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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 |
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 |
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 |
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 |
LVI OM - I - Problem 7
Three spheres, pairwise externally tangent, are tangent to a certain plane at points $ A $, $ B $, $ C $. Given the lengths of segments $ BC = a $, $ CA = b $, $ AB = c $, calculate the radii of these spheres. | From the conditions of the problem, it follows that the considered spheres lie on one side of the plane to which they are tangent.
om56_1r_img_4.jpg
Let's denote by $ r_A $, $ r_B $, and $ r_C $ the radii of the spheres, tangent to a given plane at points $ A $, $ B $, and $ C $, respectively. Then, by the Pythagorean ... | Geometry | math-word-problem | Yes | Incomplete | olympiads | false | |
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 |
LIX OM - II - Task 2
In a convex pentagon $ABCDE$, the following relationships are satisfied:
Segments $BD$ and $CE$ intersect at point $S$. Prove that lines $AS$ and $CD$ are perpendicular. | From the equality
it follows (Fig. 1) that triangles $ BAD $ and $ CAE $ are similar (angle-angle-angle criterion). The similarity of these triangles
allows us to conclude that
These relationships prove that triangles $ BAC $ and $ DAE $ are similar (side-angle-side criterion). Consequently, we obtain
Fu... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
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 |
XV OM - II - Task 4
Find the real numbers $ x, y, z $ satisfying the system of equations
where $ a, b, c $ are given real numbers. | Multiplying the equations of the system (1), we obtain
We see from this that the system (1) has a real solution only when
Assume that condition (2) is satisfied. Then,
We will distinguish 2 cases:
I. $ abc > 0 $, i.e., $ a \ne 0 $, $ b \ne 0 $, $ c \ne 0 $. From equations (1) and (3), we obtain a system of fir... | notfound | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXXI - I - Problem 6
Prove that for different integers $ a, b, c $ and any natural number $ n $ the number
is integer. | Let's represent the considered number in the form of
The numerator of this fraction is the value of the polynomial in three variables $ f(x, y, z) = x^n(y - z) +y^n (z - x) + z^n (x- y) $ at the point $ (a, b, c) $. Since
the polynomial $ f(x,y, z) $ is divisible by $ x- y $. Similarly, we conclude that $ f(x,y, ... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
LVIII OM - I - Problem 6
Prove that if the numbers $ a,b,c $ are positive, then
The translation is provided as requested, maintaining the original text's line breaks and format. | Using the inequality between the arithmetic mean and the geometric mean, we obtain
and similarly $ b+bc+bca\ge3\sqrt[3]{abc} \sqrt[3]{b^2c} $ and $ c+ca+cab\ge 3\sqrt[3]{abc}\sqrt[3]{c^2a} $. Therefore, it follows that
Thus, the problem will be solved if we prove the inequality
Introducing the n... | proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
XLIII OM - I - Problem 3
Given are natural numbers $ a $ and $ b $. Prove that natural numbers $ c $ and $ d $
satisfying the equation
exist if and only if $ ab $ is an even number. | We rewrite the given equation in the form
If the product $ab$ is an odd number, then both $a$ and $b$ are odd. Therefore, the sum $a + b$ is even, and its square is divisible by $4$; the left side of (1) is divisible by $2$ but not by $4$. On the other hand, the right side of (1) is, for any choice of numbers $c$, $d$... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
III OM - II - Task 4
Prove that if the numbers $ a $, $ b $, $ c $ satisfy the equation
then two of them are opposite numbers. | We write equality (1) in the form
From this, we obtain the equality
which, after performing the multiplication and reduction, gives
We arrange the left side according to the powers of $ a $:
We factor out $ b + c $:
and finally we obtain
From this equality, it follows that one of the... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
IV OM - I - Problem 9
Prove that if $ a < b < c < d $, then
The translation is provided as requested, maintaining the original text's line breaks and format. | Let's organize the expression $ m = (a + b + c + d)^2 - 8ac - 8bd $ according to the powers of one of the letters, for example, the letter $ a $:
and calculate the discriminant of the resulting quadratic function:
When $ b < c < d $ or $ d < c < b $, then $ \Delta < 0 $, hence $ m > 0 $. | proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
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 |
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 |
XII OM - I - Problem 9
Prove that when $ n $ is a natural number, the number
is divisible by $ 21 $. | We will apply the method of complete induction.
When $ n = 1 $, the theorem is true, since $ 2^4 + 5 = 21 $. Suppose it is true for some value of $ n $, i.e., that $ 2^{4^n} + 5 = 21k $, where $ k $ is some integer. Then $ 2^{4^n} = 21 k - 5 $, hence
where $ m $ is some integer, therefore
The number $ 2^{4^{n+1... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
XVII OM - III - Problem 1
Solve in integers the equation
Note: The equation is not provided in the original text, so it is not included in the translation. | Equation (1) obviously has the solution $(0,0,0,0)$. We will show that this is the only solution of the equation in integers.
\spos{1} The proof is based on the following lemma:
If $k_1, k_2, \ldots, k_n$ are distinct natural numbers, and $x_1, x_2, \ldots, x_n$ are integers not divisible by the natural number $c$, the... | (0,0,0,0) | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
XLVI OM - I - Zadanie 2
Dana jest liczba naturalna $ n \geq 2 $. Rozwiązać układ równań
|
Załóżmy, że liczby $ x_1, x_2, \ldots, x_n $ spełniają podany układ równań. Przepisujemy ten układ w postaci
przyjmując, że niewiadome są numerowane cyklicznie ($ x_{n+1} = x_1 $). Dodajemy wszystkie równania stronami, redukujemy występujące po obu stronach składniki $ x_1 |x_1| $, $ x_2 |x_2| $, $ \ldots $, $ ... | x_1=x_2=\ldots=x_n=1 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
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 |
IX OM - I - Problem 5
Prove that the number
is divisible by $ 78 $. | Notice that in the expansion of $ (a \pm 1)^n $, where $ a $ and $ n $ are natural numbers, all terms except the last one are divisible by $ a $, and the last term equals $ (\pm 1)^m $. Therefore,
Hence,
The number $ 53^{103} + 103^{53} $ is therefore divisible by $ 6 $.
Similarly,
the number $ 53^... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXXV OM - I - Problem 3
Determine all values of $ p, q $ for which the inequality
holds for every $ x $ in the interval $ [0; 1] $. | We are looking for linear functions $ f(x) = px + q $, such that for every $ x_0 $ from the interval $ [0; 1] $, the value $ f(x_0) $ differs from the value of the function $ g(x) = \sqrt{1 - x^2} $ at point $ x_0 $ by no more than $ \frac{1}{2}(\sqrt{2}-1) $. First, we will show that the function $ f(x) = - x + \frac{... | p=-1,q=\frac{1}{2}(\sqrt{2}+1) | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
VII OM - II - Problem 1
For what value of $ m $ is the polynomial $ x^3 + y^3 + z^3 + mxyz $ divisible by $ x + y + z $? | The expression $ x^3 + y^3 + z^3 + mxyz $ is a polynomial of the third degree with respect to $ x $; this polynomial is divisible by $ x + y + z $ if and only if it has a root $ x = - y - z $, i.e., when substituting $ - (y + z) $ for $ x $ we obtain identically $ 0 $. Hence the condition
which can be written in ... | Algebra | math-word-problem | Yes | Incomplete | olympiads | false | |
VI OM - III - Task 1
What conditions should the real numbers $ a $, $ b $, and $ c $ satisfy so that the equation
has three distinct real roots forming a geometric progression? | The task can be formulated as follows: Find the necessary and sufficient conditions that the real numbers $a$, $b$, $c$ should satisfy so that there exist numbers $x_1$ and $q$ with the following properties:
a) the numbers $x_1$, $x_1 q$, $x_1 q^2$ satisfy equation (1),
b) the numbers $x_1$ and $q$ are real,
c) $x_1 \n... | ^2+2am-3m^2>0^3=--\frac{}{3}<< | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XLIV OM - I - Problem 7
In space, the points are given
Let $ P_{ij} $ ($ i,j \in \{0,1,2,3\} $) be the points determined by the equality $ \overrightarrow{A_0P_{ij}} = \overrightarrow{A_iA_j} $. Calculate the volume of the smallest convex polyhedron containing all points $ P_{ij} $. | Let's consider the points $ B_1 $, $ B_2 $, $ B_3 $, which are symmetric to $ A_1 $, $ A_2 $, $ A_3 $, respectively, with respect to the point $ A_0 $ (the origin of the coordinate system):
The points under consideration
and
lie on the boundary of a cube with vertices $ (\pm1,\pm1,\pm1) $. Of ... | \frac{10}{3} | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
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 |
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 |
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 |
XIX OM - III - Task 2
Prove that for every natural $ n $ | Let us introduce the notation
Notice that
It seems plausible that for every natural number $ n $
Equality (2) can be easily proved by induction. Assuming that equality (2) holds for some natural number $ k $, it follows that
i.e., equality (2) also holds when $ n = k + 1 $. Therefore, and since equality (2)... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
XXXVI OM - I - Problem 11
Provide an example of a convex polyhedron having 1985 faces, among which there are 993 faces such that no two of them share a common edge. | It is easy to obtain a number of faces significantly greater than $993$, and not only without common edges, but completely disjoint. For this purpose, consider any convex polyhedron $W$ having $5$ faces and vertices; we assume that no two faces lie in the same plane, i.e., that all dihedral angles are less than a strai... | 993 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
X OM - II - Task 5
On a plane, $ n \geq 3 $ segments are placed in such a way that any $ 3 $ of them have a common point. Prove that there exists a point common to all segments. | Since every $3$ of the given segments have a common point, then any $2$ of them also have a common point. Suppose among the given segments $a_1, a_2, \ldots, a_n$ there are two segments, say $a_1$ and $a_2$, that do not lie on the same line, so they have only one common point $A$. If $a_k$ is any of the remaining segme... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
LIII OM - III - Problem 5
In space, there is a triangle $ABC$ and a sphere $s$ disjoint from the plane $ABC$. Through each of the points $A$, $B$, $C$, a tangent line to this sphere has been drawn. The points of tangency are denoted as $K$, $L$, $M$ respectively. The point $P$ lies on the sphere $s$ and satisfies the ... | Let $ \lambda $ denote the common value of the fractions given in the problem. Let $ A, $ B, $ C $ be the second points of intersection of the lines $ PA $, $ PB $, $ PC $ with the sphere $ s $ (Fig. 2 and 3). If one of these lines, say $ PA $, is tangent to the sphere $ s $, then we take $ A. In this case, $ AK^2 = AP... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
XXXVI OM - I - Problem 2
Prove that if in a triangle, the lines perpendicular to the sides, drawn through the points of intersection of these sides with the angle bisectors of the opposite angles, intersect at one point, then the triangle is isosceles. | In triangle $ABC$, let points $A_1$, $B_1$, and $C_1$ be the points of intersection of the sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ with the angle bisectors of the opposite angles, respectively. Suppose that the perpendiculars to the sides drawn from these points intersect at point $K$ (Figure 1). De... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
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 |
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 |
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 |
XXXIII OM - I - Problem 6
Prove that for every real number $ x $ and every natural number $ n $ the following inequality holds:
(Note: The original text did not provide the specific inequality to be proven. If you have the specific inequality, please provide it for a complete translation.) | Since $ \cos 2t = 2 \cos^2 t - 1 $, it follows from the assumption $ |\cos 2^{k} x| < \frac{1}{2} $ that
Let $ k_1, k_2, \ldots, k_r $ ($ 1 \leq k_1 < k_2 < \ldots < k_r \leq n $) be all the exponents for which $ |\cos 2^{k} x| < \frac{1}{2} $. The exponents $ k_i $, $ k_i + 1 $ cannot be consecutive natural numbers f... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
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 |
V OM - II - Task 4
Provide the conditions under which the equation
has roots, assuming that the numbers $ a $, $ b $, $ c $ are pairwise distinct. | Suppose that the number $ x $ is a root of a given equation. In this case, $ x > a $, $ x > b $, $ x > c $; besides, $ \sqrt{x - a} < \sqrt{x} - c $, from which it follows that $ x - a < x - c $, hence $ c < a $ and similarly $ c < b $. The given equation can therefore have roots only when the numbers $ a $, $ b $, $ c... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XX OM - I - Task 8
Prove the theorem:
If | We multiply equations (1) and (2) side by side; applying known formulas, we get:
We square equations (1) and (2) side by side and subtract them:
We multiply equations (1) and (2) by $ \cos x $ and $ \sin x $ respectively and subtract them:
Similarly, we obtain the equations
Therefore,
We m... | proof | Other | proof | Incomplete | Yes | olympiads | false |
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 |
XXXIX OM - I - Problem 11
Given is a sequence defined by the formulas $ a_0 = 0 $, $ a_1 = 1 $, $ a_2 = 2 $, $ a_3 = 6 $, $ a_{n+4} = 2a_{n+3} + a_{n+2} - 2 a_{n+1} - a_n $ for $ n > 0 $. Prove that $ a_n $ is divisible by $ n $. | Let $ (v_n) $ be the Fibonacci sequence, that is, the sequence of integers defined by the conditions: $ v_0 = 0 $, $ v_1 = 1 $, $ v_{n+2} = v_n + v_{n+1} $ (see preparatory problem $ F $; this time it is convenient to number the terms of the sequence starting from zero, taking $ v_0 = 0 $). We will show that $ a_n = nv... | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
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 |
XXV - I - Problem 10
Each diagonal of a certain convex pentagon cuts off a triangle of area 1 from it. Calculate the area of this pentagon. | Let $ABCDE$ be a given convex pentagon, and $A$ - a regular pentagon such that the area of triangle $A$ is equal to $1$. Since triangles $ABC$ and $ABE$ have equal areas, the distances from points $C$ and $E$ to the line $AB$ are equal. Therefore, $CE \parallel AB$, and similarly, each diagonal of the pentagon $ABCDE$ ... | Geometry | math-word-problem | Yes | Incomplete | olympiads | false | |
VIII OM - I - Task 1
Solve the system of equations
| Considering the formula
we can transform the binomial $ x^5 - y^5 $ as follows:
If in the system of equations (1) we replace the left side of the first equation with the expression obtained above, and then substitute the number $ 2 $ for $ x - y $ in it, we get (after simplifying the first equation) an equi... | x=4,y=2;x=-2,y=-4;x=1+i\sqrt{11},y=-1+i\sqrt{11};x=1-i\sqrt{11},y=-1-i\sqrt{11} | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
II OM - I - Problem 12
Prove that if $ A + B + C = 180^\circ $, and $ n $ is an odd number, then
The translation is provided as requested, maintaining the original text's line breaks and format. | Based on known trigonometric formulas we have
Let $ n = 2k + 1 $, where $ k $ is an integer. From the equality $ A + B + C = 180^\circ $ it follows that
Therefore
and similarly
Substituting expressions (3) and (4) into the right side of equation (2) we get
and after transforming the expression in parentheses we ha... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
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 |
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 |
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 |
LVII OM - II - Problem 1
Positive integers $ a, b, c, x, y, z $ satisfy the equations
as well as the inequalities $ |x-a|\leq 1 $, $ |y-b|\leq 1 $.
Prove that the sets $ \{a,b\} $ and $ \{x,y\} $ are equal. | Due to the symmetry of the triples $ (a,b,c) $ and $ (x,y,z) $, we can assume that $ z \geq c $. Looking at the points
$ A=(a,b) $, $ X =(x,y) $ on the plane with the origin $ O = (0,0) $, we see that
Thus, we have two possibilities: $ z=c+1 $ or $ z=c $.
If $ z=c+1 $, then $ x^2+y^2 > a^2+b^2 $. Therefore, $ x > a... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
LVI OM - III - Problem 5
Let $ k $ be a natural number greater than 1 and let $ m = 4k^2 - 5 $.
Prove that there exist positive integers $ a $, $ b $, such that each term of the sequence $ (x_n) $ defined by the formulas
is coprime with the number $ m $. | We will show that the conditions of the problem are satisfied by the numbers $ a = 1 $, $ b = 2k^2 + k - 2 $. We will prove by induction that $ x_n \equiv b^{n-1}(\mathrm{mod}\ m) $ for $ n = 1,2,3,... $. For $ n = 1 $ and $ n = 2 $, this relation is obviously satisfied.
Using the relation $ 4k^2\equiv 5(\mathrm{mod}\... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
V OM - I - Problem 9
Points $ A, B, C, D, \ldots $ are consecutive vertices of a certain regular polygon, and the following relationship holds
How many sides does this polygon have? | \spos{1} Let $ r $ denote the radius of the circle circumscribed around a regular polygon $ ABCD\ldots $, and $ 2x $ - the central (convex) angle of this circle corresponding to the chord $ AB $. Then (Fig. 25)
Substituting these expressions into equation (1), we get
To solve equation (2), we multiply it by... | 7 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
XXII OM - III - Problem 4
Prove that if the natural numbers $ x, y, z, n $ satisfy the equation
then $ \min (x, y) > n $. | Suppose that the natural numbers $ x $, $ y $, $ z $, $ n $ satisfy the equation
(1). Without loss of generality, we can assume that $ x \leq y $. Since $ z^n = x^n + y^n > y^n $, we have $ z > y $ and thus $ z \geq y + 1 $. Therefore, by the binomial theorem we have
From (1) and (2), we obtain that $ x^n \geq ny... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
XX OM - I - Task 6
Solve the equation
Note: The symbol $ [a] $ denotes the greatest integer not greater than $ a $. | \spos{1} We can write the given equation in the form
Suppose that the number $ x $ satisfies equation (2). In this case, the number
is an integer. Substituting $ x = 2y+1 $ into (2), we get
One of the following cases applies:
a) $ y \equiv 0 \pmod 3 $.
Then $ \left[ \frac{2y+2}{3} \right] = \frac{2y}{... | 1,3,5 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
XXXIII OM - I - Problem 8
Given natural numbers $ l, m, p, q, n $ satisfying the conditions: $ \frac{l}{m} < \frac{p}{q} $ and $ m, q $ are relatively prime. Prove the equivalence of the following conditions.
(1) There exists a natural number $ k $ such that
(2) There exist natural numbers $ r, s $ such that ,
... | Assume that condition (1) is satisfied. Since the inequality $ \frac{l}{m} < \frac{k}{n} $ holds, then $ ln < km $. For some natural number $ s $, we have the equality $ km = ln + s $. We can write:
Similarly, from the inequality $ \frac{k}{n} < \frac{p}{q} $, it follows that $ kq < pn $. There exists a natural number... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
LVII OM - I - Problem 11
In quadrilateral $ABCD$, the measure of the internal angle at vertex $A$ is greater than $180^{\circ}$, and the following equality holds:
Point $P$ is symmetric to point $A$ with respect to line $BD$. Prove that $\measuredangle PCB = \measuredangle ACD$. | If $ AB=AD $, then from the given equality it follows that $ CD =BC $. Then points $ A $ and $ P $ lie on the bisector of angle $ BCD $,
from which we directly conclude that $ \measuredangle PCB = \measuredangle ACD $.
Let us assume in the further part of the solution that $ AB \neq AD $ and let for the sake of argume... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XVII OM - II - Task 2
Prove that if two polynomials of the third degree with integer coefficients have a common irrational root, then they have another common root. | Suppose the polynomials
which have integer coefficients $ a_1 $ and $ b_1 $ ($ i = 0, 1, 2, 3 $), with $ a_0 \ne 0 $, $ b_0 \ne 0 $, have a common irrational root $ \alpha $.
The number $ \alpha $ is also a root of the polynomial
since $ R(\alpha) = b_0 \cdot P(\alpha)- a_0 \cdot Q(\alpha)=0 $. One of two possibiliti... | proof | Algebra | proof | Yes | Incomplete | olympiads | false |
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 |
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 |
XIII OM - III - Problem 1
Prove that if the numbers $ a_1, a_2,\ldots, a_n $ ($ n $ - a natural number $ \geq 2 $) form an arithmetic progression, and none of them is zero, then | If the difference of progression $ d $ equals zero, then equation (1) expresses the identity $ \frac{n-1}{a_1^2} = \frac{n-1}{a_1^2} $.
If $ d \ne 0 $, then for any natural $ k < n $
Therefore
| proof | Algebra | proof | Yes | Incomplete | olympiads | false |
XIV OM - I - Problem 2
Prove that the product of binomials
where $ m $ is a natural number, is divisible by the product of binomials
| Because
since
In the factorization (1) of the polynomial $ Q(x) $, the factors
are pairwise relatively prime. Indeed, the polynomial $ (x - 1)^5 $ is relatively prime to each of the remaining polynomials (2), as it is a power of the irreducible polynomial $ x - 1 $, which is not a divisor of any... | proof | Combinatorics | proof | Incomplete | Incomplete | olympiads | false |
XXX OM - I - Task 4
In a convex polyhedron $ Q $, a convex polyhedron $ P $ is placed, in which an internal point $ O $ is chosen. For every point $ X $ lying on the surface of the polyhedron $ Q $, denote by $ P_X $ the point of intersection of the segment $ \overline{OX} $ with the surface of the polyhedron $ P $. P... | If point $ X $ belongs to the wall (or edge) of polyhedron $ Q $, then there exist points $ A $ and $ B $ belonging to the edges of polyhedron $ Q $ (respectively: to the set of its vertices), such that point $ X $ belongs to segment $ \overline{AB} $. Let $ A $ and $ B $ be such points that $ A, $ B, $ A $ and $ P_X \... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
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 |
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 |
XXVII OM - I - Problem 2
Prove that | We will prove that for every natural number $ n $, the formula holds.
For $ n = 25 $, the formula given in the problem follows from it. By the binomial formula, we have
Grouping the terms of the obtained sum in pairs and using the formula
we conclude that
We write this last sum as the sum of two sums. The terms of ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
II OM - I - Task 10
A circle is circumscribed around triangle $ ABC $. Given the radius $ R $ of this circle, calculate the radius of the circle passing through the centers of the three excircles (inscribed in the angles) of triangle $ ABC $. | Let $ S $, $ T $, $ U $ (Fig. 44) denote the centers of the excircles of triangle $ ABC $. These points are the vertices of a triangle formed by the external angle bisectors of triangle $ ABC $.
To calculate the radius of the circumcircle of triangle $ STU $, we will first determine the angles of this triangle. Since
... | 2R | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
XLIII OM - II - Problem 2
Given a natural number $ n \geq 2 $. Let $ a_1, a_2, \ldots , a_n $, $ b_1, b_2, \ldots , b_n $ be real numbers. Prove that the following conditions are equivalent:
For any real numbers $ x_1 \leq x_2 \leq \ldots \leq x_n $
the inequality holds
For every natural number $ k\in \{1,2,\ld... | Proof of implication (a) $ \Longrightarrow $ (b). In the inequality assumed in condition (a), we substitute $ x_1 = x_2 = \ldots = x_n = 1 $ and obtain
while substituting $ x_1 = x_2 = \ldots = x_n = -1 $ we obtain
Hence
Let us fix a number $ k \in \{1,2,\ldots,n- 1\} $ and assume $ x_1 = \ldots = x_k... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
I OM - B - Task 7
In the plane, two intersecting lines
$ a $ and $ b $ and a point $ P $ not lying on them are given. Draw a line through point $ P $
intersecting lines $ a $ and $ b $ at points $ A $ and $ B $, such that | Let $ AB $ (Fig. 2) be the line sought.
Draw through point $ P $ a line parallel to line $ b $; it will intersect line $ a $ at point $ Q $.
om1_Br_img_2.jpg
According to Thales' theorem we have
Thus, the construction is carried out by determining point $ Q $ by measuring $ QA = 2 \cdot OQ $. Line $ AP $ will b... | notfound | Geometry | math-word-problem | Incomplete | Incomplete | olympiads | false |
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 |
II OM - I - Task 5
Prove that if $ n $ is an even natural number, then the number $ 13^n + 6 $ is divisible by $ 7 $. | It is enough to prove that when $ n $ is even, $ 13^n $ gives a remainder of $ 1 $ when divided by $ 7 $.
We will use the method of complete induction.
Let $ n = 2k $. When $ k = 1 $, the statement is true, because
Suppose that for some $ k $ the equality holds
where $ m $ is a natural number.
Then
From thi... | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
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