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VII OM - I - Problem 12
Prove that: a) the sum of the distances from any point $ M $ inside an equilateral triangle to its three sides is constant, i.e., it does not depend on the position of point $ M $ inside the triangle; b) the sum of the distances from any point inside a regular tetrahedron to its four faces is c... | \spos{1}
1°. Let us choose a point $ M $ inside an equilateral triangle $ ABC $ (Fig. 12). Let $ a $ denote the length of the side of this triangle, $ h $ - its height, $ x $, $ y $, $ z $ the distances from point $ M $ to the sides $ BC $, $ CA $, $ AB $, respectively. Since
then
from which
If the po... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,463 |
XXXI - I - Problem 5
A well-known mathematician, Professor X, is playing a game with his assistant, Master Y, as follows. X gives Y a finite sequence of vectors $ \overrightarrow{a_1}, \overrightarrow{a_2}, \ldots, \overrightarrow{a_n} $; all these vectors lie in the same plane and each has a length of 1. Knowing thes... | om31_1r_img_9.jpg
om31_1r_img_10.jpg
We will describe Igr's strategy inductively. $ \varepsilon_1 $ can be chosen arbitrarily, for example, $ \varepsilon_1 = 1 $. Then, $ \varepsilon_2 $ should be chosen so that the vector $ \varepsilon_1 \overrightarrow{a_1} + \varepsilon_2 \overrightarrow{a_2} $ has a length less tha... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,464 |
XII OM - I - Problem 8
Two equilateral triangles are inscribed in a circle. The vertices $ A $, $ B $, $ C $ of one triangle follow each other on the circle in the same direction as the vertices $ A_1 $, $ B_1 $, $ C_1 $ of the other triangle. The lines $ AB $ and $ A_1B_1 $ intersect at point $ P $, the lines $ BC $ ... | Let's rotate the entire figure (Fig. 8) around the center $ O $ of the given circle by $ 120^\circ $. Points $ A $, $ B $, $ C $ will move to the positions of points $ B $, $ C $, $ A $, respectively, and points $ A_1 $, $ B_1 $, $ C_1 $ will move to the positions of points $ B_1 $, $ C_1 $, $ A_1 $, respectively. The ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,466 |
LV OM - II - Task 6
At a party, $ n $ people met $ (n \geq 5) $. It is known that among any three people, some two know each other. Prove that among the party attendees, one can select at least $ n/2 $ people and seat them at a round table so that each person sits between two of their acquaintances. | Let $ X $ be the set of people present at the party. Let $ k $ be the largest natural number with the property: There exist $ k $ people $ A_1, A_2, \ldots, A_k $ such that for $ i = 1, 2, \ldots, k-1 $, people $ A_i $ and $ A_{i+1} $ know each other. Denote:
In the further solution, we will consider three cases:... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,467 |
XVII OM - II - Problem 6
Prove that the sum of the squares of the orthogonal projections of the sides of a triangle onto a line $ p $ in the plane of the triangle does not depend on the position of the line $ p $ if and only if the triangle is equilateral. | Let $ S $ denote the sum of the squares of the orthogonal projections of the sides of an equilateral triangle $ ABC $ with side length $ a $ onto a line $ p $ lying in the plane of the triangle. Since the projections of a segment onto parallel lines are equal, for the calculation of $ S $ we can assume that the line $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,470 |
LX OM - III - Task 4
Let $ x_1,x_2,\cdots ,x_n $ be non-negative numbers whose sum is 1. Prove that there exist
numbers $ a1,a2,\cdots ,an \in \{0, 1, 2, 3, 4\} $ such that $ (a_1,a_2,\cdots ,a_n) \neq (2, 2,\cdots , 2) $ and | Consider all possible sequences $ t =(t_1,t_2,\cdots ,t_n) $ such that $ t_1,t_2,\cdots ,t_n \in \{0, 1, 2\} $.
There are $ 3^n $ of them. For each of them, let $ S_t $ denote the sum $ t_1x_1 +t_2x_2 +\cdots +t_nx_n $.
Since the numbers $ x_1,x_2,\cdots ,x_n $ are non-negative and their sum is 1, the following inequal... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,471 |
XLV OM - II - Task 6
Given a prime number p. Prove the equivalence of the following statements:
(a) There exists an integer $ n $ such that the number $ n^2 - n + 3 $ is divisible by $ p $.
(b) There exists an integer $ m $ such that the number $ m^2 - m + 25 $ is divisible by $ p $. | The desired equivalence follows easily from the observation that
Implication (a) $ \Longrightarrow $ (b) is immediate: if (for some $ n $) the number $ n^2 -n + 3 $ is divisible by $ p $, it suffices to take $ m = 3n-1 $; according to (1), the number $ m^2 -m + 25 $ is also divisible by $ p $.
To prove the implication... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,472 |
XLVI OM - I - Problem 3
A quadrilateral with sides $a$, $b$, $c$, $d$ is inscribed in a circle with radius $R$. Prove that if $a^2 + b^2 + c^2 + d^2 = 8R^2$, then one of the angles in the quadrilateral is a right angle or the diagonals are perpendicular. | Suppose that none of the interior angles of the considered quadrilateral $ABCD$ is a right angle; we need to show that in this case $AC \bot BD$. The quadrilateral is inscribed in a circle, so $|\measuredangle DAB| + |\measuredangle BCD| = 180^\circ$. Without loss of generality, assume that angle $DAB$ is acute, and an... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,473 |
XXX OM - III - Task 1
Given is a set $ \{r_1, r_2, \ldots, r_k\} $ of natural numbers, which when divided by a natural number $ m $ give different remainders. Prove that if $ k > \frac{m}{2} $, then for every integer $ n $ there exist such $ i $ and $ j $ ($ i $ and $ j $ not necessarily different), that the number $ ... | The set $ A $ of remainders when dividing $ n-r_1, n-r_2, \ldots, n-r_k $ by $ m $ contains more than $ \displaystyle \frac{m}{2} $ elements. Similarly, the set $ B $ of remainders when dividing $ r_1, r_2, \ldots, r_k $ by $ m $ also contains more than $ \displaystyle \frac{m}{2} $ elements. Sets $ A $ and $ B $ are c... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,474 |
XL OM - III - Task 1
An even number of people sat down at a round table for the meeting. After the lunch break, the participants took their seats at the table in any order. Prove that there are two people who are separated by the same number of people as before the break. | Let's denote the number of people by $ n $. We number the seats at the table consecutively with integers from $ 0 $ to $ n - 1 $; seat number $ 0 $ and the direction of circulation are chosen arbitrarily. The person occupying seat $ k $ before the break takes seat $ m_k $ after the break; the sequence $ (m_0, \ldots, m... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,476 |
XV OM - II - Problem 5
Given is a trihedral angle with edges $SA$, $SB$, $SC$, where all dihedral angles are acute, and the dihedral angle along the edge $SA$ is right. Prove that the section of this trihedral angle by any plane perpendicular to any of the edges, at a point different from the vertex $S$, is a right tr... | First, note that if the angles of the tetrahedron $SABC$ are acute, then a plane perpendicular to any edge of the tetrahedron at a point different from $S$ intersects the other edges at points different from $S$, so the section of the tetrahedron by this plane is a triangle. For example, a plane passing through point $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,479 |
XXVI - II - Problem 6
Let $ f(x) $ and $ g(x) $ be polynomials with integer coefficients. Prove that if for every integer value $ n $ the number $ g(n) $ is divisible by the number $ f(n) $, then $ g(x) = f(x) \cdot h(x) $, where $ h(x) $ is a polynomial. Show by example that the coefficients of the polynomial $ h(x) ... | In the solution to the preparatory task D of Series III, we proved that if polynomials $f$ and $g$ have integer coefficients, $\mathrm{st} f \leq \mathrm{st} g$, and for every natural number $n$, the number $f(n)$ is a divisor of the number $g(n)$, then there exists an integer $c$ such that $g(x) = c \cdot f(x)$. There... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,480 |
XIV OM - III - Task 6
Through the vertex of a trihedral angle, in which no edge is perpendicular to the opposite face, a line perpendicular to the opposite edge is drawn in the plane of each face. Prove that the three obtained lines lie in one plane. | Let $SM$, $SN$, and $SP$ denote the lines perpendicular to the edges $SA$, $SB$, and $SC$ of the tetrahedron $SABC$ and lying in the planes of the faces $BSC$, $CSA$, and $ASB$, respectively. Given that no edge of the tetrahedron is perpendicular to the opposite face, the lines $SM$, $SN$, and $SP$ are uniquely determi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,481 |
XLIII OM - I - Problem 2
In square $ABCD$ with side length $1$, point $E$ lies on side $BC$, point $F$ lies on side $CD$, the measures of angles $EAB$ and $EAF$ are $20^{\circ}$ and $45^{\circ}$, respectively. Calculate the height of triangle $AEF$ drawn from vertex $A$. | The measure of angle $ FAD $ is $ 90^\circ - (20^\circ + 45^\circ) = 25^\circ $. From point $ A $, we draw a ray forming angles of $ 20^\circ $ and $ 25^\circ $ with rays $ AE $ and $ AF $, respectively, and we place a segment $ AG $ of length $ 1 $ on it (figure 2).
From the equality $ |AG| =|AB| = 1 $, $ | \measureda... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,482 |
LVIII OM - I - Problem 5
Given is an acute triangle $ ABC $, where $ \angle ACB=45^\circ $. Point $ O $ is the center of the circumcircle of triangle $ ABC $, and point $ H $ is the orthocenter of triangle $ ABC $. The line passing through point $ O $ and perpendicular to line $ CO $ intersects lines $ AC $ and $ BC $... | Let $D$ be the foot of the altitude from vertex $B$, and let $G$ be the point symmetric to point $H$ with respect to the line $AC$ (Fig. 2); segments $BG$ and $AC$ intersect at point $D$.
Notice that point $G$ lies on the circumcircle of triangle $ABC$. Indeed, due to the perpendicularities $BH \perp AC$ and $CH \perp ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,483 |
XXXI - I - Problem 7
On a sphere of radius 1, 13 points are chosen. Prove that among these points, there are two whose distance is less than $ \frac{2\sqrt{3}}{3} $. | Consider a regular dodecahedron inscribed in a sphere bounded by a given sphere. Among the radii of the sphere connecting certain points with the center, two of them intersect one face of the dodecahedron. The angle formed by these radii is no greater than the angle formed by the radii passing through two non-adjacent ... | \frac{2\sqrt{3}}{3} | Geometry | proof | Yes | Yes | olympiads | false | 1,484 |
I OM - B - Task 3
Someone has an unbalanced two-pan scale (i.e., a scale
whose arms are unequal). Wanting to weigh 2 kg of sugar, they proceed
as follows: they place a 1 kg weight on the left pan of the scale,
and pour sugar onto the right pan until it balances; then:
after emptying both pans, they place the 1 kg wei... | Let's denote the amount of sugar weighed for the first time by $ x $, and for the second time by $ y $. The lengths of the arms of the balance are $ p $ and $ q $, where $ p \neq q $.
According to the lever law (Fig. l) we have:
Therefore,
since
Answer: More than 2 kg of sugar was weighed in total. | More\than\2\ | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 1,485 |
III OM - II - Task 5
A vertical mast located on a tower is seen at the largest angle from such a point on the ground, the distance of which from the axis of the mast is $ a $; this angle equals the given angle $ \alpha $. Calculate the height of the tower and the height of the mast. | Let segment $AB$ (Fig. 31) represent a tower, and segment $BC$ - a mast. From point $P$ on the ground, the mast is seen at an angle $BPC$, which is an inscribed angle in a circle passing through points $B$, $C$, and $P$. This angle is larger the smaller the radius of that circle. Among all circles passing through point... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,487 | |
XV OM - I - Problem 2
Prove the following two divisibility rules for 7:
a) A natural number $ N $ is divisible by 7 if and only if the number $ P $ formed in the following way is divisible by 7: multiply the first digit from the left of the number $ N $ by 3 and add the second digit; multiply the resulting number by 3... | Let $ a_0, a_1, \ldots, a_n $ denote the consecutive digits of the number $ N $ (from left to right). Then
a) According to the problem, the number $ P $ is the $(n+1)$-th term of the sequence
therefore
Since for a natural number $ k $
then each term on the right side of equation (1) is divisible by $ 7 $. The numbe... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,488 |
XXXVII OM - I - Problem 11
We roll a die once, and then as many times as the number of dots that came up on the first roll. Calculate the expected value of the sum of the dots rolled (including the first roll). | Let $ X_0 $ denote the number of eyes rolled in the first throw. If $ X_0 = j $, then $ j $ additional throws are performed. Denote their results sequentially by $ X_1, \ldots, X_j $ and assume $ X_k = 0 $ for $ k > j $. The random variable considered in the task is then the sum $ X = \sum_{k=0}^6 X_k $.
Let us fix $ k... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,491 |
XXI OM - II - Problem 4
Prove that if triangle $ T_1 $ contains triangle $ T_2 $, then the perimeter of triangle $ T_1 $ is not less than the perimeter of triangle $ T_2 $. | At the vertices of triangle $ T_2 $, we erect lines perpendicular to its sides (Fig. 11). These lines determine points $ A_1 $, $ B_1 $, $ B_2 $, $ C_2 $, $ C_3 $, $ A_3 $ on the boundary of triangle $ T_1 $. Since $ AB \bot AA_1 $ and $ AB \bot BB_1 $, the length of segment $ \overline{AB} $ is not greater than the le... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,493 |
II OM - II - Task 1
In a right triangle $ ABC $ with the right angle at vertex $ C $, a height $ CD $ is drawn from vertex $ C $, and a circle is inscribed in each of the triangles $ ABC $, $ ACD $, and $ BCD $. Prove that the sum of the radii of these circles is equal to the height $ CD $. | We adopt the notations indicated in Figure 52a; let the radii of the circles inscribed in triangles $ABC$, $BCD$, $CAD$ be denoted by $r$, $r_1$, $r_2$, respectively.
Since the right triangles $ABC$, $CBD$, $ACD$ are similar, the radii of the circles inscribed in these right triangles are proportional to their hypotenu... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,496 |
XXXII - II - Problem 2
Two circles are internally tangent at point $ P $. A line tangent to one of the circles at point $ A $ intersects the other circle at points $ B $ and $ C $. Prove that line $ PA $ is the angle bisector of $ \angle BPC $. | om32_2r_img_11.jpg
Consider a homothety with center $ P $ and scale equal to the ratio of the radius of the larger circle to the radius of the smaller one (Fig. 11). This homothety transforms the smaller circle into the larger one, and the tangent $ BC $ into the line $ B_1C_1 $ tangent to the larger circle at point $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,497 |
XIII OM - III - Problem 2
Inside a given convex quadrilateral, find a point such that the segments connecting this point to the midpoints of the sides of the quadrilateral divide the quadrilateral into four parts of equal area. | The area of a convex quadrilateral $ABCD$ equals the area of triangle $B$, where $B$ is the intersection point of line $DA$ with a line parallel to diagonal $AC$ and drawn from point $B$. Let $K$ be the midpoint of segment $DB$ (Fig. 32), $P$ and $Q$ the midpoints of sides $CD$ and $DA$, and $O$ the point to be found. ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,502 |
LVII OM - II - Problem 2
Given a triangle $ABC$ where $AC + BC = 3AB$. The incircle of triangle $ABC$ with center $I$ is tangent to sides $BC$ and $CA$ at points $D$ and $E$, respectively. Let $K$ and $L$ be the points symmetric to $D$ and $E$ with respect to point $I$. Prove that points $A, B, K, L$ lie on the same c... | Let P be the point of intersection of the angle bisector of angle ACB with the circumcircle of triangle ABC (Fig. 1). Then AP = BP. Moreover,
from which it follows that $ AP = IP $.
om57_2r_img_1.jpg
om57_2r_img_2.jpg
The points $ A, B $, and $ I $ therefore lie on a circle with center $ P $. We will show that po... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,504 |
XXXIII OM - I - Problem 7
Prove that in any convex pentagon, all of whose sides have length \( a \), an equilateral triangle with sides of length \( a \) can be placed. | om33_1r_img_2.jpg
Let $ABCDE$ be any convex pentagon whose all sides have length $a$. Without loss of generality, we can assume that $\overline{AC}$ is the diagonal of maximum length. We will show that an equilateral triangle with side length $a$ can be placed inside the quadrilateral $ACDE$. For this purpose, we will ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,506 |
V OM - II - Task 5
Given points $ A $, $ B $, $ C $, and $ D $ not lying in the same plane. Construct a plane through point $ A $ such that the orthogonal projection of quadrilateral $ ABCD $ onto this plane is a parallelogram. | Let $ B, $ C, $ D be the orthogonal projections of points $ B $, $ C $, $ D $ onto the plane a passing through point $ A $. The projection of the midpoint $ M $ of segment $ AC $ is the midpoint $ M of segment $ AC, and the projection of the midpoint $ N $ of segment $ BD $ is the midpoint $ N of segment $ B. The plana... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,508 |
XX OM - I - Problem 7
Square $ABCD$ is rotated around its center $S$ by an acute angle $\alpha$, and then the rotated square is translated to the position $EFGH$ by a vector of the length of the side of the square, perpendicular to the plane of the square. Calculate the volume of the solid bounded by the squares $ABCD... | We will assume that the square has been rotated in the direction indicated by the sequence of its vertices $ A $, $ B $, $ C $, $ D $.
The solution is illustrated by Figure 4, which shows the figure in orthogonal projections onto two planes: the horizontal plane on which the square $ ABCD $ lies and the vertical plane.... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,509 | |
XXXIX OM - I - Problem 10
Knowing that $ x \in \langle 0; \pi/4\rangle $, determine which of the numbers is greater: | We will show that the first of these numbers is greater (equality holds only for $ x=0 $).
Let us introduce the notation:
We calculate the derivative of the function $ f $ in the interval $ (0; \pi/2) $:
We will need the following inequality in the further course of the proof:
We prove it as follows:... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,510 |
XXXVIII OM - I - Problem 11
Prove that a sphere can be illuminated by four point light sources placed outside it, but cannot be illuminated by three. | The reasoning is based on the observation that the smaller of the two hemispheres into which any plane dividing the interior of a sphere but not passing through its center can be illuminated by a single point light source placed outside the sphere; however, such a source cannot illuminate the entire hemisphere.
We insc... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,511 |
XXV - I - Problem 9
A set of documents has been divided into $ n $ parts, each given to one of $ n $ people, each of whom has a phone. Prove that for $ n \geq 4 $, it suffices to have $ 2n - 4 $ phone calls, after which each of the $ n $ people knows the content of all the documents. | For four people $ A $, $ B $, $ C $, $ D $, four conversations suffice. Specifically, first $ A $ and $ B $, as well as $ C $ and $ D $, inform each other about the documents they possess. Then $ A $ and $ C $, as well as $ B $ and $ D $, exchange the information they have.
If the number of people $ n $ exceeds $ 4 $, ... | 2n-4 | Combinatorics | proof | Yes | Yes | olympiads | false | 1,512 |
XIX OM - III - Problem 1
What is the largest number of regions into which a plane can be divided by $ n $ pairs of parallel lines? | Let $ f(n) $ denote the greatest number of regions into which $ n $ pairs of parallel lines can divide the plane.
Consider $ k + 1 $ pairs of parallel lines on the plane.
Suppose that $ (p_1, q_1), (p_1, q_2),\ldots (p_k,q_k) $ divide the plane into $ l $ regions, and that the line $ p_{k+1} $ intersects the lines of t... | 2n^2-n+1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,514 |
XXXV OM - III - Task 5
A regular hexagon with side length 1 is contained within the union of six circles, each having a diameter of 1. Prove that no vertex of the hexagon belongs to two of these circles. | Suppose a certain vertex of a hexagon belongs to two circles. If neither of these two circles contained any other vertex of the hexagon, then each of the remaining five vertices would belong to one of the remaining four circles. It follows that there is a circle $ k_1 $ containing two vertices of the hexagon. The diame... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,516 |
XLVII OM - I - Problem 10
Prove that the equation $ x^x = y^3 + z^3 $ has infinitely many solutions in positive integers $ x, y, z $. | We will present a broad class of solutions to this equation.
Let $ m, n $ be positive integers satisfying the condition:
For any integer $ c $, the number $ c^3-c = c(c-1)(c+1) $ is divisible by $ 3 $. From the identity $ m^3 + n^3 -1 = (m^3 - m) + (n^3 - n) + (m + n -1) $, it follows that condition (1) implies c... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,517 |
LIV OM - I - Problem 11
Given is a convex quadrilateral $ABCD$. Points $P$ and $Q$, different from the vertices of the quadrilateral, lie on sides $BC$ and $CD$ respectively, such that $ \measuredangle BAP = \measuredangle DAQ $. Prove that triangles $ABP$ and $ADQ$ have equal areas if and only if their orthocenters l... | Let $ K $, $ L $ denote the orthogonal projections of points $ A $, $ B $ onto the lines $ BC $, $ AP $ (Fig. 3). Let $ H $ denote the orthocenter of triangle $ ABP $, and let $ X $ be the orthogonal projection of point $ H $ onto the line $ AC $.
Using the similarity of right triangles $ AHL $ and $ APK $, we obtain
... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,522 |
LIII OM - III - Problem 4
Prove that for every natural number $ n \geq 3 $ and for every sequence of positive numbers $ x_1,x_2,\ldots,x_n $, at least one of the inequalities holds
(we assume $ x_{n+1}=x_{1},\ x_{n+2}=x_{2} $ and $ x_{0}=x_{n},\ x_{-1}=x_{n-1} $). | If none of the given inequalities held, then by adding them side by side, we would have $ \displaystyle \sum_{i=1}^{n}\frac{x_{i-1}+x_{i+2}}{x_{i}+x_{i+1}} < n $. It is therefore sufficient to prove that
By substituting $ a_{i}=\displaystyle \frac{x_{i-1}+x_{i}}{x_{i}+x_{i+1}} $, the last inequality takes the form
an... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,523 |
XXVI - I - Task 2
In a convex quadrilateral $ABCD$, points $P$ and $B$ are chosen on the opposite sides $\overline{AB}$ and $\overline{GD}$, and points $Q$ and $S$ are chosen on the opposite sides $\overline{AD}$ and $\overline{BC}$ such that $\frac{AP}{PB} = \frac{DR}{RC} = a$, $\frac{AQ}{QD} = \frac{BS}{SC} = b$. Pr... | We place masses of $1$, $a$, $ab$, and $b$ at points $A$, $B$, $C$, and $D$ respectively, and calculate the center of mass $T$ of the system $U$ composed of these four points.
From the conditions of the problem, it follows that the centers of mass of the systems $\{A, B\}$ and $\{C, D\}$ are the points $P$ and $R$ resp... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,529 |
I OM - B - Task 6
Prove that the number which, in the decimal positional system,
is expressed using 91 ones, is a composite number. | Since $ 91=7\cdot 13 $, the digits of the given number $ L $ can be divided into 13 groups
with 7 ones in each group:
From this, it is clear that the number is divisible by $ 1111111 $; in the quotient, we obtain the number
To put it more precisely, let's denote $ 1111111=N $ for short. The given number ca... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,530 |
XI OM - II - Problem 6
Calculate the volume of the tetrahedron $ABCD$ given the edges $AB = b$, $AC = c$, $AD = d$ and the angles $\measuredangle CAD = \beta$, $\measuredangle DAB = \gamma$ and $\measuredangle BAC = \delta$. | Let's draw the height $DH$ of the tetrahedron $ABCD$ (Fig. 24). The volume $V$ of the tetrahedron is given by the formula:
---
The volume $V$ of the tetrahedron is expressed by the formula:
---
The volume $V$ of the tetrahedron is given by the formula:
---
The volume $V$ of the tetrahedron is expressed by the for... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,531 | |
IX OM - I - Problem 7
Prove that if a plane figure of finite size has a center of symmetry $ O $ and an axis of symmetry $ s $, then the point $ O $ lies on the line $ s $. | Suppose the theorem is not true, i.e., there exists a figure $ F $ of finite size having an axis of symmetry $ s $ and a center of symmetry $ O $ not lying on the line $ s $. Let $ OS = d $ (Fig. 7) denote the distance from point $ O $ to the line $ s $, and let $ A_1 $ be any point of figure $ F $, and $ A_1T = a $ it... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,533 |
IV OM - I - Task 1
Prove that the ratio of the sum of the medians of any triangle to its perimeter lies in the interval $ \left( \frac{3}{4}, 1 \right) $ and that no narrower interval has this property. | Let the sides of the triangle be $BC = a$, $CA = b$, $AB = c$, and the medians to these sides be $m_a$, $m_b$, $m_c$. Let $S$ denote the point of intersection of the medians, i.e., the centroid of triangle $ABC$ (Fig. 20).
In triangle $BSC$, we have the inequality $BS + SC > BC$, which is $\frac{2}{3} m_b + \frac{2}{3... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,535 |
VII OM - I - Problem 7
In space, there lies a rectangle $ABCD$ and a point $M$. Calculate the distance $MD$ given the distances $MA$, $MB$, and $MC$. | We will rely on the auxiliary theorem:
If $ O $ is the midpoint of segment $ AB $, and $ M $ is any point, then
We can justify the above equality as follows. If point $ M $ does not lie on line $ AB $, then from triangles $ AOM $ and $ BOM $ (Fig. 5) we have
Adding equations (4) side by side and considering... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,537 |
VIII OM - I - Problem 12
Prove that:
1) $ n $ lines lying on a plane, each two of which intersect, but no three pass through the same point, divide the plane into $ \frac{1}{2} (n^2 + n + 2) $ parts;
2) $ n $ planes, each three of which have one and only one common point, but no four pass through the same point, divid... | 1. We will apply complete induction. One straight line divides the plane into $2$ parts. Since the expression $\frac{1}{2} (n^2 + n + 2)$ has for $n = 1$ the value $\frac{1}{2} (1^2 + 1 + 2) = 2$, the theorem is true in the case $n = 1$. Let us assume that it is true when $n = k$ and consider $k + 1$ lines $l_1, l_2, \... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,539 |
XXXV OM - I - Problem 1
Determine whether a regular hexagon can be covered by six circles with a diameter equal to the length of the hexagon's side. | om35_1r_img_1.jpg
The desired coverage is obtained by taking circles whose diameters are the segments connecting the center of the hexagon with each of its vertices. The circle having diameter $ \overline{AO} $, where $ A $ is a vertex of the hexagon, contains the halves of the sides meeting at vertex $ A $. This follo... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,541 |
XXVII OM - I - Problem 7
Prove that any pair of non-intersecting circles can be transformed by inversion into a pair of concentric circles.
Note: Inversion with respect to a circle with center $ O $ and radius $ r $ is a transformation that assigns to each point $ X $ of the plane containing this circle, different fro... | Let line $ m $ contain the centers of given circles and let segments $ \overline{AB} $ and $ \overline{CD} $, where $ A, B, C, D \in m $, be diameters of the given circles. From the conditions of the problem, it follows that points $ A $, $ B $, $ C $, $ D $ are distinct and either segments $ \overline{AB} $ and $ \ove... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,545 |
XII OM - I - Problem 11
Given is an isosceles triangle $ ABC $ ($ AB = AC $). Construct an isosceles triangle $ A_1B_1C_1 $ ($ A_1B_1 = A_1C_1 $) such that the triangles $ AB_1C_1 $, $ A_1BC_1 $, and $ A_1B_1C $ are: a) located outside the triangle $ A_1B_1C_1 $, b) isosceles with bases $ B_1C_1 $, $ C_1A_1 $, and $ A... | In construction tasks, it is often convenient to use the method of the inverse problem, which involves building figure $G$ when figure $F$ is given, instead of building figure $F$ when figure $G$ is given. We then consider the relative positions of figures $F$ and $G$ to infer how to perform the construction in the rev... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,546 |
XLVI OM - I - Problem 4
In a certain school, 64 students participate in five subject olympiads. In each of these olympiads, at least 19 students from this school participate; none of them participate in more than three olympiads. Prove that if any three olympiads have a common participant, then some two of them have a... | Let's denote the Olympiads by the symbols $ A $, $ B $, $ C $, $ D $, $ E $. Imagine a table with $ 64 $ rows, corresponding to students, and $ 25 $ columns with the following headers:
In the column with the header $ X $, representing the Olympiad $ X \in \{A,B,C,D,E\} $, we write $ 1 $ if the student parti... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,548 |
XXX OM - III - Task 6
Polynomial $ w $ of degree $ n $ ($ n>1 $) has $ n $ distinct real roots $ x_1, x_2, \ldots, x_n $. Prove that | From the definition, it follows that
For $ i = 1, 2, \ldots, n $, let $ w_i(x) $ be the polynomial $ \displaystyle \frac{w(x)}{x-x_i} $, that is, the polynomial obtained from $ w(x) $ by removing the factor $ x - x_i $ in representation (1).
Since the numbers $ x_1, x_2, \ldots, x_n $ are distinct, $ w_i(x_i) \ne 0 $ ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,549 |
XLV OM - II - Task 1
Determine all polynomials $ P(x) $ of degree at most fifth with real coefficients, having the property that the polynomial $ P(x) + 1 $ is divisible by $ (x - 1)^3 $ and the polynomial $ P(x) - 1 $ is divisible by $ (x + 1)^3 $. | Suppose that the polynomial $ P(x) $ satisfies the given conditions. By assumption, there exist polynomials $ Q(x) $ and $ R(x) $, of degree at most two, satisfying for all real values of the variable $ x $ the equations
The coefficient of $ x^2 $ in each of the polynomials $ Q(x) $, $ R(x) $ is equal to the coefficie... | P(x)=-\frac{1}{8}x^5+\frac{5}{8}x^3-\frac{15}{8}x | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,550 |
XII OM - I - Problem 4
Prove that if $ A $, $ B $, $ C $, $ D $ are vertices of a tetrahedron, and $ S $ is the centroid of triangle $ ABC $, then | \spos{I} The proof of a spatial theorem can often be conducted analogously to the proof of a corresponding planar theorem. This is also the case here. Indeed, the following theorem holds: the median $CS$ of triangle $ABC$ is less than the arithmetic mean of sides $AC$ and $BC$ (Fig. 3).
For the proof, it suffices to co... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,551 |
XVII OM - III - Task 3
Prove that the sum of the squares of the areas of the rectangular projections of a cuboid onto one plane does not depend on the position of this plane if and only if the cuboid is a cube. | Let $ \delta $ denote the sum of the squares of the areas of the rectangular projections of a rectangular parallelepiped onto the plane $ \pi $.
When $ \pi $ is the plane of one of the faces of the parallelepiped, the sum $ \sigma $ equals twice the square of the area of that face. If the parallelepiped is not a cube, ... | 2a^4 | Geometry | proof | Yes | Yes | olympiads | false | 1,552 |
XVIII OM - I - Problem 5
Find such natural numbers $ p $ and $ q $, so that the roots of the equations $ x^2 - qx + p = 0 $ and $ x^2 - px + q = 0 $ are natural numbers. | Let $ x_1 $, $ x_2 $ and $ y_1 $, $ y_2 $ denote the roots of the given equations, respectively. Then the following equalities hold:
From equalities (1) and (2), it follows that
Hence
Since each of the numbers $ x_1-1 $, $ x_2-1 $, $ y_1-1 $, $ y_2-1 $ is $ \geq 0 $, one of the following cases must hold:
a) $ (x_1-1... | p=4,\p= | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,553 |
LIX OM - I - Task 12
Given an integer $ m \geqslant 2 $. Determine the smallest such integer
$ n \geqslant m $, such that for any partition of the set $ \{m,m+1,\dots ,n\} $ into two subsets, at least one of
these subsets contains numbers $ a, b, c $ (not necessarily distinct), such that $ ab = c $. | Answer: $ n = m^5 $.
Assume that there is a partition of the set $ \{m,m+1,\dots ,m^5\} $ into two subsets $ S $ and $ T $,
neither of which contains two numbers (not necessarily distinct) along with their product. Suppose,
moreover, that $ m \in S $. The number $ m^2 = m \cdot m $ is the product of two elements of th... | ^5 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,554 |
XV OM - I - Problem 10
Find the $ n $-th term of the sequence of numbers $ a_1, a_2, a_n, \ldots $, where $ a_1 = 1 $, $ a_2 = 3 $, $ a_3 = 6 $, and for every natural number $ k $ | We can write equality (1) in the form
Let's denote $ a_{i+1} - a_i = r_i $ for $ i = 1, 2, 3, \ldots $; according to the above equality, $ r_{k+2} - 2r_{k+1} + r_k = 0 $, from which $ r_{k+2} - r_{k+1}= r_{k+1} - r_k $.
The sequence of differences $ r_k $ is therefore an arithmetic progression. Since $ r_1 = a_2... | \frac{n(n+1)}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,556 |
XLVI OM - III - Problem 1
Determine the number of subsets of the set $ \{1,2, \ldots , 2n\} $ in which the equation $ x + y = 2n+1 $ has no solutions. | A set $ A $, contained in the set $ \{1,2,\ldots,2n\} $, will be called good if it has the required property; that is, if the equation $ x + y = 2n + 1 $ has no solution $ x,y \in A $.
The set $ \{1,2,\ldots,2n\} $ is the union of the following disjoint two-element subsets (i.e., pairs of numbers):
the symbol $ P_j $... | 3^n | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,557 |
I OM - I - Task 2
Prove that the sum of the cubes of $ n $ consecutive terms of an arithmetic progression is divisible by the sum of these terms. | In the calculation to be performed, it will be convenient to use the known formulas for the sum of the first, second, and third powers of the first $ (n - 1) $ natural numbers. We will denote these sums by $ \sigma_1, \sigma_2, \sigma_3 $:
Notice that
Let $ a $ denote the first of the $ n $ consecutive terms of the a... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,558 |
XV OM - I - Problem 5
Prove the following theorems, where $ m $ and $ n $ denote natural numbers.
1. If $ m > 1 $, and $ m^n-1 $ is a prime number, then $ n $ is a power of 2 with an integer exponent.
2. If $ n > 1 $, and $ m^n- 1 $ is a prime number, then $ m = 2 $, and $ n $ is a prime number. | 1. Suppose that $ n $ is not a power of the number $ 2 $ with an integer exponent. Then $ n $ has an odd divisor greater than $ 1 $, for example, $ 2p+1 $; where $ p $ is a natural number, so $ n = k (2p + 1) $, where $ k $ is a natural number smaller than $ n $. Then
therefore $ m^n+1 $ is divisible by $ m^k+1 $. But... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,562 |
XXXI - I - Problem 12
Let $ D $ be the region on the plane bounded by the half-line $ \{(x,y) : x \geq 0, y = 0\} $ and the parabola with the equation $ y = x^2 $, i.e., $ D = \{(x,y) : x > 0, 0 < y < x^2\} $. In the region $ D $, a billiard ball (treated as a material point) moves without friction, reflecting off the... | Let $A_1, A_2, \ldots$ be the consecutive points of reflection of the ball from the line with the equation $y = 0$, and $B_1, B_2, \ldots$ be the consecutive points of reflection of the ball from the parabola with the equation $y = x^2$. To trace the path of the ball, we will construct the following. We build a triangl... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,564 |
XIX OM - I - Problem 11
In the plane, a polygon is given, all of whose angles are equal. Prove that the sum of the distances from any point lying inside the polygon to the lines containing the sides of the polygon is constant. | For a regular polygon, the thesis of the theorem is true. Indeed, the distance $ x_i $ of a point $ P $ of the polygon $ A_1A_2\ldots A_n $ from the line $ A_iA_{i+1} $ (Fig.7) is equal to the quotient of twice the area $ S_i $ of the triangle $ A_iP A_{i+1} $ *) by the length of the side $ A_iA_{i+1} $ of the polygon.... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,567 |
XXXVII OM - I - Problem 4
In a given circle, we inscribe an acute triangle $ABC$. Let $A_1, B_1, C_1$ be the midpoints of the arcs $BC, CA, AB$ contained within semicircles, respectively. Prove that the area of triangle $ABC$ is not greater than the area of triangle $A_1B_1C_1$, and that the areas are equal if and onl... | We introduce the following notations:
$ O $ and $ R $ - the center and the length of the radius of a given circle
$ S $ and $ S_1 $ - the areas of triangles $ ABC $ and $ A_1B_1C_1 $
$ \alpha $, $ \beta $, $ \gamma $ - the measures of angles $ \measuredangle A $, $ \measuredangle B $, $ \measuredangle C $ of tria... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,569 |
XXIII OM - II - Zadanie 2
W prostokącie o bokach długości 20 i 25 umieszczono 120 kwadratów o boku długości 1. Dowieść, że istnieje koło o średnicy 1 zawarte w tym prostokącie i nie mające punktów wspólnych z żadnym z tych kwadratów.
|
Koło o średnicy $ 1 $ ma punkt wspólny z kwadratem $ ABCD $ o boku długości $ 1 $ wtedy i tylko wtedy, gdy środek $ P $ tego koła jest odległy od pewnego boku kwadratu nie więcej niż o $ \displaystyle \frac{1}{2} $. Punkt $ P $ należy więc do figury $ A (rys. 11) złożonej z kwadratu $ ABCD $, czterech prostokątów o ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,570 |
XXII OM - I - Problem 1
Prove the theorem: The binomial coefficient $ \binom{n}{k} $ has, for every natural $ k < n $, the same common factor with $ n $ greater than one if and only if $ n $ is a power of a prime number. | We will prove that if a number $ n $ is a power of a prime number $ p $, $ n = p^r $, then each of the binomial coefficients, where $ 1 \leq k < n $, is divisible by $ p $.
We have
Therefore, $ k\binom{n}{k}=n \binom{n-1}{k-1} $, so the number $ n $ is a divisor of the number $ k\binom{n}{k} $. Since $ k < n $, $ n $ ... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,572 |
LIV OM - III - Task 5
A sphere inscribed in the tetrahedron $ABCD$ is tangent to the face $ABC$ at point $H$. Another sphere is tangent to the face $ABC$ at point $O$ and is tangent to the planes containing the other faces of the tetrahedron at points that do not belong to the tetrahedron. Prove that if $O$ is the cen... | Let $ K $ and $ L $ be the points of tangency of the inscribed sphere in the tetrahedron $ ABCD $ with the faces $ ACD $ and $ BCD $, respectively. Let $ P $ and $ Q $ be the points of tangency of the second sphere considered in the problem, with the planes $ ACD $ and $ BCD $, respectively. Then the triangles $ KCP $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,574 |
XXI OM - I - Problem 9
Prove that for every odd number $ n > 1 $ there exists a natural number $ d < n $ such that the number $ 2^d - 1 $ is divisible by $ n $. | None of the numbers $ 2, 2^2, \ldots, 2^n $ give a remainder of $ 0 $ when divided by $ n $, because $ n > 1 $ and $ n $ is an odd number.
Since there are $ n $ such numbers, and there are $ n - 1 $ possible non-zero remainders, two numbers $ 2^r $ and $ 2^s $, where $ 1 \leq r < s \leq n $, must give the same remainde... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,575 |
II OM - I - Task 6
Prove that if the sum of positive numbers $ a $, $ b $, $ c $ is equal to $ 1 $, then | Dividing both sides of the equality $ 1=a+b+c $ successively by $ a $, $ b $, $ c $ we obtain
\[ \frac{1}{a} = 1 + \frac{b}{a} + \frac{c}{a} \]
\[ \frac{1}{b} = \frac{a}{b} + 1 + \frac{c}{b} \]
\[ \frac{1}{c} = \frac{a}{c} + \frac{b}{c} + 1 \]
Therefore,
Indeed,
From this, it immediately follows that,
Consequent... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,578 |
XVIII OM - I - Problem 4
In a circle with center $ O $, a triangle $ ABC $ is inscribed, and points $ A_1 $ and $ B_1 $ are determined as the points symmetric to vertices $ A $ and $ B $ with respect to point $ O $, and point $ P $ is the intersection of the line passing through point $ A_1 $ and the midpoint $ M $ of... | Since $ M $ and $ N $ are the midpoints of sides $ BC $ and $ AC $, $ MN \parallel AB $ and $ MN = \frac{1}{2} AB $, thus $ MN \parallel A_1B_1 $ and $ MN = \frac{1}{2} A_1B_1 $, because segments $ AB $ and $ A_1B_1 $ are symmetric with respect to $ O $, so they are parallel and equal (Fig. 2). Therefore, in triangle $... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,580 |
LVII OM - I - Problem 7
Given a prime number $ p > 3 $ and positive integers $ a, b, c $ such that $ a + b + c = p + 1 $ and the number
$ a^3 + b^3 + c^3 - 1 $ is divisible by $ p $. Prove that at least one of the numbers $ a, b, c $ is equal to 1. | Notice that
According to the conditions of the problem, the numbers $ a+b+c $ and $ a^3 +b^3 +c^3 $ give a remainder of 1 when divided by $ p $. Therefore, by the above identity, the number
is divisible by $ p $. Since $ p $ is a prime number different from 3, one of the factors $ a + b $, $ b + c $, or $... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,581 |
LX OM - I - Task 6
Given is an isosceles triangle $ ABC $, where $ AB = AC $. On the rays
$ AB $ and $ AC $, points $ K $ and $ L $ are chosen outside the sides
of the triangle such that
Point $ M $ is the midpoint of side $ BC $. Lines $ KM $ and $ LM $ intersect
the circumcircle of triangle $ AKL $ again at po... | From the dependency (1) we have (Fig. 3)
Moreover, from the assumptions of the problem, it follows that triangle $ BAC $ is
isosceles, from which we obtain the equality of angles $ \measuredangle KBM = \measuredangle MCL $.
Together with condition (2), this means that triangles $ KBM $ and $ MCL $ are
similar (si... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,583 |
XIII OM - II - Task 3
Prove that the four segments connecting the vertices of a tetrahedron with the centroids of the opposite faces have a common point. | Let's consider two of the segments mentioned in the task. Let $ S $ be the centroid of the face $ ABC $, and $ T $ be the centroid of the face $ ABD $ of the tetrahedron $ ABCD $ (Fig. 26).
The point $ S $ lies on the median $ CM $ of triangle $ ABC $, and the point $ T $ lies on the median $ DM $ of triangle $ ABD $. ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,585 |
XLV OM - I - Problem 10
Positive numbers $ p $ and $ q $ satisfy the condition $ p+q=1 $. Prove that for any natural numbers $ m $ and $ n $ the following inequality holds
| Let's imagine a rectangular chessboard with $ m $ horizontal rows and $ n $ vertical rows. Assume that the fields are colored randomly: for each field, the probability of being painted white is $ p $, and the probability of being painted black is $ q $.
Let's fix a vertical row $ C $. The probability that all fields in... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,586 |
V OM - I - Task 10
Calculate $ x^{13} + \frac{1}{x^{13}} $ given that $ x + \frac{1}{x} = a $, where $ a $ is a given number. | Let's generalize the task and show that for any natural $ k $, the expression $ x^k + \frac{1}{x^k} $ can be calculated in terms of $ x + \frac{1}{x} $. To this end, notice that
therefore
Equality (1) allows reducing the calculation of the expression $ x^k + \frac{1}{x^k} $ to calculating expressions of the same form... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,588 | |
XLIX OM - I - Problem 4
Given a positive number $ a $. Determine all real numbers $ c $ that have the following property: for every pair of positive numbers $ x $, $ y $, the inequality $ (c-1)x^{a+1} \leq (cy - x)y^a $ is satisfied. | Assume that a real number $ c $ has the considered property:
Substituting $ y=tx $ and dividing both sides by $ x^{a+1} $, we obtain the relation
Notice that the converse is also true: condition (2) implies (1). Suppose that $ c $ is a real number satisfying condition (2). Let $ x $, $ y $ be any positive numbers. In... | \frac{}{+1} | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 1,591 |
L OM - I - Problem 11
In an urn, there are two balls: a white one and a black one. Additionally, we have 50 white balls and 50 black balls at our disposal. We perform the following action 50 times: we draw a ball from the urn, and then return it to the urn along with one more ball of the same color as the drawn ball. ... | Let $ P(k,n) $, where $ 1 \leq k\leq n-1 $, denote the probability of the event that when there are $ n $ balls in the urn, exactly $ k $ of them are white. Then
Using the above relationships, we prove by induction (with respect to $ n $) that $ P(k,n) = 1/(n-1) $ for $ k = 1,2,\ldots,n-1 $. In particular
T... | 51 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,592 |
XXV - I - Problem 12
Prove that in any trihedral angle, in which at most one dihedral angle is right, three lines perpendicular to the vertex of the trihedral angle to its edges and contained in the planes of the opposite faces are contained in one plane. | Let $ a $, $ b $, $ c $ be non-zero vectors parallel to the edges of a given trihedral angle. No two of these vectors are therefore parallel. From the conditions of the problem, it follows that at least two of the numbers $ \lambda = bc $, $ \mu = ca $, $ \nu = ab $ are different from zero, since at most one pair of ve... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,595 |
VII OM - III - Problem 6
Given a sphere with radius $ R $ and a plane $ \alpha $ that has no points in common with the sphere. A point $ S $ moves on the plane $ \alpha $, which is the vertex of a cone tangent to the sphere along a circle with center $ C $. Find the geometric locus of the point $ C $. | The task boils down to a certain planimetric problem. Let's choose on the plane $ \alpha $ a point $ S $ different from the point $ S_0 $, where the perpendicular dropped from the center $ O $ of the sphere to the plane $ \alpha $ intersects this plane (Fig. 21).
The plane $ S_0OS $ perpendicular to the plane $ \alpha ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,599 |
XXXIX OM - I - Problem 5
Prove that there are infinitely many natural numbers $ n $ for which $ [n\sqrt{2}] $ is a power of 2.
Note: $ [x] $ is the greatest integer not greater than $ x $.
| The number $ \sqrt{2} $ can be replaced by any number $ x $ from the interval $ (1; 2) $.
Let $ x \in (1; 2) $ and let $ M $ be the set of all natural numbers $ n $ for which $ [nx] $ is a power of two (with a non-negative integer exponent). We will show in several ways that the set $ M $ is infinite; for $ x = \sqrt{2... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,600 |
XLVII OM - II - Zadanie 1
Rozstrzygnąć, czy każdy wielomian o współczynnikach całkowitych jest sumą trzecich potęg wielomianów o współczynnikach całkowitych.
|
Jeśli $ P(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n $ jest wielomianem o współczynnikach całkowitych, to $ (P(x))^3 = b_0 + b_1x + b_2 x^2 + b_3 x^3 +\ldots + b_{3n}x^{3n} $; przy tym $ b_0 = a_0^3 $, $ b_1= 3a_0^2 a_1 $. Tak więc trzecia potęga wielomianu o współczynnikach całkowitych ma współczynnik przy $ x $ p... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,601 |
XXXVI OM - II - Problem 5
Prove that for a natural number $ n $ greater than 1, the following conditions are equivalent:
a) $ n $ is an even number,
b) there exists a permutation $ (a_0, a_1, a_2, \ldots, a_{n-1}) $ of the set $ \{0,1,2,\ldots,n-1\} $ such that the sequence of remainders when dividing by $ n $ of the ... | We prove that condition a) implies condition b). We assume that $ n $ is an even number and we take
Let $ b_k=a_0+a_1+\ldots+a_k $ for $ k = 0,1,\ldots,n-1 $. Then for $ j<n $ the following equalities hold
(also for $ j = 0 $) and
Therefore, when divided by $ n $:
$ b_0 $ gives a remainder of $ 0 $
$ b_{2j} $ gi... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,602 |
VIII OM - III - Task 1
Through the midpoint $ S $ of the segment $ MN $ with endpoints lying on the legs of an isosceles triangle, a line parallel to the base of the triangle is drawn, intersecting its legs at points $ K $ and $ L $. Prove that the orthogonal projection of the segment $ MN $ onto the base of the trian... | Since the broken lines $ MNL $ and $ MKL $ have common ends $ M $ and $ L $ (Fig. 19), their projections on the base of the triangle are equal, which we will write as
Considering that the projection of a broken line equals the sum of the projections of its sides, we obtain from this
By drawing $ MP \paralle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,604 |
XIX OM - II - Problem 2
Given a circle $ k $ and a point $ H $ inside it. Inscribe a triangle in the circle such that point $ H $ is the orthocenter of the triangle. | Suppose the sought triangle is triangle $ABC$ (Fig. 11).
Since, by assumption, point $H$ lies inside the circle $k$, triangle $ABC$ is acute-angled and point $H$ lies inside triangle $ABC$. Let's call $H'$ the second point of intersection of line $AH$ with circle $k$; it lies on the opposite side of line $BC$ from poin... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,606 |
XXI OM - I - Problem 8
On a billiard table in the shape of a square, there is a ball. The ball was hit so that it bounced off the cushion (the angle of incidence equals the angle of reflection) at an angle $ \alpha $. Prove that the ball will move along a closed broken line if and only if $ \tan \alpha $ is a rational... | Let's assume for simplicity that the billiard table is a square with vertices at points $A(0, 0)$, $B(1, 0)$, $C(1, 1)$, $D(0, 1)$ (Fig. 6). Suppose that at the initial moment, the ball is located at point $P(a, b)$, where $0 < a < 1$, $0 < b < 1$.
Let $v$ be the speed at which it moves at the initial moment, and $\alp... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,608 |
LIV OM - I - Task 4
We consider the set of all $ k $-term sequences with terms from the set $ \{1, 2, 3,\ldots , m\} $. From each such sequence, we select the smallest term and sum the selected terms. Prove that the resulting sum is equal to | Let $ L_i $ $ (i =1,2,\ldots,m) $ denote the number of $ k $-term sequences with terms from the set $ \{1, 2, 3, \ldots , m\} $, where each term is not less than $ i $. Such sequences are $ k $-term sequences with terms from a $ (m-i+1) $-element set.
Hence, $ L_i = (m-i+1)^k $. Therefore, the number of sequences with... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,609 |
XI OM - I - Task 2
Prove that if the sum $ a + b $ of two integers is divisible by an odd number $ n $, then $ a^n + b^n $ is divisible by $ n^2 $.
Does this theorem hold for even $ n $? | It is known that when $ n $ is an odd number, the binomial $ a^n + b^n $ is divisible by the binomial $ a + b $, because the following equality holds:
To prove that if $ a + b $ is divisible by $ n $, then $ a^n + b^n $ is divisible by $ n^2 $, it suffices to show that the number
is divisible by $ n $.
For this... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,610 |
XXII OM - I - Problem 2
Prove that for any positive numbers $ p $ and $ q $ satisfying the condition $ p + q = 1 $, the inequality holds
保留了源文本的换行和格式，但请注意，最后一句“保留了源文本的换行和格式”是中文，应删除或翻译为英文：“Note that the last sentence '保留了源文本的换行和格式' is in Chinese and should be deleted or translated to English: 'Note that the l... | From the conditions of the problem, it follows that $0 < p < 1$ and $q = 1 - p$. Therefore, the problem reduces to showing that the function $f(p) = p \log p + (1 - p) \log (1 - p)$ in the interval $(0; 1)$ takes values no less than $-\log 2$. We calculate the derivative of this function:
Solving the equation $f'(p) =... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,611 |
X OM - I - Problem 11
On the same side of the line $ AB $, three semicircles were constructed with diameters $ AB = a + b $, $ AC = a $, $ CB = b $. Calculate the radius of the circle inscribed in the figure bounded by these semicircles, given $ a $ and $ b $. | Let $ O $, $ S $, $ T $ denote the centers of semicircles with diameters $ AB $, $ AC $ and $ CB $, $ U $ - the center of the circle inscribed in the figure bounded by these semicircles, and $ x $ - its diameter (Fig. 17).
The lengths of all segments in Fig. 17 can easily be expressed in terms of $ a $, $ b $, $ x $. U... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,612 |
XLVIII OM - I - Problem 5
The angle bisectors of the interior angles $ A $, $ B $, $ C $ of triangle $ ABC $ intersect the opposite sides at points $ D $, $ E $, $ F $, and the circumcircle of triangle $ ABC $ at points $ K $, $ L $, $ M $. Prove that | Given the geometric inequality to be proven, we will reduce it to a simple algebraic inequality. Notice that $ |AD| = |AK|-|DK| $, $ |BE| = |BL| -|EL| $, $ |CF| = |CM|-|FM| $, and therefore
\[ \frac{|AD|}{|DK|} = \frac{|AK|}{|DK|}-1, \quad \frac{|BE|}{|EL|}=\frac{|BL|}{|EL|}-1, \quad \frac{|CF|}{|FM|}=\frac{|CM|}{|FM|... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,613 |
XL OM - I - Problem 11
Points $ A_1 $, $ A_2 $, $ A_3 $, $ A_4 $ lie on the surface of a sphere, such that the distance between any two of them is less than the length of the edge of a regular tetrahedron inscribed in this sphere. Prove that all these points can be illuminated by a single point light source placed out... | Let $ C_1C_2C_3C_4 $ be any regular tetrahedron inscribed in a given sphere; let point $ O $ be its center, and $ r $ be the length of the radius. Consider the vectors
\[
\mathbf{u}_1 = \overrightarrow{OA_1}, \quad \mathbf{u}_2 = \overrightarrow{OA_2}, \quad \mathbf{u}_3 = \overrightarrow{OA_3}, \quad \mathbf{u}_4 = \... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,614 |
LVI OM - II - Task 6
Numbers $ a $, $ b $, $ c $ belong to the interval $ \langle 0;1\rangle $. Prove that | We will show that for any numbers $ a $, $ b $, $ c $ belonging to the interval $ \langle 0; 1\rangle $, the following inequality holds:
Indeed, if $ a = 0 $, then the above inequality is satisfied. For $ a>0 $, inequality (1) is equivalent to the inequality $ a + b + c \leq 2bc + 2 $, which can be written as
The las... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,615 |
XVI OM - I - Problem 9
Prove that if the value of the function $ y = ax^2 + bx + c $ for three consecutive integer values of the variable $ x $ is an integer, then it is also an integer for every integer value of $ x $. | Suppose that when $ x $ takes integer values $ k - 1 $, $ k $, $ k + 1 $, then $ y $ takes integer values $ m $, $ n $, $ p $. From the equality
we obtain by subtracting the equality
From (4) and (5) it follows similarly that
From (6) the number $ 2a $ is an integer, and from (4) and (5) we conclude that the numbers... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,616 |
XLVI OM - III - Problem 2
The diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal areas? | om46_3r_img_12.jpg
Let's denote the considered pentagon by $ABCDE$, and the pentagon formed by the intersection points of the diagonals by $KLMNP$ so that the following triangles are those mentioned in the problem:
$\Delta_0$: triangle $LEM$; $\quad \Delta_1$: triangle $EMA$;
$\Delta_2$: triangle $MAN$; $\quad \Delta... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,617 |
XXVII OM - III - Task 5
A fishing vessel is fishing in the territorial waters of a foreign state without permission. Each cast of the net results in a catch of the same value. During each subsequent cast, the probability of the vessel being intercepted by the border guard is $ \frac{1}{k} $, where $ k $ is a fixed nat... | The probability of an event where a ship is not caught with a certain cast of the net is $1 - \frac{1}{k}$. Since events involving catching or not catching a ship with each subsequent cast of the net are independent, the probability of the event where the ship is not caught after $m$-times casting the net is $\left(1 -... | n=k-1n=k | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,620 |
XLIV OM - III - Problem 4
A convex polyhedron is given, all of whose faces are triangles. The vertices of this polyhedron are colored with three colors. Prove that the number of faces having vertices of all three colors is even. | For a tetrahedron, the following statement holds: the number of faces with vertices of three different colors is $0$ or $2$ (an easy exercise).
Now consider any polyhedron $W$ with $n$ triangular faces (and vertices colored with three colors). Choose any point $P$ inside $W$. Each face of the polyhedron $W$ serves as t... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,621 |
XXVIII - II - Task 4
Given a pyramid with a quadrilateral base such that each pair of circles inscribed in adjacent faces has a common point. Prove that the points of tangency of these circles with the base of the pyramid lie on one circle. | We will first prove a few auxiliary facts that we will use to solve the problem. We will first show that if in a quadrilateral $ABCD$ (not necessarily convex) the sums of the lengths of opposite sides are equal:
if
then the angle bisectors of the angles of this quadrilateral intersect at one point.
Indeed, from condi... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,622 |
I OM - B - Problem 11
A vertical pole of height $a$, illuminated by sunlight, casts a shadow on a horizontal plane; at one moment the shadow has a length of $a$, at another moment the shadow had a length of $2a$, and at a third moment the shadow's length was $3a$. Prove that the sum of the angles of inclination of the... | The angles mentioned in the problem are $\angle ACB$, $\angle ADB$, $\angle AEB$ in Figure 5, where $AB = a$, $BC = a$, $BD = 2a$, $BE = 3a$, and $\angle B = 90^{\circ}$. Since $\angle ACB = 45^{\circ}$, it is necessary to prove that
$\angle ADB + \angle AEB = 45^{\circ}$.
om1_Br_img_5.jpg
To do this, rotate triangle ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,623 |
XLII OM - I - Problem 9
Let $ f $ be a transformation of the plane having the following property: for any pair of points $ P $, $ Q $, whose distance $ |PQ| $ is a rational number, the distance of their images $ P', $ Q' satisfies the equality $ |P'Q'| = |PQ| $. Prove that $ f $ is an isometry. | We will first show that the transformation $ f $ is injective.
Suppose this is not the case and that there exist different points $ U $, $ V $ that are mapped to the same point $ U $. Take any rational numbers $ q, r > 0 $ satisfying the conditions $ q + r > |UV| > q - r > 0 $. We can then construct a triangle $ UVW $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,625 |
XXXVI OM - I - Problem 6
Calculate the lengths of the parallel sides of a trapezoid given the lengths of the other sides and the diagonals. | Let $ABCD$ be the considered trapezoid, $AB\parallel CD$, $|AB| = a$, $|BC| = b$, $|CD| = c$, $|DA| = d$, $|AC| = p$, $|BD| = q$ (Figure 2). Given $b$, $d$, $p$, $q$, we need to find $a$ and $c$.
om36_1r_img_2.jpg
First, note that if $p = q$, then it must be $b = d < p$ and in this case, the problem does not have a uni... | \frac{2(p^2-q^2)}{b^2-^2+p^2-q^2},\quad\frac{2(q^2-p^2)}{b^2-^2+q^2-p^2} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,630 |
VIII OM - III - Task 3
Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa. | Suppose that when the number $ x $ is an integer, the number $ f (x) = ax^2 + bx + c $ is also an integer. Then
1) $ f (0) = c $, so $ c $ is an integer
2) $ f (1) = a + b + c $, hence $ a + b = f (1) - c $, so $ a + b $ is an integer
3) $ f (2) = 4a + 2b + c $, hence $ 2a = f (2) - 2 (a + b) - c $, so $ 2a $ is an int... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,632 |
XXXVI OM - II - Problem 3
Let $ L $ be the set of all broken lines $ ABCDA $, where $ A, B, C, D $ are distinct vertices of a fixed regular 1985-gon. A broken line is randomly chosen from the set $ L $. Calculate the probability that it is the boundary of a convex quadrilateral. | Let $ W $ be the set of all vertices of a given regular polygon. Each broken line from the set $ L $ can be associated with a set of its vertices, i.e., a certain four-point subset of the set $ W $. Conversely, every four points from $ W $ are the set of vertices of exactly three different broken lines from the set $ L... | \frac{1}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,635 |
XLII OM - I - Problem 11
For a given natural number $ n>10^3 $ and each $ k \in \{1,\ldots, n\} $, consider the remainder $ r_k $ from the division of $ 2^n $ by $ k $. Prove that $ r_1+\ldots r_n > 2n $. | For $ j = 0,1,2,3,4 $, let $ K_j $ denote the set of positive integers not exceeding $ n $, which are divisible by $ 2^j $ but not by $ 2^{j+1} $, and have at least one odd divisor greater than $ 1 $. Thus,
Let $ j \in \{0,1,2,3,4\} $ and consider any number $ k \in K_j $. Then $ k = 2^j m $, where $ m $ is odd a... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,637 |
XXVIII - II - Task 6
What is the maximum number of parts that the edges of $ n $ squares can divide a plane into? | The common part of the boundaries of two squares is either a finite set of points (and in this case, there are at most $8$ points, since each side of one square can intersect the boundary of the other square in at most two points), or it contains a segment (and in this case, it is a segment, the union of two segments -... | (2n-1)^2+1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,639 |
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