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LIV OM - III - Task 2
The number $ a $ is positive and less than 1. Prove that for every finite, strictly increasing sequence of non-negative integers ($ k_1,\ldots,k_n $) the following inequality holds
| We apply induction on $ n $. For $ n =1 $, the inequality takes the form
and is obviously satisfied. Fix $ n \geq 2 $ and assume that the inequality holds for any increasing sequence of length $ n-1 $. Consider any increasing sequence of length $ n $: $ 0 \leq k_1 < k_2 < \ldots < k_n $. By dividing both sides of t... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,112 |
XLVII OM - I - Problem 1
Determine all integers $ n $ for which the equation $ 2 \sin nx = \tan x + \cot x $ has solutions in real numbers $ x $. | Assume that the given equation has solutions in real numbers: let $ \alpha $ be one of these solutions. Of course, $ \alpha $ cannot be an integer multiple of $ \pi/2 $, because for such numbers the right side loses its meaning. According to the equation,
hence $ 2\sin n\alpha \cdot \sin 2\alpha=2 $, which means
... | 8k+2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,115 |
XXXVI OM - III - Task 3
Prove that if the function $ f: \mathbb{R} \to \mathbb{R} $ satisfies for every $ x \in \mathbb{R} $ the equation $ f(3x) = 3f(x) - 4(f(x))^3 $ and is continuous at 0, then all its values belong to the interval $ \langle -1;1\rangle $. | We will first prove a lemma.
Lemma. If $ |f(x)| \leq 1 $, then $ |f(3x)| \leq 1 $.
Proof.
If $ |f(x)| \leq 1 $, then for some $ y $ we have $ f(x) = \sin y $. Therefore, $ f(3x) = 3 \sin y - (\sin y)^3 = \sin 3y $, which implies that $ |f(3x)| \leq 1 $.
Let's calculate $ f(0) $. Substituting $ x = 0 $ in the given form... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,116 |
XLII OM - I - Problem 5
Given a segment $ AD $. Find points $ B $ and $ C $ on it, such that the product of the lengths of the segments $ AB $, $ AC $, $ AD $, $ BC $, $ BD $, $ CD $ is maximized. | We consider such positions of points $ B $ and $ C $, for which
this means, we assume that points $ A $, $ B $, $ C $, $ D $ lie on a line in this exact order (we do not lose generality, as the product being studied in the problem does not change value when the roles of $ B $ and $ C $ are swapped); the weak inequalit... | \frac{1}{5}\sqrt{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,118 |
XXVIII - I - Problem 4
Inside the tetrahedron $ABCD$, a point $M$ is chosen, and it turns out that for each of the tetrahedra $ABCM$, $ABDM$, $ACDM$, $BCDM$, there exists a sphere tangent to all its edges. Prove that there exists a sphere tangent to all the edges of the tetrahedron $ABCD$. | We will use the following theorem, which was given as problem 4 at the VI Mathematical Olympiad.
Theorem. There exists a sphere tangent to all edges of the tetrahedron $ PQRS $ if and only if the sums of the lengths of opposite edges of this tetrahedron are equal, i.e., if
Applying this theorem to the tetrahedra $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,121 |
LX OM - I - Task 12
Given a prime number $ p $. On the left side of the board, the numbers $ 1, 2, 3, \cdots, p - 1 $ are written,
while on the right side, the number $ 0 $ is written. We perform a sequence of $ p - 1 $ moves, each of which proceeds
as follows: We select one of the numbers written on the left side of ... | Answer: For all prime numbers $p$ except 2 and 3.
Let $ t_i $ for $ i=1,2, \cdots,p-1 $ denote the initial value of the number that
was erased from the board in the $ i $-th move. Thus, the sequence $ (t_1,t_2,t_3, \cdots,t_{p-1}) $ is a permutation
of the sequence $ (1,2,3, \cdots,p-1) $ and uniquely determines the wa... | Forallpexcept23 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,124 |
XVIII OM - II - Problem 5
On a plane, there are two triangles, one outside the other.
Prove that there exists a line passing through two vertices of one triangle such that the third vertex of this triangle and the other triangle lie on opposite sides of this line. | Let's denote the vertices of the given triangles by $A_1, A_2, A_3$ and $B_1, B_2, B_3$. Let $MN$ be a segment such that: $1^\circ$ point $M$ lies on the perimeter of triangle $A_1A_2A_3$, $2^\circ$ point $N$ lies on the perimeter of triangle $B_1B_2B_3$, $3^\circ$ segment $MN$ is not longer than any segment connecting... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,126 |
XII OM - I - Problem 3
On a plane, a circle $ K $ and a point $ A $ are given. Draw a circle $ L $ passing through point $ A $, so that the lens, which is the common part of both circles, has a given diameter $ a $ and a given thickness $ b $. | Since the size of the lens determined by the circles $K$ and $L$ does not change when circle $L$ is rotated around the center $O$ of circle $K$, we will solve the problem by drawing any circle that forms a lens of diameter $a$ and thickness $b$ with the given circle, and then rotating this circle around point $O$ so th... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,127 |
LX OM - II - Task 2
Given integers $ a $ and $ b $ such that $ a > b > 1 $ and the number $ ab + 1 $ is divisible by $ a + b $,
while the number $ ab - 1 $ is divisible by $ a - b $.
Prove that $ a < b\sqrt{3} $. | First, let's note that
Thus, from the divisibility data in the problem, the number $ b^2 - 1 $ is divisible by
$ a+b $ and by $ a-b $.
The numbers $ a $ and $ b $ are relatively prime. If $ d $ is a positive common divisor of them,
then the numbers $ a + b $ and $ ab $ are divisible by $ d $; on the other hand, the... | b\sqrt{3} | Number Theory | proof | Yes | Yes | olympiads | false | 1,129 |
LVII OM - III - Problem 5
Given is a tetrahedron $ABCD$ where $AB = CD$. The inscribed sphere of this tetrahedron is tangent to the faces $ABC$ and $ABD$ at points $K$ and $L$, respectively. Prove that if points $K$ and $L$ are the centroids of the faces $ABC$ and $ABD$, then the tetrahedron $ABCD$ is regular. | Let $s$ be the sphere inscribed in the tetrahedron $ABCD$. We will start by showing that triangles $ABC$ and $ABD$ are congruent.
Let $E$ be the midpoint of edge $AB$. Since $K$ and $L$ are the points of tangency of the sphere $s$ with the tetrahedron $ABCD$, we have $AK = AL$ and $BK = BL$, which implies that triangle... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,130 |
III OM - I - Problem 7
Let $ a $, $ b $ denote the legs of a right triangle, $ c $ - its hypotenuse, $ r $ - the radius of the inscribed circle, and $ r_a $, $ r_b $, $ r_c $ - the radii of the excircles of this triangle. Prove that:
1) $ r + r_a + r_b = r_c $
2) the radii $ r $, $ r_a $, $ r_b $, $ r_c $ are simultan... | In this task, we are dealing with figure 16, which is a special case of figure 11, as angle \( C \) is a right angle. We see that \( r = CM \), \( r_a = CP \), \( r_b = CR \), \( r_c = CQ \).
By denoting \( a + b + c = 2p \) and applying the formulas given in Note II to problem 16, we obtain
From this, the first ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,133 |
LI OM - II - Task 5
Let $ \mathbb{N} $ denote the set of positive integers. Determine whether there exists a function $ f: \mathbb{N}\to \mathbb{N} $ such that for every $ n\in \mathbb{N} $ the equality $ f(f(n)) =2n $ holds. | Such a function does exist. Here is an example.
Let $ A $ be the set of positive integers in whose prime factorization the number "3" appears an odd number of times. We define the function $ f $ by the formula:
Then, if $ n \in A $, we have $ f(n) \in \mathbb{N}\setminus A $, which gives
Similarly, if $ n \... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,135 |
XXXIX OM - I - Problem 1
For each positive number $ a $, determine the number of roots of the polynomial $ x^3+(a+2)x^2-x-3a $. | Let's denote the considered polynomial by $ F(x) $. A polynomial of the third degree has at most three real roots. We will show that the polynomial $ F $ has at least three real roots - and thus has exactly three real roots (for any value of the parameter $ a > 0 $).
It is enough to notice that
If a continuous func... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 1,136 |
XXXIV OM - III - Problem 1
In the plane, a convex $ n $-gon $ P_1, \ldots, P_n $ and a point $ Q $ inside it are given, which does not lie on any diagonal. Prove that if $ n $ is even, then the number of triangles $ P_iP_jP_k $ ($ i,j,k= 1,2,\ldots,n $) that contain the point $ Q $ is even. | om34_3r_img_11.jpg
First, consider a convex quadrilateral $ABCD$ and a point $K$ inside it that does not lie on any diagonal. Point $K$ belongs to one of the two half-planes with edge $AC$. It follows that point $K$ belongs to exactly two of the triangles $ABC$, $ABD$, $ACD$, $BCD$ (in Fig. 11, point $K$ belongs to tri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,137 |
XLVIII OM - II - Problem 4
Determine all triples of positive integers having the following property: the product of any two of them gives a remainder of $ 1 $ when divided by the third number. | Let $ a $, $ b $, $ c $ be the numbers we are looking for. From the given conditions, these numbers are greater than $ 1 $ and are pairwise coprime. The number $ bc + ca + ab - 1 $ is divisible by $ a $, by $ b $, and by $ c $, so it is also divisible by the product $ abc $:
After dividing both sides by $ abc $:
The ... | (5,3,2) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,140 |
XXXIII OM - II - Problem 4
Let $ A $ be a finite set of points in space having the property that for any of its points $ P, Q $ there is an isometry of space mapping the set $ A $ onto the set $ A $ and the point $ P $ onto the point $ Q $. Prove that there exists a sphere passing through all points of the set $ A $. | Let $ S $ be the center of gravity of the system of points of set $ A $. Each isometry transforming set $ A $ into set $ A $ also transforms point $ S $ into $ S $. For any $ P, Q \in A $, an isometry transforming $ A $ into $ A $ and $ P $ into $ Q $ also transforms the segment $ \overline{PS} $ into the segment $ \ov... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,142 |
LVIII OM - I - Problem 2
Determine all pairs of positive integers $ k $, $ m $, for which each of the numbers $ {k^2+4m} $, $ {m^2+5k} $ is a perfect square. | Suppose the pair $ (k,m) $ satisfies the conditions of the problem.
If the inequality $ m\geq k $ holds, then
$$(m+3)^2=m^2+6m+9>m^2+5m\geq m^2+5k>m^2,$$
and since $ m^2+5k $ is a square of an integer, it follows that one of the equalities $ m^2+5k=(m+1)^2 $ or $ m^2+5k=(m+2)^2 $ must hold.
If $ m^2+5k=(m+1)^2=m^2+2... | (1,2),(9,22),(8,9) | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,144 |
XXXI - II - Problem 4
Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal. | Suppose that the positive numbers $ s $, $ t $, $ u $ are roots of the polynomial $ ax^3 - ax^2 + 9bx - b $. Therefore, $ ax^3 - ax^2 + 9bx - b = a (x - s) (x - t)(x - u) $, and by expanding the right side and equating the coefficients of the successive powers of the variable $ x $, we obtain the so-called Vieta's form... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,145 |
XXXII - I - Problem 9
In space, there is a set of $3n$ points, no four of which lie on the same plane. Prove that this set can be divided into $n$ three-element sets $\{A_i, B_i, C_i\}$ such that the triangles $A_iB_iC_i$ are pairwise disjoint. | Proof by induction. For $ n = 1 $, the thesis of the theorem is obviously satisfied. Assume that the thesis is satisfied for some $ n $ and consider $ 3(n+1) $ points in space, no four of which lie on the same plane. The smallest convex set containing the considered points is a polyhedron, each face of which is a trian... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,148 |
XVI OM - III - Problem 3
On a circle, $ n > 2 $ points are chosen and each of them is connected by a segment to every other. Can all these segments be traced in one continuous line, i.e., so that the end of the first segment is the beginning of the second, the end of the second is the beginning of the third, etc., and... | To facilitate pronunciation, we will introduce a certain convention. Let $ Z $ be a finite set of points on a circle (the condition that the points of set $ Z $ lie on a circle can be replaced by the weaker assumption that no $ 3 $ points of this set lie on the same straight line. The same remark applies to the further... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,149 |
LVI OM - II - Problem 5
Given is a rhombus $ABCD$, where $ \measuredangle BAD > 60^{\circ} $. Points $E$ and $F$ lie on sides $AB$ and $AD$ respectively, such that $ \measuredangle ECF = \measuredangle ABD $. Lines $CE$ and $CF$ intersect diagonal $BD$ at points $P$ and $Q$ respectively. Prove that | From the equality $ \measuredangle FDQ = \measuredangle PCQ $, it follows that triangles $ FDQ $ and $ PCQ $ are similar. Since points $ A $ and $ C $ are symmetric with respect to the line $ BD $, triangles $ PCQ $ and $ PAQ $ are congruent, and
This equality means that points $ A $, $ P $, $ Q $, and $ F $ lie on th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,151 |
XXVI - I - Problem 7
Let $ Z $ be the set of all finite sequences with terms $ a, b, c $ and let $ Z_1 $ be a subset of $ Z $ containing for each natural number $ k $ exactly one $ k $-term sequence.
We form the smallest set $ Z_2 $ with the property that $ Z_2 \supset Z_1 $ and if a certain $ k $-term sequence belong... | Every $ n $-term sequence belonging to $ Z_2 $ arises from an $ (n - k) $-term sequence (where $ 1 \leq k \leq n $) belonging to $ Z_1 $ by appending a certain $ k $-term sequence at the beginning or end, or it is an $ n $-term sequence belonging to $ Z_1 $. The number of such $ k $-term sequences is $ 3^k $, because e... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,153 |
XIII OM - I - Problem 7
Given on a plane are segments $ AB $ and $ CD $. Find on this plane a point $ M $ such that triangles $ AMB $ and $ CMD $ are similar, and at the same time, angle $ AMB $ is equal to angle $ CMD $. | In the similarity of triangles $ AMB $ and $ CMD $, vertex $ M $ of one triangle must correspond to vertex $ M $ of the other triangle according to the conditions of the problem. Vertex $ C $ of triangle $ CMD $ can correspond to either vertex $ A $ or vertex $ B $ of triangle $ AMB $, which leads to two types of solut... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,154 |
XX OM - II - Task 2
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit, and the tens digit is equal to the units digit, and which are squares of integers. | Suppose the number $ x $ satisfies the conditions of the problem and denote its consecutive digits by the letters $ a, a, b, b $. Then
The number $ x $ is divisible by $ 11 $, so as a square of an integer, it is divisible by $ 11^2 $, i.e., $ x = 11^2 \cdot k^2 $ ($ k $ - an integer), hence
Therefore,
... | 7744 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,157 |
XLVIII OM - I - Problem 4
Prove that a natural number $ n \geq 2 $ is composite if and only if there exist natural numbers $ a,b,x,y \geq 1 $ satisfying the conditions: $ a+b=n $, $ \frac{x}{a}+\frac{y}{b}=1 $. | Let $ n $ be a composite number: $ n = qr $, $ q \geq 2 $, $ r \geq 2 $. Assume:
These are positive integers with the properties:
We have thus shown that if $ n \geq 2 $ is a composite number, then there exist integers $ a, b, x, y \geq 1 $ satisfying conditions (1).
The remaining part to prove is the converse implic... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,158 |
Let $ a_1,a_2, \ldots, a_n $, $ b_1,b_2, \ldots, b_n $ be integers. Prove that | Sums appearing on both sides of the given inequality in the task will not change if we arbitrarily change the order of numbers $ a_i $ and numbers $ b_i $. Without loss of generality, we can therefore assume that $ a_1 \leq a_2 \leq \ldots \leq a_n $ and $ b_1 \leq b_2 \leq \ldots \leq b_n $. In this case | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,159 |
XXV OM - III - Problem 6
A convex $ n $-gon was divided into triangles by diagonals in such a way that
1° an even number of diagonals emanate from each vertex,
2° no two diagonals have common interior points.
Prove that $ n $ is divisible by 3. | We will first prove the
Lemma. If $F$ is a figure on a plane and it is divided into parts by $r$ lines, then these parts can be painted with two colors in such a way that any two parts sharing a segment have different colors.
Proof. We will use induction with respect to $r$. In the case of $r = 1$, the thesis is obviou... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,160 |
XXXVI OM - II - Problem 6
In space, there are distinct points $ A, B, C_0, C_1, C_2 $, such that $ |AC_i| = 2 |BC_i| $ for $ i = 0,1,2 $ and $ |C_1C_2|=\frac{4}{3}|AB| $. Prove that the angle $ C_1C_0C_2 $ is a right angle and that the points $ A, B, C_1, C_2 $ lie in the same plane. | The proof will be based on the following theorem of Apollonius:
On the plane $\pi$, there are two points $A$ and $B$; in addition, there is a positive number $\lambda \ne 1$. The set of points $X$ on the plane $\pi$ that satisfy the condition $|AX| = \lambda |BX|$ forms a circle, and the center of this circle lies on t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,161 |
XXXVIII OM - III - Problem 6
A plane is covered with a grid of regular hexagons with a side length of 1. A path on the grid is defined as a sequence of sides of the hexagons in the grid, such that any two consecutive sides have a common endpoint. A path on the grid is called the shortest if its endpoints cannot be con... | Attention. The phrase "consecutive sides have a common end" used in the text should be understood to mean that directed segments are considered, and the end of the previous one is the start of the next (thus, it does not form a path, for example, a star of segments with one common end). Despite the lack of precision, t... | 6\cdot2^{30}-6 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,163 |
XXXIX OM - II - Problem 5
Determine whether every rectangle that can be covered by 25 circles of radius 2 can also be covered by 100 circles of radius 1. | The answer is affirmative. Let $ R $ be a rectangle that can be covered by 25 circles of radius 2. The axes of symmetry of the rectangle $ R $ divide it into four rectangles similar to it on a scale of $ 1 /2 $. Each of them can be covered by a family of 25 circles of radius 1; it is enough to transform the given 25 ci... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,164 |
XXXV OM - II - Problem 1
For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $. | For $ n = 1 $, the given equation has no solutions in natural numbers. For $ n > 1 $, each pair of numbers $ x_1 = 1^2, y_1 = (n-1)^2, x_2 = 2^2, y_2 = (n-2)^2, \ldots, x_{n-1} = (n-1)^2, y_{n-1} = 1^2 $ is obviously a solution to the given equation, since $ \sqrt{k^2} + \sqrt{(n - k)^2} = k + (n - k) = n $ for $ 0 < k... | n-1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,165 |
VII OM - III - Problem 5
Prove that any polygon with a perimeter equal to $2a$ can be covered by a disk with a diameter equal to $a$. | Let $W$ be a polygon whose perimeter has length $2a$. Choose two points $A$ and $B$ on its perimeter that divide the perimeter into halves, i.e., into two parts of length $a$; then $AB < a$ (Fig. 20).
We will prove that the disk of radius $\frac{a}{2}$, whose center lies at the midpoint $O$ of segment $AB$, completely ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,167 |
V OM - I - Task 2
Investigate when the sum of the cubes of three consecutive natural numbers is divisible by $18$. | Let $ a - 1 $, $ a $, $ a + 1 $, be three consecutive natural numbers; the sum of their cubes
can be transformed in the following way:
Since one of the numbers $ a - 1 $, $ a $, $ a + 1 $ is divisible by $ 3 $, then one of the numbers $ a $ and $ (a + 1) (a - 1) + 3 $ is also divisible by $ 3 $. Therefore, the sum $ ... | 3 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,168 |
LI OM - I - Problem 9
Given positive integers $ m $ and $ n $ such that $ mn | m^2 + n^2 + m $. Prove that $ m $ is a perfect square. | Let $ d $ be the greatest common divisor of the numbers $ m $ and $ n $. Then $ d^2 | mn $, and thus $ d^2 | m^2 + n^2 + m $. Therefore, given the divisibility $ d^2 | m^2 $ and $ d^2 | n^2 $, we obtain $ d^2 | m $.
On the other hand, $ m | m^2 + n^2 + m $, which gives us the divisibility $ m|n^2 $. Therefore, $ m $ is... | ^2 | Number Theory | proof | Yes | Yes | olympiads | false | 1,169 |
IX OM - II - Task 3
Prove that if the polynomial $ f(x) = ax^3 + bx^2 + cx + d $ with integer coefficients takes odd values for $ x = 0 $ and $ x = 1 $, then the equation $ f(x) = 0 $ has no integer roots. | According to the assumption, the numbers $ f(0) = d $ and $ f(1) = a + b + c + d $ are odd. We will show that $ f(x) $ is then an odd integer for every integer $ x $. Indeed, if $ x $ is an even number, then $ f(x) = ax^3 + bx^2 + cx + d $ is the sum of even numbers $ ax^3 $, $ bx^2 $, $ cx $, and the odd number $ d $.... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,171 |
XLVII OM - III - Problem 2
Inside a given triangle $ABC$, a point $P$ is chosen satisfying the conditions: $|\measuredangle PBC|=|\measuredangle PCA| < |\measuredangle PAB|$. The line $BP$ intersects the circumcircle of triangle $ABC$ at points $B$ and $E$. The circumcircle of triangle $APE$ intersects the line $CE$ a... | Let's denote the circumcircle of triangle $ABC$ by $\Omega$, and the circumcircle of triangle $APE$ by $\omega$ (Figure 10). Through point $E$, we draw a tangent line to circle $\omega$ and choose an arbitrary point $Q$ on this line, lying on the same side of line $AE$ as points $B$, $C$, and $P$ (it could be, for exam... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,174 |
XLV OM - I - Problem 9
In a conference, $2n$ people are participating. Each participant has at least $n$ acquaintances among the other participants. Prove that all participants of the conference can be accommodated in double rooms so that each participant shares a room with an acquaintance. | In the problem, it is tacitly assumed that if one person knows another, then the latter also knows the former, and that no one is their own acquaintance.
Let $ m $ be the maximum number of disjoint pairs of acquaintances that can be formed among the conference participants. The task reduces to proving that $ m = n $. S... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,175 |
I OM - B - Task 14
Prove that the sum of natural numbers is divisible by 9 if and only if the sum of all digits of these numbers is divisible by 9. | We will first prove that the difference between a natural number $ a $ and the sum $ s $ of its digits is divisible by 9. Let $ C_O, C_1, C_2, C_3, \dots $ denote the units, tens, hundreds, thousands, and further place values of this number. Then
Subtracting these equations side by side, we get
Since each term on the... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,176 |
XXXII - III - Task 4
On the table lie $ n $ tokens marked with integers. If among these tokens there are two marked with the same number, for example, the number $ k $, then we replace them with one token marked with the number $ k+1 $ and one with the number $ k-1 $. Prove that after a finite (non-negative) number of... | We will first prove the following lemma:
Lemma. Let $m_0$ be the minimum number among the numbers written on the tokens on the table before any changes are made. If after $k$ changes, the numbers $m_1^{(k)}, m_2^{(k)}, \ldots, m_{j_k}^{(k)}$ are all the numbers not greater than $m_0$, with $m_{j_k}^{(k)} \leq m_{j_k-1}... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,178 |
XII OM - III - Task 1
Prove that every natural number which is not an integer power of the number $ 2 $, is the sum of two or more consecutive natural numbers. | Let $ N $ be a natural number which is not an integer power of two, hence
where $ m $ and $ n $ are integers, with $ m \geq 1 $, $ n \geq 0 $.
We need to prove that there exist integers $ a $ and $ k $ such that
and
i.e.,
Condition (2) is satisfied if $ 2a + k - 1 = 2^{n+1} $, $ k = 2m + 1 $, i... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,179 |
XLIII OM - I - Problem 9
Prove that among any $ n+2 $ integers, there exist two such that their sum or difference is divisible by $ 2n $. | Let the considered numbers be denoted by $ k_1, \ldots, k_{n+2} $ and let $ r_i $ be the remainder of the division of $ k_i $ by $ 2n $.
Suppose that among these remainders there are two equal ones:
In this case, the difference $ k_i - k_j $ is divisible by $ 2n $; the thesis holds.
Suppose, on the other hand, that am... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,181 |
XLII OM - III - Problem 6
Prove that for every triple of real numbers $ x,y,z $ such that $ x^2+y^2+z^2 = 2 $, the inequality $ x+y+z\leq 2+xyz $ holds, and determine when equality occurs. | Let's adopt the notations
From the assumption that $ x^2 + y^2 + z^2 = 2 $, the following equalities follow:
The left sides are non-negative numbers. Hence, $ u, v, w \leq 1 $. Therefore, we have the inequality
From the adopted notations, it follows that
which means
Furthermore,
and... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 1,183 |
XXXIX OM - I - Problem 6
From the numbers $1, 2, \ldots, 3n$ ($n > 2$), we randomly select four different numbers. Each selection of four numbers is equally probable. Calculate the probability that the sum of the selected numbers is divisible by 3. | The sum of four integers is divisible by $3$ if and only if the system of remainders of these numbers when divided by $3$ has (up to permutation) one of the following five forms:
Among the natural numbers from $1$ to $3n$, exactly $n$ numbers give a remainder of $0$ when divided by $3$; also, $n$ numbers give a r... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,184 |
XXXVIII OM - II - Problem 3
On a chessboard with dimensions 1000 by 1000 and fields colored in the usual way with white and black, there is a set A consisting of 1000 fields. Any two fields of set A can be connected by a sequence of fields from set A such that consecutive fields share a common side. Prove that set A c... | A finite sequence of chessboard squares (of arbitrary dimensions) having the property that any two consecutive squares share a side, we will call a path, and the number of squares in this sequence decreased by $1$ - the length of the path (a single square constitutes a path of length $0$).
A non-empty set $A$ composed ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,185 |
XXIV OM - III - Problem 4
On a line, a system of segments with a total length $ < 1 $ is given. Prove that any system of $ n $ points on the line can be moved along it by a vector of length $ \leq \frac{n}{2} $ so that after the move, none of the points lies on any of the given segments. | Let $ P_i $, where $ i = 1, 2, \ldots, n $, be given points and let $ A $ be the union of given segments with a total length less than $ 1 $. For $ i = 1, 2, \ldots, n $, let $ I_i $ be a segment of length $ n $, whose center is the point $ P_i $.
Of course, $ A \cap I_i $ is the union of segments with a total length l... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,186 |
IX OM - III - Problem 6
Prove that among all quadrilaterals circumscribed around a given circle, the one with the smallest perimeter is a square. | Since the area of a polygon circumscribed about a circle is proportional to its perimeter, the theorem we need to prove is equivalent to the theorem that among all quadrilaterals circumscribed about a given circle, the square has the smallest area.
Let $Q$ be a square circumscribed about the circle $K$, $ABCD$ a quadri... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,188 |
XXV OM - III - Problem 3
Let $ r $ be a natural number. Prove that the quadratic polynomial $ x^2 - rx - 1 $ is not a divisor of any non-zero polynomial $ p(x) $ with integer coefficients, all of which are less in absolute value than $ r $. | \spos{1} Suppose the polynomial
where $ a_n \ne 0 $, has integer coefficients, each in absolute value less than $ r $, and is divisible by the quadratic polynomial $ f(x) = x^2 - rx - 1 $. We then have
where the polynomial $ h(x) = b_0 + b_1x + \ldots + b_{n-2}x^{n-2} $ has integer coefficients, which follo... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,189 |
L OM - II - Task 2
A cube $ S $ with an edge of $ 2 $ is built from eight unit cubes. A block will be called a solid obtained by removing one of the eight unit cubes from the cube $ S $. A cube $ T $ with an edge of $ 2^n $ is built from $ (2^n)^3 $ unit cubes. Prove that after removing any one of the $ (2^n)^3 $ unit... | We will conduct an inductive proof. For $ n = 1 $, the thesis of the problem is obviously true. Assume, therefore, that the thesis is true for some $ n $ and consider a cube $ C $ with edge $ 2^{n+1} $ and a distinguished unit cube $ D $. The cube $ C $ is divided into eight congruent cubes $ C_1, C_2, \ldots, C_8 $, e... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,190 |
V OM - II - Task 1
The cross-section of a ball bearing consists of two concentric circles $ C $ and $ C_1 $, between which there are $ n $ small circles $ k_1, k_2, \ldots, k_n $, each of which is tangent to two adjacent ones and to both circles $ C $ and $ C_1 $. Given the radius $ r $ of the inner circle $ C $ and t... | In figure 29, representing a part of the cross-section of a ball bearing, $ S $ is the center of one of the small circles, $ L $ and $ N $ are the points of tangency of this circle with the adjacent small circles, and $ M $ and $ P $ are the points of tangency with the circles $ C $ and $ C_1 $, which have a common cen... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,194 | |
LX OM - I - Task 3
The incircle of triangle $ABC$ is tangent to sides $BC$, $CA$, $AB$ at points $D$, $E$, $F$ respectively. Points $M$, $N$, $J$ are the centers of the incircles of triangles $AEF$, $BDF$, $DEF$ respectively. Prove that points $F$ and $J$ are symmetric with respect to the line $MN$. | om60_1r_img_2.jpg
First, we will prove that points $M$ and $N$ are the midpoints of the shorter arcs $FE$ and $FD$ of the incircle of triangle $ABC$.
Indeed, let $M$ be the midpoint of the shorter arc $FE$ of this
circle. The line $AC$ is tangent to it at point $E$, so (Fig. 2)
Since point $M$ is the midpoint of ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,196 |
XLVI OM - III - Problem 5
Let $ n $ and $ k $ be natural numbers. From an urn containing 11 slips numbered from 1 to $ n $, we draw slips one by one, without replacement. When a slip with a number divisible by $ k $ appears, we stop drawing. For a fixed $ n $, determine those numbers $ k \leq n $ for which the expecte... | Let's establish a natural number $ k \in \{1,2,\ldots,n\} $. Consider the following events:
Let's denote the probabilities of these events: $ p_j = P(\mathcal{A}_j) $, $ q_j = P(\mathcal{B}_j) $. Of course, $ \mathcal{B}_0 $ is a certain event. Therefore, $ q_0 = 1 $.
The occurrence of event $ \mathcal{A}_j $ is ... | k | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,208 |
XVII OM - I - Problem 3
Find the geometric locus of the centroids of acute triangles inscribed in a given circle. | First, we establish that every point $S$ inside a circle is the centroid of some triangle inscribed in that circle. Indeed, if point $S$ lies on the radius $OC = r$ (Fig. 1), and point $M$ lies on the extension of $CS$ beyond point $S$, such that $CM = \frac{3}{2} CS$, hence $CM < 2r$, then by drawing a chord $AB$ of t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,209 |
X OM - I - Problem 12
Prove that if in a triangle $ ABC $ the numbers $ \tan A $, $ \tan B $, and $ \tan C $ form an arithmetic progression, then the numbers $ \sin 2A $, $ \sin 2B $, and $ \sin 2C $ also form an arithmetic progression. | From the assumptions of the problem, it follows that
thus
therefore
so
and finally
which was to be proved.
Note. The converse statement is also true: if the angles of triangle $ABC$ satisfy equation (2), then they also satisfy equation (1). Indeed, based on the transformations performed ab... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,212 |
LIV OM - I - Task 7
At Aunt Reni's, $ n \geq 4 $ people met (including Aunt Reni). Each of the attendees gave at least one gift to at least one of the others. It turned out that everyone gave three times as many gifts as they received, with one exception: Aunt Reni gave only $ \frac{1}{6} $ of the number of gifts she ... | Let $ x_i $ ($ i = 1,2, \ldots ,n-1 $) denote the number of gifts received by the $ i $-th guest of Aunt Renia, and let $ x_R $ denote the number of gifts Aunt Renia received. Then the total number of gifts is
Therefore, the number $ x_R $ is divisible by $ 12 $, say $ x_R = 12k $, where $ k $ is some non-negative int... | 12[(n+3)/5] | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,214 |
XIX OM - II - Task 3
Prove that if at least 5 people are sitting around a round table, then they can be rearranged so that each person has two different neighbors than before. | \spos{1} Let us denote the given people by natural numbers from $1$ to $n$ ($n \geq 5$) and assume that person $1$ is seated next to persons $n$ and $2$, person $2$ is seated next to persons $1$ and $3$, and so on, finally person $n$ is seated next to persons $n-1$ and $1$. We will simply say that the numbers from $1$ ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,218 |
VIII OM - III - Task 2
Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle, the following relationship holds | Substituting in equation (1)
($ R $ - the radius of the circle circumscribed around the triangle), after dividing both sides by $ R^2 $, we get:
It suffices to justify formula (3), since formula (1) immediately follows from formulas (3) and (2). We will transform the right side of formula (3)
Th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,219 |
XXXV OM - III - Problem 1
Determine the number of functions $ f $ mapping an $ n $-element set to itself such that $ f^{n-1} $ is a constant function, while $ f^{n-2} $ is not a constant function, where $ f^k = f\circ f \circ \ldots \circ f $, and $ n $ is a fixed natural number greater than 2. | Suppose that $ A = \{a_1, a_2, \ldots, a_n \} $ is a given $ n $-element set, and $ f $ is a function satisfying the conditions of the problem. Consider the sets $ A, f(A), f^2(A), \ldots, f^{n-1}(A) $. From the assumption, it follows that the set $ f^{n-1}(A) $ is a singleton, while the set $ f^{n-2}(A) $ has more tha... | n! | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,220 |
XLVII OM - II - Problem 2
A circle with center $ O $ inscribed in a convex quadrilateral $ ABCD $ is tangent to the sides $ AB $, $ BC $, $ CD $, $ DA $ at points $ K $, $ L $, $ M $, $ N $, respectively, and the lines $ KL $ and $ MN $ intersect at point $ S $. Prove that the lines $ BD $ and $ OS $ are perpendicular... | Let the orthogonal projections of points $ B $ and $ D $ onto the line $ OS $ be denoted by $ U $ and $ V $, respectively. The angles $ OKB $, $ OLB $, and $ OUB $ are right angles, and thus the points $ K $, $ L $, and $ U $ lie on a circle whose diameter is the segment $ OB $ (Figure 7). (The point $ U $ may coincide... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,221 |
XXXVI OM - II - Problem 2
Prove that for a natural number $ n > 2 $, the number $ n! $ is the sum of $ n $ distinct divisors of itself. | We prove by induction a slightly stronger thesis: For every natural number $ n > 2 $, the number $ n! $ is the sum of $ n $ distinct divisors of itself, the smallest of which is $ 1 $. For $ n = 3 $, this is true: $ 3! = 6 = 1+2+3 $. Assume the thesis is true for some $ n $:
where the natural numbers $ k $ are divisor... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,222 |
XVIII OM - I - Problem 8
Given a sphere $ k $, a circle $ c $ lying on its surface, and a point $ S $ not lying on this surface, such that every line passing through $ S $ and through some point $ M $ of the circle $ c $ intersects the surface of the sphere $ k $ at another point $ N $ different from $ M $. Prove that... | When point $ S $ lies in the plane of circle $ c $, the proof of the theorem's thesis is immediate, for $ S $ is then inside circle $ c $ (since points lying outside $ c $ do not satisfy the adopted assumption) and the set of all points $ N $ is circle $ c $.
Assume that point $ S $ does not lie in the plane of circle ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,223 |
XXIX OM - III - Problem 4
Let $ X $ be an $ n $-element set. Prove that the sum of the number of elements in the sets $ A \cap B $ extended over all ordered pairs $ (A, B) $ of subsets of $ X $ equals $ n \cdot 4^{n-1} $. | Let $ a $ be a fixed element of the set $ X $. The number of subsets of the set $ X $ containing the element $ a $ is equal to the number of all subsets of the $(n-1)$-element set $ X- \{a\} $, which is $ 2^{n-1} $. Therefore, the number of such pairs $ (A, B) $ of subsets of the set $ X $, such that $ a \in A \cap B $... | n\cdot4^{n-1} | Combinatorics | proof | Yes | Yes | olympiads | false | 1,229 |
XII OM - I - Problem 12
Prove that if all the faces of a tetrahedron are congruent triangles, then these triangles are acute. | Suppose each face of the tetrahedron $ABCD$ (Fig. 11) is a triangle with sides $a$, $b$, $c$; let, for example, $BC = a$, $CA = b$, $AB = c$. There are 3 cases, which we will consider in turn. 1° - None of the lengths $a$, $b$, $c$ are equal. Two edges emanating from the same vertex cannot then be equal, since they are... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,231 |
XXXIV OM - II - Problem 3
Point $ P $ lies inside triangle $ ABC $, such that $ \measuredangle PAC = \measuredangle PBC $. Points $ L $ and $ M $ are the projections of $ P $ onto lines $ BC $ and $ CA $, respectively, and $ D $ is the midpoint of segment $ AB $. Prove that $ DL = DM $. | Let $ Q $ and $ R $ be the midpoints of segments $ AP $ and $ BP $, respectively. These points are the centers of the circles circumscribed around triangles $ APM $ and $ BPL $, which implies that
as central angles subtending the same arcs as the corresponding inscribed angles. Angles $ PAM $ and $ PBL $ are equal by ... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,233 |
VII OM - I - Problem 8
Given an acute triangle $ ABC $ and a point $ M $ inside one of its sides. Construct a triangle with the smallest perimeter, one of whose vertices is the point $ M $, and the other two lie on the two remaining sides of the triangle. | Let's consider triangle $MNP$ whose vertices lie on the sides of an acute triangle $ABC$, as indicated in Fig. 6.
Let $M'$ be the point symmetric to point $M$ with respect to the line $AB$, and $M''$ be the point symmetric to point $M$ with respect to the line $AC$. Then $PM'$ and $NM''$. Therefore, the length of the b... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,234 |
LII OM - II - Problem 6
For a given positive integer $ n $, determine whether the number of $ n $-element subsets of the set $ \{1,2,3,\ldots,2n-1,2n\} $ with an even sum of elements is the same as the number of $ n $-element subsets with an odd sum of elements. If not, determine which is greater and by how much. | Let's connect the numbers from the set $ S = \{1,2,3,\ldots,2n-1,2n\} $ in pairs:
Let $ X $ be any $ n $-element subset of the set $ S $. Choose (if possible) the smallest number $ x $ from the set $ X $ with the property that the number $ y $ paired with $ x $ does not belong to the set $ X $. Then remove the e... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,235 |
IX OM - I - Problem 8
What plane figure is the cross-section of a cube with edge $ a $ by a plane passing through the midpoints of three pairwise skew edges? | Each edge of a cube meets four other edges, is parallel to three further ones, and is skew to four remaining pairs of parallel edges. If we arbitrarily choose one edge of the cube $ABCDA$ (Fig. 8), for example, $AB$, then we can select two other edges that are skew to $AB$ and to each other in two ways: $AB$, $CC_1$, $... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,236 |
XXXIII OM - III - Problem 2
In a cyclic quadrilateral $ABCD$, a line passing through the midpoint of side $\overline{AB}$ and the intersection of the diagonals is perpendicular to side $\overline{CD}$. Prove that sides $\overline{AB}$ and $\overline{CD}$ are parallel or the diagonals of the quadrilateral are perpendic... | Let $O$ be the point of intersection of the diagonals, $E$ - the midpoint of side $\overline{AB}$, $F$ - the point of intersection of side $\overline{CD}$ and line $EC$. Since quadrilateral $ABCD$ is inscribed in a circle, therefore
Thus
By applying the Law of Sines to triangles $AOE$ and $BOF$ in sequence, we get
S... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,237 |
XI OM - III - Task 3
On a circle, 6 different points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $. | The given points can lie on the circle in various ways. To solve the problem correctly, we need to provide a proof that is independent of the specific arrangement of the points. Let's reason without any diagram.
First, note that if two chords of a circle are parallel, for example, \( MN \parallel PQ \), then the point... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,238 |
XXI OM - I - Problem 12
Find the smallest positive number $ r $ with the property that in a regular tetrahedron with edge length 1, there exist four points such that the distance from any point of the tetrahedron to one of them is $ \leq r $. | Let $ O $ be the center of the sphere circumscribed around the regular tetrahedron $ ABCD $ with edge $ 1 $ (Fig. 9). The radius of the sphere circumscribed around $ ABCD $ is $ R = \frac{1}{4} \sqrt{6} $. The tetrahedron $ ABCD $ is divided into tetrahedra with vertices: $ O $, one of the vertices of the tetrahedron, ... | \frac{\sqrt{6}}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,239 |
XLVIII OM - I - Problem 9
Determine all functions $ f: \langle 1,\ \infty) \to \langle 1,\ \infty) $ that satisfy the following conditions:
[(i)] $ f(x+1) = \frac{(f(x))^2-1}{x} $ for $ x \geq 1 $;
[(ii)] the function $ g(x)=\frac{f(x)}{x} $ is bounded. | It is easy to notice that the given conditions are satisfied by the function $ f(x) = x + 1 $. We will show that there are no other such functions.
Suppose, therefore, that $ f: \langle 1,\ \infty) \to \langle 1,\ \infty) $ is any function satisfying conditions (i) and (ii). We define the function $ h: \langle 1,\ \inf... | f(x)=x+1 | Algebra | proof | Yes | Yes | olympiads | false | 1,240 |
XXIII OM - II - Problem 5
Prove that in a convex quadrilateral inscribed in a circle, the lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point. | Let the convex quadrilateral $ABCD$ be inscribed in a circle. Denote by $P$, $Q$, $R$, $S$ the midpoints of the sides of this quadrilateral, and by $O$ - the midpoint of the segment $PR$ (Fig. 14).
Thus *)
Since $\displaystyle \frac{1}{2} (Q + S) = \frac{1}{4} (A + B + C + D) = O$, the point $O$ is also the midpoint o... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,241 |
XXIV OM - I - Problem 1
Prove that for every triangle, there exists a line dividing the triangle into two figures with equal perimeters and equal areas. | In an isosceles triangle, the line containing the bisector of the angle formed by its legs satisfies the conditions of the problem. Let the side lengths of triangle $ABC$ be different, for example, $a < b < c$, where $a = BC$, $b = AC$, $c = AB$ (Fig. 6). Let $p = \frac{1}{2} (a + b + c)$.
If $p - a \leq x \leq c$, the... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,242 |
XXXIV OM - I - Problem 8
Determine whether 20 regular tetrahedra with edge length 1, having pairwise disjoint interiors, can fit inside a sphere of radius 1. | Consider a regular icosahedron inscribed in a sphere. Each face of this icosahedron is an equilateral triangle with a side length greater than 1 (compare with preparatory problem E from the 31st Mathematical Olympiad). Therefore, in the spherical triangle bounded by the arcs of three great circles passing through pairs... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,243 |
LVIII OM - I - Problem 9
Let $ F(k) $ be the product of all positive divisors of the positive integer $ k $. Determine whether there exist different positive integers $ m $, $ n $, such that $ {F(m)=F(n)} $. | Answer: Such numbers $ m $ and $ n $ do not exist.
Let $ 1=d_1<d_2<\ldots<d_k=n $ be all the positive divisors of a fixed positive integer $ n $. Then $ n/d_1 $, $ n/d_2 $, $ \ldots $, $ n/d_k $ are also all the positive divisors of the number $ n $, so we can write
Thus, we have
where $ d(n) $ denotes the number... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,246 |
XXXI - I - Problem 11
Prove that $ \tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7} = \sqrt{7} $. | We will express the trigonometric functions of the angle $7 \alpha$ in terms of the trigonometric functions of the angle $\alpha$. This can be done using known trigonometric formulas or by utilizing properties of complex numbers. We will use the following de Moivre's formula:
For $n = 7$ and $x_k = \frac{k\pi}{7}\ (k ... | \sqrt{7} | Algebra | proof | Yes | Yes | olympiads | false | 1,247 |
IV OM - I - Task 4
Prove that the number $ 2^{55} + 1 $ is divisible by $ 11 $. | It is known that if $ n $ is a natural number, and $ a $ and $ b $ are any numbers, then
if we substitute in this equality $ a = 2^5 $, $ b = - 1 $, $ n = 11 $, we get
where $ C $ is an integer. The number $ 2^{55} + 1 $ is therefore divisible by $ 33 $, and thus also by $ 11 $.
Note 1. In the same way, we can genera... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,248 |
LVI OM - I - Task 2
Determine all natural numbers $ n>1 $ for which the value of the sum $ 2^{2}+3^{2}+\ldots+n^{2} $ is a power of a prime number with a natural exponent. | From the formula $ 1+2^{2}+3^{2}+ \ldots+n^{2}=\frac{1}{6}n(n+1)(2n+1) $, true for any natural number $ n $, we obtain
The task thus reduces to finding all natural numbers $ n>1 $, prime numbers $ p $, and positive integers $ k $ for which the equality
is satisfied.
Since the factor $ 2n^2 +5n+6 $ is greater tha... | 2,3,4,7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,251 |
XL OM - II - Task 3
Given is a trihedral angle $ OABC $ with vertex $ O $ and a point $ P $ inside it. Let $ V $ be the volume of the parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ OA^{\rightarrow} $, $ OB^{\rightarrow} $, $ OC^{\rightarrow} $. Calculate the m... | Let $ \mathbf{Q} $ denote the considered parallelepiped with volume $ V $. Let $ OA $, $ OB $, $ OC $ be its three edges. We draw a plane through point $ P $ intersecting the rays $ OA^\rightarrow $, $ OB^\rightarrow $, $ OC^\rightarrow $ at points $ K $, $ L $, $ M $ respectively (thus forming the tetrahedron $ OKLM $... | \frac{9}{2}V | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,252 |
XXVI - III - Task 4
In the decimal expansion of a certain natural number, the digits 1, 3, 7, and 9 appear. Prove that by permuting the digits of this expansion, one can obtain the decimal expansion of a number divisible by 7. | By performing an appropriate permutation of the digits of a given natural number, we can assume that its last four digits are $1$, $3$, $7$, and $9$. Thus, the considered natural number $n$ is the sum of the number $1379$ and some non-negative integer $a$, whose last four digits are zeros. We will prove that by perform... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 1,255 |
VI OM - II - Task 4
Inside the triangle $ ABC $, a point $ P $ is given; find a point $ Q $ on the circumference of this triangle such that the broken line $ APQ $ divides the triangle into two parts of equal area. | Let's consider triangles $ APB $ and $ APC $. There are three possible cases:
a) Each of the triangles $ APB $ and $ APC $ has an area less than half the area of triangle $ ABC $. This case occurs when the distances from point $ P $ to the sides $ AB $ and $ BC $ are less than half the corresponding heights of the tria... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,257 |
XXIII OM - II - Problem 3
The coordinates of the vertices of a triangle in the Cartesian coordinate system $ XOY $ are integers. Prove that the diameter of the circumcircle of this triangle is not greater than the product of the lengths of the sides of the triangle. | If the lengths of the sides of a triangle are $a$, $b$, $c$, and the length of the radius of the circle circumscribed around the triangle is $R$, then the area of the triangle is given by the formula
If the coordinates of the vertices of the triangle are $A = (a_1, a_2)$, $B = (b_1, b_2)$, $C = (c_1, c_2)$, then its a... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,258 |
XLIX OM - III - Problem 6
We consider unit squares on the plane, whose vertices have both integer coordinates. Let $ S $ be a chessboard whose fields are all unit squares contained in the circle defined by the inequality $ x^2+y^2 \leq 1998^2 $. On all the fields of the chessboard, we write the number $ +1 $. We perfo... | Answer: It is not possible.
Suppose it is possible to obtain a configuration with exactly one field $ P_1 $, on which the number $ -1 $ is written. Due to the standard symmetries of the chessboard $ S $, we can assume that the field $ P_1 $ has its center at $ (a-\frac{1}{2},b-\frac{1}{2}) $, where $ a \geq b \geq 1 $ ... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,259 |
XLVIII OM - I - Problem 7
Calculate the upper limit of the volume of tetrahedra contained within a sphere of a given radius $ R $, one of whose edges is the diameter of the sphere. | Let $ABCD$ be one of the considered tetrahedra. Assume that the edge $AB$ is the diameter of the given sphere. Denote the distance from point $C$ to the line $AB$ by $w$, and the distance from point $D$ to the plane $ABC$ by $h$. The volume of the tetrahedron $ABCD$ is given by
Of course, $w \leq R$, $h \leq R$, and t... | \frac{1}{3}R^3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,260 |
IX OM - I - Problem 1
Prove that if each of the functions
[ (1) \qquad (a_1x + b_1)^2 + (a_2x + b_2)^2, \]
[ (2) \qquad (b_1x + c_1)^2 + (b_2x + c_2)^2 \]
is the square of a linear function that is not constant, then the function
[ (3) \qquad (c_1x + a_1)^2 + (c_2x + a_2)^2 \]
is the square of a linear ... | First, let us note that if the function $ Ax^2 + 2Bx + C $ is the square of a linear function $ mx + n $, then its discriminant vanishes, i.e., $ B^2 - AC = 0 $. Indeed, from the identity $ Ax^2 + 2Bx + C = m^2x^2 + 2mnx + n^2 $, it follows that $ B^2 - AC = (mn)^2 - m^2n^2 = 0 $.
Conversely, if $ B^2 - AC = 0 $ and $ ... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,261 |
XXI OM - I - Problem 10
Find the largest natural number $ k $ with the following property: there exist $ k $ different subsets of an $ n $-element set, such that any two of them have a non-empty intersection. | Let the sought number be denoted by $ k(n) $. We will prove that $ k(n) = 2^{n-1} $. To each $ n $-term sequence with values $ 0 $ or $ 1 $, we assign the set of indices such that the corresponding terms of the sequence are equal to $ 1 $. This assignment establishes a one-to-one correspondence between the set of all $... | 2^{n-1} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,263 |
LIX OM - II - Task 1
Determine the maximum possible length of a sequence of consecutive integers, each of which can be expressed in the form $ x^3 + 2y^2 $ for some integers $ x, y $. | A sequence of five consecutive integers -1, 0, 1, 2, 3 satisfies the conditions of the problem: indeed, we have
On the other hand, among any six consecutive integers, there exists a number, say
$ m $, which gives a remainder of 4 or 6 when divided by 8. The number $ m $ is even; if there were a representation
in... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 1,264 |
XV OM - I - Problem 11
In triangle $ ABC $, angle $ A $ is $ 20^\circ $, $ AB = AC $. On sides $ AB $ and $ AC $, points $ D $ and $ E $ are chosen such that $ \measuredangle DCB = 60^\circ $ and $ \measuredangle EBC = 50^\circ $. Calculate the angle $ EDC $. | Let $ \measuredangle EDC = x $ (Fig. 9). Notice that $ \measuredangle ACB = \measuredangle $ABC$ = 80^\circ $, $ \measuredangle CDB = 180^\circ-80^\circ-60^\circ = 40^\circ $, $ \measuredangle CEB = 180^\circ - 80^\circ-50^\circ = \measuredangle EBC $, hence $ EC = CB $. The ratio $ \frac{DC}{CE} $ of the sides of tr... | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,265 |
XLVI OM - II - Problem 6
A square with side length $ n $ is divided into $ n^2 $ unit squares. Determine all natural numbers $ n $ for which such a square can be cut along the lines of this division into squares, each of which has a side length of 2 or 3. | If $ n $ is an even number, then a square of dimensions $ n \times n $ can be cut into $ 2 \times 2 $ squares. Let's assume, then, that $ n $ is an odd number. The square $ n \times n $ consists of $ n $ rows, with $ n $ cells (unit squares) in each row. We paint these rows alternately black and white (in a "zebra" pat... | nisdivisible2or3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 1,266 |
XXVI - III - Task 2
On the surface of a regular tetrahedron with edge length 1, a finite set of segments is chosen in such a way that any two vertices of the tetrahedron can be connected by a broken line composed of some of these segments. Can this set of segments be chosen so that their total length is less than $1 +... | On the surface of a regular tetrahedron with edge length $1$, one can choose a finite set of segments such that any two vertices of the tetrahedron can be connected by a broken line composed of some of these segments, and the total length of the segments belonging to this set is less than $1 + \sqrt{3}$.
Consider, in p... | \sqrt{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,267 |
XVI OM - I - Problem 7
On the side $ AB $ of triangle $ ABC $, a point $ K $ is chosen, and points $ M $ and $ N $ are determined on the lines $ AC $ and $ BC $ such that $ KM \perp AC $, $ KN \perp BC $. For what position of point $ K $ is the area of triangle $ MKN $ the largest? | Let $ BC = a $, $ AC = b $, $ \measuredangle ACB = \gamma $, $ KM = p $, $ KN = q $ (Fig. 2).
Notice that $ \measuredangle MKN = 180^\circ - \gamma $.
Indeed, if from a point $ K_1 $ chosen on the bisector of angle $ ACB $ we drop perpendiculars $ K_1M_1 $ and $ K_1N_1 $ to the sides of this angle, then the rays $ KM $... | KisthemidpointofsideAB | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,268 |
I OM - I - Task 3
Construct an equilateral triangle whose vertices lie on three given parallel lines. | Analyze. Let $ABC$ be an equilateral triangle, whose vertices $A, B, C$ lie on the given parallel lines $a, b, c$ (Fig. 13a).
om1_Br_img_13a.jpg
Rotate the entire figure around point $A$ by the angle $BAC = 60^{\circ}$. Point $B$ will end up at point $C$ after the rotation, and line $b$ will take the position $b_l$. Si... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,270 |
XV OM - I - Problem 4
In the plane, there are 5 points, no three of which are collinear. Prove that among them, there are four points that are the vertices of a convex quadrilateral. | Let $A$, $B$, $C$, $D$, $E$ denote given points. Since no three of them are collinear, points $A$, $B$, $C$ are the vertices of a triangle, and each of the points $D$ and $E$ lies either inside or outside this triangle. Therefore, one of the following cases occurs:
a) Points $D$ and $E$ lie inside the triangle $ABC$. T... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,274 |
XLIV OM - III - Task 6
Determine whether the volume of a tetrahedron can be calculated knowing the areas of its four faces and the radius of the circumscribed sphere (i.e., whether the volume of the tetrahedron is a function of the areas of its faces and the radius of the circumscribed sphere). | A tetrahedron is determined, up to isometry, by six parameters (for example, the lengths of the six edges). The areas of the faces and the radius of the circumscribed sphere - this is only five pieces of information (equations). Hence the conjecture that the answer to the posed question should be negative. (We emphasiz... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,275 |
XXXIII OM - III - Problem 6
Prove that in any tetrahedron, the sum of all dihedral angles is greater than $2\pi$. | We will first prove the following lemma.
Lemma. The sum of the measures of the planar angles at the vertices of a trihedral angle is less than $2\pi$.
Proof. Let $O$ be the vertex of the trihedral angle, $A$, $B$, $C$ - points lying on the respective edges at equal distance from $O$.
om33_3r_img_8.jpg
Let $O'$ be the o... | |\alpha|+|\beta|+|\gamma|+|\delta|+|\varepsilon|+|\zeta|>2\pi | Geometry | proof | Yes | Yes | olympiads | false | 1,277 |
LVIII OM - I - Problem 7
Given is a tetrahedron $ABCD$. The angle bisector of $\angle ABC$ intersects the edge $AC$ at point $Q$. Point $P$ is symmetric to $D$ with respect to point $Q$. Point $R$ lies on the edge $AB$, such that $BR = \frac{1}{2}BC$. Prove that a triangle can be formed with segments of lengths $BP$, ... | Let $ S $ be the midpoint of edge $ BC $, and $ T $ - the point symmetric to point $ C $ with respect to point $ Q $ (Fig. 3).
om58_1r_img_3.jpg
Since $ BS={1\over 2}BC=BR $, points $ R $ and $ S $ are symmetric with respect to the angle bisector $ BQ $ of angle $ ABC $, and thus $ QR=QS $. Furthermore, triangle $ CTB... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,278 |
XXXI - II - Problem 1
Students $ A $ and $ B $ play according to the following rules: student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length 1 on the plane, then student $ B $ gives a number $ s_1 $, equal to $ 1 $ or $ -1 $; then student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and ... | We will prove by induction that student $ B $ can choose the numbers $ \varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n $ such that the length of the vector $ \overrightarrow{\omega_n} = \sum_{k=1}^n \varepsilon_k \overrightarrow{a_k} $ is not less than $ \sqrt{n} $.
For $ n = 1 $, the vector has a length of $ 1 $ r... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,279 |
II OM - III - Problem 6
Given a circle and a segment $ MN $. Find a point $ C $ on the circle such that the triangle $ ABC $, where $ A $ and $ B $ are the points of intersection of the lines $ MC $ and $ NC $ with the circle, is similar to the triangle $ MNC $. | First, let us note that if points $M$ and $N$ lie on a given circle $k$, then any point $C$ on the circle, except for points $M$ and $N$, satisfies the condition of the problem; points $A$ and $B$ then coincide with points $M$ and $N$, and triangle $ABC$ coincides with triangle $MNC$. We will henceforth omit this case ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 1,281 |
XXIX OM - III - Task 2
On a plane, there are points with integer coordinates, at least one of which is not divisible by A. Prove that these points cannot be paired in such a way that the distance between the points of each pair is equal to 1; this means that an infinite chessboard with fields cut out at coordinates di... | Consider the set $ A $ of points on the plane with integer coordinates $ (i, j) $, where $ 0 \leq i, j \leq 4k $. Notice that if we pair the points according to the conditions of the problem, the sum of the coordinates of one point in each pair will be odd, and the other will be even.
The set $ A $ has $ (4k + 1)^2 $ p... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 1,283 |
XLV OM - I - Problem 12
Prove that the sums of opposite dihedral angles of a tetrahedron are equal if and only if the sums of opposite edges of this tetrahedron are equal. | For a quadruple of non-coplanar points $ U $, $ V $, $ X $, $ Y $, we will denote by $ |\measuredangle X(UV)Y| $ the measure of the convex dihedral angle formed by the half-planes with common edge $ UV $, passing through points $ X $ and $ Y $.
Let us consider a tetrahedron $ ABCD $. We will prove the equivalence of th... | proof | Geometry | proof | Yes | Yes | olympiads | false | 1,286 |
XLVI OM - I - Problem 5
Given are positive numbers $ a $, $ b $. Prove the equivalence of the statements:
保留了源文本的换行和格式，但最后一句是中文，翻译成英文应为: "The statements are to be proven equivalent while retaining the original text's line breaks and format." 但为了保持格式的一致性，这里不添加额外的句子。 | Assume the truth of statement (2). Substituting $ x = 1+\frac{1}{\sqrt{a}} $, we obtain the inequality
thus $ a + \sqrt{a} + \sqrt{a} + 1 > b $, or equivalently,
Since $ b $ is a positive number, it follows that condition (1) holds.
Conversely, from condition (1), we derive (2) as follows. Let $ x $ be any number gre... | proof | Algebra | proof | Yes | Yes | olympiads | false | 1,288 |
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