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 |
|---|---|---|---|---|---|---|---|---|
XLIV OM - II - Problem 3
On the edge $ |OA_1| $ of the tetrahedron $ |OA_1B_1C_1| $, points $ A_2 $, $ A_3 $ are chosen such that $ |OA_1| > |OA_2| > |OA_3| > O $. Let $ B_2 $, $ B_3 $ be points on the edge $ |OB_1| $, and $ C_2 $, $ C_3 $ be points on the edge $ |OC_1| $, such that the planes $ |A_1B_1C_1| $, $ |A_2B... | Let $ B_1, $ B_2 be the orthogonal projections of points $ B_1, $ B_2 onto the plane $ OA_1C_1 $, and $ C_1, $ C_3 be the projections of points $ C_1, $ C_3 onto the plane $ OA_1B_1 $ (Figure 7; it is convenient to imagine the tetrahedron each time so that the face we are projecting onto is its "horizontal" base). From... | V_1+V_2+V_3\geq3V | Geometry | proof | Yes | Yes | olympiads | false |
XXXII - I - Problem 7
Let $ (x_n) $ be a sequence of natural numbers satisfying the conditions
a) $ x_1 < x_2 < x_3 < \ldots $,
b) there exists $ j \in \mathbb{N} $, $ j > 1 $, such that $ x_j = j $,
c) $ x_{kl} = x_kx_l $, for coprime $ k,l \in \mathbb{N} $.
Prove that $ x_n = n $ for every $ n $. | Since the number of natural numbers less than a given natural number $n$ is $n-1$, if for some $j$ we have $x_j = j$, then by condition a) it must be that $x_k = k$ for $k \leq j$. If the number $j$ mentioned in condition b) is greater than $2$, then the number $j-1$ is greater than $1$ and is relatively prime to the n... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXVI - I - Problem 4
Given are pairwise disjoint spheres $ K_1, K_2, K_3 $ with pairwise distinct radii. Let $ A_{ij} $ be the vertex of the cone circumscribed around the spheres $ K_i $ and $ K_j $. Prove that the points $ A_{12} $, $ A_{23} $, $ A_{31} $ are collinear. | If $ A $ is the vertex of a cone circumscribed around spheres $ K $ and $ L $ of different radii, then there exists exactly one such homothety $ j $ such that $ j(K) = L $. Its center is the point $ A $, and the coefficient is the ratio of the radii of the spheres $ L $ and $ K $.
Without loss of generality, we can ass... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVI - I - Problem 9
Calculate the limit
| For any natural numbers $ k $ and $ n $, where $ n \leq k $, we have
In particular, $ \displaystyle \binom{2^n}{n} \leq (2^n)^n = 2^{n^2} $. By the binomial formula, we have
and hence $ \displaystyle 2^n > \frac{n^2}{24} $. Therefore,
Since $ \displaystyle \lim_{n \to \infty} \frac{1}{n}= 0 $, it follows that $ \dis... | 0 | Calculus | math-word-problem | Incomplete | Yes | olympiads | false |
XIII OM - I - Problem 9
Solve the system of equations
The text above has been translated into English, maintaining the original text's line breaks and format. | Since only even powers of the numbers $a$, $b$, $c$ appear in the equations, we can assume without loss of generality that these numbers are non-negative.
1. Let us first consider the case where none of the numbers $a$, $b$, $c$ is zero. One solution to the system (1) is $x = y = z = 0$. It is easy to prove that any so... | notfound | Algebra | math-word-problem | Incomplete | Incomplete | olympiads | false |
XLV OM - I - Problem 1
Prove that the system of equations
has no solutions in positive integers $ a $, $ b $, $ c $, $ d $. | Suppose that the integers $ a,b,c,d > 0 $ satisfy the given system. The differences $ a^2 - c^2 = b $ and $ b^2 - d^2 = a $ are then positive numbers. Therefore, the inequalities $ a > c $ and $ b > d $ hold, which means $ a \geq c + 1 \geq 2 $ and $ b \geq d + 1 \geq 2 $. Hence,
as well as
Adding these ine... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XV OM - III - Task 2
Prove that if $ a_1 < a_2 < \ldots < a_n $ and $ b_1 < b_2 < \ldots < b_n $, where $ n \geq 2 $, then
The translation is provided as requested, maintaining the original formatting and line breaks. | \spos{1} a) When $ n = 2 $, the theorem is true, because
and since $ (a_1 - a_2) (b_1 - b_2) > 0 $, therefore
b) Suppose the theorem is true for some natural number $ n $ and let $ a_1 < a_2 < \ldots < a_{n+1} $ and $ b_1 < b_2 < \ldots < b_{n+1} $.
Then
Notice that
similarly, we conclude that t... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
LIX OM - I - Task 3
A sequence of integers $ a_1, a_2, a_3, \dots $ is defined by the conditions: $ a_1 = 1, a_2 = 2 $,
Determine whether there exists an integer $ k \geqslant 2 $ such that the number $ a_k $ is a divisor of
the product $ a_{k+1}a_{k+2} $. | From the condition (1) given in the problem, it follows that
for $ k \geqslant 2 $. Therefore, the number $ a_k $ is a divisor of the product $ a_{k+1}a_{k+2} $ if and only if it is a divisor of the number $ 5a_{k+1}^2 $.
Notice that no term in the sequence $ a_1, a_2, a_3, \dots $ is divisible by 5. Indeed, for ever... | proof | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
LVII OM - I - Problem 1
Determine all non-negative integers $ n $ for which the number
$ 2^n +105 $ is a perfect square of an integer. | If $ n $ is an odd number, then the number $ 2^n +105 $ gives a remainder of 2 when divided by 3. Since a number that is the square of an integer cannot give a remainder of 2 when divided by 3, the number $ n $ satisfying the conditions of the problem must be even.
Let's assume that $ n =2k $ for some non-negative int... | 4,6,8 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
LVI OM - I - Problem 11
A circle with center $ I $ is inscribed in a convex quadrilateral $ ABCD $, with the point $ I $ not lying on the line $ AC $. The diagonals $ AC $ and $ BD $ intersect at point $ E $. The line passing through point $ E $ and perpendicular to the line $ BD $ intersects the lines $ AI $, $ CI $ ... | Let $ o_1 $ be a circle with center $ P $ and tangent to the lines $ AD $ and $ AB $ at points $ S $ and $ T $, respectively. Similarly, let $ o_2 $ be a circle with center $ Q $ and tangent to the lines $ CB $ and $ CD $ at points $ U $ and $ W $, respectively (Fig. 5). Let $ o $ be the incircle of quadrilateral $ ABC... | PE=EQ | Geometry | proof | Yes | Yes | olympiads | false |
XVI OM - II - Task 4
Find all prime numbers $ p $ such that $ 4p^2 +1 $ and $ 6p^2 + 1 $ are also prime numbers. | To solve the problem, we will investigate the divisibility of the numbers \( u = 4p^2 + 1 \) and \( v = 6p^2 + 1 \) by \( 5 \). It is known that the remainder of the division of the product of two integers by a natural number is equal to the remainder of the division of the product of their remainders by that number. B... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XL OM - I - Task 2
For a fixed natural number $ n $, find the number of solutions to the equation
satisfying the condition $ 1 \leq x \leq n $.
Note. $ [t] $ is the greatest integer not greater than $ t $. | Let's denote the left and right side of the given equation by $ L(x) $ and $ P(x) $. Representing the real number $ x $ as $ x = k + r $, where $ k = [x] $, $ r = x - [x] \in\langle 0; 1) $, we can write:
Thus, $ L(x) = P(x) $ if and only if $ [2kr + r^2] = 2kr $. This equality is equivalent to the statement that $ 2k... | n^2-n+1 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
X OM - I - Task 4
Prove that if point $ P $ moves in the plane of triangle $ ABC $, then the triangle $ S_1S_2S_3 $, whose vertices are the centroids $ S_1 $, $ S_2 $ and $ S_3 $ of triangles $ PBC $, $ PCA $ and $ PAB $ does not change its shape or size. | If the vertex $ P $ of the triangle $ PAB $ is moved to the point $ P' $ (Fig. 6a), then the centroid $ S_3 $ of this triangle will move to the point $ S $.
The vector $ S_3S $ has the same direction as the vector $ PP' $ and is three times shorter, since these vectors are similar (directly) relative to the point $ N $... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIX OM - I - Problem 7
Given are positive integers $ m $ and $ n $. Let $ A=\{1,2,3,\ldots,n\} $. Determine the number of functions $ f:A \to A $ that take exactly $ m $ values and satisfy the condition:
保留了源文本的换行和格式，但最后一句“保留了源文本的换行和格式”是中文，应翻译为英文如下：
Preserving the line breaks and format of the original te... | Way I} A function $ f $ satisfying the given conditions will be called {\it good
. It is clear that if $ m > n $, then good functions do not exist. It is enough to consider the case where the given natural numbers $ n $ and $ m $ satisfy the inequality $ n \geq m \geq 1 $.
Let $ f:A \to A $ be a good function and let t... | \binom{n+-1}{2m-1} | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false |
IX OM - II - Task 1
Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number. | The expression $ a^4 + 4 $ can be represented in the following way as a product of $ 2 $ factors:
Since
thus when $ a \ne 1 $ and $ a \ne - 1 $, each of these factors is positive and greater than $ 1 $. Therefore, it follows that when $ a $ is an integer different from $ 1 $ and $ - 1 $, $ a^4 + 4 $ i... | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
LI OM - I - Task 7
Prove that for any positive integer $ n $ and any number $ t\in(\frac{1}{2},1) $ there exist such numbers $ a,b \in (1999,2000) $, that | Let $ c = 3999/2 $. We seek numbers $ a $, $ b $ in the form $ a = c+x $, $ b = c-x $, where $ x \in (0, \frac{1}{2}) $. Then
where $ s = 2t - 1 > 0 $. After expanding using the binomial theorem, the last expression takes the form
where $ p(x) $ is some polynomial.
We want to show that for some number $ y \in (0,\fra... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
VII OM - III - Task 3
On a straight line, there are three different points $ M $, $ D $, $ H $. Construct a right-angled triangle for which $ M $ is the midpoint of the hypotenuse, $ D $ - the intersection point of the angle bisector of the right angle with the hypotenuse, and $ H $ - the foot of the altitude on the h... | Let $ABC$ be the sought right-angled triangle. Since points $M$, $D$, $H$ are distinct, the legs $AC$ and $BC$ are not equal. Let, for example, $AC > BC$. Then point $D$ lies between points $M$ and $B$, and point $H$ lies between points $D$ and $B$, so point $D$ lies between points $M$ and $H$ (Fig. 19).
Since
an... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXVIII OM - II - Problem 5
Determine all prime numbers $ p $ and natural numbers $ x, y $, for which $ p^x - y^3 = 1 $. | Suppose that the numbers $ p $, $ x $, $ y $ satisfy the given conditions. Then
Let's consider two cases:
1. $ y = 1 $. Then (1) reduces to the equation $ p^x = 2 $, from which $ p = 2 $, $ x = 1 $.
2. $ y > 1 $. Then both factors of the product on the right side of (1) are numbers greater than $ 1 $ and as divis... | (2,1,1)(3,2,2) | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXXV OM - II - Problem 3
Given are sequences $ (x_1, x_2, \ldots, x_n) $, $ (y_1, y_2, \ldots, y_n) $ with positive terms. Prove that there exists a permutation $ p $ of the set $ \{1, 2, \ldots, n\} $ such that for any real $ t $ the sequence
has the following property: there exists a number $ k $ such that $ 1... | We calculate the ratios $ \frac{x_1}{y_1}, \frac{x_2}{y_2}, \ldots, \frac{x_n}{y_n} $, find such $ j $ that $ \frac{x_j}{y_j} $ is not greater than all other ratios, and set $ p(1) = j $. Then we find such $ k $ that $ \frac{x_k}{y_k} $ is not greater than all ratios $ \frac{x_i}{y_i} $ different from $ \frac{x_j}{y_j}... | proof | Combinatorics | proof | Incomplete | Yes | olympiads | false |
XXV OM - II - Task 2
Prove that for every $ n = 2, 3, \ldots $ and any real numbers $ t_1, t_2, \ldots, t_n $, $ s_1, s_2, \ldots, s_n $, if | When performing the same permutation of sequences $ t_1, t_2, \ldots, t_n $ and $ s_1, s_2, \ldots, s_n $, we do not change the conditions of the problem. Therefore, we can assume that the sequence $ s_1, s_2, \ldots, s_n $ is non-decreasing.
Let $ a_k = s_{k+1} - s_k $ for $ k = 1, 2, \ldots, n-1 $. Then for $ i < j $... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXXIV OM - III - Task 4
Prove that if the natural numbers $ a, b, c, d $ satisfy the equation $ ab = cd $, then | We will first prove that for any natural numbers $ k $, $ j $, $ m $, $ n $
If a prime number $ p $ satisfies the relations $ p^\alpha | \mathrm{GCD} (k, l) $ and $ p^\beta | \mathrm{GCD} (m,n) $, then $ p^\alpha | k $, $ p^\alpha | l $, $ p^\beta | m $, $ p^\beta | n $, so $ p^{\alpha+\beta} $ divides each of the nu... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XLIII OM - I - Problem 7
In the plane, a convex polygon $ Q $ and a line $ l $ are given.
(a) Prove that there exists exactly one polygon $ R $ symmetric with respect to $ l $ such that for every line $ s $ perpendicular to $ l $, the intersections $ s \cap R $ and $ s \cap Q $ are segments of equal length, or single-... | (a) Let's take a line $ l $ as the z-axis of a Cartesian coordinate system, and let the interval $ \langle a;\ b \rangle $ of this axis be the orthogonal projection of a polygon $ Q $. Let $ x_0, x_1, \ldots, x_n $ be the abscissas of all vertices of the polygon $ Q $, numbered such that $ a = x_0 < x_1 < \ldots < x_n ... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
III OM - I - Task 2
Solve the system of equations
| Multiplying the second equation of the given system:
by the product $ xyz $, we obtain the equation
If in the system of equations (1) instead of equation (*) we take equation (**), we obtain the system of equations
We will show that the system of equations (2) is equivalent to system (1). Indeed, ever... | Algebra | math-word-problem | Incomplete | Yes | olympiads | false | |
XLII OM - III - Problem 4
In the plane with a Cartesian coordinate system, we consider the set $ V $ of all free vectors, both of whose coordinates are integers. Determine all functions $ f $, defined on the set $ V $, with real values, satisfying the conditions:
(a) $ f(v) = 1 $ for each of the four vectors $ v\in V ... | Suppose that the function $ f $ satisfies the given conditions. Using these conditions, let's list the values of $ f(\overrightarrow{\mathbf{v}}) $ for a few vectors with small integer coordinates:
(When calculating the last of the found values, we used the orthogonality of the vectors $ [3,0] $ and $ [0,1] $ and the ... | f([x,y])=x^2+y^2 | Algebra | proof | Yes | Yes | olympiads | false |
XXVIII - III - Task 3
Consider the set $ A = \{0, 1, 2, 3, \ldots, 2^{2n-1}\} $. Given is a function $ f: A \to A $ such that for every sequence $ (x_0, x_1, \ldots, x_{2n-1}) $ with terms equal to 0 or 1, the following equality holds
Prove that if the numbers $ a_1, a_2, \ldots, a_9 \in A $ are consecutive term... | Let any number $a \in A$ be written in binary, appending the necessary number of zeros to the left so that the representation has $2n$ digits. From the conditions of the problem, it follows that the representation of the number $f(a)$ is obtained from the representation of the number $a$ by swapping zeros and ones on e... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XII OM - II - Problem 5
Prove that if the real numbers $ a $, $ b $, $ c $ satisfy the inequalities
[ (1) \qquad a + b + c > 0, \]
[ (2) \qquad ab + bc + ca > 0, \]
[ (3) \qquad abc > 0, \]
then $ a > 0 $, $ b > 0 $, $ c > 0 $. | From inequalities (1) and (2), it can be inferred that at least two of the numbers $a$, $b$, $c$ are positive. For from inequality (1) it follows first that at least one of these numbers is positive, let's say $c > 0$. Indeed, for any $a$ and $b$
and from inequality (2) it follows that
therefore
transferring t... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXIX OM - II - Problem 3
Given a sequence of natural numbers $ (a_i) $, such that for every natural number $ n $, the sum of the terms of the sequence that are not greater than $ n $ is not less than $ n $. Prove that for every natural number $ k $, one can choose a finite subsequence from $ (a_i) $ whose sum of terms... | We will apply induction with respect to $k$. Consider the case $k = 1$. By assumption, the sum of those terms of the given sequence that are not greater than $1$ (and thus are equal to $1$) is not less than $1$. Therefore, there exists a term in the given sequence that is equal to $1$. Hence, the sum of the terms of a ... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XIV OM - I - Problem 11
Inside the tetrahedron $ABCD$, a point $S$ is chosen. The lines $AS$, $BS$, $CS$, $DS$ intersect the opposite faces of the tetrahedron at points $A'$, $B'$, $C'$, $D'$. Prove that | Let's draw the heights $ SK $ and $ DH $ of the tetrahedrons $ SABC $ and $ DABC $ with the common base $ ABC $ (Fig. 15). The ratio $ \frac{SK}{DH} $ equals the ratio of the volumes of these tetrahedrons. The right triangles $ DHD $ and $ SKD $ are similar, so $ \frac{SD.
Therefore,
Similarly,
Adding the a... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XLVII OM - II - Problem 6
Inside a parallelepiped, whose edges have lengths $ a $, $ b $, $ c $, there is a point $ P $. Prove that there exists a vertex of the parallelepiped whose distance from point $ P $ does not exceed $ \frac{1}{2}\sqrt{a^2 + b^2 + c^2} $. | The walls of the parallelepiped define six planes. Let $ \pi $ be the plane whose distance from point $ P $ is the smallest; if there are two such planes (or more), we choose any one of them and denote it by $ \pi $. Let $ ABCD $ be the face of the parallelepiped contained in the plane $ \pi $ and let $ N $ be the orth... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXIX OM - I - Zadanie 2
Dla danej liczby naturalnej $ k>7 $ skonstruować ciąg $ S = (a_0, a_1, \ldots, a_{k-1}) $ o tej własności, że dla każdego $ i $ ($ 0 < i < k - 1 $) $ a_i $ równe liczbie wyrazów ciągu $ S $ równych $ i $.
|
Dla dowolnej liczby naturalnej $ k $ znajdziemy wszystkie ciągi $ S $ spełniające warunki zadania.
Z warunków zadania wynika, że w ciągu $ S $ jest $ a_0 $ wyrazów równych $ 0 $, $ a_1 $ wyrazów równych $ 1 $, $ \ldots $ , $ a_{k-1} $ wyrazów równych $ k - 1 $ i innych wyrazów nie ma. Zatem liczba wszystkich wyrazów c... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XLV OM - I - Problem 7
Outside a convex quadrilateral $ABCD$, we construct similar triangles $APB$, $BQC$, $CRD$, $DBA$ in such a way that
Prove that if the quadrilateral $PQRB$ is a parallelogram, then the quadrilateral $ABCD$ is also a parallelogram. | If quadrilateral $PQRS$ is a parallelogram, then its diagonals $PR$ and $QS$ have a common midpoint $O$. Let the midpoints of the diagonals $AC$ and $BD$ of quadrilateral $ABCD$ be denoted by $M$ and $N$, respectively. To prove that quadrilateral $ABCD$ is a parallelogram, it suffices to show that points $M$ and $N$ co... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XVII OM - I - Problem 7
In a given triangle, two triangles are inscribed in such a way that the vertices of one of them are symmetric to the vertices of the other with respect to the midpoints of the sides of the given triangle. Prove that the areas of these triangles are equal. | The area of a triangle with vertices $ K $, $ L $, $ M $ will be briefly denoted by the symbol $ (KLM) $.
Let $ (M_1,M_2) $, $ (N_1,N_2) $, $ (P_1, P_2) $ be pairs of points lying on the sides $ BC $, $ CA $, $ AB $ of triangle $ ABC $ and symmetric with respect to the midpoints of these sides (Fig. 6).
The task is to ... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXVI - I - Problem 5
Determine all integers $ m $ for which the polynomial $ x^3-mx^2+mx-(m^2+1) $ has an integer root. | Suppose that the integers $ x $ and $ m $ satisfy the equation $ x^3 - mx^2+mx-(m^2+1) = 0 $, which is $ (x^2+m)(x-m) = 1 $. Then $ x^2+m = 1 $ and $ x- m = 1 $ or $ x^2+m = -1 $ and $ x-m = -1 $. Adding the equations in each of these systems, we get $ x^2+x = 2 $ or $ x^2+x = - 2 $. The roots of the first equation are... | =0=-3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
XLIV OM - I - Problem 12
Prove that the polynomial $ x^n + 4 $ is a product of two polynomials of lower degree with integer coefficients if and only if $ n $ is divisible by $ 4 $. | \spos{I} Suppose that the polynomial $ x^4 + 4 $ is the product of two polynomials,
with the properties under consideration:
In the product $ F(x)G(x) $, the coefficient of $ x^n $ is $ a_kb_m $, and the constant term equals $ a_0b_0 $. Therefore, the following equations hold:
The numbers ... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XIII OM - I - Problem 5
Prove that all powers of a number whose last eight digits are $12890625$ also end with the digits $12890625$. | We need to prove that if $ n $ and $ l $ are integers, with $ n > 0 $, $ l \geq 0 $, then
where $ m $ is a non-negative integer.
First, observe that in the expansion of the left side of the above equality according to the binomial theorem for the power of a binomial, all terms except for $ (12890625)^n $ have a factor... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XV OM - III - Task 6
Given is a pyramid SABCD, whose base is a convex quadrilateral $ABCD$ with perpendicular diagonals $AC$ and $BD$, and the orthogonal projection of vertex S onto the base is point O, the intersection of the diagonals of the base. Prove that the orthogonal projections of point O onto the lateral fac... | Let $M$, $N$, $P$, $Q$ be the orthogonal projections of point $O$ onto the planes $ASB$, $BSC$, $CSD$, $DSA$ (Fig. 23).
The plane $SOM$ is perpendicular to the plane of quadrilateral $ABCD$ and to the plane $ASB$, as it contains the perpendiculars $SO$ and $OM$ to these planes. Therefore, the plane $SOM$ is perpendicul... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LIX OM - I - Task 6
Determine all polynomials $ W(x) $ with real coefficients such that for every real number
$ x $ the equality is satisfied | First, note that the only constant polynomials satisfying the conditions of the problem are $ W(x) \equiv 0 $ and $ W(x) \equiv 1 $. Let us assume from now on that the polynomial $ W(x) $ is not constant.
If the polynomial $ W(x) $ is of the form $ W(x)=cx^n $ for some real number $ c \neq 0 $ and integer $ n \geqslan... | proof | Algebra | math-word-problem | Yes | Yes | olympiads | false |
XVII OM - I - Problem 5
Given positive numbers $ p $ and $ q $. Prove that a rectangular prism, in which the sum of the edges equals $ 4p $, and the surface area equals $ 2q $, exists if and only if $ p^2 \geq 3q $. | a) Suppose that the desired rectangular parallelepiped exists. Then there are numbers $ x $, $ y $, $ z $ satisfying the equations
In such a case, the numbers $ p $ and $ q $ satisfy the inequality $ p^2 \geq 3q $, since from (1) and (2) it follows that
b) Suppose that given positive numbers $ p $ and... | proof | Algebra | proof | Yes | Yes | olympiads | false |
LVI OM - II - Task 3
In the data space, there are $ n $ points ($ n\geq 2 $) of which no four lie in the same plane. Some of these points have been connected by segments. Let $ K $ be the number of segments drawn ($ K\geq 1 $), and $ T $ the number of triangles formed. Prove that | Let us number from 1 to $ m $ those points from which at least one segment starts, and assume that from the point numbered $ i $ ($ i=1,2,\ldots,m $) exactly $ k_i $ segments start. Then $ k_i>0 $. Furthermore, let $ t_i $ ($ i=1,2,\ldots,m $) denote the number of those triangles, one of whose vertices is the point num... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XL OM - I - Task 6
Calculate the sum of the series $ \sum_{n\in A}\frac{1}{2^n} $, where the summation runs over the set $ A $ of all natural numbers not divisible by 2, 3, 5. | A natural number $ n $ is not divisible by $ 2 $, $ 3 $, and $ 5 $ if and only if, when divided by 30, it gives one of the following eight remainders:
Therefore, the sum we need to calculate is equal to
where $ s_i $ denotes the sum of an analogous series formed from terms corresponding to those values of $... | \frac{1}{1-2^{-30}}(\frac{1}{2^1}+\frac{1}{2^7}+\frac{1}{2^{11}}+\frac{1}{2^{13}}+\frac{1}{2^{17}}+\frac{1}{2^{19}}+\frac{1}{2^{23}}+\frac{1}{2^{} | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
III OM - III - Task 6
In a round tower with an internal diameter of $2$ m, there are spiral stairs with a height of $6$ m. The height of each step is $0.15$ m. In a horizontal projection, the steps form adjacent circular sectors with an angle of $18^\circ$. The narrower ends of the steps are attached to a round pillar... | For one revolution of the tower, there are $ \frac{360}{18} = 20 $ steps with a total height of $ 20 \cdot 0.15 \textrm{m} = 3 \textrm{m} $. The step with number $ 20 + k $ is therefore exactly above the step with number $ k $ at a distance of $ 3 \textrm{m} $. The orthogonal projection of the staircase on a horizontal... | 4.47\textrm{} | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
XL OM - I - Task 1
Prove that if the numbers $ k $, $ n $ ($ k < n $) are coprime, then the number $ \binom{n-1}{k-1} $ is divisible by $ k $. | Let's denote:
These are integers. The equality
holds, that is, $ M \cdot n = N \cdot k $. Since $ k $ is relatively prime to $ n $, it must be a divisor of $ M $; which is exactly what we needed to prove. | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XVI OM - III - Task 1
Prove the theorem: the lengths $ a $, $ b $, $ c $ of the sides of a triangle and the radian measures $ \alpha $, $ \beta $, $ \gamma $ of the opposite angles satisfy the inequalities | a) Proof of the left part of formula (1).
Assume that $ a \geq b \geq c $, hence $ \alpha \geq \beta \geq \gamma $. In this case
Transforming the left side of the above inequality and taking into account that $ \alpha + \beta + \beta = \pi $, we obtain successively
and finally
b) Proof of ... | proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
IV OM - II - Task 5
Calculate the volume $ V $ of the tetrahedron $ ABCD $ given the length $ d $ of the edge $ AB $ and the area $ S $ of the projection of the tetrahedron onto a plane perpendicular to the line $ AB $. | Let's lead through the vertex $A$ of the tetrahedron $ABCD$ a plane $\alpha$ perpendicular to the edge $AB$ (Fig. 37) and through points $C$ and $D$ draw lines parallel to the line $AB$ until they intersect the plane $\alpha$ at points $C'$ and $D'$. The orthogonal projection of the tetrahedron $ABCD$ onto the plane $\... | Geometry | math-word-problem | Yes | Incomplete | olympiads | false | |
XXXI - II - Problem 6
Prove that if point $ P $ runs along the incircle of triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite vertices $ A, B, C $, respectively). | The center $ O $ of the circle inscribed in triangle $ ABC $ is the intersection point of the angle bisectors $ \overline{AA}, $ \overline{BB}, $ \overline{CC}.
om31_2r_img_17.jpg
Since
therefore
and thus
Since $ PO $ is equal to the radius $ r $ of the inscribed circle, and $ a $,... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXVIII - III - Task 1
Given is a tetrahedron $ABCD$, about whose dihedral angles it is known that
Prove that the edges $AB$ and $CD$ are perpendicular. | Since the sum of the angles in triangle $ABC$ is $180^\circ$, we have $\measuredangle ACB = 180^\circ - \measuredangle ABC - \measuredangle BAC = 180^\circ - 70^\circ - 40^\circ = 70^\circ$. It follows that triangle $ABC$ is isosceles: $AB = AC$.
As we know, the dot product of two vectors is equal to the product of the... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
LVIII OM - II - Task 2
Given is a convex pentagon $ABCDE$, in which
Prove that a triangle can be constructed from segments of lengths $AC$, $CE$, $EB$. Determine the measures of its angles, given the measure of $\alpha$ of angle $ACE$ and the measure of $\beta$ of angle $BEC$. | Let $ P $ be the image of point $ B $ under a $ 90^\circ $ rotation about point $ E $ (Fig. 9); then, of course, lines $ BE $ and $ EP $ are perpendicular and $ BE=EP $.
Under this rotation, point $ A $ maps to point $ D $, so triangles $ ABE $ and $ DPE $ are congruent. Hence, $ AB=DP $ and $ \measuredangle EAB=\measu... | 90-\beta,\alpha+\beta,90-\alpha | Geometry | proof | Incomplete | Yes | olympiads | false |
XLIV OM - III - Zadanie 1
Rozwiązać w liczbach wymiernych $ t $, $ w $, $ x $, $ y $, $ z $ układ równań
|
Załóżmy, że liczby wymierne $ t $, $ w $, $ x $, $ y $, $ z $ spełniają podany układ równań. Odejmujemy stronami równanie trzecie od pierwszego: otrzymujemy równość $ 2xy - 2yz = z^2 - x^2 $, czyli
Dodajemy stronami drugie i trzecie równanie układu: otrzymujemy równość $ 2xz + 2yz = 2t^2+x^2-y^2 $, czyli
W... | =w=x=y=z=0 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXXII - I - Problem 11
Determine all natural numbers $ n $ for which the following statement is true: If in a convex polygon with $ 2n $ sides inscribed in a circle there are $ n-1 $ pairs of parallel sides, then the remaining two sides are also parallel. | om32_1r_img_9.jpgom32_1r_img_10.jpg
Two chords of a circle are parallel if and only if there exists a common perpendicular bisector (passing through the center $ O $ of the circle). Let $ \overline{A_iA_{i+1}} $, $ \overline{B_iB_{i+1}} $ be parallel sides of a polygon inscribed in a circle for $ i = 1,2,\ldots,2k $. W... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXVIII OM - III - Problem 4
Among the set of tetrahedrons with a base area of 1, a total surface area of 4, and having equal angles of inclination of the lateral faces to the base, find the tetrahedron with the maximum volume. | Let $ABCD$ be any tetrahedron satisfying the given conditions:
From condition (2), it follows that the point $D$, which is the projection of point $D$ onto the plane $ABC$, is at the same distance from the lines $BC$, $CA$, $AB$ (equal to $r = h \ctg \phi$, where $h = |DD'|$ and $\phi$ is the dihedral angle... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXIX OM - II - Problem 3
Inside an acute triangle $ ABC $, consider a point $ P $ and its projections $ L, M, N $ onto the sides $ BC, CA, AB $, respectively. Determine the point $ P $ for which the sum $ |BL|^2 + |CM|^2 + |AN|^2 $ is minimized. | Let $ P $ be any point inside the triangle $ ABC $. Considering the right triangles $ ANP $, $ BLP $, $ CMP $ (Figure 7), we observe that
Similarly, from the triangles $ AMP $, $ BNP $, $ CLP $ we get
Thus, by equating the right sides of the obtained equalities:
The left side of equation (1) is precis... | PisthecircumcenteroftriangleABC | Geometry | math-word-problem | Yes | Yes | olympiads | false |
VIII OM - III - Task 4
Prove that if $ a \geq 0 $ and $ b \geq 0 $, then
保留了源文本的换行和格式。请注意，最后一句“保留了源文本的换行和格式”是说明性的，不应包含在翻译文本中。正确的翻译结果如下：
VIII OM - III - Task 4
Prove that if $ a \geq 0 $ and $ b \geq 0 $, then
（此处为空白，保留了原文的格式） | The right side of inequality (1) can be written in the form $ (\sqrt{a} - \sqrt{b})^2 + \sqrt{2ab} $, from which it is clear that it is a non-negative number. Therefore, by squaring both sides of inequality (1), we obtain an equivalent inequality
Inequality (2) can be rewritten as
or
Inequality (3) is obviously true... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XXV OM - III - Task 4
Prove that for every natural number $ n $ and sequence of real numbers $ a_1, a_2, \ldots, a_n $, there exists a natural number $ k \leq n $ such that | We will first prove the
Lemma. If $ \varepsilon \geq 0 $ and the real numbers $ s_1, s_2, \ldots, s_n $ satisfy the conditions
Proof. If $ \varepsilon = 0 $, then from (1) it follows that $ s_1 = s_2 = \ldots = s_n = 0 $, and the thesis of the lemma obviously holds. Let us assume, therefore, that $ \varepsi... | proof | Combinatorics | proof | Incomplete | Incomplete | olympiads | false |
XLVII OM - II - Problem 4
Given is a sequence $ a_1, a_2, \ldots , a_{99} $ of numbers from the set $ \{0,1,2,3,4,5,6,7,8,9\} $. We assume that for $ i =1,2, \ldots, 98 $ the implications hold:
Prove that for some two different numbers $ k,l \in \{1,2, ... ,98\} $ the equalities $ a_k = a_l $ and $ a_{k+1} = a_{... | Let $ P_k $ denote the ordered pair $ (a_k,a_{k+1}) $. It is required to prove that in the sequence $ P_1,P_2,\ldots,P_{98} $, some pair repeats. According to the condition of the problem, the pairs $ (1,2) $ and $ (3,4) $ are not allowed in this sequence.
Suppose, for the sake of contradiction, that in the sequence $ ... | proof | Combinatorics | proof | Incomplete | Yes | olympiads | false |
XLVIII OM - I - Problem 2
Point $ P $ lies inside the parallelogram $ ABCD $, and the equality $ |\measuredangle ABP| = |\measuredangle ADP| $ holds. Prove that $ |\measuredangle PAB| = |\measuredangle PCB| $. | We translate the triangle $ ADP $ in parallel such that the image of side $ AD $ is segment $ BC $. Let the image of vertex $ P $ be denoted by $ Q $. Therefore, we have the equality $ |\measuredangle ADP| = |\measuredangle BCQ| $ (Figure 1). Segments $ AB $ and $ PQ $ are parallel, so $ |\measuredangle ABP| = |\measur... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XI OM - I - Problem 7
A triangle $ABC$ is inscribed in a given circle. On the sides $AC$ and $BC$, segments $AM$ and $BN$ of a given length $d$ are measured. Find the geometric locus of the midpoint $S$ of segment $MN$, when points $A$ and $B$ are fixed, and point $C$ moves along the given circle. | We draw from point $ S $ parallel to the lines $ AC $ and $ BC $ and from points $ A $ and $ B $ parallel to the line $ MN $; we obtain parallelograms $ AMSP $ and $ BNSR $ (Fig. 5).
If triangle $ ABC $ is not isosceles, for example if (as in Fig. 5) $ AC > BC $, then $ MC > NC $, so $ \measuredangle CMN < \measuredang... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
L OM - III - Task 3
Prove that there exist natural numbers $ n_1 < n_2 < \ldots < n_{50} $, such that
where $ S(n) $ is the sum of the digits of the number $ n $. | Let's introduce the following notations:
Then $ a_1(k)+ S (a_1(k)) = a_2(k)+ S (a_2(k)) = 10^{10^k+k+1} +1 $. We define the numbers $ k_0,k_1,k_2,\ldots $ by the formulas:
For any sequence $ \varepsilon=(\varepsilon_0,\varepsilon_1,\ldots,\varepsilon_5) $, where $ \varepsilon_0,\varepsilon_1,\ldots,\vareps... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXI OM - I - Problem 3
On the sides $ \overline{BC} $, $ \overline{CA} $, $ \overline{AB} $ of triangle $ ABC $, points $ K, L, M $ are chosen in such a way that the segments $ \overline{AK} $, $ \overline{BL} $, $ \overline{CM} $ intersect at a point $ S $ lying inside the triangle and $ \frac{AS}{SK}=\frac{BS}{SL}=\... | From the given ratio equality, it follows that triangles $ KLM $ and $ ABC $ (Fig. 3) are similar with respect to point $ S $, hence the lines $ KL $, $ LM $, $ MK $ are respectively parallel to the lines $ AB $, $ BC $, $ CA $. Applying Thales' theorem successively to angles $ \measuredangle A $, $ \measuredangle B $,... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXVI OM - I - Zadanie 9
W urnie jest 1985 kartek z napisanymi liczbami 1,2,3,..., 1985, każda lczba na innej kartce. Losujemy bez zwracania 100 kartek. Znaleźć wartość oczekiwaną sumy liczb napisanych na wylosowanych kartkach.
|
Losowanie $ 100 $ kartek z urny zawierającej $ 1985 $ kartek można interpretować jako wybieranie $ 100 $-elementowego podzbioru zbioru $ 1985 $-elementowego. Zamiast danych liczb $ 1985 $ i $ 100 $ weźmy dowolne liczby naturalne $ n $ i $ k $, $ n \geq k $. Dla dowolnego $ k $-elementowego zbioru $ X $ będącego podzbi... | 99300 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XX OM - I - Problem 12
Prove that if a figure in space has $ n $ axes of symmetry, then the number $ n $ is odd. | The proof of the theorem will be based on two lemmas.
a) If the axes $ OX $ and $ OY $ of a rectangular coordinate system are axes of symmetry of a figure $ F $, then the axis $ OZ $ is also an axis of symmetry of this figure (Fig. 7).
Let $ A = (x, y, z) $ be a point of the figure $ F $. The coordinates of the point $... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
LII OM - I - Task 4
Determine whether 65 balls with a diameter of 1 can fit into a cubic box with an edge of 4. | Answer: It is possible.
The way to place the balls is as follows.
At the bottom of the box, we place a layer consisting of 16 balls. Then we place a layer consisting of 9 balls, each of which is tangent to four balls of the first layer (Fig. 1 and 2). The third layer consists of 16 balls that are tangent to the balls o... | 66 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXX OM - II - Task 2
Prove that if $ a, b, c $ are non-negative numbers, then
| We have the identity
Therefore, to solve the problem, it suffices to prove that for any non-negative numbers $a$, $b$, $c$, the inequality
holds.
We have
Therefore, at most one of the numbers $a + b - c$, $a - b + c$, $-a + b + c$ is negative. If exactly one of them is negative, then inequality (1) holds, since the... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XII OM - II - Task 2
Prove that all altitudes of a tetrahedron intersect at one point if and only if the sums of the squares of opposite edges are equal. | The solution to the problem will help us observe that the segment connecting the midpoints of two edges of a tetrahedron belonging to the same face is equal to half of the third edge of that face (and is parallel to it). Let's denote the midpoints of the edges of the tetrahedron $ABC$ as shown in Fig. 13 with the lette... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVII OM - I - Zadanie 12
Ciąg $ (x_n) $ określony jest wzorami $ x_0 = 25 $, $ x_n = x_{n-1} + \frac{1}{x_{n-1}} $ ($ n = 1, 2, \ldots $) Dowieść, że $ x_n > 1975 $ dla $ n > 1950000 $.
|
Udowodnimy ogólniejsze
Twierdzenie. Jeżeli $ \displaystyle x_0 > \frac{1}{2} $ i $ x_{n} = x_{n-1} + \frac{1}{x_{n-1}} $ dla $ n = 1, 2, \ldots $, to dla każdej liczby naturalnej $ N $ zachodzi nierówność
Dowód. Wyrazy ciągu $ (x_{n}) $ są oczywiście liczbami dodatnimi. Wobec tego $ x_n = x_{n-1} + \frac{... | proof | Algebra | proof | Yes | Yes | olympiads | false |
II OM - III - Task 2
What digits should be placed instead of zeros in the third and fifth positions in the number $ 3000003 $ to obtain a number divisible by $ 13 $? | Let the sought digits be denoted by $ x $ and $ y $, and write the number $ 30x0y03 $ in the form
We need to find such integers $ x $ and $ y $ that the number $ N $ is divisible by $ 13 $ and that the inequalities
are satisfied.
The numbers $ 10^6 $, $ 10^4 $, $ 10^2 $ give remainders $ 1 $, $ 3 $, $ 9 $ respective... | 3080103,3040203,3020303,3090503,3060603,3030703,3000803 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
III OM - I - Task 5
Prove that the polynomial
is divisible by the polynomial $ x^4 + x^3 + x^2 + x + 1 $. | Let's multiply each of the given polynomials
and
by $ x - 1 $; we obtain the polynomials
and
To prove that the polynomial $ f(x) $ is divisible by the polynomial $ g(x) $, it suffices to show that the polynomial $ F(x) $ is divisible by the polynomial $ G(x) $, i.e., by $ x^5 - 1 $.
Indeed,
Therefore,
... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLIII OM - III - Problem 6
Prove that for every natural number $ k $, the number $ (k!)^{k^2+k+1} $ is a divisor of the number $ (k^3)! $. | For every pair of natural numbers $ n,l \geq 1 $, the following equality holds:
Let us fix natural numbers $ n,m \geq 1 $. Substitute $ l = 1,2,\ldots,m $ into (1) and multiply the resulting equations side by side:
We transform the left side of equation (2):
and after reducing the repeated factors in the numerators ... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LX OM - I - Task 2
Given an integer $ n \geq 2 $. Let $ r_1, r_2, r_3, \cdots, r_{n-1} $ be the respective
remainders of the division of numbers
by $ n $. Find all such values of $ n $ for which the sequence $ (r_1,r_2,\cdots,r_{n-1}) $ is a permutation
of the sequence $ (1, 2, \cdots,n-1) $. | Answer: $ n =2^k $ for $ k =1,2,3,\cdots $.
First, we will show that powers of two satisfy the conditions of the problem.
To this end, it suffices to prove that if $ n =2^k $ for some integer $ k \geq 1 $, then the remainders
$ r_1, r_2, r_3, \cdots, r_n-1 $ are pairwise distinct and none of them is equal to zero. If a... | n=2^kfork=1,2,3,\cdots | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
LVII OM - III - Problem 3
Given a convex hexagon $ABCDEF$, in which $AC = DF$, $CE = FB$ and $EA = BD$. Prove that the lines connecting the midpoints of opposite sides of this hexagon intersect at one point. | Let $P, Q, R$ be the midpoints of the diagonals $AD, BE, CF$, respectively.
First, assume that two of the points $P, Q, R$ coincide; let, for example, $P = Q$ (Fig. 1).
Then the quadrilateral $ABDE$ is a parallelogram. Moreover, triangle $ACE$ is congruent to triangle $DFB$, which implies that
$\measuredangle EAC = \me... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LV OM - I - Task 3
Let $ \mathbb{Q} $ denote the set of all rational numbers. Determine all functions $ f: \mathbb{Q} \to \mathbb{Q} $ satisfying the condition
for every pair of rational numbers $ x $, $ y $. | Let $ n $ be a positive integer. We will prove by induction that for any rational number $ y $ and any positive integer $ k $, the following equality holds:
For $ k = 1 $, the equality (2) is directly obtained from the relation (1) by substituting $ x = 1/n $. Now assume that the equality (2) holds for the number $ k ... | f(x)=ax | Algebra | proof | Incomplete | Yes | olympiads | false |
XVII OM - III - Problem 6
In the plane, there are 6 points. Prove that the ratio of the longest among the segments connecting these points in pairs to the shortest of these segments is not less than $ \sqrt{3} $. | We will adopt the notation: $ \{A_i\} $ as the set of data points $ A_1, A_2, \ldots, A_6 $, $ \alpha $ - the maximum distance $ A_iA_k $, $ \delta $ - the minimum distance $ A_iA_k $ ($ i, k=1, 2, \ldots, 6 $).
We need to prove that $ d \geq \delta \sqrt{3} $.
The truth of the theorem can be easily verified in the fol... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XII OM - I - Problem 1
$k$ parallel lines were drawn and intersected by $n$ parallel lines. How many parallelograms were formed? | Every parallelogram defines (by extending its sides) one pair $ (a, b) $ among the given $ k $ parallel lines and one pair $ (c, d) $ among the $ n $ parallel lines; conversely, any two such pairs $ (a, b) $ and $ (c, d) $ of parallel lines define one parallelogram of the figure. Therefore, the figure has as many paral... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | |
XLVI OM - I - Problem 9
Let $ a $ and $ b $ be real numbers whose sum is equal to 1. Prove that if $ a^3 $ and $ b^3 $ are rational numbers, then $ a $ and $ b $ are also rational numbers. | We raise the equality $ a + b = 1 $ to the second and third powers on both sides and obtain the relations: $ a^2 +2ab + b^2 = 1 $, that is,
and $ a^3 + 3a^2b + 3ab^2 + b^3 = 1 $, that is,
If, therefore, the numbers $ a^3 $ and $ b^3 $ are rational, then from equation (2) it follows that the product $ ab = (... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XIV OM - I - Problem 12
In a circle with center $ O $ and radius $ r $, a regular pentagon $ A_1A_2A_3A_4A_5 $ is inscribed, and on the smaller arc with endpoints $ A_1 $, $ A_5 $, a point $ M $ is chosen. Prove that | The equality (1) we need to prove is a relationship between the lengths of certain chords of a circle. Another relationship of the same kind is the well-known Ptolemy's theorem: The product of the diagonals of a cyclic quadrilateral equals the sum of the products of its opposite sides1). It suggests using this theorem ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LX OM - II - Task 4
Segment $ AB $ is the diameter of the circle $ o $ circumscribed around the convex quadrilateral $ ABCD $, whose diagonals
intersect at point $ E $. The tangent lines to the circle $ o $ at points $ C $ and $ D $ intersect at point
$ P $.
Prove that $ PC = PE $. | Let $ Q $ be the point of intersection of the lines $ AD $ and $ BC $.
First, note that by the conditions of the problem, the lines $ AC $ and $ BD $ are altitudes of triangle $ ABQ $.
Thus, the line $ QE $ is its third altitude and therefore intersects the line $ AB $ at some point $ F $
at a right angle.
In triangl... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXII OM - I - Problem 12
Prove that every convex polyhedron has a triangular face or a trihedral angle. | Assume that there exists a convex polyhedron $W$ without any triangular faces or trihedral angles. Let $w$ be the number of vertices, $k$ the number of edges, and $s$ the number of faces of this polyhedron, and let $\varphi$ be the sum of the measures of all dihedral angles at the vertices of the polyhedron $W$. Since ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
V OM - III - Problem 6
Inside a hoop of radius $2r$, a disk of radius $r$ rolls without slipping along the hoop. What line does a point chosen arbitrarily on the edge of the disk trace? | When a disk rolls along the circumference, the points on the edge of the disk become points of contact with the circumference one after another. The condition of rolling without slipping means that the length of the arc $PQ$ between two points $P$ and $Q$ on the edge of the disk is equal to the length of the arc on the... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXI OM - III - Task 6
Find the smallest real number $ A $ such that for every quadratic trinomial $ f(x) $ satisfying the condition
the inequality $ f.
holds. | Let the quadratic trinomial $ f(x) = ax^2 + bx + c $ satisfy condition (1). Then, in particular, $ |f(0)| \leq 1 $, $ \left| f \left( \frac{1}{2} \right) \right| \leq 1 $, and $ |f(1)| \leq 1 $.
Since
and
thus
Therefore, $ A \leq 8 $.
On the other hand, the quadratic trinomial $ f(x) = -8x^2 + 8x - 1 ... | 8 | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false |
IX OM - I - Problem 10
Prove that for every natural $ n $ | \spos{1} Let us denote the left side of the above equality by the letter $ S $; since its last term $ \sin \frac{2n\pi}{n} $ equals zero, therefore
By changing the order of terms in the sum $ S $, we have
Adding both equalities side by side, we obtain
Each of the expressions contained in the parenthes... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XLVIII OM - II - Problem 2
Point $ P $ lies inside triangle $ ABC $ and satisfies the conditions:
Prove that: | Let $ K $ be the intersection point of lines $ BP $ and $ AC $, and let $ L $ be the intersection point of lines $ CP $ and $ AB $. According to the conditions of the problem, we have:
which proves that $ KP = KC $. Similarly, $ LP = LB $. Triangles $ ABK $ and $ ACL $ are similar, so we get
which is the eq... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
L OM - I - Task 5
Find all pairs of positive integers $ x $, $ y $ satisfying the equation $ y^x = x^{50} $. | We write the given equation in the form $ y = x^{50/x} $. Since for every $ x $ being a divisor of $ 50 $, the number on the right side is an integer, we obtain solutions of the equation for $ x \in \{1,2,5,10,25,50\} $. Other solutions of this equation will only be obtained when $ x \geq 2 $ and for some $ k \geq 2 $,... | 8 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXXIII OM - II - Problem 6
Given is a finite set $ B $ of points in space, such that any two distances between points of this set are different. Each point of the set $ B $ is connected by a segment to the nearest point of the set $ B $. In this way, we obtain a set of segments, one of which (chosen arbitrarily) is pa... | Suppose the segment $ A_1A_2 $ is red, and the rest are green. If there existed a broken line composed of green segments connecting points $ A_1 $ and $ A_2 $, then there would be points $ A_3, \ldots, A_n $ being the successive endpoints of the segments forming this broken line. The notation $ A_i \to A_j $ is read as... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXIV OM - II - Task 1
Prove that if the positive numbers $ x, y, z $ satisfy the inequality
then they are the lengths of the sides of some triangle. | \spos{1} As is known, positive numbers $ x $, $ y $, $ z $ are the lengths of the sides of a certain triangle if and only if each of them is less than the sum of the other two, i.e., if
Suppose that this condition is not satisfied, i.e., that one of the numbers $ x, y, z $ is not less than the sum of the remainin... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
VIII OM - I - Problem 8
In the rectangular prism $ ABCDA_1B_1C_1D_1 $, the lengths of the edges are given as $ AA_1 = a $, $ AB = b $, $ AD = c $. On the face $ A_1B_1C_1D_1 $, a point $ M $ is chosen at a distance $ p $ from the side $ A_1B_1 $, and at a distance $ q $ from the side $ A_1D_1 $, and a parallelepiped i... | Accepting the notation indicated in figure $5$, we calculate the area of the parallelogram $AMQD$ which is a face of the parallelepiped $ABCDMNPQ$.
The parallelogram $AMQD$ and the rectangle $AKLD$ have a common base $AD$ and height $AK$, thus
But $AK = \sqrt{AA_1^2 + A_1K^2} = \sqrt{a^2 + q^2},\ KL = AD = c$; he... | \sqrt{^2+} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LI OM - II - Task 3
On an $ n \times n $ chessboard, $ n^2 $ different integers are placed, one on each square. In each column, the square with the largest number is painted red. A set of $ n $ squares on the chessboard is called admissible if no two of these squares are in the same row or the same column. Among all a... | We will conduct an indirect proof.
Assume that in the chosen admissible set (denoted by $ A $) there is no red field.
We define a sequence of fields $ D_1, E_1, D_2, E_2, \ldots $ of the given chessboard according to the rule described below. (For $ n = 8 $, the process of selecting fields $ D_1, E_1, D_2, E_2, \ldots ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXIII OM - III - Problem 1
Polynomials $ u_1(x) = a_ix + b_i $ ($ a_i, b_i $ - real numbers; $ i = 1, 2, 3 $) satisfy for some natural $ n > 2 $ the equation
Udowodnić, że istnieją takie liczby rzeczywiste $ A, B, c_1, c_2, c_3 $, że $ u_i(x)=c_i(Ax+B) $ for $ i = 1, 2, 3 $.
Prove that there exist real numbers $ A, ... | If $ a_1 = a_2 = 0 $, then the polynomials $ u_1 $ and $ u_2 $ are constant. Therefore, the polynomial $ u_3 $ is also constant, i.e., $ a_3 = 0 $. In this case, it suffices to take $ c_i = b_i $ for $ i = 1, 2, 3 $ and $ A = 0 $ and $ B = 1 $.
Let then at least one of the numbers $ a_1, a_2 $ be different from zero, f... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXXVI OM - III - Problem 5
Let $ P $ be a polynomial in two variables such that for every real number $ t $ the equality $ P(\cos t, \sin t) = 0 $ holds. Prove that there exists a polynomial $ Q $ such that the identity holds
保留了源文本的换行和格式，但最后一句“保留了源文本的换行和格式”是中文，应该翻译成英文如下：
Preserving the line breaks and f... | Let's arrange the polynomial $ P(x,y) $ in decreasing powers of the variable $ x $. $ P(x,y) = P_n(y) \cdot x^n + P_{n-1}(y) \cdot x^{n-1} + \ldots + P_1(y) \cdot x + P_0(y) $, where $ P_n, P_{n-1}, \ldots, P_1, P_0 $ are polynomials in the variable $ y $. Treating $ y $ as a fixed value, we can compute the quotient an... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXXIV OM - I - Problem 6
Prove that for any polynomial $ P $, the polynomial $ P\circ P \circ P \circ \ldots \circ P(x)-x $ is divisible by $ P(x)-x $. | Putting $ P(x)= a_nx^n + a_{n-1}x^{n-1} + \ldots +a_1x+a_0 $ we get
Since each term of the last sum can be factored:
thus
Substituting in the last relation $ P(x) $ for $ x $ we get $ P \circ P(x)-P(x) | P \circ P \circ P(x) - P \circ P(x) $, from which by the transitivity of the divisibility relation it foll... | proof | Algebra | proof | Yes | Yes | olympiads | false |
VI OM - I - Problem 12
In what part of an equilateral triangle should point $ P $ lie so that from segments equal to the distances of this point from the sides of the triangle, a triangle can be constructed? | Let $ k_1 $, $ k_2 $, $ k_3 $ denote the distances from a point $ P $ inside an equilateral triangle $ ABC $ to the sides $ BC $, $ CA $, $ AB $ of the triangle, respectively. We can form a triangle with segments of lengths $ k_1 $, $ k_2 $, $ k_3 $ if and only if they satisfy the following inequalities:
Draw a line t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XIX OM - I - Problem 8
Given is a tetrahedron $ABCD$ where $AB = BC = CD = DA = 1$. Prove that the volume $V$ of the tetrahedron $ABCD$ satisfies the inequality
(Note: The original text did not provide the specific inequality to be proven. If you have the complete problem statement, please provide it for... | From the equality of the sides of quadrilateral $ABCD$, it follows that the plane passing through line $BD$ and through the midpoint $M$ of segment $AC$ is a plane of symmetry of the tetrahedron $ABCD$. Indeed, the median $DM$ of the isosceles triangle $ADC$ is perpendicular to its base $AC$, just as $BM$ is perpendicu... | V\leq\frac{\sqrt{2}}{12} | Geometry | proof | Incomplete | Yes | olympiads | false |
LII OM - I - Problem 7
Given is an isosceles right triangle $ABC$ with a right angle at vertex $A$. Points $D$ and $E$ lie on the hypotenuse $BC$, such that $ \measuredangle DAE = 45^\circ $. The circumcircle of triangle $ADE$ intersects sides $AB$ and $AC$ at points $P$ and $Q$, respectively. Prove that $BP + CQ = PQ... | Points $A$, $P$, $E$, $D$ lie on the same circle, so $\measuredangle ADE = \measuredangle BPE$ (Fig. 1). Moreover, $\measuredangle DAE = 45^\circ = \measuredangle PBE$. From these equalities, it follows that triangles $ADE$ and $BPE$ are similar, and in particular
We similarly prove that
Let $A'$, $P'$, $Q'$ be... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XIX OM - I - Zadanie 1
Rozwiązać układ równań
|
Układ równań (1) jest równoważny układowi
Drugie równanie układu (2) jest spełnione wtedy i tylko, wtedy, gdy liczby $ x $ i $ y $ są tego samego znaku, lub gdy któraś z nich jest zerem, tzn. gdy $ xy \geq 0 $. Układ (2) jest więc równoważny układowi
Układ (3) jest równoważny alternatywie
lub
U... | (x,y)={\begin{pmatrix}(x,1-x)& | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXXVII OM - II - Problem 2
In a chess tournament, 66 players participate, each playing one game against every other player, and the matches take place in four cities. Prove that there exists a trio of players who play all their games against each other in the same city. | Let's choose one of the players, let's call him $Z_1$. He has to play $65$ games, so he plays at least $17$ games in one city. Let's denote this city by $M_1$. Consider the opponents of $Z_1$ in the matches played in $M_1$. There are at least $17$ of them. If there is a pair among them who play a game against each othe... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
LIV OM - II - Task 6
Each pair of non-negative integers $ (x,y) $ is assigned a number $ f (x,y) $ according to the conditions:
Let $ n $ be a fixed non-negative integer and let $ a $, $ b $ be such non-negative integers that $ f (a,b) = n $. Determine how many numbers $ x $ satisfy the equation
Note:... | We will prove the following lemma giving an alternative description of the function $ f $.
Lemma
Let $ k $ be a positive integer and let $ x $, $ y $ be non-negative integers less than $ 2^k $. Let
be the binary representations of $ x $ and $ y $, respectively, with leading zeros allowed. Then
(Thus, $ f (x... | 2^n | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XI OM - I - Problem 8
In the plane, there are three rays $ OX $, $ OY $, $ OZ $, dividing the plane into three convex angles, and points $ A $, $ B $, $ C $ lying inside the angles $ YOZ $, $ ZOX $, $ XOY $, respectively. Construct a triangle whose vertices lie on $ OX $, $ OY $, $ OZ $, and whose sides pass through $... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | |
XXIII OM - III - Problem 6
Prove that the sum of the digits of the number $ 1972^n $ tends to infinity as $ n $ tends to infinity. | We will prove in general that if $a$ is an even natural number not divisible by $5$ and $s_n$ denotes the sum of the digits of the number $a^n$ for $n = 1, 2, \ldots$, then the sequence $s_n$ tends to infinity.
Let $a_r, a_{r-1}, \ldots, a_2, a_1$, where $a \ne 0$, be the consecutive digits of the number $a^n$, i.e.,
... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XVI OM - I - Problem 3
Through each edge of a trihedral angle, a plane containing the bisector of the opposite planar angle has been drawn. Prove that the three planes intersect along a single line. | Consider a trihedral angle with vertex $O$. We measure three equal segments $OA = OB = OC$ on its edges. Consider the plane $\alpha$ passing through the edge $OA$ and the bisector of the angle $BOC$. Since the triangle $BOC$ is isosceles, the bisector of the angle $BOC$ passes through the midpoint $M$ of the side $BC$.... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVIII OM - II - Problem 3
Two circles are internally tangent at point $ A $. A chord $ BC $ of the larger circle is tangent to the smaller circle at point $ D $. Prove that $ AD $ is the angle bisector of $ \angle BAC $. | The point of tangency $ A $ of the given circles is their center of homothety. In this homothety, the tangent $ BC $ of the smaller circle corresponds to the tangent to the larger circle at point $ E $, which corresponds to point $ D $ (Fig. 6). As homothetic lines, these tangents are parallel. Therefore, the arcs $ BE... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLVI OM - II - Problem 2
In a convex hexagon $ABCDEF$, the following equalities hold: $|AB| = |BC|$, $|CD| = |DE|$, $|EF| = |FA|$. Prove that the lines containing the altitudes of triangles $BCD$, $DEF$, and $FAB$ drawn from vertices $C$, $E$, and $A$, respectively, intersect at a single point. | Let's adopt the following notations:
$ k_1 $ - the circle with center $ D $ and radius $ |DC| = |DE| $;
$ k_2 $ - the circle with center $ F $ and radius $ |FE| = |FA| $;
$ k_3 $ - the circle with center $ B $ and radius $ |BA| = |BC| $.
Circles $ k_2 $ and $ k_3 $ intersect at point $ A $; let's denote the second ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XIII OM - I - Problem 2
How many digits do all natural numbers with at most $ m $ digits have in total? | The number of one-digit natural numbers is $ c_1 = 9 $; the number of two-digit numbers is $ c_2= 10^2 -10= 10 \cdot 9 $, they have $ 2 \cdot 10 \cdot 9 $ digits; the number of three-digit numbers is $ c_3=10^3-10^2= 10^2 \cdot 9 $, the number of their digits is $ 3 \cdot 10^2 \cdot 9 $, etc. Finally, the number of $ m... | 9\cdot\frac{10^-1}{9}\cdot\frac{(+1)}{2} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.