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XXIV OM - II - Problem 5
Prove that if in a tetrahedron $ABCD$ we have $AB = CD$, $AC = BD$, $AD = BC$, then all faces of the tetrahedron are acute triangles. | We will first prove the
Lemma. Let $ \alpha, \beta, \gamma $ be numbers in the interval $ (0; \pi) $. There exists a trihedral angle whose dihedral angles have measures equal to $ \alpha, \beta, \gamma $ if and only if each of the numbers $ \alpha, \beta, \gamma $ is less than the sum of the other two.
Proof. Without l... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVII OM - I - Problem 1
Present the polynomial $ x^5 + x + 1 $ as a product of two polynomials of lower degree with integer coefficients. | Suppose the desired factorization exists. None of the factors of this factorization can be of the first degree. For if for every $x$ the equality held
where the numbers $m, n, a, b, \ldots$ are integers, then it would follow that $ma = 1$, so $m = a = \pm 1$, hence the polynomial $x^5 + x + 1$ would have an integer... | x^5+x+1=(x^2+x+1)(x^3-x^2+1) | Algebra | math-word-problem | Yes | Yes | olympiads | false |
XL OM - I - Problem 10
Positive numbers $ \alpha $, $ \beta $, $ \gamma $ satisfy the inequalities
Prove that a triangle can be constructed from segments of lengths $ \sin \alpha $, $ \sin \beta $, $ \sin \gamma $ and that the area of this triangle is not greater than
保留了源文本的换行和格式，直接输出了翻译结果。希望这符合您的要求！如... | We will show that there exists a trihedral angle $OABC$ in space such that
We can assume that $ \gamma \geq \alpha $, $ \gamma \geq \alpha $. Of course, $ \gamma < \pi/2 $ (if $ \gamma \geq \pi/2 $, then we would have $ \alpha + \beta < \pi - \gamma \leq \pi/2 \leq \gamma $, contrary to the assumption). Therefore, $ 2... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
LVI OM - III - Task 1
Determine all triples $ (x,y,n) $ of positive integers satisfying the equation
Note: The equation is missing in the provided text. If you have the equation, please provide it for a complete translation. | For $ n = 1 $, the given equation takes the form $ (x + 1)(y - 1) = -1 $, so there are no solutions in positive integers. Let us assume, therefore, that $ n \geq 2 $.
Let $ z = x - y $. The given equation takes the form $ x^{2}-xz =z^{n} $. After multiplying by 4 and a simple transformation, we obtain
Therefore, the f... | proof | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XVI OM - I - Problem 8
On the lateral edges of a triangular prism with base $ABC$, segments $AA_1 = p_1$, $BB_1 = p_2$, $CC_1 = p_3$ were measured. A line parallel to the lateral edges intersects the base $ABC$ at point $D$ and the plane $A_1B_1C_1$ at point $D_1$. Prove that
where $S_1, S_2, S_3$ denote the areas ... | The length $ DD_1 = x $ can be calculated using the fact that the volume $ V $ of the truncated triangular prism $ ABCA_1B_1C_1 $ is the sum of the volumes $ V_1 $, $ V_2 $, $ V_3 $ of the truncated prisms $ DBCD_1B_1C_1 $, $ DCAD_1C_1A_1 $, and $ DABD_1A_1B_1 $ (Fig. 3):
It is known that the volume of a truncated tri... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
XLIV OM - I - Problem 8
For a fixed natural number $ n \geq 2 $, determine the maximum value of the sum of natural numbers $ k_1, k_2, \ldots , k_n $ satisfying the condition | Consider any sequence $ (k_1,\ldots,k_n) $ of natural numbers satisfying the given condition; we can assume that $ k_1 \geq \ldots \geq k_n $. Suppose that $ k_1 - k_n \geq 2 $. Assume that in the sequence $ (k_1, \ldots ,k_n) $ there are $ \alpha $ numbers equal to $ k_1 $ and $ n - \beta + 1 $ numbers equal to $ k_n ... | [13n/7] | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XLVI OM - III - Zadanie 3
Dana jest liczba pierwsza $ p \geq 3 $. Określamy ciąg $ (a_n) $ wzorami
Wyznaczyć resztę z dzielenia liczby $ a_{p^3} $ przez $ p $.
|
Podany wzór rekurencyjny
możemy zastosować do każdego ze składników jego prawej strony (pod warunkiem, że numer $ n - p $ jest nie mniejszy od $ p $, czyli że $ n \geq 2p $); dostajemy zależność
Stosując ponownie wzór (1) do każdego ze składników ostatniej sumy otrzymujemy równość
(jeśli tylko $ n \g... | p-1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXVIII - II - Task 3
In a hat, there are 7 slips of paper. On the $ n $-th slip, the number $ 2^n-1 $ is written ($ n = 1, 2, \ldots, 7 $). We draw slips randomly until the sum exceeds 124. What is the most likely value of this sum? | The sum of the numbers $2^0, 2^1, \ldots, 2^6$ is $127$. The sum of any five of these numbers does not exceed $2^2 + 2^3 + 2^4 + 2^5 + 2^6 = 124$. Therefore, we must draw at least six slips from the hat.
Each of the events where we draw six slips from the hat, and the seventh slip with the number $2^{n-1}$ ($n = 1, 2, ... | 127 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXVII OM - III - Problem 6
A function $ f $ increasing on the set of natural numbers satisfies the following condition
for any pair of natural numbers $ (k, l) $.
Prove that there exists a real number $ p > 1 $ such that
(Note: The last part of the text appears to be incomplete or missing in the origina... | From the formula $ f(k \cdot l)=f(k) + f(l) $, it easily follows by induction that
for any natural numbers $ m $ and $ s $. Hence, in particular, we obtain $ f(1) =f(1^2) = 2f(1) $, which means $ f(1)=0 $. Since the function $ f $ is increasing, it follows that $ f(2) > f(1) = 0 $.
We define the number $ p $ by the fo... | f(n)=\log_pn | Algebra | proof | Incomplete | Yes | olympiads | false |
XLIV OM - III - Problem 5
Determine all functions $ f: \mathbb{R} \to \mathbb{R} $ that satisfy the following conditions: | Assume that the function $ f: \mathbb{R} \to \mathrm{R} $ satisfies the given equations. Let $ t $ be a real number different from $ 0 $ and $ -1 $.
In the first equation, we substitute $ x = 1/(t + 1) $; we get the relation
In the second equation, we substitute $ x = 1/t $, $ x = -1/(t + 1) $, $ x = t $; we obta... | f()= | Algebra | proof | Yes | Yes | olympiads | false |
I OM - B - Task 12
How many sides does a polygon have if each of its angles is $ k $ times larger than the adjacent angle? What values can $ k $ take? | If each angle of a polygon is $ k $ times larger than the adjacent angle,
then the sum of all $ n $ angles of the polygon, which - as
we know - equals $ (n - 2) \cdot 180^{\circ} $, is also $ k $ times larger
than the sum of the corresponding adjacent angles (i.e., the exterior angles),
which means it is $ k $ times la... | \frac{1}{2},1,\frac{3}{2},2,\dots | Geometry | math-word-problem | Yes | Yes | olympiads | false |
IV OM - III - Task 3
Through each vertex of a tetrahedron of a given volume $ V $, a plane parallel to the opposite face of the tetrahedron has been drawn. Calculate the volume of the tetrahedron formed by these planes. | To make it easier to find a solution, let's first consider an analogous problem on the plane:
Let's draw through each vertex of the triangle $ABC$ a line parallel to the opposite side of the triangle. These lines form a triangle $A_1B_1C_1$ similar to triangle $ABC$ (Fig. 46). The center of similarity is the point $S$,... | 27V | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLII OM - I - Problem 8
Determine the largest natural number $ n $ for which there exist in space $ n+1 $ polyhedra $ W_0, W_1, \ldots, W_n $ with the following properties:
(1) $ W_0 $ is a convex polyhedron with a center of symmetry,
(2) each of the polyhedra $ W_i $ ($ i = 1,\ldots, n $) is obtained from $ W_0 $ by ... | Suppose that polyhedra $W_0, W_1, \ldots, W_n$ satisfy the given conditions. Polyhedron $W_1$ is the image of $W_0$ under a translation by a certain vector $\overrightarrow{\mathbf{v}}$ (condition (2)). Let $O_0$ be the center of symmetry of polyhedron $W_0$ (condition (1)); the point $O_1$, which is the image of $O_0$... | 26 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXV OM - III - Task 3
Given is a regular octahedron $ W $ with center $ O $. In the plane $ P $ passing through the point $ O $, circles $ K(O, r_1) $ and $ K (O, r_2) $ with center $ O $ and radii $ r_1 $ and $ r_2 $, respectively, are chosen. Prove that if
then
保留了源文本的换行和格式，直接输出了翻译结果。 | Consider the section of an octahedron by a plane $P$. Point $O$ is the center of symmetry of this section, which implies that the section is a polygon with an even number of sides, and that opposite sides are parallel and of equal length. Depending on whether the plane $P$ passes through a vertex of the octahedron or n... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XLVII OM - I - Problem 12
Determine whether there exist two congruent cubes with a common center such that each face of the first cube has a point in common with each face of the second cube. | We will prove that there do not exist two cubes $\mathcal{C}$ and $\mathcal{C}$ for which the following condition is satisfied:
$\quad (*) \quad$ each face of the cube $\mathcal{C}$ has points in common with each face of $\mathcal{C}$.
The additional assumptions given in the problem (congruence, common center) are not ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVI OM - II - Problem 4
Prove that if for natural numbers $ a, b $ the number $ \sqrt[3]{a} + \sqrt[3]{b} $ is rational, then $ a, b $ are cubes of natural numbers. | Let's denote the numbers $ \sqrt[3]{a} $ and $ \sqrt[3]{b} $ by $ x $ and $ y $, respectively, and their sum by $ s $. We need to prove that $ x $ and $ y $ are natural numbers. By assumption, the number $ s = x + y $ is rational. Since
\[
xy \text{ is a rational number.}
\]
Further, we have
\[
\text{from the ration... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXVII OM - I - Problem 2
Prove that if $ x, y, z $ are non-negative numbers such that $ x + y + z = 1 $, then | If $ a $, $ b $, $ c $ are any positive numbers, then by the inequality between the arithmetic mean and the geometric mean
Taking $ a = x+1 $, $ b=y+1 $, $ c = z+1 $ we get
Hence
| proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
XIX OM - II - Problem 1
Prove that a polynomial in the variable $ x $ with integer coefficients, whose absolute value for three different integer values of $ x $ equals 1, does not have integer roots. | Suppose that $ f(x) $ is a polynomial in the variable $ x $ with integer coefficients and that
where $ a $, $ b $, $ c $ are three distinct integers.
Assume that $ x_0 $ is an integer root of the polynomial $ f(x) $, so for every $ x $
where $ \varphi (x) $ is a polynomial with integer coefficients.
From (1) and (2),... | proof | Algebra | proof | Yes | Yes | olympiads | false |
VIII OM - II - Task 6
Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e., an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other. | Suppose the convex quadrilateral $ABCD$ (Fig. 17) satisfies the conditions mentioned above.
1. From the assumption that there exists an inscribed circle in the quadrilateral, it follows that the sums of the opposite sides of the quadrilateral are equal, i.e.,
2. From the assumption that there exists a circle tangen... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXI OM - I - Problem 7
In parallelogram $ABCD$, point $A$ is connected to the midpoint $A$ of side $\overline{BC}$, point $B$ is connected to the midpoint $B$ of side $\overline{CD}$, point $C$ is connected to the midpoint $C$ of side $\overline{DA}$, and point $D$ is connected to the midpoint $D$ of side $\overline{A... | Consider triangles $ ADD and $ CBB (Fig. 5). They are congruent because $ AD, $ AD = CB $ and $ \measuredangle D.
Therefore, $ \measuredangle AD. It follows that lines $ DD and $ BB are parallel. Similarly, we prove that lines $ AA and $ CC are also parallel. Therefore, quadrilateral $ PQRS $ is a parallelogram. Consid... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
LIV OM - I - Task 5
A natural number $ n_1 $ is written in the decimal system using 333 digits, none of which are zero. For $ i = 1 , 2, 3, \ldots , 332 $, the number $ n_{i+1} $ is formed from the number $ n_i $ by moving the units digit to the beginning. Prove that either all the numbers $ n_1, n_2, n_3, \ldots , n_... | Let $ j_i $ $ (i = 1,2,3,\ldots ,333) $ denote the units digit of the number $ n_i $. Then for $ i =1,2,\ldots,332 $ we have
The number $ 10^{333} - 1 = (10^3 - 1) \cdot (10^{330} + 10^{327} + 10^{324} +\ldots + 10^3 + 1) $ is divisible by 333, and the numbers 10 and 333 are relatively prime. From the second equ... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
LIII OM - III - Problem 6
Given a natural number $ k $. We define the sequence $ (a_n) $ by the formulas
Prove that if $ m \ne n $, then the numbers $ a_m $ and $ a_n $ are coprime. | We will first prove by induction that for $ n \geq 1 $ the equality holds:
From the given recursive relation in the problem, we calculate
which proves the statement (1) for $ n = 1 $. The inductive step reduces to the sequence of equalities:
Now suppose that $ p $ is a common prime divisor of the numb... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXXVI OM - I - Problem 5
Given are infinite sequences of real numbers $ (x_j) $, $ (y_j) $. Let
Prove that if for some natural number $ m $ it holds that $ z_j = 0 $ for $ j < m $ and $ z_m = 1 $, then there exist natural numbers $ p, q $ such that $ x_py_q = m $. | If one of the sequences $ (x_j) $, $ (y_j) $ consisted entirely of zeros, then all terms of the sequence $ (z_n) $ would also be zeros, contradicting the fact that $ z_m = 1 $. Therefore, each of these sequences contains non-zero terms. Let $ p $ and $ q $ denote the smallest indices such that $ x_p \neq 0 $ and $ y_q ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXII OM - I - Problem 3
Through a point $ P $, belonging to the plane of triangle $ ABC $, three lines perpendicular to the lines $ BC $, $ AC $, and the line containing the median $ \overline{CE} $ have been drawn. Prove that they intersect the line containing the altitude $ CD $ at points $ K, L, M $, respectively, ... | We will first prove a few auxiliary facts.
Lemma. If the corresponding sides of two triangles are parallel, then the triangles are similar.
Proof. Let $ AB\parallel A', $ BC \parallel B', $ CA \parallel C'. $ Then the corresponding angles of triangles $ ABC $ and $ A'B'C' $ have sides that are parallel. Therefore, the ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
X OM - I - Problem 10
Prove that if a circle can be circumscribed around each of the quadrilaterals $ABCD$ and $CDEF$, and the lines $AB$, $CD$, $EF$ intersect at a single point $M$, then a circle can be circumscribed around the quadrilateral $ABEF$.
| We apply the theorem about secants of a circle passing through one point: 1° to lines $ AB $ and $ CD $, 2° to lines $ CD $ and $ EF $ (Fig. 16).
We obtain
From this last equality, it follows that points $ A $, $ B $, $ E $, $ F $ lie on a circle. Indeed, due to this equality, the proportion $ MA \colon ME ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLVIII OM - I - Problem 6
The polynomial $ P(x) $ of degree $ n $ satisfies the condition
Calculate $ P(0) $.
| Let $ P(x) = a_0 + a_1x + \dots + a_nx^n $. Consider the polynomial
Substituting the variable $ x $ with the numbers $ k = 1,2,4,\dots ,2^n $, we obtain, based on the condition of the problem, the equalities
Thus, these numbers are roots of the polynomial $ T(x) $. This is a polynomial of degree $ n+1 $, and it can b... | Algebra | math-word-problem | Incomplete | Yes | olympiads | false | |
XI OM - I - Task 3
Each side of a triangle with a given area $ S $ is divided into three equal parts, and the points of division are connected by segments, skipping one point to form two triangles. Calculate the area of the hexagon that is the common part of these triangles. | The solution to the problem can be easily read from Fig. 1. The area of the hexagon $123456$ equals $\frac{2}{3}S$, as this hexagon is obtained from the triangle $ABC$ by cutting off three equal corner triangles with an area of $\frac{1}{9}S$. The sides of triangles $135$ and $246$ bisect each other into equal segments... | \frac{2}{9}S | Geometry | math-word-problem | Yes | Yes | olympiads | false |
V OM - I - Task 11
Given are two intersecting lines $ a $ and $ b $. Find the geometric locus of point $ M $ having the property that the distance between the orthogonal projections of point $ M $ on lines $ a $ and $ b $ is constant, equal to a given segment $ d $. | Let $A$ and $B$ denote the projections of point $M$ onto lines $a$ and $b$ intersecting at point $O$, and let $\alpha$ denote the convex angle $AOB$ (Fig. 27). The circle with diameter $OM$ passes through points $A$ and $B$, since angles $OAM$ and $OBM$ are right angles; this circle is thus the circumcircle of triangle... | The\geometric\locus\is\the\circle\with\center\O\\radius\\frac{}{\sin\alpha} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LI OM - I - Problem 11
Given a positive integer $ n $ and a set $ M $, consisting of $ n^2 + 1 $ different positive integers and having the following property: among any $ n+1 $ numbers chosen from the set $ M $, there is a pair of numbers, one of which divides the other. Prove that in the set $ M $ there exist distin... | Let $ k \in M $. Denote by $ d(k) $ the largest natural number $ \ell $ for which there exist different numbers $ a_1 = k, a_2, a_3, \ldots, a_\ell $ satisfying the condition: for $ i = 1,2,\ldots,\ell-1 $, the number $ a_i $ is divisible by $ a_{i+1} $.
Suppose the thesis of the problem is not satisfied. This means th... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XX OM - I - Task 1
Prove that if the numbers $ a_1, a_2, \ldots, a_n $ and $ b_1, b_2, \ldots, b_n $ are positive, then | The left side $ L $ of inequality (1) is transformed as follows.
Since for any positive numbers $ x $, $ y $, the inequality holds
thus
which means
Note. The above solution can be presented in the following form. Both the left side $ L $ and the right side $ P $ of inequality (1) are equal to a ... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
L OM - I - Problem 12
All vertices of a cube with edge $ a $ lie on the surface of a regular tetrahedron with edge 1. Determine the possible values of $ a $. | From the conditions of the problem, it follows that a given cube $ABCDA$ lies inside a given tetrahedron $KLMN$. There are two possible cases:
- There exists a face of the tetrahedron $KLMN$ (for example, $KLM$) on which at least three vertices of the cube lie;
- On each face of the tetrahedron $KLMN$, exactly two ver... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
X OM - I - Task 8
The sides of a triangle are the medians of another triangle. Calculate the ratio of the areas of both triangles. Can a triangle be constructed from the medians of any triangle? | Let $ S $ be the centroid of triangle $ ABC $. Extend the median $ AD $ by a segment $ DM = SD $ (Fig. 15). We obtain triangle $ BSM $, in which $ BS = \frac{2}{3} BE $, $ SM = \frac{2}{3} AD $, and $ BM = SC = \frac{2}{3} CF $ (since quadrilateral $ SBMC $ is a parallelogram). Triangle $ BSM $ is thus similar to trian... | \frac{3}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXIII OM - I - Problem 5
In a certain workplace, each employee is a member of exactly one of 100 trade unions. The employees are to elect a director from among two candidates. Members of each union agree on whether to abstain from voting or which of the two candidates they will vote for. Prove that there exists a uni... | Let $ a_1, a_2, \ldots, a_{100} $ be the number of members of each of the respective unions. We can assume that one of the numbers $ a_1, \ldots, a_{100} $ (for example, $ a_j $) is odd, because otherwise we could divide all $ a_i $ by the highest power of two that divides them all. If now the sum $ a_1+ \ldots+ a_{100... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XLIX OM - I - Problem 3
Sequences $ (a_n) $, $ (b_n) $, $ (c_n) $ are defined by the conditions:
for $ n = 1,2,3,\ldots $. Prove that the sequence $ (c_n) $ is bounded. | We will prove by induction that
For $ n = 1 $, the inequalities $ 2^1 < a_1 \leq 2^2 $ are satisfied. Assuming that the inequalities (1) hold for some natural number $ n $, we obtain for $ n+1 $ similar estimates:
and - using the fact that $ a_n $ is an integer, so the left inequality (1) can be strengthened to the f... | 0\leqc_n<1 | Algebra | proof | Incomplete | Yes | olympiads | false |
XXIV OM - III - Task 2
Let $ p_n $ be the probability that a series of 100 consecutive heads will appear in $ n $ coin tosses. Prove that the sequence of numbers $ p_n $ is convergent and calculate its limit. | The number of elementary events is equal to the number of $n$-element sequences with two values: heads and tails, i.e., the number $2^n$. A favorable event is a sequence containing 100 consecutive heads. We estimate the number of unfavorable events from above, i.e., the number of sequences not containing 100 consecutiv... | 1 | Combinatorics | proof | Yes | Yes | olympiads | false |
IX OM - III - Task 4
Prove that if $ k $ is a natural number, then
where $ m $ is a natural number dependent on $ k $; determine $ m $.
| We state that for $ k = 1, 2, 3, \ldots $ the formula (1) holds, where $ m = 1, 3, 7, \ldots $, that is, $ m = 2 - 1, 4 - 1, 8 - 1, \ldots $. We apply complete induction, denoting the left side of equation (1) by $ P_k $; suppose that for some natural $ k $
Taking into account that $ P_{k+1} = P_k (1 + x^{2^{k+1}... | 2^{k+1}-1 | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXV - I - Problem 11
Let $ X_n $ and $ Y_n $ be independent random variables with the same distribution $ \left{ \left(\frac{k}{2^n}, \frac{1}{2^n}\right) : k = 0, 1, \ldots, 2^n-1\right} $. Denote by $ p_n $ the probability of the event that there exists a real number $ t $ satisfying the equation $ t^2 + X_n \cdot t... | A favorable event consists in the discriminant of the quadratic polynomial $ t^2 + X_n t + Y_n $ being a non-negative number. Let $ \displaystyle X_n = \frac{r}{2^n} $, $ \displaystyle Y_n = \frac{s}{2^n} $, where $ r, s = 0, 1, \ldots, 2^n - 1 $. The inequality $ X_n^2 - 4 Y_n \geq 0 $ is equivalent to the inequality ... | \frac{1}{12} | Algebra | math-word-problem | Yes | Yes | olympiads | false |
XXXIX OM - I - Problem 4
A triangle of maximum perimeter is inscribed in an ellipse. Prove that at each vertex, the angles between the tangent to the ellipse and the sides emanating from that vertex are equal. | Let $ABC$ be a triangle of maximum perimeter inscribed in a given ellipse. Suppose the statement of the problem is not true and, for example, the line $l$ tangent to the ellipse at point $B$ forms different angles with the sides $AB$ and $BC$. Draw a line $m$ through point $B$ that forms equal angles with these sides (... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVIII OM - II - Task 1
From an urn containing one ball marked with the number 1, two balls marked with the number 2, ..., $ n $ balls marked with the number $ n $, we draw two balls without replacement. We assume that drawing each ball from the urn is equally probable. Calculate the probability that both drawn balls... | To make the content of the task meaningful, we need to assume that $ n \geq 2 $.
In total, there are $ N = 1+2+ ... +n = n(n+1)/2 $ balls in the urn. Two balls can be chosen from $ N $ balls in $ q $ ways, where
Let us fix $ k \in \{2, \ldots,n\} $. From the $ k $ balls marked with the number $ k $, we can choose a pa... | \frac{4}{3(n+2)} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXXII - III - Task 2
The perpendicular bisectors of sides $ \overline{AB} $ and $ \overline{AC} $ of triangle $ ABC $ intersect the line containing side $ \overline{BC} $ at points $ X $ and $ Y $. Prove that $ BC = XY $ if and only if $ \tan B \cdot \tan C= 3 $ or $ \tan B\cdot \tan C=-1 $. | Let $ M $, $ N $, be the midpoints of sides $ \overline{AB} $ and $ \overline{AC} $ (Fig. 14).
(if $ B = \frac{\pi}{2} $ or $ C = \frac{\pi}{2} $, then one of the perpendicular bisectors does not intersect the line $ BC $, so this case can be disregarded). Therefore,
and the equality $ BC = XY $ holds... | \tanB\cdot\tanC=3or\tanB\cdot\tanC=-1 | Geometry | proof | Yes | Yes | olympiads | false |
LVIII OM - III - Problem 5
In the tetrahedron $ABCD$, the following relationships are satisfied:
Prove that the center of the sphere circumscribed around this tetrahedron lies on the line passing through the midpoints of edges $AB$ and $CD$. | Consider the hexagon $A_1BA_2CA_3D$ being the net of the tetrahedron $ABCD$ (Fig. 18).
om58_3r_img_18.jpg
om58_3r_img_19.jpg
By rewriting the first of the given equalities in the problem statement as
we see that the sum of the measures of angles $A_1BA_2$ and $A_2CA_3$ (considered as internal angles of the hexagon $A_... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXIX OM - III - Problem 1
In a given convex angle on a plane, a light ray runs, reflecting from the sides of the angle according to the principle that the angle of incidence equals the angle of reflection. A ray that hits the vertex of the angle is absorbed. Prove that there exists a natural number $ n $ such that eac... | If a light ray emanating from point $C$ reflects off one of the sides of a given angle at point $P$, and then off the other side at point $Q$, then by performing a symmetry $\varphi$ with respect to the line $OP$, where $O$ is the vertex of the given angle, we observe that points $C$, $P$, and $\varphi(Q)$ are collinea... | [\frac{\pi}{\theta}]+1 | Geometry | proof | Yes | Yes | olympiads | false |
XXX OM - I - Task 1
Given are seven numbers belonging to the interval $ [1, 10] $. Prove that three of them are the lengths of the sides of some triangle. | Three positive numbers are the lengths of the sides of a certain triangle if and only if the largest of these numbers is less than the sum of the other two. Let's number the given numbers $a_1, a_2, \ldots, a_7$ so that they form a non-decreasing sequence. Then $1 \leq a_1 \leq a_2 \leq \ldots \leq a_7 \leq 10$. Suppos... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
II OM - I - Task 7
Given a circle $ C $ and points $ A $ and $ B $ lying at unequal distances from the center of this circle. Prove that the common chords of circle $ C $ with circles passing through points $ A $ and $ B $ lie on lines that have one point in common. | When one of the given points, for example, point $A$, lies on the circle $C$, the theorem is obvious, as in this case all the chords mentioned in the theorem have a common endpoint $A$.
We will prove the theorem for the case when neither of the points $A$ and $B$ lies on the given circle $C$.
Consider two circles $K_1$... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
LIV OM - I - Task 3
Three different points $ A $, $ B $, $ C $ lie on the circle $ o $. The tangents to the circle $ o $ at points $ A $ and $ B $ intersect at point $ P $. The tangent to the circle $ o $ at point $ C $ intersects the line $ AB $ at point $ Q $. Prove that | Let $ M $ be the midpoint of segment $ AB $ (Fig. 1). Then triangle $ PMQ $ is a right triangle. By the Pythagorean theorem and the similarity of triangles $ QBC $ and $ QCA $ we get
which was to be proved. | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
LVI OM - III - Problem 6
Prove that every convex polygon with an area of 1 contains a convex hexagon with an area of at least 3/4. | From the vertices of a given polygon $\mathcal{W}$, we select three $A$, $B$, and $C$ which are the vertices of a triangle with the largest area, equal to $s$. Let $X$ be a point of the polygon $\mathcal{W}$, lying on the opposite side of the line $BC$ from point $A$ and farthest from the line $BC$ (Fig. 4). We define ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVIII OM - I - Problem 3
In a convex quadrilateral $ABCD$, the midpoint $M$ of side $AB$ is connected to vertices $C$ and $D$, and the midpoint $N$ of side $CD$ is connected to vertices $A$ and $B$. Segments $AN$ and $DM$ intersect at point $P$, and segments $BN$ and $CM$ intersect at point $Q$. Prove that the area of... | The area of a polygon $ABC\ldots$ will be denoted by the symbol $ (ABC\ldots) $. We state that (Fig. 1)
Thus,
therefore,
But,
and from (1) and (2) it follows that,
Similarly,
Indeed,
From (3), (4), and (5) we obtain,
| proof | Geometry | proof | Yes | Yes | olympiads | false |
LVII OM - I - Problem 8
Tetrahedron $ABCD$ is circumscribed around a sphere with center $S$ and radius 1, such that $SA \geq SB \geq SC$. Prove that $SA > \sqrt{5}$. | We will first prove the following lemma.
Lemma
Triangle $ABC$ is contained in a circle of radius $R$. A circle of radius $r$ is inscribed in triangle $ABC$. Then $R \geq 2r$.
Proof
Let $D, E, F$ be the midpoints of sides $BC, CA, AB$ respectively (Fig. 3). Furthermore, let $S$ be the point of intersection of the median... | SA>\sqrt{5} | Geometry | proof | Yes | Yes | olympiads | false |
LX OM - I - Task 5
For each integer $ n \geqslant 1 $, determine the largest possible number of different subsets of the set $ \{1,2,3, \cdots,n\} $ with the following property: Any two of these subsets are either disjoint or one is contained in the other. | Answer: $ 2n $ subsets.
Let the number of sought subsets be denoted by $ a_n $.
It is not difficult to observe that for $ n =1,2,3,\cdots $ the following $ 2n $
subsets of the set $ \{1,2,3, \cdots,n\} $ have the property described in the problem statement:
Indeed, any of the first $ n+1 $ sets has at most one el... | 2n | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
LX OM - III - Problem 6
Let $ n \geqslant $ 3 be a natural number. The sequence of non-negative numbers
$ (c_0,c_1,\cdots ,c_n) $ satisfies the condition
for all $ p, r, s, t \geqslant 0 $ such that $ p + r + s + t = n $. Determine all possible
values of $ c_2 $, if $ c_1 =1 $. | Method I.
We will show that for every natural number $ n \geqslant 3 $ there exists exactly one sequence satisfying
the conditions of the problem, and the only possible value of $ c_2 $ is $ 2 \cos(\pi/n) $.
Notice that for $ p = n $ and $ r = s = t = 0 $, condition (1) takes the form
$ c_nc_0 + c^2_0 = c_nc_0 $, fro... | 2\cos(\pi/n) | Algebra | math-word-problem | Yes | Incomplete | olympiads | false |
XIV OM - I - Task 7
Given a triangle $ ABC $, divide a given line $ p $ parallel to it into two parts of equal areas. | Let's draw through points $A$, $B$, $C$ the lines $a$, $b$, $c$ parallel to line $p$. Suppose these are three different lines; then one of them lies in the strip bounded by the other two. Let this be line $a$; it intersects side $BC$ at some point $D$ (Fig. 4). If $BD = DC$, the desired line is $AD$; suppose $BD < DC$;... | CN=\sqrt{\frac{1}{2}CB\cdotCD} | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
I OM - B - Task 16
Form a rectangle from nine squares with sides 1, 4, 7, 8, 9, 10, 14, 15, and 18. | From the given squares, a rectangle can be formed in only one way. We will demonstrate this by conducting the following reasoning: First of all, it is clear that the squares filling the rectangle must have sides parallel to the sides of the rectangle. The area of the rectangle must equal the sum of the areas of all the... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
XIII OM - II - Task 2
What conditions should the real numbers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ satisfy so that the quadratic polynomial
is the product of two linear polynomials with real coefficients? | Considering the problem statement, we only deal with polynomials with real coefficients.
First, we will consider the case when in a given polynomial (1) the coefficient $ a \ne 0 $ and write the polynomial in the form
If the polynomial (2) equals the product of two factors of the first degree with respect to $ x $ and... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XLV OM - I - Problem 11
A triangle with perimeter $ 2p $ is inscribed in a circle with radius $ R $ and circumscribed around a circle with radius $ r $. Prove that $ p < 2(R + r) $. | Let $ABC$ be the considered triangle; we adopt the usual notations
assuming (without loss of generality) that $a$ is the largest angle of this triangle. Thus, $a \geq 60^\circ$. Let $I$ be the center of the inscribed circle of triangle $ABC$, and let $A'$, $B'$, $C'$ be the points of tangency of this circle with the... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LIV OM - I - Problem 9
Find all polynomials $ W $ with real coefficients that have the following property: if $ x+ y $ is a rational number, then
is a rational number. | First, note that the conditions of the problem are satisfied by any polynomial of the form $ W(x) = ax + b $, where $ a $ and $ b $ are rational numbers.
We will prove that these are all the polynomials with the property given in the problem.
Let $ W $ be a polynomial satisfying the conditions of the problem. For a rat... | W(x)=ax+b | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XX OM - III - Task 2
Given are numbers $ a_1, a_2, \ldots, a_n $, each two of which are different. Find the smallest value of the function defined by the formula
Note: The formula or expression that should follow is missing in the original text. If you have the complete formula, please provide it for a full ... | Let's assume that the given numbers are arranged in an increasing sequence, i.e.,
1. Note that according to the definition of the absolute value of a number,
the function $ y = |x-c| $ is defined and continuous for every $ x $ in the interval $ - \infty < x \leq c $, it is a decreasing linear function, and in the int... | Algebra | math-word-problem | Incomplete | Yes | olympiads | false | |
LIII OM - III - Problem 2
On the sides $ AC $ and $ BC $ of the acute-angled triangle $ ABC $, rectangles $ ACPQ $ and $ BKLC $ of equal areas are constructed on its external side. Prove that the midpoint of segment $ PL $, point $ C $, and the center of the circumcircle of triangle $ ABC $ lie on one straight line. | Let $ O $ be the center of the circle circumscribed around triangle $ ABC $ (Fig. 1). Complete triangle $ PCL $ to parallelogram $ PCLX $. It suffices to prove that points $ X $, $ C $, $ O $ are collinear.
From the equality of the areas of the given rectangles, we obtain
Moreover, $ \measuredangle XLC = 180^\cir... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVI OM - I - Problem 1
Prove that if an integer divisible by 3 is the sum of five squares of integers, then at least two of these squares are divisible by 9. | The square of an integer not divisible by $3$ gives a remainder of $1$ when divided by $3$. Therefore, if a certain number is the sum of five squares and none of these squares is divisible by $3$, then the given number gives a remainder of $2$ when divided by $3$; if only one of these squares is divisible by $3$, then ... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXII OM - I - Problem 7
On a plane, there are five lattice points (i.e., points with integer coordinates). Prove that the midpoint of one of the segments connecting these points is also a lattice point. | Let the lattice points be $ P_i = (x_i, y_i) $, where $ i = 1, 2, 3, 4, 5 $. Among the numbers $ x_1 $, $ x_2 $, $ x_3 $, $ x_4 $, $ x_5 $, there are at least three of the same parity (i.e., three even or three odd), because if, for example, at most two are even, then the remaining ones are odd. Let's assume, for examp... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
X OM - II - Task 2
Under what relationship between the sides of a triangle is it similar to the triangle formed by its medians? | Let's denote the medians drawn to the sides $a$, $b$, $c$ of triangle $ABC$ by $m_a$, $m_b$, $m_c$ respectively. We need to investigate in which triangles the medians are proportional to the sides of the triangle.
First, it is important to note that the larger of two unequal medians is drawn to the shorter side. To v... | ^2=2b^2-^2 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLIX OM - II - Problem 5
Non-negative numbers $ a_1, a_2,\ldots, a_7 $, $ b_1, b_2,\ldots, b_7 $ satisfy the condition
Prove that for some two different numbers $ k, m \in \{1, 2,\ldots, 7\} $ the inequality $ |a_k - a_m| + |b_k - b_m| \leq 1 $ holds. | Let's mark points $ P_i = (a_i,b_i) $ $ (i = 1, 2,\ldots, 7) $ in the coordinate system. From the conditions of the problem, it follows that these points lie inside or on the boundary of the triangle with vertices $ (0, 0) $, $ (2, 0) $, $ (0,2) $. Let's divide this triangle into six regions: two squares and four right... | proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
XLVII OM - I - Problem 8
A light ray emanates from the center of a square, reflecting off the sides of the square according to the principle that the angle of incidence equals the angle of reflection. After some time, the ray returns to the center of the square. The ray never hit a vertex or passed through the center ... | On the plane with a rectangular coordinate system $Oxy$, we draw all lines with equations $x = c$ ($c$ integers) and $y = c$ ($c$ integers); in this way, the entire plane is covered by an infinite grid of unit squares. In each pair of squares sharing a common side, we identify points that are symmetric with respect to ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVI OM - I - Problem 12
Prove that for every natural number $ n $ the following equality holds
Note. $ [x] $ is the greatest integer less than or equal to $ x $. | We will first show that for every natural number $ n > 2 $, the double inequality holds:
Raising all terms to the square, we obtain the equivalent inequality:
And this, in turn, is equivalent to the following:
The right part of the last inequality is obvious, while the left part, when squared, takes the equivalent f... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXIII OM - I - Problem 8
On a plane, there are two congruent equilateral triangles $ABC$ and $A'B'C'$. Prove that if the midpoints of segments $\overline{AA'},$ $\overline{BB'},$ $\overline{CC'}$ are not collinear, then they are the vertices of an equilateral triangle. | Let points $O$ and $O'$ be the centers of the circumcircles of triangles $ABC$ and $A'B'C'$, respectively. Denote by $T$ the translation of the plane by the vector $\overrightarrow{OO'}$, and let $A'$, $B'$, $C'$ be the images of $A$, $B$, $C$ under this translation. Denote by $A_1$, $B_1$, $C_1$ the midpoints of segme... | proof | Geometry | proof | Yes | Yes | olympiads | false |
VI OM - I - Task 3
Draw a line through the midpoint of one of the non-parallel sides of a trapezoid, which divides the trapezoid into two parts of equal area. | Let $ABCD$ be a trapezoid, in which the parallel sides are $AB$ and $DC$, and let $M$ be the midpoint of side $AD$ (Fig. 1). Notice that $\text{area of }ABM = \frac{1}{4} AB \cdot h$, $\text{area of }DCM = \frac{1}{4} DC \cdot h$, $\text{area of }ABCD = \frac{1}{2}(AB + CD) \cdot h$, where $h$ denotes the height of the... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XIX OM - II - Problem 5
Tetrahedra $ABCD$ and $A_1B_1C_1D_1$ are positioned such that the midpoints of segments $AA_1$, $BB_1$, $CC_1$, $DD_1$ are the centroids of triangles $BCD$, $ACD$, $ABD$, and $ABC$, respectively. In what ratio are the volumes of these tetrahedra? | Let $ M $ denote the centroid of triangle $ BCD $, and $ S $ - the centroid of tetrahedron $ ABCD $. Point $ S $ lies on segment $ AM $ and $ SA = 3 SM $. Point $ A_1 $ lies on the extension of segment $ AM $ and $ AM = MA_1 $ (Fig. 14). Therefore, point $ S $ lies on segment $ AA_1 $ and divides it in the ratio
Simi... | (\frac{5} | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
XXXIX OM - III - Zadanie 5
Ciąg $ (a_n) $ określony Jest warunkami $ a_1 = a_2 = a_3 = 1 $, $ a_{n+1} = a_{n+2}a__{n+1}+a_n $ ($ n \geq 1 $). Wykazać, że dla każdej liczby naturalnej $ r $ istnieje taka liczba $ s $, że $ a_s $ jest podzielne przez $ r $.
|
Gdy $ r=1 $, nie ma czego dowodzić. W dalszym ciągu zakładamy, że $ r \geq 2 $.
Przyjmijmy $ a_0 = 0 $ i rozważajmy ciąg $ (a_n) $ indeksowany nieujemnymi liczbami całkowitymi. Dana w zadaniu zależność rekurencyjna jest zachowana; możemy ją przepisać w równoważnej postaci
Zajmiemy się ciągiem $ (x_0, x_1 x_2, x_... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
VIII OM - II - Task 2
Prove that if $ M $, $ N $, $ P $ are the feet of the altitudes of an acute triangle $ ABC $, then the ratio of the perimeter of triangle $ MNP $ to the perimeter of triangle $ ABC $ is equal to the ratio of the radius of the inscribed circle of triangle $ ABC $ to the radius of the circumscribed... | Accepting the notation indicated in Fig. 14, we will first show that $ \measuredangle 1 = \measuredangle 2 $. Indeed, the vertices of the quadrilateral $ ANHP $ lie on a circle, since the angles of this quadrilateral at vertices $ N $ and $ P $ are right angles. Thus, $ \measuredangle 1 = \measuredangle AHP $, as inscr... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXVIII - I - Problem 11
From the numbers $ 1, 2, \ldots, n $, we choose one, with each of them being equally likely. Let $ p_n $ be the probability of the event that in the decimal representation of the chosen number, all digits: $ 0, 1, \ldots, 9 $ appear. Calculate $ \lim_{n\to \infty} p_n $. | Let the number $ n $ have $ k $ digits in its decimal representation, i.e., let $ 10^{k-1} \leq n < 10^k $, where $ k $ is some natural number. Then each of the numbers $ 1, 2, \ldots, n $ has no more than $ k $ digits. We estimate from above the number $ A_0 $ of such numbers with at most $ k $ digits that do not cont... | 0 | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false |
LVIII OM - III - Problem 4
Given an integer $ n \ge 1 $. Determine the number of possible values of the product $ k \cdot m $, where $ k $, $ m $ are integers
satisfying the inequalities
(Note: The inequalities are not provided in the original text, so they are left blank in the translation.) | Suppose that two different pairs $ (k,m) $ and $ (k,
satisfying a given condition, give the same product $ {km=k.
Let us denote
Then $ {4q=s^2-r^2} $, which means $ {r=\sqrt{s^2-4q}} $; the difference $ r
can be expressed analogously. If the equality $ {s=s,
were to hold, it would also imply the equality $ {r=r... | 2n^2+5n+2 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
I OM - B - Task 8
Prove that the altitudes of an acute triangle are the angle bisectors of the triangle whose vertices are the feet of these altitudes. | Let $ S $ be the point of intersection of the altitudes $ AM $, $ BN $, and $ CP $ of an acute triangle $ ABC $ (Fig. 3).
om1_Br_img_3.jpg
The quadrilateral $ SMCN $, in which the angles at vertices $ M $ and $ N $ are right angles, is a cyclic quadrilateral with diameter $ SC $; therefore,
(angles subtended by t... | proof | Geometry | proof | Yes | Yes | olympiads | false |
II OM - I - Task 4
Given are two intersecting planes $ A $ and $ B $ and a line $ m $ intersecting planes $ A $ and $ B $. Find the geometric locus of the midpoints of segments parallel to line $ m $, whose ends lie on planes $ A $ and $ B $. | Let $M$ and $N$ be the points of intersection of the line $m$ with the planes $A$ and $B$, $S$ the midpoint of the segment $MN$, and $s$ the line of intersection of the planes $A$ and $B$. We are to find the figure formed by the midpoints of all segments $XY$ parallel to the segment $MN$ and having their endpoints $X$ ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLI OM - III - Task 4
Inside a square with a side length of 1, a triangle was drawn, each side of which has a length of at least 1. Prove that the center of the square belongs to the triangle. | We apply proof by contradiction.
Suppose that the center $O$ of the given square $ABCD$ does not belong to the drawn triangle $A$. There then exists a line $l$ passing through $O$ such that triangle $A$ lies entirely on one side of this line (i.e., it is contained in one of the two open half-planes determined by the li... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLII OM - I - Problem 12
For any natural number $ n $, consider the rectangular prism $ K_n $ with edge lengths $ 1,1,n $ and the set $ R_n $ consisting of $ 4n+1 $ points: the vertices of the rectangular prism $ K_n $ and the points dividing its longer edges into unit segments. We randomly select three different poin... | Let $ Q(n) $ denote the number of all three-element subsets of the set $ R_n $, $ T(n) $ - the number of non-degenerate triangles with vertices in the set $ R_n $, and $ P(n) $ - the number of obtuse triangles. The probability of interest is the fraction $ p_n = P(n)/Q(n) $. Notice that
(Justification of the second ... | \frac{15}{16} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
LVII OM - I - Problem 5
Let $ a, b $ be real numbers. Consider the functions $ f(x)= ax+b|x| $ and $ g(x)= ax-b|x| $.
Prove that if $ f(f(x)) = x $ for every real number $ x $, then $ g(g(x)) = x $ for every real number $ x $. | Notice that for any real number $ x $ we have
Therefore, for every real number $ x $ we obtain
which was to be proved. | proof | Algebra | proof | Yes | Incomplete | olympiads | false |
XVI OM - I - Problem 12
In space, there are two polyhedra. Prove that the longest of the segments connecting points of one polyhedron with points of the other has its ends at the vertices of the polyhedra. | First, let us note that if all vertices of a polyhedron lie on one side of a certain plane, then all other points of the polyhedron lie on that same side of the plane.
Suppose, for instance, that all vertices of the polyhedron $ W $ are located in an open half-space (i.e., a half-space without the points of the boundin... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIV OM - I - Zadanie 4
Wyznaczyć wszystkie funkcje $ f: \mathbb{R} \to \mathbb{R} $ takie, że
|
Załóżmy, że funkcja $ f $ spełnia dane równanie funkcyjne. Podstawiając $ x = y = 0 $ dostajemy równość $ f(0) = 0 $. Podstawiając następnie w równaniu $ x = 0 $ (i wiedząc, że $ f(0) = 0 $) dostajemy zależność
To znaczy, że $ f $ jest funkcją parzystą.
Weźmy dowolną liczbę $ y \in \mathbb{R} $ i podstawmy w dan... | f(x)=0 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XIII OM - II - Task 1
Prove that if the numbers $ x $, $ y $, $ z $ satisfy the equation
then at least one of them is equal to $ a $. | If $ x $, $ y $, $ z $ satisfy equations (1) and (2), then
Therefore
The left side $ L (x, y, z) $ of equation (3) can be expressed as a polynomial in the variable $ x $. By substitution, we check that when $ x = -y $, this polynomial is identically zero; it is therefore divisible by $ x + y $. In the same way, we de... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXVI - II - Task 1
Given the polynomial $ W(x) = x^4 + ax^3 + bx + cx + d $. Prove that for the equation $ W(x) = 0 $ to have four real roots, it is necessary and sufficient for there to exist an $ m $ such that $ W(x+m) = x^4+px^2+q $, that the sum of some two roots of the equation $ W(x) = 0 $ equals the sum of the ... | If the roots of the polynomial $W(x)$ are the numbers $x_1, x_2, x_3, x_4$ and
then by taking $ \displaystyle m = \frac{1}{2} (x_1 + x_2) $, we obtain that the roots of the polynomial $W(x+m)$ are the numbers $ y_i = x_i - m $ for $ i = 1, 2, 3, 4 $, i.e., the numbers
Thus,
It follows from this that
We have thus sh... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XVII OM - II - Problem 3
In the plane, 6 points are chosen, no three of which lie on the same line, and all segments connecting these points in pairs are drawn. Some of these segments are drawn in red, and others in blue. Prove that some three of the given points are vertices of a triangle with sides of the same color... | Let $ A_1 $, $ A_2 $, $ A_3 $, $ A_4 $, $ A_5 $, $ A_6 $ be given points. Among the five segments $ A_1A_i $ ($ i = 2, 3, 4, 5, 6 $) at least three are of the same color. Suppose, for example, that the segments $ A_1A_2 $, $ A_1A_3 $, $ A_1A_4 $ are red and consider the segments $ A_2A_3 $, $ A_3A_4 $, $ A_4A_2 $. If t... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
Points $ A $, $ B $, $ C $, $ D $ lie, in that exact order, on circle $ o $. Point $ S $ lies inside circle $ o $ and satisfies the conditions
The line containing the bisector of angle $ ASB $ intersects circle $ o $ at points $ P $ and $ Q $. Prove that $ PS = QS $. | Let us assume that the lines $AS$, $BS$, $CS$ intersect the circle $o$ for the second time at points $E$, $F$, $G$ (Fig. 1). From the given equalities in the problem, it follows that triangles $ASD$ and $CSB$ are similar, and thus $\measuredangle ASC = \measuredangle DSB$. These equalities prove that triangles $ASC$ an... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LVII OM - I - Problem 12
Let $ a_0 $ be a positive integer and let
for $ i=0,1,2,\dots $. Prove that if $ n $ is a positive integer satisfying the condition $ a_n = a_0 $, then $ 2^n > a_0 $. | Let $ a_{\ell} $ be the largest among the terms $ a_0,a_1,\dots ,a_{n-1} $. Then the sequence $ b_i = a_{\ell+i} $ for $ i =0,1,2,... $ also satisfies the given recursive condition and the equality $ b_n = b_0 $. Moreover, $ b_0 $ is the largest number among $ b_0,b_1,\dots,b_{n-1} $.
Therefore, without loss of genera... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
LX OM - I - Task 9
Given is a $2008 \times 2008$ board. Two players take turns
making moves, each of which consists of choosing a white or
black pawn and placing it on a selected free field. The player wins
whose move results in a sequence of 5 consecutive
pawns of the same color in a vertical, horizontal, or diagonal... | Answer: It does not exist.
We will indicate a strategy for the player who does not start
(called the second player in the further part of the solution), which
will prevent the starting player (the first player) from winning.
This strategy can be described as follows: If the second player can
make a move leading to the ... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XIV OM - I - Problem 3
Prove that if vertices $ A $ and $ B $ of triangle $ ABC $ are fixed, and vertex $ C $ moves in such a way that angle $ ABC $ remains constant, then the segment connecting the feet $ H $ and $ K $ of the altitudes from vertices $ A $ and $ B $ has a constant length. | When $ \measuredangle ACB $ is a right angle, the theorem is obvious, as $ H $ and $ K $ coincide with point $ C $ and $ HK = 0 $. Assume that $ \measuredangle ACB = \gamma \ne 90^\circ $. Points $ H $ and $ K $ lie on the circle with diameter $ AB $ (Fig. 1); let $ O $ be its center. Based on the inscribed angle theor... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XIII OM - II - Task 6
Find a three-digit number with the property that the number represented by these digits and in the same order, but in a different base of numeration than $ 10 $, is twice as large as the given number. | If $ x $, $ y $, $ z $ denote the consecutive digits of the sought number, and $ c $ - the new base of numeration, then
the equation can be replaced by the equation
The task is to solve equation (1) in integers $ x $, $ y $, $ z $, $ c $, satisfying the conditions $ 1 \leq x \leq 9 $, $ 0 \leq y \leq 9 $, $ 0 \leq z ... | 145,150,295 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXVII OM - I - Problem 1
Factorize into factors of the lowest possible degree:
a) $ (a + b + c)^3 - a^3 - b^3 - c^3 $,
b) $ 2a^2b^2 + 2b^2c^2 + 2a^2c^2 - a^4 - b^4 - c^4 $. | a) By virtue of the identity
we obtain that the expression given in the task is equal to
b) We transform the expression given in the task
翻译完成，但根据您的要求，我保留了原文的换行和格式。请注意，最后一部分似乎没有提供完整的数学表达式或句子。如果您需要进一步的帮助或有其他内容需要翻译，请告诉我。 | notfound | Algebra | math-word-problem | Yes | Incomplete | olympiads | false |
XXXII - II - Task 4
Given are natural numbers $ k, n $. We define inductively two sequences of numbers $ (a_j) $ and $ (r_j) $ as follows:
First step: divide $ k $ by $ n $ and obtain the quotient $ a_1 $ and the remainder $ r_1 $,
j-th step: divide $ k + r_{j-1} $ by $ n $ and obtain the quotient $ a_j $ and the rema... | According to the definition
Adding these equalities we get
thus
The number $ r_n $ as the remainder of the division of $ k+r_{n-1} $ by $ n $ satisfies the condition $ 0 \leq r_n < n $, but from the last equality, it follows that $ r_n $ is a number divisible by $ n $. Therefore, $ r_n = 0 $ and
... | k | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXX OM - I - Task 5
Find a real number $ m $ for which the product of some two real roots of the equation
is 2. | If $ a $ and $ b $ are real roots of the polynomial
then by the theorem of Bézout, this polynomial can be written in the form
where $ c $ and $ d $ are certain real numbers. By comparing the coefficients of the same powers of $ x $ in both representations of the polynomial $ f $, we get
By the conditi... | -9 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XLIII OM - II - Problem 3
Through the centroid of an acute triangle $ABC$, lines perpendicular to the sides $BC$, $CA$, $AB$ are drawn, intersecting them at points $P$, $Q$, $R$ respectively. Prove that if $|BP| \cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB|$, then triangle $ABC$ is isosceles.
Note: According to C... | om43_2r_img_7.jpg
Figure 7 shows an "arbitrary" acute triangle $ABC$; it is not isosceles, and thus (if the thesis of the problem is to be valid) - this figure is not an illustration of the problem. Nevertheless, or rather precisely because of this, such a figure is "safe"; it does not suggest using the equality of ang... | proof | Geometry | proof | Yes | Yes | olympiads | false |
IV OM - III - Task 4
Prove that if $ n $ is a natural number, then the following equality holds
where $ m $ is a natural number. | We will first prove that for every natural number $ n $, the equality can be written as
where $ a $ and $ b $ are natural numbers satisfying the equality
Proof. For $ n = 1 $, the theorem is true, specifically, we need to take $ a = b = 1 $. Suppose the theorem is true for some $ n $; then
where $ a_1 $ and $ b_1 $ ... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
II OM - I - Task 11
In a given square, inscribe a square such that one of its sides or its extension passes through a given point $ K $. | A square inscribed in a square $ABCD$ is defined as any square whose vertices lie on the perimeter of the square $ABCD$. According to this definition, the square $ABCD$ itself is also considered an inscribed square.
When a given point $K$ lies on the perimeter of the square $ABCD$, the problem is solved immediately. Sp... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXIV OM - III - Task 6
Prove that for any convex polygon with a center of symmetry, there exists at most one ellipse of the smallest area containing the given polygon.
Note. The area of an ellipse with semi-axes of lengths $a$ and $b$ is given by the formula $\pi ab$. | Let $\varphi$ be a mapping of the plane onto itself defined by the formula $\varphi (x, y) = (x, ky)$, where $k$ is a fixed positive number. Such a mapping is called a rectangular affinity; more precisely, a rectangular affinity with the $x$-axis and coefficient $k$. Of course, the inverse mapping to the rectangular af... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXV - I - Problem 7
Prove that if in an acute triangle each altitude can be moved inside the triangle by a rigid motion so that its foot takes the place of the vertex, and the vertex takes the place of the foot, then the triangle is equilateral. | If $ h $ is the length of a certain altitude of triangle $ ABC $, then from the conditions of the problem, it follows that in triangle $ ABC $, a segment of length $ h $ can be placed having any predetermined direction. If triangle $ ABC $ is not equilateral, then the lengths of some of its altitudes are different, e.g... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXIX OM - I - Problem 8
For a given cube with edge length 1, find the set of segments with a total length not exceeding $1 + 3\sqrt{3}$, having the property that any two vertices of the cube are the endpoints of some broken line composed of segments from this set. | We will provide an example of a set of segments with the required property. This example can be considered the spatial counterpart of the "optimal road network connecting the vertices of a rectangle," constructed in preparatory task $C$. (A hint for the construction is the observation that it is purposeful to search fo... | 1+3\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LI OM - I - Problem 12
In an acute triangle $ABC$, points $D$, $E$, $F$ lie on sides $BC$, $CA$, $AB$ respectively. The circumcircles of triangles $AEF$, $BFD$, $CDE$ intersect at point $P$. Prove that if
then $AD$, $BE$, $CF$ are the altitudes of triangle $ABC$. | We will use the following
Theorem
The altitudes $ AD $, $ BE $, $ CF $ of triangle $ ABC $ intersect at point $ H $. Points $ K $, $ L $, $ M $ are the midpoints of sides $ BC $, $ CA $, $ AB $, respectively. Points $ X $, $ Y $, $ Z $ are the midpoints of segments $ AH $, $ BH $, $ CH $, respectively. Then the followi... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
V OM - I - Problem 8
Given are the mutual distances of four points $ A $, $ B $, $ C $, $ D $ in space. Calculate the length of the segment connecting the midpoint of segment $ AB $ with the midpoint of segment $ CD $. | Assume that points $A$, $B$, $C$, $D$ do not lie on the same plane. The midpoints $M$, $N$, $P$, $Q$, $R$, $S$ of segments $AB$, $BC$, $AC$, $AD$, $BD$, $CD$ (Fig. 24) are then distinct points. By the theorem on the segment connecting the midpoints of two sides of a triangle, we have $MR \parallel AD \parallel PS$ and ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXVIII OM - I - Problem 9
In a convex pentagon $ KLMNP $, we draw diagonals which, intersecting, determine disjoint segments: on diagonal $ KM $ in the order from $ K $ to $ M $ segments of lengths $ a_1, b_1, c_1 $; similarly $ a_2, b_2, c_2 $ on diagonal $ LN $, $ a_3, b_3, c_3 $ on diagonal $ MP $, $ a_4, b_4, c_4... | We change the notation of vertices $ L $, $ M $, $ N $, $ P $, $ K $ to $ A_1 $, $ A_2 $, $ A_3 $, $ A_4 $, $ A_5 $, respectively. We introduce further notations (Figure 9):
for $ i = 1, 2, 3, 4, 5 $; the addition $ i\pm 1 $, $ i\pm 2 $ in the indices should be understood modulo $ 5 $.
om38_1r_img_9.jpg... | proof | Geometry | proof | Yes | Yes | olympiads | false |
V OM - II - Task 3
Given: point $ A $, line $ p $, and circle $ k $. Construct triangle $ ABC $ with angles $ A = 60^\circ $, $ B = 90^\circ $, where vertex $ B $ lies on line $ p $, and vertex $ C $ lies on circle $ k $. | (method of geometric loci).
The task boils down to finding a point $ C $ that satisfies two conditions:
1° Point $ C $ lies on a given circle $ k $;
2° point $ C $ is a vertex of triangle $ ABC $ with a given vertex $ A $, whose vertex $ B $ lies on a given line $ p $ and in which $ \measuredangle A = 60^\circ $, $ \me... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XX OM - I - Task 5
Prove that for any positive numbers $ a $ and $ b $ and natural $ n $ | We apply complete induction
a) When $ n = 1 $, the thesis of the theorem takes the form $ \frac{a+b}{2} \geq \frac{a+b}{2} $, which is therefore true.
b) If for some natural $ n $ the inequality (1) holds, then
thus
indeed
since
From (2) and (3) we obtain
i.e., the thesis of the theorem is... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
LIII OM - II - Problem 6
Determine all natural numbers \( n \) such that for any real numbers \( x_{1}, x_{2}, \ldots, x_{n}, \) \( y_{1}, y_{2}, \ldots, y_{n} \) the following inequality holds
保留了源文本的换行和格式，但请注意，最后一句“保留了源文本的换行和格式”是额外的说明，不是翻译的一部分。正确的翻译结果如下：
LIII OM - II - Problem 6
Determine all natural n... | For $ n = 1 $, the given inequality takes the form
which is not true for arbitrary real numbers $ x $, $ y $ - it suffices to take $ x_1 = y_1 = 1 $.
We will show that inequality (1) is true if $ n \geq 2 $.
If one of the factors on the right-hand side of inequality (1), for example,
is equal to $ 0 $, then bot... | proof | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
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