problem stringlengths 0 45.3k | solution stringlengths 0 26.8k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 9
values | question_type stringclasses 5
values | problem_is_valid stringclasses 5
values | solution_is_valid stringclasses 5
values | source stringclasses 11
values | synthetic bool 2
classes |
|---|---|---|---|---|---|---|---|---|
XLV OM - III - Task 5
Points $ A_1, A_2, \ldots , A_8 $ are the vertices of a parallelepiped with center $ O $. Prove that | Let's assume that one of the faces of a given parallelepiped is the parallelogram $A_1A_2A_3A_4$, and the opposite face is the parallelogram $A_5A_6A_7A_8$, with segments $A_1A_5$, $A_2A_6$, $A_3A_7$, $A_4A_8$ being four edges of the parallelepiped (Figure 17). Let's denote the distances from the vertices to the point ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVI - III - Task 6
Given are functions $ S $ and $ T $, $ S(x) = 1 - x $, $ T(x) = -\frac{1}{2} $ for $ 0\leq x \leq 1 $. Does there exist a function $ f $ of the form
for $ k = 1,2, \ldots, n $ for some natural number $ n $ such that
Please note that the mathematical expressions and symbols have bee... | We will prove the following:
Theorem. Every rational number $ \displaystyle \frac{m}{n} $ in the interval $ (0 ;\ 1) $, where $ m $ is an odd number, and $ n $ is a power of two, is equal to $ f\left( \frac{1}{2} \right) $ for some function $ f $ of the form (1).
Proof. We will use induction with respect to the number ... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
LIX OM - III - Task 4
Each point in the plane with both integer coordinates has been painted either white or black. Prove that from the set of all painted points, an infinite subset can be selected which has a center of symmetry and all of whose points have the same color. | Suppose the thesis of the problem is false.
Consider the central symmetry with respect to the point $ (0,0) $. Since there does not exist an infinite set symmetric with respect to this point and composed of points of the same color, only finitely many points with integer coordinates pass to points of the same color und... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXVII OM - II - Problem 5
Prove that if $ \cos \pi x =\frac{1}{3} $ then $ x $ is an irrational number. | Let $ \cos t = \frac{1}{3} $. We will prove by induction that for every natural number $ n $ the following formula holds:
where $ a_n $ is an integer not divisible by $ 3 $.
For $ n=1 $, it suffices to take $ a_1 = 1 $. For $ n = 2 $ we have
Thus, $ a_2 = -7 $. Suppose next that for some natural number $ k $ the foll... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LVIII OM - I - Problem 11
For each positive integer $ n $, determine the number of permutations $ (x_1,x_2,\ldots,x_{6n-1}) $ of the set $ \{1,2,\ldots,6n{-}1\} $, satisfying the conditions: | The sought number is
For any permutation satisfying the conditions of the problem, the following sequences of inequalities hold
Notice that each of the elements $ x_1 $, $ x_2 $, $ \ldots $, $ x_{6n-1} $ has been listed exactly once in the considered rows. Indeed, in the inequalities (1), all odd indice... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXVIII - I - Zadanie 2
Dane są trzy różne liczby całkowite $ a, b, c $. Udowodnić, że nie istnieje wielomian $ w(x) $ o współczynnikach całkowitych taki, że
|
Zauważmy najpierw, że jeżeli $ w(x) = p_0 + p_1x + p_2x^2 + \ldots + p_nx^n $, gdzie $ p_0, p_1, \ldots, p_n $ są liczbami całkowitymi, to dla dowolnych liczb całkowitych $ r $ i $ s $ liczba $ w(r) - w(s) $ jest podzielna przez $ r - s $. Mamy bowiem
a każda z liczb $ r - s, r^2 - s^2, \ldots, r^n - s^n $ jest ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLII OM - I - Problem 3
Real numbers $ a $, $ x $, $ y $, $ z $ satisfy the equation
Prove that
(Note: The original text had blank lines and symbols that were not fully provided, so the translation retains these elements as closely as possible.) | For convenience, let's write $ x_1 $, $ x_2 $, $ x_3 $ instead of $ x $, $ y $, $ z $; assume $ x_4 = x_1 $ and $ w = x_1+x_2+x_3 $. The assumption takes the form:
Hence
| proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XV OM - I - Problem 8
On three pairwise skew edges of a cube, choose one point on each in such a way that the sum of the squares of the sides of the triangle formed by them is minimized. | Let $ABCD$ be the base of a cube and $AA_1$, $BB_1$, $CC_1$, $DD_1$ its edges perpendicular to $ABCD$. Let $X$, $Y$, $Z$ be points chosen on three pairwise skew edges of the cube. It is known that by rotating the cube around certain axes passing through its center, the cube can be superimposed onto itself in such a way... | \frac{9}{2}^2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
III OM - III - Task 5
Prove that none of the digits $ 2 $, $ 4 $, $ 7 $, $ 9 $ can be the last digit of the number
where $ n $ is a natural number. | If the last digit of a given number is $ x $, then
so
Since $ n $ is an integer, the discriminant of the above equation, i.e., the number
is a square of an integer. The last digit of this discriminant is the last digit of the number $ 8x + 1 $. When $ x $ equals $ 2 $, $ 4 $, $ 7 $, $ 9 $, then $ 8x +... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XI OM - I - Problem 10
Find the formula expressing the product
in dependence on $ n $. | Denoting the given product by $ P $ we can write
thus
We propose to the readers to verify the above result using mathematical induction. | notfound | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
LII OM - I - Problem 9
Prove that among any $12$ consecutive positive integers, there exists a number that is not the sum of $10$ fourth powers of integers. | A number that is the fourth power of an integer gives a remainder of $0$ or $1$ when divided by $16$. Indeed: for even numbers of the form $2k$ we have $(2k)^4 = 16k^4$, whereas for odd numbers $2k + 1$ we obtain
(The number $\frac{1}{2} k(3k+1)$ is an integer for any integer $k$).
Hence, a number that is the sum of $... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXVII OM - II - Problem 4
Natural numbers $ x, y, z $, whose greatest common divisor is 1, satisfy the equation
Prove that $ x + y $ is a square of a natural number. | We rewrite the equation in the form
Let $ k $ denote the greatest common divisor of the numbers $ x $ and $ y $: $ k = \gcd (x,y) $. Therefore, we have
Notice that
The first of the equalities (2) follows from the fact that $ \gcd (m, n) = 1 $ (any prime divisor of the product $ mn $ must divide $ m $ or $ n $,... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
VI OM - II - Task 2
Find the natural number $ n $ knowing that the sum
is a three-digit number with identical digits. | A three-digit number with identical digits has the form $ 111 \cdot c = 3 \cdot 37 \cdot c $, where $ c $ is one of the numbers $ 1, 2, \ldots, 9 $, and the sum of the first $ n $ natural numbers is $ \frac{1}{2} n (n + 1) $, so the number $ n $ must satisfy the condition
Since $ 37 $ is a prime number, one of the n... | 36 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XIX OM - I - Zadanie 10
Dowieść, że wielomian $ W $ zmiennej $ x $ określony wzorem
gdzie $ a $ i $ b $ oznaczają liczby całkowite, nie jest iloczynem dwóch wielomianów zmiennej $ x $ stopnia dodatniego mających współczynniki całkowite.
|
Przypuśćmy, że twierdzenie, które mamy udowodnić, nie jest prawdziwe, tj. że dla każdego $ x $
gdzie $ P(x) $ i $ Q(x) $ są wielomianami stopnia dodatniego o współczynnikach całkowitych. Lewa strona wzoru (2) jest dla każdego $ x $ dodatnia, więc wielomian $ P(x) $ i tak samo wielomian $ Q(x) $ nie ma pierwiastk... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
XXXIV OM - I - Problem 4
Inside the circle $ K $, there are $ n $ circles whose centers lie on the diameter of $ K $. Prove that the perimeter of the union of these circles does not exceed the perimeter of $ K $. | We will first prove two lemmas.
Lemma 1. The perimeter of the union of two externally tangent circles is equal to the perimeter of the circle circumscribed around this union, that is, a circle to which the given circles are internally tangent and whose center is collinear with their centers.
om34_1r_img_1.jpg
Proof. Th... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
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 |
LIX OM - III -Zadanie 2
Funkcja $ f(x, y, z) $ trzech zmiennych rzeczywistych spełnia dla dowolnych liczb rzeczywistych $ a, b, c, d, e $zależność
Dowieść, że dla dowolnych liczb rzeczywistych $ x_1, x_2, \dots, x_n $ ($ n \geqslant 5 $) prawdziwa jest równość
|
Podstawiając $ a = b = c = d = e = 0 $ w danym w treści zadania warunku dostajemy $ f(0, 0, 0) = 0 $.
Niech teraz $ s, t, u $ będą dowolnymi liczbami rzeczywistymi. Wówczas
Odejmując powyższe dwie równości stronami i wykorzystując udowodnioną wcześniej równość $ f (0, 0, 0) = 0 $ otrzymujemy
Wypisując r... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXVII OM - III - Task 1
Determine whether the number
is rational. | Using the formula $ \sin 2a = 2 \sin a \cdot \cos a $ three times, we get
Since $ \cos a = \sin \left( \frac{\pi}{2} - a \right) $, it follows from the proven equality that
Therefore
Since $ \sin \frac{3 \pi}{18} = \sin \frac{\pi}{6} = \frac{1}{2} $ and
$ \sin \frac{9 \pi}{18} = \sin \frac{\pi}{2} = 1... | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false | |
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 |
XXXI - II - Problem 3
Given in space is a sphere $ K $ and points $ A, B $ outside the sphere such that the segment $ AB $ intersects the interior of the sphere. Prove that the set of points $ P $ for which the segments $ AP $ and $ BP $ are tangent to the sphere $ K $, is contained in a certain plane. | Let $ O $ be the center of the sphere $ K $, and $ r $ the length of its radius. We have
since the vector $ \overrightarrow{OR} $ is perpendicular to $ \overrightarrow{RP} $ and to $ \overrightarrow{RA} $. Vectors $ \overrightarrow{RP} $ and $ \overrightarrow{RA} $ are parallel, so
om31_2r_img_16.jpg
Similarly,
Mult... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XLVI OM - II - Problem 4
Positive numbers $ x_l, x_2, \ldots , x_n $ satisfy the condition
Prove that for every real number $ t $ greater than 1, the inequality holds | Let's denote briefly:
Knowing that $ f(1) \geq 0 $, we need to prove that $ f(t) \geq 0 $ for $ t > 1 $. For this purpose, it suffices to show that
This inequality follows from the following transformation:
For every index $ i = 1,\ldots,n $ and for every value $ t > 1 $, both differences in the par... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
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 |
XVI OM - I - Problem 5
Find the integers $ x $ and $ y $, satisfying the equation
(Note: The equation is missing in the original text, so it is also missing in the translation.) | Suppose that the integers $ x $ and $ y $ satisfy equation (1). Then $ 2^y $ is an integer, so $ y \geq 0 $. Equation (1) can be written in the form
Since $ 2^y > 0 $ and $ 1 + x^2 > 0 $, then $ 1 + x > 0 $, i.e., $ x > -1 $. We will distinguish two cases:
a) $ x = 0 $, then $ 2^y = 1 $ and $ y = 0 $,
b) $ x > 0 $, th... | x=0,y=0x=1,y=2 | Other | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
XIV OM - II - Task 2
In the plane, a quadrilateral $ABCD$ and a point $M$ are given. Construct a parallelogram with center $M$ and vertices lying on the lines $AB$, $BC$, $CD$, $DA$. | Suppose that the parallelogram $PQRS$ satisfies the conditions of the problem, with $P$ being the vertex that lies on the line $AB$. The vertex $R$, symmetric to it with respect to the point $M$, lies on one of the lines $BC$, $CD$, $DA$. Let us first consider the case where $R$ lies on the line $BC$. We can then assum... | notfound | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
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 |
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 |
LV OM - I - Task 5
For positive integers $ m $, $ n $ let $ N(m,n) $ denote the number of non-decreasing $ m $-term sequences with terms from the set
of
Prove that $ N(m, n+1) =N(n, m+1) $. | To any non-decreasing sequence $a_1, a_2, \ldots, a_m$ with terms from the set $\{1,2,3,\ldots,n+1\}$, we assign the sequence $b_1, b_2, \ldots, b_n$ defined as follows: for $\ell = 1,2,\ldots,n$
The sequence $b_1, b_2, \ldots, b_n$ thus defined has terms from the set $\{1,2,\ldots,m+1\}$ and is non-decreasing. More... | proof | Combinatorics | proof | Incomplete | Yes | olympiads | false |
II OM - III - Task 4
Determine the coefficients of the equation
in such a way that the roots of the equation are the numbers $ a $, $ b $, $ c $. | We know from algebra that the numbers $ x_1 $, $ x_2 $, $ x_3 $ are roots of the equation $ x^3 + mx^2 + nx + p = 0 $ if and only if the following conditions are satisfied:
Numbers $ a $, $ b $, $ c $ are therefore roots of equation (1) if and only if they satisfy the equations
The first of these equations ... | x^3-ax^2=0withroots,0,0x^3+x^2-x-1=0withroots-1,-1,1 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
XXX OM - I - Task 12
To qualify for the team, a player must pass at least four out of seven tests with a positive result. If the player prepares for all tests, the probability of success in each of these tests is the same and equals $ \frac{1}{2} $. If the player prepares for five tests, the corresponding probability ... | Of course, if a player does not prepare for a certain test, they do not take that test.
First, let's consider the case where the player prepares for all seven tests. The probability of obtaining any sequence of seven outcomes (successes or failures) in these tests is the same and equals $ \displaystyle \binom{1}{2}^7 $... | \frac{81}{128} | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false |
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 |
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 |
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 |
XXIV OM - II - Problem 3
Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the conditions:
1. $ f(x+1) = f(x) + 1 $ for every $ x \in \mathbb{R} $,
2. there exists an integer $ p $ such that $ f(f(f(0))) = p $. Prove that for every real number $ x $
where $ x_1 = x $ and $ x_n = f(x_{n-1}) $ ... | We will first provide several properties of the function $f$ satisfying condition $1^\circ$ of the problem. Let $f_n(x)$ be the $n$-fold composition of the function $f$, i.e., let $f_1(x) = f(x)$ and $f_{n+1}(x) = f_n(f_1(x))$ for $n = 1, 2, \ldots$. It follows that if $n = k + m$, where $k$ and $m$ are natural numbers... | 3 | Algebra | proof | Incomplete | Yes | olympiads | false |
VIII OM - I - Problem 11
Through the vertices $ A $, $ B $, $ C $ of an acute triangle and through the center $ O $ of the circumcircle of this triangle, lines $ AO $, $ BO $, $ CO $ are drawn, which intersect the opposite sides of the triangle at points $ M $, $ N $, $ P $. Prove that
where $ R $ denotes the radiu... | We will calculate the lengths $ AM $, $ BN $, $ CP $ in dependence on $ R $ and the angles $ A $, $ B $, $ C $ of the triangle (Fig. 9).
Notice that in triangle $ AMC $, side $ AC = 2R \sin B $, and $ \measuredangle MAC = 90^\circ - B $, because in the right triangle $ ACM $, where $ AM $ is the diameter of the circumc... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
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 |
L OM - I - Problem 7
Given a natural number $ n \geq 2 $. Determine all polynomials $ P(x) = a_0 + a_1x + \ldots + a_nx^n $ that have exactly $ n $ roots not greater than $ -1 $ and satisfy the condition
Note: Roots are counted with multiplicity: if a number $ x_0 $ is a root of multiplicity $ k $ of the polynom... | Let $ -p_1 \leq -p_2 \leq \ldots \leq -p_n $ be the roots of a polynomial satisfying the conditions of the problem. Then $ p_1 \geq p_2 \geq \ldots \geq p_n \geq 1 $ and $ a_n \neq 0 $. Furthermore,
and $ a_0 = a_np_1p_2\ldots p_n $. The condition $ a_0^2 + a_1a_n = a_n^2 + a_0a_{n-1} $ can thus be rewritten as
... | P(x)=(x+1)^{n-1}(x+b) | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
XXII OM - II - Task 2
Prove that if $ A, B, C $ are angles of a triangle, then
The text above has been translated into English, preserving the original text's line breaks and format. | Since $ C = \pi - (A + B) $, therefore
From this
Moreover, considering that $ \left| \frac{A-B}{2} \right| \leq \frac{\pi}{2} $, we obtain
Please note that the original text contains incomplete mathematical derivations or statements. If you need a more detailed or complete translation, please pro... | proof | Algebra | proof | Incomplete | Incomplete | olympiads | false |
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 |
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 |
LVIII OM - II - Problem 4
Prove that if $ a $, $ b $, $ c $, $ d $ are positive integers and $ \ ad=b^2+bc+c^2\;,\; $ then the number
is composite. | From the conditions of the problem, we obtain the equality $2ad = b^2 + c^2 + (b+c)^2$, and thus
On the other hand, from the factorization of the difference of squares, we have
Combining the above dependencies, we get
The factor $a + b + c + d$ is an integer greater than $1$. If it were equal to $a^2 + b^2 + c^2 + d... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XLIV OM - III - Problem 3
Let $ g(k) $ denote the greatest odd divisor of the positive integer $ k $, and let us assume
The sequence $ (x_n) $ is defined by the relations $ x_1 = 1 $, $ x_{n+1} = f(x_n) $. Prove that the number 800 appears exactly once among the terms of this sequence. Determine $ n $ for which ... | Let's list the first fifteen terms of the sequence $ (x_n) $, grouping them into blocks consisting of one, two, three, four, and five terms respectively:
We have obtained five rows of the infinite system (U), which can be continued according to the following rules: the $ j $-th row consists of $ j $ numbers, the ... | 166 | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
IV OM - III - Task 1
Investigate whether the equation
where $ a $, $ b $, $ c $ denote given real numbers, has real roots. | Every number $ x $ satisfying equation (1) must also satisfy the equation
obtained from equation (1) by multiplying both sides by $ (x - a) (x - b) (x - c) $.
Conversely, every root $ x $ of equation (2), different from $ a $, $ b $, and $ c $, is a root of equation (1), since dividing equality (2) by the non-zero num... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
Given a natural number $ n $ and a finite sequence of real numbers $ A^{(1)}= (a_1^{(1)}, a_2^{(1)}, \ldots, a_n^{(1)}) $. For each natural number $ m $, we choose two natural numbers $ i $, $ j $ satisfying the condition $ 1 \leq i < j \leq n $ and define the sequence $ A^{(m+1)}= (a_1^{(m+1)}, a_2^{(m+1)}, \ldots, a_... | Let's adopt the notation:
Let's denote the sum $ s^{(1)} $ simply by $ s $. The thesis of the problem states that the sequence $ \left(a_1^{(m)}\right)^\infty_{m=1} $ converges to the number $ s/n $, and the sequence $ \left(a_2^{(m)}\right)^\infty_{m=1} $ also converges to this number, and the same can be ... | proof | Algebra | proof | Incomplete | Incomplete | olympiads | false |
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 |
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 |
Given an integer $ n \geqslant 1 $. To each non-empty subset $ A $ of the set $ \{1, 2,\dots ,n\} $
we assign a number $ w(A) $ as follows: If $ a_1 > a_2 > \dots > a_k $ are all the elements
of the set $ A $, then
Obtain the sum of all $ 2n-1 $ numbers $ w(A) $. | For $ n = 1 $, the considered sum is equal to 1. Let us then assume that $ n \geqslant 2 $.
Let $ B_1, B_2, \dots , B_m $ be all the non-empty subsets of the set $ \{1, 2,\dots ,n-1\} $.
Then $ m =2^{n-1} -1 $ and
are all the non-empty subsets of the set $ \{1, 2,\dots ,n\} $.
Fix a value $ i\in \{1,2,\dots ,m\} $... | n\cdot2^{n-1} | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false |
XXI OM - I - Problem 5
Given are real numbers $ M $, $ a_1, a_2, \ldots, a_{10} $. Prove that if $ a_1x_1 + a_2x_2 + \ldots + a_{10}x_{10} \leq M $ for all $ x_i $ such that $ |x_i| = 1 $ ($ i = 1, 2, \ldots, 10 $), then | If $ a_i \geq 0 $, then we substitute $ x_i = 1 $ into the given inequality, and if $ a_i < 0 $, then we substitute $ x_i = - 1 $ for $ i = 1, 2, \ldots, 10 $. We will then obtain
From this inequality, it follows that $ |a_i| \leq M $ for $ i = 1, 2, \ldots, 10 $. Hence, $ |a_i|^2 \leq |a_i|M $, and therefore
T... | Algebra | proof | Yes | Incomplete | olympiads | false | |
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 |
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 |
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 |
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 |
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 |
XXIII OM - I - Problem 2
For a given prime number $ p $, determine all irreducible fractions $ \frac{a}{b}>0 $, such that | Given equality is transformed into the form
Since the first two terms on the left side are divisible by $ p $, it follows that $ b^2 $ is divisible by $ p $, and thus $ b $ is divisible by $ p $. Substituting $ b = pc $ into the above equality and dividing both sides by $ p^2 $, we obtain
and from this
Ther... | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false | |
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 |
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 |
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 |
XXXIV OM - III - Task 2
Given an irrational number $ a $ belonging to the interval $ (0,1) $ and a natural number $ N $. Prove that there exist natural numbers $ p, q, r, s $, such that
The translation preserves the original text's line breaks and formatting. | Let $ k $ be the largest natural number satisfying the inequality $ \frac{1}{k} > a $. Therefore,
if we take $ p_0 = 1 $, $ q_0= k+1 $, $ r_0 = 1 $, $ s_0=k $, we have the inequality
satisfied,
as well as the condition $ r_0q_0-p_0s_0 = 1 $, since $ 1 \cdot (k+1)-1 \cdot k = 1 $. If, in addition, $ \frac{1}{k} - \fr... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
X OM - I - Task 6
Prove that if $ n $ is a natural number greater than $ 1 $, and $ d $ is any positive number, then
untranslated part remains the same as the source text, maintaining the original format and line breaks. | a) According to the binomial formula of Newton
\[
(1 + 1)^n = \sum_{k=0}^{n} \binom{n}{k} 1^{n-k} 1^k = \sum_{k=0}^{n} \binom{n}{k}
\]
where \( \binom{n}{0} = 1 \), and \( \binom{n}{k} = \frac{n (n - 1) \cdot \ldots \cdot (n - k + 1)}{1 \cdot 2 \cdot \ldots \cdot k} \) for positive integer \( k \).
Since all terms on... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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 |
XVII OM - I - Problem 9
Solve the equation in natural numbers
Note: The equation to be solved is not provided in the original text. If you have the specific equation, please provide it for a complete solution. | It is enough to determine such solutions of equation (1) in natural numbers, that
All other solutions will then be found by creating permutations of the found values of $ x $, $ y $, $ z $.
\spos{1} Suppose that the natural numbers $ x $, $ y $, $ z $ satisfy equation (1) and condition (2). Then
from which ... | (1,2,3),(1,3,2),(2,1,3),(2,3,1),(3,1,2),(3,2,1),(-1,-2,-3),(-1,-3,-2),(-2,-1,-3),(-2,-3,-1),(-3,-1,-2 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XVIII OM - I - Problem 1
Solve in integers the equation
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”是说明性的，不是原文的一部分，因此在最终翻译中应被移除。以下是最终的翻译结果：
XVIII OM - I - Problem 1
Solve in integers the equation
（注意：最后一句说明性文字已移除） | Suppose that the integers $ x $, $ y $, $ z $ satisfy equation (1). If one of the numbers $ x $, $ y $, $ z $ is not zero, then there exists an integer $ d \geq 1 $ which is their greatest common divisor, i.e.,
where $ u $, $ v $, $ w $ are integers whose greatest common divisor is $ 1 $. By equation (1),
after divid... | (0,0,0) | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
XXVIII - III - Task 5
Prove that for any convex polygon, there exists a circle on whose circumference lie three consecutive vertices of the polygon and which contains the polygon. | If the segment $ \overline{AB} $ is contained in the boundary of the half-plane $ \pi $, then the set of points belonging to the half-plane $ \pi $, from which the segment $ \overline{AB} $ is seen at an angle $ \alpha $, is an arc of a certain circle $ K $.
From the points of the half-plane $ \pi $ lying inside the ci... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
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 |
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 |
XXIX OM - II - Problem 5
Prove that there is no such inclined plane that any tetrahedron placed on it with any of its faces will not tip over. This means the following. Given a plane \( \pi \) and a line \( l \) not perpendicular to it, prove that there exists a tetrahedron \( T \) such that for each of its faces \( S... | We will prove that for every inclined plane, there exists a tetrahedron such that for each of its faces, the tetrahedron can be placed on that face against the inclined plane in such a way that the tetrahedron will tip over.
Consider a tetrahedron $ABCD$ whose vertices have coordinates $A = (-1, 0, a)$, $B = (1, 0, a)$... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
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 |
LIV OM - I - Task 1
Find all pairs of positive integers $ x $, $ y $ satisfying the equation
| For any real numbers $ x,y \geq 2 $
We have
Therefore, for real numbers $ x,y \geq 2 $, the following inequalities are true
This means that the given equation has no solutions in natural numbers (or even real numbers) $ x,y \geq 2 $.
Thus, if a pair $(x,y)$ of positive integers satisfies the given equation, then one... | x=1,y=2x=2,y=1 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
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 |
XXXV OM - II - Task 5
Calculate the lower bound of the areas of convex hexagons, all of whose vertices have integer coordinates. | We will use the following lemma, which was the content of a competition problem in the previous Mathematical Olympiad.
Lemma. Twice the area of a triangle, whose all vertices have integer coordinates, is an integer.
Proof. The area of triangle $ABC$ is equal to half the absolute value of the determinant formed from the... | 3 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
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 |
XXXIII OM - I - Problem 3
Calculate the limit
where
保留了源文本的换行和格式，但最后一句是中文，翻译成英文如下：
where
the original text's line breaks and formatting have been preserved. | When factoring the numerator and the denominator, we obtain
Therefore, in the product,
each number $ m $ satisfying the inequality $ 9 \leq m \leq n - 4 $ appears exactly four times in the numerator and four times in the denominator. For example, the number $ 10 $ appears in the numerator in the factors $ a_6 =\d... | \frac{1}{14} | Calculus | math-word-problem | Incomplete | Yes | olympiads | false |
XLIX OM - I - Problem 5
Given is an integer $ n \geq 1 $. Solve the equation
The text has been translated while preserving the original line breaks and formatting. | Since $ 2\cot 2x=2 \cdot \frac{\cos 2x}{\sin 2x} = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x}=\cot x - \tan x $, the given equation is equivalent to the following:
When $ n = 1 $, equation (1) is identically satisfied in the set of numbers that are not integer multiples of $ \pi/2 $.
We further consider the case when $... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XLIX OM - I - Zadanie 5
Dana jest, liczba całkowita $ n \geq 1 $. Rozwiązać równanie
|
Ponieważ $ 2\cot 2x=2 \cdot \frac{\cos 2x}{\sin 2x} = \frac{\cos^2 x - \sin^2 x}{\sin x \cos x}=\cot x - \tan x $, dane równanie jest równoważne następującemu:
Gdy $ n = 1 $, równanie (1) jest spełnione tożsamościowo w zbiorze liczb, które nie są całkowitymi wielokrotnościami liczby $ \pi/2 $.
Dalej rozważamy pr... | proof | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
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 | |
XX OM - I - Task 3
Prove that no figure can have exactly two centers of symmetry. | Suppose that points $O_1$ and $O_2$ are centers of symmetry of figure $F$, and let $O_3$ be the point symmetric to $O_1$ with respect to the center $O_2$ (Fig. 1).
We will prove that point $O_3$ is also a center of symmetry of figure $F$.
Let $A$ be any point of figure $F$, $B$ - the point symmetric to $A$ with respect... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
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 |
XIV OM - I - Problem 9
Prove that if two non-identical polynomials of degree three
have a common factor of degree two, then
(Note: The last part of the original text is incomplete and has been translated as is.) | Suppose that the polynomials (1) are not identical and that they have a common factor of degree two. They are then divisible by the same quadratic trinomial of the form $ x^2 + px + q $, so there exist numbers $ \alpha $ and $ \alpha_1 $ such that for every $ x $
The numbers $ \alpha $ and $ \alpha_1 $ are not equal,... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XIII OM - II - Problem 4
Prove that if the sides $a$, $b$, $c$ of a triangle satisfy the inequality
then the angle bisectors $d_a$, $d_b$, $d_c$ of the opposite angles satisfy the inequality
(Note: The original text contains blank lines and symbols that are not filled in, which are preserved in the tr... | We can solve the problem by calculating the angle bisectors of triangle $ABC$ in dependence on its sides and angles. First, notice that the segments $BD$ and $DC$, into which the angle bisector $AD$ divides $BC$ (Fig. 27), are proportional to the sides $AB$ and $AC$:
\[
\frac{BD}{DC} = \frac{AB}{AC}
\]
From this, we ... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
LVII OM - I - Problem 10
Positive real numbers $ a, b, c $ satisfy the condition
Prove that
(Translation note: The original text had blank spaces where the conditions and the inequality to prove were supposed to be. These have been preserved in the translation.) | We will first show that for any positive numbers \(a\), \(b\), \(c\), the inequality
\[
\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} \geq \frac{3}{2}
\]
is satisfied.
Multiplying out the brackets, we rewrite inequality (1) as
\[
2(a^3 + b^3 + c^3) + 6abc \geq (a+b)(b+c)(a+c)
\]
and then, by grouping appropriate ... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.