these thirteen identical expressions, we obtain a sum of many (specifically $ 3^{13} $) terms of the form
It suffices to show that each such term is divisible by the product $ abc $. This is obvious when none o... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 696 |
XLVII OM - III - Problem 3
Given a natural number $ n \geq 2 $ and positive numbers $ a_1, a_2, \ldots , a_n $, whose sum equals 1.
(a) Prove that for any positive numbers $ x_1, x_2, \ldots , x_n $ whose sum equals 1, the following inequality holds:
(b) Determine all systems of positive numbers $ x_1, x_2, \ldots , ... | From the given conditions ($ n \geq 2 $; $ a_i>0 $; $ \sum a_i =1 $), it follows that the numbers $ a_1, a_2, \ldots, a_n $ belong to the interval $ (0;\ 1) $. Let us assume
these are also numbers in the interval $ (0;\ 1) $. For them, the equality
holds.
Let $ x_1, x_2,\ldots, x_n $ be any real numbers whose sum is... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 698 |
XXIX OM - I - Problem 1
The sequence of numbers $ (p_n) $ is defined as follows: $ p_1 = 2 $, $ p_n $ is the largest prime divisor of the sum $ p_1p_2\ldots p_{n-1} + 1 $. Prove that the number 5 does not appear in the sequence $ (p_n) $. | We have $ p_1 = 2 $ and $ p_2 = 3 $, because $ p_2 $ is the largest prime divisor of the number $ p_1 + 1 = 3 $. Therefore, for $ n > 2 $, the number $ p_1p_2 \ldots p_{n-1} + 1 = 6p_3 \ldots p_n + 1 $ is not divisible by either $ 2 $ or $ 3 $. If, for some natural number $ n $, the largest prime divisor of the number ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 699 |
XXIX OM - II - Problem 4
From the vertices of a regular $2n$-gon, 3 different points are chosen randomly. Let $p_n$ be the probability that the triangle with vertices at the chosen points is acute. Calculate $\lim_{n\to \infty} p_n$.
Note. We assume that all choices of three different points are equally probable. | Let $ A_1, A_2, \ldots , A_{2n} $ be the consecutive vertices of a regular $ 2n $-gon, and $ O $ - the center of the circle circumscribed around this polygon. We will investigate how many triangles $ A_1A_iA_j $, where $ i < j $, have the angle $ \measuredangle A_iA_1A_j $ not acute.
Since the inscribed angle in a circ... | \frac{1}{4} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 700 |
XXXIV - III - Task 3
We consider a single-player game on an infinite chessboard based on the following rule. If two pieces are on adjacent squares and the next square is empty (the three squares being discussed lie on the same horizontal or vertical line), we can remove these pieces and place one piece on the third of... | By introducing a coordinate system on the plane, we can assign a pair of integers to each square of the chessboard, such that two adjacent squares in the same column are assigned the numbers $(m, n)$ and $(m, n+1)$, and two adjacent squares in the same row are assigned the numbers $(m, n)$ and $(m+1, n)$. We can assign... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 705 |
XXXV OM - I - Problem 12
In a trihedral angle $ W $, a ray $ L $ passing through its vertex is contained. Prove that the sum of the angles formed by $ L $ with the edges of $ W $ does not exceed the sum of the dihedral angles of $ W $. | Consider a sphere of radius 1 with its center at the vertex of a trihedral angle. The measure of each plane angle, whose sides are rays originating from the center of the sphere, is equal to the length of the corresponding arc of a great circle (the shorter of the two arcs defined by the points of intersection of the g... | proof | Geometry | proof | Yes | Yes | olympiads | false | 706 |
V OM - I - Task 3
Prove that if in a quadrilateral $ABCD$ the equality $AB + CD = AD + BC$ holds, then the incircles of triangles $ABC$ and $ACD$ are tangent. | If the incircles of triangles $ABC$ and $ACD$ touch their common side $AC$ at points $M$ and $N$ respectively (Fig. 20), then according to known formulas we have
from which
If $AB + CD = AD + BC$, then from the above equality it follows that $AM = AN$, i.e., that points $M$ and $N$ coincide; both circles are tangent ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 707 |
IX OM - II - Task 2
Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to... | Let circle $K$ with center $O$ and radius $r$ roll without slipping on a circle with center $S$ and radius $R$ (Fig. 16). The rolling without slipping means that different points of one circle successively coincide with different points of the other circle, and in this correspondence, the length of the arc between two ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 711 |
X OM - I - Task 7
On a plane, there are four different points $ A $, $ B $, $ C $, $ D $.
a) Prove that among the segments $ AB $, $ AC $, $ AD $, $ BC $, $ BD $, $ CD $, at least two are of different lengths,
b) How are the points $ A $, $ B $, $ C $, $ D $ positioned if the segments mentioned above have only two di... | a) To prove that the segments $ AB $, $ AC $, $ AD $, $ BC $, $ BD $, $ CD $ cannot all be equal to each other, it suffices to show that if five of them have the same length, then the sixth has a different length. Let's assume, for example, that $ AB = AC = AD = BC = BD $. In this case, $ ABC $ and $ ABD $ are equilate... | notfound | Geometry | proof | Yes | Yes | olympiads | false | 712 |
XXI OM - I - Problem 6
Prove that a real number $ a $ is rational if and only if there exist integers $ p > n > m \leq 0 $ such that $ a + m, a + n, a + p $ form a geometric progression. | Let $ a $ be a rational number. We choose a natural number $ m $ such that $ a + m > 0 $. Then
where $ r $ and $ q $ are natural numbers.
Consider a geometric sequence with a common ratio of $ 1 + q $ and the first term $ \frac{r}{q} $. Thus, the subsequent terms of the sequence will be
Assuming then $ n = m + r $ an... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 714 |
XXXVI OM - III - Problem 1
Determine the largest number $ k $ such that for every natural number $ n $ there are at least $ k $ natural numbers greater than $ n $, less than $ n+17 $, and coprime with the product $ n(n+17) $. | We will first prove that for every natural number $n$, there exists at least one natural number between $n$ and $n+17$ that is coprime with $n(n+17)$.
In the case where $n$ is an even number, the required property is satisfied by the number $n+1$. Of course, the numbers $n$ and $n+1$ are coprime. If a number $d > 1$ we... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 718 |
XXV OM - III - Task 5
Prove that if natural numbers $ n $, $ r $ satisfy the inequality $ r + 3 \leq n $, then the numbers $ \binom{n}{r} $, $ \binom{n}{r+1} $, $ \binom{n}{r+2} $, $ \binom{n}{r+3} $ are not consecutive terms of any arithmetic sequence. | We will first prove two lemmas.
Lemma 1. For a fixed natural number $n$, there are at most two natural numbers $k \leq n - 2$ such that the numbers $\binom{n}{k}$, $\binom{n}{k+1}$, $\binom{n}{k+2}$ are consecutive terms of an arithmetic sequence.
Proof. If the numbers $\displaystyle \binom{n}{k}, \binom{n}{k+1}, \bino... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 719 |
XXXIX OM - II - Problem 4
Prove that for every natural number $ n $ the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1)^3 $. | Let us denote:
Assume that $ n \geq 2 $, hence $ m \geq 1 $. According to the binomial formula,
where $ A $, $ B $, $ C $ are integers. We transform further:
Thus, $ N = Dm^3 = D(n - 1)^3 $, where $ D $ is an integer (this is also true for $ n = 1 $).
We have thus shown that $ N $ is divisible by $ ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 720 |
LVIII OM - I - Problem 1
Problem 1.
Solve in real numbers $ x $, $ y $, $ z $ the system of equations | Subtracting the second equation of the system from the first, we get
Similarly, by subtracting the third equation from the first, we obtain the equation $ 0=(x-z)(x+z-2y+5) $. Therefore, the system of equations given in the problem is equivalent to the system
The product of two numbers is equal to zero if and ... | (x,y,z)=(-2,-2,-2),(\frac{1}{3},\frac{1}{3},\frac{1}{3}),(2,2,-3),(-\frac{1}{3},-\frac{1}{3},-\frac{16}{3}),(2,-3,2),(-\frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 722 |
XLIV OM - III - Problem 2
Point $ O $ is the center of the circle $ k $ inscribed in the non-isosceles trapezoid $ ABCD $, whose longer base $ AB $ has midpoint $ M $. The shorter base $ CD $ is tangent to the circle $ k $ at point $ E $, and the line $ OM $ intersects the base $ CD $ at point $ F $. Prove that $ |DE|... | Since the equality $ |DE| = |FC| $ (considered in the thesis of the problem) holds if and only if $ |DF| = |EC| $, the conditions of the assumptions and the thesis will not change when the roles of the legs $ AD $ and $ BC $ of the given trapezoid are swapped. Therefore, we can assume, without loss of generality, that ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 723 |
XXXII - I - Problem 8
Given a point $ P $ inside a sphere $ S $. The transformation $ f: S\to S $ is defined as follows: for a point $ X \in S $, the point $ f(X) \in S $, $ f(X) \neq X $, and $ P\in \overline{Xf(X)} $. Prove that the image of any circle contained in $ S $ under the transformation $ f $ is a circle. | Let $ k $ be a circle contained in the sphere $ S $, and let $ S_1 $ be a sphere containing $ k $ and the point $ P $, $ P_1 $ be such a point that the segment $ \overline{PP_1} $ is a diameter of $ S_1 $. For every $ X \in k $, the angle $ PXP_1 $ is a right angle. Let $ X $ be the orthogonal projection of the point $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 724 |
LVIII OM - II - Problem 3
From $ n^2 $ tiles in the shape of equilateral triangles with a side of $ 1 $, an equilateral triangle with a side of $ n $ was formed. Each tile is white on one side and black on the other. A move consists of the following actions: We select a tile $ P $ that shares sides with at least two o... | Let's call a boundary segment the common side of two tiles whose visible sides have different colors. Of course, the number of boundary segments is non-negative and does not exceed the number $ m $ of all segments that are the common side of two tiles. We will investigate how the number of boundary segments changes as ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 725 |
XXXI - II - Task 5
We write down the terms of the sequence $ (n_1, n_2, \ldots, n_k) $, where $ n_1 = 1000 $, and $ n_j $ for $ j > 1 $ is an integer chosen randomly from the interval $ [0, n_{j-1} - 1] $ (each number in this interval is equally likely to be chosen). We stop writing when the chosen number is zero, i.e... | For a given non-negative integer $ n $, consider the random variable $ X_n $ being the length $ k $ of the sequence ($ n_1, n_2, \ldots, n_k $), where $ n_1 = n $, $ n_j $ for $ j > 1 $ is an integer chosen randomly from the interval $ [0, n_{j-1} - 1] $, and $ n_k = 0 $. Let $ E_n $ denote the expected value of this r... | 8.48547 | Combinatorics | proof | Yes | Yes | olympiads | false | 726 |
IV OM - II - Task 6
Given a circle and two tangents to this circle. Draw a third tangent to the circle in such a way that the segment of this tangent contained between the given tangents has a given length $ d $. | When the tangent lines $ m $ and $ n $ intersect, the task coincides with the already solved problem No. 11.
When the lines $ m $ and $ n $ are parallel, the solution is immediate. From any point $ A $ on the line $ m $ as the center, we draw a circle with radius of length $ d $.
If this circle has a common point $ B $... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 727 |
XVI OM - III - Task 2
Prove that if the numbers $ x_1 $ and $ x_2 $ are roots of the equation
$ x^2 + px - 1 = 0 $, where $ p $ is an odd number, then for every natural $ n $ the numbers $ x_1^n + x_2^n $ and $ x_1^{n+1} + x_2^{n+1} $ are integers and relatively prime. | We will apply the method of induction. Since $ p $ is an odd number, the numbers
are integers and relatively prime. The theorem is thus true for $ n = 0 $.
Suppose that for some natural number $ n \geq 0 $ the numbers
are integers and relatively prime. We will prove that then $ x_1^{n+2} + x_2^{n+2} $ is an... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 728 |
LVI OM - II - Problem 4
Given is the polynomial $ W(x)=x^2+ax+b $, with integer coefficients, satisfying the condition:
For every prime number $ p $ there exists an integer $ k $ such that the numbers $ W(k) $ and $ W(k + 1) $ are divisible by $ p $. Prove that there exists an integer $ m $ for which | The condition $ W(m) = W(m +1) = 0 $ means that $ W(x)=(x-m)(x-m+1) $, i.e., $ W(x)=x^{2}-(2m+1)x+m^{2}+m $. Therefore, it is necessary to show that there exists an integer $ m $ such that
Let us fix a prime number $ p $. Then for some integer $ k $, the numbers
are divisible by $ p $. Therefore, the following nu... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 729 |
XIII OM - I - Problem 6
Factor the quadratic polynomial into real factors
where $ p $ and $ q $ are real numbers satisfying the inequality
Please note that the mathematical expressions and symbols are kept as they are, only the text has been translated. | From the inequality $ p^2 - 4q < 0 $, it follows that the quadratic polynomial $ y^2 + py + q $ has no real roots; therefore, the quartic polynomial $ x^4 + px^2 + q $ also has no real roots. If the number $ x $ were a real root of the second polynomial, then the number $ y = x^2 $ would be a real root of the first pol... | x^4+px^2+(x^2+\alphax+\beta)(x^2-\alphax+\beta) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 732 |
VIII OM - III - Task 6
Given a cube with base $ABCD$, where $AB = a$ cm. Calculate the distance between the line $BC$ and the line passing through point $A$ and the center $S$ of the opposite face. | Since $ BC\parallel AD $, the line $ BC $ is parallel to the plane $ ADS $. The sought distance between the lines $ BC $ and $ AS $ is equal to the distance from the line $ BC $ to the plane $ ADS $ (Fig. 25).
Consider a parallelepiped whose base is the square $ ABCD $, and one of its side faces is the rectangle $ AMND... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 735 | |
XXXVIII OM - III - Zadanie 5
Wyznaczyć najmniejszą liczbę naturalną $ n $, dla której liczba $ n^2-n+11 $ jest iloczynem czterech liczb pierwszych (niekoniecznie różnych).
|
Niech $ f(x) = x^2-x+11 $. Wartości przyjmowane przez funkcję $ f $ dla argumentów całkowitych są liczbami całkowitymi niepodzielnymi przez $ 2 $, $ 3 $, $ 5 $, $ 7 $. Przekonujemy się o tym badając reszty z dzielenia $ n $ i $ f(n) $ przez te cztery początkowe liczby pierwsze:
\begin{tabular}{lllll}
&\multicolum... | 132 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 736 |
XXV OM - III - Task 2
Salmon swimming upstream must overcome two waterfalls. The probability that a salmon will overcome the first waterfall in a given attempt is $ p > 0 $, and the probability of overcoming the second waterfall in a given attempt is $ q > 0 $. We assume that successive attempts to overcome the waterf... | Let $ A_n $ be the event that the salmon does not overcome the first waterfall in $ n $ attempts, and $ B_n $ - the event that the salmon does not overcome both waterfalls in $ n $ attempts. Since the probability that the salmon does not overcome the first waterfall in one attempt is $ 1 - p $, and the attempts are ind... | \max(0,1-\frac{p}{q}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 739 |
XXXVI OM - I - Problem 8
Prove that if $ (a_n) $ is a sequence of real numbers such that $ a_{n+2}=|a_{n+1}|-a_n $ for $ n = 1,2,\ldots $, then $ a_{k+9} = a_k $ for $ k = 1,2,\ldots $. | Let's consider the transformation $ F $ of the plane into itself given by the formula $ F(x,y) = (y,|y|-x) $. Let $ a $ be a positive number. Let $ I_0 $ be the segment with endpoints $ (0,-a) $ and $ (-a,0) $, let $ I_1 $ be the image of the segment $ I_0 $ under the transformation $ F $, and further, inductively, let... | proof | Algebra | proof | Yes | Yes | olympiads | false | 743 |
XX OM - I - Problem 11
In a convex quadrilateral $ABCD$, the sum of the distances from each vertex to the lines $AB$, $BC$, $CD$, $DA$ is the same. Prove that this quadrilateral is a parallelogram. | Let's denote the measures of the angles of a quadrilateral by $A$, $B$, $C$, $D$, and the lengths of the sides $AB$, $BC$, $CD$, $DA$ by $a$, $b$, $c$, $d$ respectively (Fig. 6).
The sum of the distances from vertex $A$ to the lines containing the sides of the quadrilateral is then $a \sin B + d \sin D$. Denoting this ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 744 |
LII OM - I - Problem 5
Prove that for any natural number $ n \geq 2 $ and any prime number $ p $, the number $ n^{p^p} + p^p $ is composite. | If $ p $ is an odd prime, then by the identity
in which we set $ x = n^{p^{p-1}} $ and $ y = p $, the number $ n^{p^p} + p^p $ is composite. (For the above values of $ x $, $ y $, the inequalities $ x^p + y^p > x + y > 1 $ hold, so each of the factors on the right-hand side of formula (1) is greater than 1).
On the ot... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 745 |
XXI OM - I - Problem 4
In a square $ABCD$ with side length 1, there is a convex quadrilateral with an area greater than $\frac{1}{2}$. Prove that this quadrilateral contains a segment of length $\frac{1}{2}$ parallel to $\overline{AB}$. | We draw through the vertices of a given quadrilateral lines parallel to line $AB$ (Fig. 4). In the case where these lines are different, the quadrilateral is divided into $3$ parts: a triangle with base $p$ and height $h_1$, a trapezoid with bases $p$ and $q$ and height $h_2$, and a triangle with base $q$ and height $h... | proof | Geometry | proof | Yes | Yes | olympiads | false | 746 |
XIII OM - I - Problem 4
Prove that the line symmetric to the median $CS$ of triangle $ABC$ with respect to the angle bisector of angle $C$ of this triangle divides the side $AB$ into segments proportional to the squares of the sides $AC$ and $BC$. | Let $ CM $ (Fig. 4) be the line symmetric to the median $ CS $ with respect to the angle bisector $ CD $ of angle $ C $ in triangle $ ABC $.
Since the areas of triangles with the same height are proportional to the bases of these triangles, therefore
From the equality $ \measuredangle ACD = \measuredangle DCB $ and... | proof | Geometry | proof | Yes | Yes | olympiads | false | 749 |
LIX OM - I - Task 7
In an $ n $-person association, there are $ 2n-1 $ committees (any non-empty set of association members
forms a committee). A chairperson must be selected in each committee. The following condition must be met: If
committee $ C $ is the union $ C = A\cup B $ of two committees $ A $ and $ B $, then ... | Answer: n!.
We will establish a one-to-one correspondence between the possible choices of committee chairs
and the ways of assigning members of the association the numbers $1, 2, \dots, n$ (where different members
are assigned different numbers). There are, of course, as many of the latter as there are permutations of ... | n! | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 750 |
XL OM - I - Problem 7
In space, a finite set of points is given, any four of which are vertices of a tetrahedron with a volume less than or equal to 1. Prove that there exists a tetrahedron with a volume no greater than 27, containing all these points. | Let $ABCD$ be a tetrahedron of maximum volume among all tetrahedra whose vertices are points of the considered set. Denote by $O$ its centroid, that is, the point of intersection of four segments, each connecting a vertex of the tetrahedron with the centroid of the opposite face. It is known that the centroid of a tetr... | proof | Geometry | proof | Yes | Yes | olympiads | false | 752 |
XVII OM - I - Problem 4
On a plane, a circle and a point $ M $ are given. Find points $ A $ and $ B $ on the circle such that the segment $ AB $ has a given length $ d $, and the angle $ AMB $ is equal to a given angle $ \alpha $. | Suppose $ A $ and $ B $ are points on a given circle satisfying the conditions $ AB = d $, $ \measuredangle AMB = \alpha $ (Fig. 5).
Let us choose any chord $ CD $ of length $ d $ in the given circle. Rotate the triangle $ AMB $ around the center $ O $ of the given circle by the angle $ HOK $, where $ H $ and $ K $ den... | 4,3,2,1,或0 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 753 |
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