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II OM - I - Task 9
Solve the system of equations
| We distinguish the following cases:
1° $ a = 0 $, $ b = 0 $. The system of equations (1) then takes the form
from which it is clear that two of the unknowns $ x $, $ y $, $ z $ must be equal to zero, while the third can have any value.
2° $ a \ne 0 $, $ b = 0 $. The system of equations (1) then takes the form
... | Algebra | math-word-problem | Incomplete | Yes | olympiads | false | |
XXXII - II - Problem 6
The areas of the bases of a given truncated triangular pyramid are $ B_1 $ and $ B_2 $. This pyramid can be cut by a plane parallel to the bases such that a sphere can be inscribed in each of the resulting parts. Prove that the lateral surface area of the given pyramid is $ (\sqrt{B_1} + \sqrt{B... | We will first derive the formula for the lateral surface area of a truncated pyramid circumscribed around a sphere. This formula will then be applied to each of the parts obtained from the given truncated pyramid as a result of the section. The volume $ V $ of a truncated triangular pyramid with base areas $ Q_1 $ and ... | (\sqrt{B_1}+\sqrt{B_2})(\sqrt[4]{B_1}+\sqrt[4]{B_2})^2 | Geometry | proof | Yes | Incomplete | olympiads | false |
XLII OM - I - Problem 10
Let $ (a_n) $ be any sequence of natural numbers satisfying the conditions:
Prove that for every natural number $ n\geq 1 $ the product $ a_na_{n+1} $ equals some term of this sequence. | The content of the task does not specify the value of the second term of the sequence $(a_n)$. Let's denote it by $c$; it is some natural number.
Let's list the first few terms of the sequence $(a_n)$, using the given recursive definition and the initial conditions $a_1 = 1$, $a_2 = c$:
We see that for $n=1,2,3,4... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXVIII - II - Problem 5
Let the polynomials $ w_n $ be defined by the formulas:
and let $ a $ be a real number. How many different real solutions does the equation $ w_n(x) = a $ have? | \spos{1} Let $ r_n(a) $ be the number of distinct real solutions of the equation $ w_n (x) = a $. We will prove by induction on $ n $ that
We check the validity of formula (1) for $ n = 1 $. The equation $ w_1(x)= a $ has the form $ x^2 = 1 + a. $ It has $ 0 $, $ 1 $, or $ 2 $ solutions when $ a < - 1 $, $ a = -1... | proof | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXXII - II - Problem 1
Prove that for any real numbers $ x_1, x_2, \ldots, x_{1981} $, $ y_1, y_2, \ldots, y_{1981} $ such that $ \sum_{j=1}^{1981} x_j = 0 $, $ \sum_{j=1}^{1981} y_j = 0 $, the inequality holds | Since all numbers under the square root signs are positive, squaring both sides will yield an equivalent inequality
Since , then
and similarly
hence
Therefore, to prove the inequality (*), it suffices to state that for any pair of numbers $ i $ and $ j $,
We will further transf... | proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
I OM - B - Task 10
To prove that for any numbers $ a, b, c $ the inequality holds
untranslated part due to lack of specific inequality in the original text. If there is a specific inequality you want translated, please provide it. | Since
therefore
and similarly
as well as
Adding the above three inequalities side by side, we get
from which | proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
LIX OM - I - Problem 8
Given is a quadrilateral pyramid $ABCDS$ with a convex quadrilateral base $ABCD$. A sphere inscribed in this
pyramid is tangent to the face $ABCD$ at point $P$. Prove that | Let $ K, L, M, N $ be the points of tangency of the considered sphere with the faces
$ SDA, SAB, SBC, SCD $, respectively. Let also $ K $ be the points of intersection of the lines
$ SK, SL, SM, SN $ with the plane $ ABCD $.
From the equality $ KS = SL $ and $ AK = AL $, it follows that triangles $ AKS $ and $ ALS $ ar... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XLIV OM - I - Problem 10
The function $ f $ mapping the cube $ \mathcal{C} $ onto $ \mathcal{C} $ satisfies the condition
Prove that $ f $ is an isometry. | Let's denote the vertices of the cube by $B_1, \ldots, B_8$ in such a way that the square $B_1B_2B_3B_4$ is one of the faces, and the segments $B_1B_5$, $B_2B_6$, $B_3B_7$, $B_4B_8$ are four parallel edges; the length of the edge is taken as the unit.
The function $f$ (by assumption) maps the cube $\mathcal{C}$ onto th... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XLVI OM - III - Problem 5
Let $ n $ and $ k $ be natural numbers. From an urn containing 11 slips numbered from 1 to $ n $, we draw slips one by one, without replacement. When a slip with a number divisible by $ k $ appears, we stop drawing. For a fixed $ n $, determine those numbers $ k \leq n $ for which the expecte... | Let's establish a natural number $ k \in \{1,2,\ldots,n\} $. Consider the following events:
Let's denote the probabilities of these events: $ p_j = P(\mathcal{A}_j) $, $ q_j = P(\mathcal{B}_j) $. Of course, $ \mathcal{B}_0 $ is a certain event. Therefore, $ q_0 = 1 $.
The occurrence of event $ \mathcal{A}_j $ is ... | k | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XVII OM - I - Problem 3
Find the geometric locus of the centroids of acute triangles inscribed in a given circle. | First, we establish that every point $S$ inside a circle is the centroid of some triangle inscribed in that circle. Indeed, if point $S$ lies on the radius $OC = r$ (Fig. 1), and point $M$ lies on the extension of $CS$ beyond point $S$, such that $CM = \frac{3}{2} CS$, hence $CM < 2r$, then by drawing a chord $AB$ of t... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XVI OM - I - Problem 10
Prove that if for every value of $ a $ the equality holds
The above text has been translated into English, preserving the original text's line breaks and formatting. | Substituting into equation (1) successively $ \alpha = \frac{1}{2}\pi $, $ \alpha = \frac{1}{3}\pi $, and $ \alpha = \frac{1}{4}\pi $, we obtain the system of equations
From the first and second equations of the system (2), we get $ z = x $, $ y = -x $; substituting into the third of these equations gives the equation... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXII OM - I - Zadanie 5
Dana jest liczba naturalna $ N $. Obliczyć liczbę takich ciągów $ (x_1, \ldots, x_N) $, że $ x_i = 7 $ albo $ x_i = —13 $ ($ i = 1, \ldots, N $) oraz
|
Przypuśćmy, że dla pewnej liczby naturalnej $ N $ istnieje ciąg $ (x_1, \ldots, x_N) $ spełniający warunki zadania. Oznaczmy przez $ t $ liczbę wyrazów tego ciągu równych $ 7 $. Wtedy z warunków zadania wynika, że $ 7t = 13(N - t) $, czyli $ 20r = 13N $. Wynika stąd, że liczba $ N $ jest podzielna przez $ 20 $.
Na odw... | \binom{20r}{13r} | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false |
X OM - I - Problem 12
Prove that if in a triangle $ ABC $ the numbers $ \tan A $, $ \tan B $, and $ \tan C $ form an arithmetic progression, then the numbers $ \sin 2A $, $ \sin 2B $, and $ \sin 2C $ also form an arithmetic progression. | From the assumptions of the problem, it follows that
thus
therefore
so
and finally
which was to be proved.
Note. The converse statement is also true: if the angles of triangle $ABC$ satisfy equation (2), then they also satisfy equation (1). Indeed, based on the transformations performed ab... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XI OM - I - Problem 5
Find all integer solutions $ x $ and $ y $ satisfying the equation
保留了源文本的换行和格式，但最后一句是中文，翻译成英文应为: "The line and format of the source text are retained." 但根据上下文，最后一句可能不需要翻译。所以，最终的翻译结果如下：
XI OM - I - Problem 5
Find all integer solutions $ x $ and $ y $ satisfying the equation
... | Since
the equation (1) can be written in the form
Substituting in (2)
we obtain the equation
If the pair of integers $ (x, y) $ is a solution to equation (2), then $ y $ is an even number, so $ (u, v) $ is a pair of integers satisfying equation (3). Conversely, for every pair $ (u, v) $ of integ... | x=-1,y=0 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
LIV OM - I - Task 7
At Aunt Reni's, $ n \geq 4 $ people met (including Aunt Reni). Each of the attendees gave at least one gift to at least one of the others. It turned out that everyone gave three times as many gifts as they received, with one exception: Aunt Reni gave only $ \frac{1}{6} $ of the number of gifts she ... | Let $ x_i $ ($ i = 1,2, \ldots ,n-1 $) denote the number of gifts received by the $ i $-th guest of Aunt Renia, and let $ x_R $ denote the number of gifts Aunt Renia received. Then the total number of gifts is
Therefore, the number $ x_R $ is divisible by $ 12 $, say $ x_R = 12k $, where $ k $ is some non-negative int... | 12[(n+3)/5] | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
LII OM - I - Problem 2
Points $ D $ and $ E $ lie on sides $ BC $ and $ AC $ of triangle $ ABC $, respectively. Segments $ AD $ and $ BE $ intersect at point $ P $. Points $ K $ and $ L $ lie on sides $ BC $ and $ AC $, respectively, such that quadrilateral $ CLPK $ is a parallelogram. Prove that | Since $ KD \parallel LP $, $ DP \parallel PA $, and $ PK \parallel AL $, triangles $ KDP $ and $ LPA $ are similar (Fig. 1). Therefore,
Similarly, using the similarity of triangles $ LEP $ and $ KPB $, we prove that $ BK \cdot EL = PL \cdot PK $. By the last two equalities, we obtain
om52_1r_img_1.jpg
Hence, | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXVI OM - I - Problem 7
Given is a family $ A $ of two-element subsets of the set of natural numbers. Prove that there exists an infinite subset $ M $ of the set of natural numbers such that either all two-element subsets of $ M $ belong to $ A $, or no two-element subset of $ M $ belongs to $ A $. | We define inductively a descending sequence $ X_0 \subset X_1 \subset X_2 \subset X_3 \subset \ldots $ of infinite subsets of the set of natural numbers $ \mathbb{N} $, as well as two subsets of the set $ \mathbb{N} $, which we will denote by $ M $ and $ M $.
Let $ X_0 = \mathbb{N} $. Fix $ n \in \mathbb{N} $ and supp... | proof | Combinatorics | proof | Yes | Incomplete | olympiads | false |
VI OM - I - Task 1
Solve the system of equations
| Since
therefore, by introducing the notation
we can replace the system of equations (1) with the equivalent system
which is equivalent to the system
The system of equations (2), and thus system (1), has $ 4 $ solutions:
[ x_1= 3, y_1 = 3, z_1 = 1, u_1 = 1; \]
[ x_2= 3, y_2 = 1, z_2 ... | \begin{aligned}&x_1=3,y_1=3,z_1=1,u_1=1;\\&x_2=3,y_2=1,z_2=1,u_2=3;\\&x_3=1,y_3=3,z_3=3,u_3=1;\\&x | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XIX OM - II - Task 3
Prove that if at least 5 people are sitting around a round table, then they can be rearranged so that each person has two different neighbors than before. | \spos{1} Let us denote the given people by natural numbers from $1$ to $n$ ($n \geq 5$) and assume that person $1$ is seated next to persons $n$ and $2$, person $2$ is seated next to persons $1$ and $3$, and so on, finally person $n$ is seated next to persons $n-1$ and $1$. We will simply say that the numbers from $1$ ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
VIII OM - III - Task 2
Prove that between the sides $ a $, $ b $, $ c $ and the opposite angles $ A $, $ B $, $ C $ of a triangle, the following relationship holds | Substituting in equation (1)
($ R $ - the radius of the circle circumscribed around the triangle), after dividing both sides by $ R^2 $, we get:
It suffices to justify formula (3), since formula (1) immediately follows from formulas (3) and (2). We will transform the right side of formula (3)
Th... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXV OM - III - Problem 1
Determine the number of functions $ f $ mapping an $ n $-element set to itself such that $ f^{n-1} $ is a constant function, while $ f^{n-2} $ is not a constant function, where $ f^k = f\circ f \circ \ldots \circ f $, and $ n $ is a fixed natural number greater than 2. | Suppose that $ A = \{a_1, a_2, \ldots, a_n \} $ is a given $ n $-element set, and $ f $ is a function satisfying the conditions of the problem. Consider the sets $ A, f(A), f^2(A), \ldots, f^{n-1}(A) $. From the assumption, it follows that the set $ f^{n-1}(A) $ is a singleton, while the set $ f^{n-2}(A) $ has more tha... | n! | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XLVII OM - II - Problem 2
A circle with center $ O $ inscribed in a convex quadrilateral $ ABCD $ is tangent to the sides $ AB $, $ BC $, $ CD $, $ DA $ at points $ K $, $ L $, $ M $, $ N $, respectively, and the lines $ KL $ and $ MN $ intersect at point $ S $. Prove that the lines $ BD $ and $ OS $ are perpendicular... | Let the orthogonal projections of points $ B $ and $ D $ onto the line $ OS $ be denoted by $ U $ and $ V $, respectively. The angles $ OKB $, $ OLB $, and $ OUB $ are right angles, and thus the points $ K $, $ L $, and $ U $ lie on a circle whose diameter is the segment $ OB $ (Figure 7). (The point $ U $ may coincide... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVI OM - II - Problem 2
Prove that for a natural number $ n > 2 $, the number $ n! $ is the sum of $ n $ distinct divisors of itself. | We prove by induction a slightly stronger thesis: For every natural number $ n > 2 $, the number $ n! $ is the sum of $ n $ distinct divisors of itself, the smallest of which is $ 1 $. For $ n = 3 $, this is true: $ 3! = 6 = 1+2+3 $. Assume the thesis is true for some $ n $:
where the natural numbers $ k $ are divisor... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XVIII OM - I - Problem 8
Given a sphere $ k $, a circle $ c $ lying on its surface, and a point $ S $ not lying on this surface, such that every line passing through $ S $ and through some point $ M $ of the circle $ c $ intersects the surface of the sphere $ k $ at another point $ N $ different from $ M $. Prove that... | When point $ S $ lies in the plane of circle $ c $, the proof of the theorem's thesis is immediate, for $ S $ is then inside circle $ c $ (since points lying outside $ c $ do not satisfy the adopted assumption) and the set of all points $ N $ is circle $ c $.
Assume that point $ S $ does not lie in the plane of circle ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVII OM - III - Zadanie 4
Dowieść, że jeżeli liczby nieujemne $ x_1, x_2,\ldots,x_n $ ($ n $ - dowolna liczba naturalna) spełniają nierówność
$ x_1 + x_2 + \ldots + x_n \leq \frac{1}{2} $, to
|
Zastosujemy metodę indukcji. Dla $ n = 1 $ twierdzenie jest prawdziwe, gdyż jeśli $ x_1 \leq \frac{1}{2} $, to $ 1 - x_1 \geq \frac{1}{2} $. Załóżmy, że jest ono prawdziwe dla pewnego naturalnego $ k $, tzn. że jeśli liczby $ x_1, x_2, \ldots, x_k $ są nieujemne i $ x_1+x_2+\ldots+x_k \leq \frac{1}{2} $, to $ (1-x_1) ... | proof | Inequalities | proof | Incomplete | Incomplete | olympiads | false |
XLV OM - II - Task 2
The numbers $ a_1, a_2, \ldots , a_n $, $ b_1, b_2, \ldots , b_n $ satisfy the conditions:
Prove that $ \sum_{i=1}^n b_i \leq \prod_{i=1}^n b_i $. | Let's assume
(These symbols denote, respectively, the product of all numbers $ a_j $ and the product of all numbers $ b_j $ with indices $ j $ different from $ i $.) From the assumption that $ a_j \leq b_j $ (for all $ j $), it follows that $ A_i \leq B_i $ (for all $ i $). In such a case,
It remains to not... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XLVI OM - I - Problem 7
(a) Determine whether there exists a differentiable function $ f: \mathbb{R} \to \mathbb{R} $ not identically zero, satisfying the conditions:
(b) Determine whether there exists a differentiable function $ f: \mathbb{R} \to \mathbb{R} $ not identically zero, satisfying the conditions: | (a) There is no such function. Proof: Suppose that the function $ f $ satisfies the given conditions. It is not identically zero, so there exists $ x_0 \in \mathbb{R} $, for which $ f(x_0) > 0 $. Let us take
From the equality $ 2f(f(x_{n-1})) = f(x_{n-1}) $, we conclude that
Thus, by obvious induction,
... | proof | Calculus | proof | Incomplete | Yes | olympiads | false |
XXX OM - III - Task 5
Prove that the product of the lengths of the sides of a convex quadrilateral inscribed in a circle of radius 1 is not greater than 4. | Let $ABC$ be a triangle inscribed in a circle and let $B$ be the midpoint of the arc $\widehat{ABC}$ (Fig. 16).
om30_3r_img_16.jpg
We will prove that $AB \cdot BC \leq AB$.
The area $S_{ABC}$ of triangle $ABC$ is not greater than the area $S_{AB}$ of triangle $AB$ because the distance from point $B$ to the line $AC$ is... | notfound | Geometry | proof | Yes | Incomplete | olympiads | false |
XXX OM - II - Task 4
Let $ S_k $ be the symmetry of the plane with respect to the line $ k $. Prove that for any lines $ a, b, c $ contained in the same plane, the following equality holds
| As is known, every isometry of the plane that is the composition of an odd number of axial symmetries is a glide reflection. If $ S $ is a glide reflection, then $ S^2 $ is a translation.
Therefore, the isometries
are translations. The composition of two translations does not depend on their order: $ TU = UT $. ... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXIX OM - III - Problem 4
Let $ X $ be an $ n $-element set. Prove that the sum of the number of elements in the sets $ A \cap B $ extended over all ordered pairs $ (A, B) $ of subsets of $ X $ equals $ n \cdot 4^{n-1} $. | Let $ a $ be a fixed element of the set $ X $. The number of subsets of the set $ X $ containing the element $ a $ is equal to the number of all subsets of the $(n-1)$-element set $ X- \{a\} $, which is $ 2^{n-1} $. Therefore, the number of such pairs $ (A, B) $ of subsets of the set $ X $, such that $ a \in A \cap B $... | n\cdot4^{n-1} | Combinatorics | proof | Yes | Yes | olympiads | false |
XXVII OM - I - Problem 10
Points $ A $ of plane $ Q $ are the parallel projections of points $ A, B, C, D $ of plane $ P $, where points $ A, B, C, D, A $ are distinct and no three of them lie on a line. Prove that the tetrahedra $ ABCD $ and $ A $ have equal volumes. | Let $E$ be the point of intersection of lines $AC$ and $BD$, and $E$ be the point of intersection of lines $A$ and $B$ (Fig. 9). Since points $A$, $B$, $C$, $D$ are the respective parallel projections of points $A$, $B$, $C$, $D$, the projections of lines $AC$ and $BD$ are lines $A$ and $B$, and therefore the projectio... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
XII OM - I - Problem 12
Prove that if all the faces of a tetrahedron are congruent triangles, then these triangles are acute. | Suppose each face of the tetrahedron $ABCD$ (Fig. 11) is a triangle with sides $a$, $b$, $c$; let, for example, $BC = a$, $CA = b$, $AB = c$. There are 3 cases, which we will consider in turn. 1° - None of the lengths $a$, $b$, $c$ are equal. Two edges emanating from the same vertex cannot then be equal, since they are... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LV OM - II - Task 2
In a convex hexagon $ ABCDEF $ all sides are of equal length and
Prove that the diagonals $ AD $, $ BE $ and $ CF $ intersect at one point. | From the given angle equality in the problem, it follows that $ \measuredangle B + \measuredangle D + \measuredangle F = 360^\circ $. The lengths of all sides of the given hexagon are equal, so from triangles $ ABC $, $ CDE $, and $ EFA $, we can form a triangle $ PCE $ congruent to triangle $ ACE $ (Fig. 1). Point $ D... | proof | Geometry | proof | Incomplete | Incomplete | olympiads | false |
XXXIV OM - II - Problem 3
Point $ P $ lies inside triangle $ ABC $, such that $ \measuredangle PAC = \measuredangle PBC $. Points $ L $ and $ M $ are the projections of $ P $ onto lines $ BC $ and $ CA $, respectively, and $ D $ is the midpoint of segment $ AB $. Prove that $ DL = DM $. | Let $ Q $ and $ R $ be the midpoints of segments $ AP $ and $ BP $, respectively. These points are the centers of the circles circumscribed around triangles $ APM $ and $ BPL $, which implies that
as central angles subtending the same arcs as the corresponding inscribed angles. Angles $ PAM $ and $ PBL $ are equal by ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
VII OM - I - Problem 8
Given an acute triangle $ ABC $ and a point $ M $ inside one of its sides. Construct a triangle with the smallest perimeter, one of whose vertices is the point $ M $, and the other two lie on the two remaining sides of the triangle. | Let's consider triangle $MNP$ whose vertices lie on the sides of an acute triangle $ABC$, as indicated in Fig. 6.
Let $M'$ be the point symmetric to point $M$ with respect to the line $AB$, and $M''$ be the point symmetric to point $M$ with respect to the line $AC$. Then $PM'$ and $NM''$. Therefore, the length of the b... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LII OM - II - Problem 6
For a given positive integer $ n $, determine whether the number of $ n $-element subsets of the set $ \{1,2,3,\ldots,2n-1,2n\} $ with an even sum of elements is the same as the number of $ n $-element subsets with an odd sum of elements. If not, determine which is greater and by how much. | Let's connect the numbers from the set $ S = \{1,2,3,\ldots,2n-1,2n\} $ in pairs:
Let $ X $ be any $ n $-element subset of the set $ S $. Choose (if possible) the smallest number $ x $ from the set $ X $ with the property that the number $ y $ paired with $ x $ does not belong to the set $ X $. Then remove the e... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
IX OM - I - Problem 8
What plane figure is the cross-section of a cube with edge $ a $ by a plane passing through the midpoints of three pairwise skew edges? | Each edge of a cube meets four other edges, is parallel to three further ones, and is skew to four remaining pairs of parallel edges. If we arbitrarily choose one edge of the cube $ABCDA$ (Fig. 8), for example, $AB$, then we can select two other edges that are skew to $AB$ and to each other in two ways: $AB$, $CC_1$, $... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXIII OM - III - Problem 2
In a cyclic quadrilateral $ABCD$, a line passing through the midpoint of side $\overline{AB}$ and the intersection of the diagonals is perpendicular to side $\overline{CD}$. Prove that sides $\overline{AB}$ and $\overline{CD}$ are parallel or the diagonals of the quadrilateral are perpendic... | Let $O$ be the point of intersection of the diagonals, $E$ - the midpoint of side $\overline{AB}$, $F$ - the point of intersection of side $\overline{CD}$ and line $EC$. Since quadrilateral $ABCD$ is inscribed in a circle, therefore
Thus
By applying the Law of Sines to triangles $AOE$ and $BOF$ in sequence, we get
S... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XI OM - III - Task 3
On a circle, 6 different points $ A $, $ B $, $ C $, $ D $, $ E $, $ F $ are chosen in such a way that $ AB $ is parallel to $ DE $, and $ DC $ is parallel to $ AF $. Prove that $ BC $ is parallel to $ EF $. | The given points can lie on the circle in various ways. To solve the problem correctly, we need to provide a proof that is independent of the specific arrangement of the points. Let's reason without any diagram.
First, note that if two chords of a circle are parallel, for example, \( MN \parallel PQ \), then the point... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXI OM - I - Problem 12
Find the smallest positive number $ r $ with the property that in a regular tetrahedron with edge length 1, there exist four points such that the distance from any point of the tetrahedron to one of them is $ \leq r $. | Let $ O $ be the center of the sphere circumscribed around the regular tetrahedron $ ABCD $ with edge $ 1 $ (Fig. 9). The radius of the sphere circumscribed around $ ABCD $ is $ R = \frac{1}{4} \sqrt{6} $. The tetrahedron $ ABCD $ is divided into tetrahedra with vertices: $ O $, one of the vertices of the tetrahedron, ... | \frac{\sqrt{6}}{8} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLVIII OM - I - Problem 9
Determine all functions $ f: \langle 1,\ \infty) \to \langle 1,\ \infty) $ that satisfy the following conditions:
[(i)] $ f(x+1) = \frac{(f(x))^2-1}{x} $ for $ x \geq 1 $;
[(ii)] the function $ g(x)=\frac{f(x)}{x} $ is bounded. | It is easy to notice that the given conditions are satisfied by the function $ f(x) = x + 1 $. We will show that there are no other such functions.
Suppose, therefore, that $ f: \langle 1,\ \infty) \to \langle 1,\ \infty) $ is any function satisfying conditions (i) and (ii). We define the function $ h: \langle 1,\ \inf... | f(x)=x+1 | Algebra | proof | Yes | Yes | olympiads | false |
XXIII OM - II - Problem 5
Prove that in a convex quadrilateral inscribed in a circle, the lines passing through the midpoints of the sides and perpendicular to the opposite sides intersect at one point. | Let the convex quadrilateral $ABCD$ be inscribed in a circle. Denote by $P$, $Q$, $R$, $S$ the midpoints of the sides of this quadrilateral, and by $O$ - the midpoint of the segment $PR$ (Fig. 14).
Thus *)
Since $\displaystyle \frac{1}{2} (Q + S) = \frac{1}{4} (A + B + C + D) = O$, the point $O$ is also the midpoint o... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXIV OM - I - Problem 1
Prove that for every triangle, there exists a line dividing the triangle into two figures with equal perimeters and equal areas. | In an isosceles triangle, the line containing the bisector of the angle formed by its legs satisfies the conditions of the problem. Let the side lengths of triangle $ABC$ be different, for example, $a < b < c$, where $a = BC$, $b = AC$, $c = AB$ (Fig. 6). Let $p = \frac{1}{2} (a + b + c)$.
If $p - a \leq x \leq c$, the... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXIV OM - I - Problem 8
Determine whether 20 regular tetrahedra with edge length 1, having pairwise disjoint interiors, can fit inside a sphere of radius 1. | Consider a regular icosahedron inscribed in a sphere. Each face of this icosahedron is an equilateral triangle with a side length greater than 1 (compare with preparatory problem E from the 31st Mathematical Olympiad). Therefore, in the spherical triangle bounded by the arcs of three great circles passing through pairs... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXVII OM - I - Zadanie 7
Rozwiązać równanie
gdzie $ a, b, c $ są danymi liczbami dodatnimi.
|
Przyjmijmy oznaczenia: $ m = \min(a, b, c) $,
Za dziedzinę funkcji $ f $ przyjmujemy przedział $ \langle 0; m \rangle $ (bo jedynie dla $ x \leq m $ wszystkie wyrażenia podpierwiastkowe są nieujemne oraz jedynie wśród nieujemnych liczb $ x $ należy szukać rozwiązania równania). Dane w zadaniu równanie przybiera ... | not | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXXIII OM - III - Problem 4
In the plane, a finite set of points is given. Prove that this set can be covered by disjoint open squares $ Q_1, Q_2, \ldots, Q_n $, such that
($ j = 1, 2,\ldots, n $) where $ N_j $ is the number of points of the set lying in the square $ Q_j $, and $ S_j $ is the area of the square $ Q_j... | We choose such a coordinate system on the plane that the set of points, whose number we denote by $N$, lies within the first quadrant and additionally, none of the coordinates of its points can be represented in the form $ \frac{k}{2n} $, where $k$, $n$ are natural numbers. Let $d$ be such a natural number that the squ... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
LVIII OM - I - Problem 9
Let $ F(k) $ be the product of all positive divisors of the positive integer $ k $. Determine whether there exist different positive integers $ m $, $ n $, such that $ {F(m)=F(n)} $. | Answer: Such numbers $ m $ and $ n $ do not exist.
Let $ 1=d_1<d_2<\ldots<d_k=n $ be all the positive divisors of a fixed positive integer $ n $. Then $ n/d_1 $, $ n/d_2 $, $ \ldots $, $ n/d_k $ are also all the positive divisors of the number $ n $, so we can write
Thus, we have
where $ d(n) $ denotes the number... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXXI - I - Problem 11
Prove that $ \tan \frac{\pi}{7} \tan \frac{2\pi}{7} \tan \frac{3\pi}{7} = \sqrt{7} $. | We will express the trigonometric functions of the angle $7 \alpha$ in terms of the trigonometric functions of the angle $\alpha$. This can be done using known trigonometric formulas or by utilizing properties of complex numbers. We will use the following de Moivre's formula:
For $n = 7$ and $x_k = \frac{k\pi}{7}\ (k ... | \sqrt{7} | Algebra | proof | Yes | Yes | olympiads | false |
IV OM - I - Task 4
Prove that the number $ 2^{55} + 1 $ is divisible by $ 11 $. | It is known that if $ n $ is a natural number, and $ a $ and $ b $ are any numbers, then
if we substitute in this equality $ a = 2^5 $, $ b = - 1 $, $ n = 11 $, we get
where $ C $ is an integer. The number $ 2^{55} + 1 $ is therefore divisible by $ 33 $, and thus also by $ 11 $.
Note 1. In the same way, we can genera... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
III OM - III - Task 3
Construct a quadrilateral $ABCD$ given the lengths of sides $AB$ and $CD$ and the angles of the quadrilateral. | Let's denote the lengths of the given sides of the quadrilateral as $ AB = a $, $ CD = c $. Consider the sums of consecutive angles of the quadrilateral:
and from these sums, form pairs:
In each of these pairs, one of the sums does not exceed $ 180^\circ $, since the sum of all four angles $ A + B + C + D $ is $ 360^... | notfound | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XV OM - I - Zadanie 6
Rozłożyć na czynniki wielomian
|
Funkcję $ P (x, y, z) $ możemy traktować jako wielomian stopnia $ 5 $ względem $ x $. Wielomian ten dla $ x = y $ równa się zeru, jest on zatem podzielny przez $ x - y $. Analogicznie stwierdzamy, że $ P (x, y, z) $ jest podzielne przez $ y - z $ i przez $ z - x $. Zatem
gdzie $ Q (x, y, z) $ jest wielomianem st... | x^2+y^2+z^2-xy-yz-zx=(x-\varepsilony)(x-\varepsilon^2y)+(y-\varepsilonz)(y-\varepsilon^2z)+(z-\varepsilonx)(z-\varepsilon^2x) | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
LVI OM - I - Task 2
Determine all natural numbers $ n>1 $ for which the value of the sum $ 2^{2}+3^{2}+\ldots+n^{2} $ is a power of a prime number with a natural exponent. | From the formula $ 1+2^{2}+3^{2}+ \ldots+n^{2}=\frac{1}{6}n(n+1)(2n+1) $, true for any natural number $ n $, we obtain
The task thus reduces to finding all natural numbers $ n>1 $, prime numbers $ p $, and positive integers $ k $ for which the equality
is satisfied.
Since the factor $ 2n^2 +5n+6 $ is greater tha... | 2,3,4,7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XL OM - II - Task 3
Given is a trihedral angle $ OABC $ with vertex $ O $ and a point $ P $ inside it. Let $ V $ be the volume of the parallelepiped with two vertices at points $ O $ and $ P $, whose three edges are contained in the rays $ OA^{\rightarrow} $, $ OB^{\rightarrow} $, $ OC^{\rightarrow} $. Calculate the m... | Let $ \mathbf{Q} $ denote the considered parallelepiped with volume $ V $. Let $ OA $, $ OB $, $ OC $ be its three edges. We draw a plane through point $ P $ intersecting the rays $ OA^\rightarrow $, $ OB^\rightarrow $, $ OC^\rightarrow $ at points $ K $, $ L $, $ M $ respectively (thus forming the tetrahedron $ OKLM $... | \frac{9}{2}V | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LIX OM - I - Problem 11
Points $ P_1, P_2, P_3, P_4, P_5, P_6, P_7 $ lie on the sides $ BC, CA, AB, BC, CA, AB, BC $ of triangle $ ABC $, respectively, and the following equalities are satisfied:
Prove that $ P_1 = P_7 $. | Let \( o \) be the circumcircle of triangle \( P_1P_2P_3 \).
From the conditions of the problem, we have the equality \( \measuredangle P_1P_2P_3 = 60^{\circ} \). This, combined with the equality
\( \measuredangle P_3P_4B = 60^{\circ} \), proves that point \( P_4 \) lies on the circle \( o \), regardless of whether it... | P_1=P_7 | Geometry | proof | Incomplete | Yes | olympiads | false |
XV OM - II - Task 1
Prove that if $ n $ is a natural number, and angle $ \alpha $ is not a multiple of $ \frac{180^{\circ}}{2^n} $, then | We will apply complete induction.
1. If $ \alpha $ is not a multiple of $ 90^\circ $, then
The theorem we need to prove is therefore true when $ n=1 $.
2. Suppose the theorem is true for some natural number $ n $, i.e., if $ \alpha $ is not a multiple of $ \frac{180^\circ}{2^n} $, then formula (1) holds.
Assume ... | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
XXVI - III - Task 4
In the decimal expansion of a certain natural number, the digits 1, 3, 7, and 9 appear. Prove that by permuting the digits of this expansion, one can obtain the decimal expansion of a number divisible by 7. | By performing an appropriate permutation of the digits of a given natural number, we can assume that its last four digits are $1$, $3$, $7$, and $9$. Thus, the considered natural number $n$ is the sum of the number $1379$ and some non-negative integer $a$, whose last four digits are zeros. We will prove that by perform... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XIX OM - I - Problem 12
Given is a tetrahedron $ A_1A_2A_3A_4 $. Prove that if points $ P_1, P_2, P_3, P_4 $ are the orthogonal projections of some point $ P $ in space onto the lines $ A_1A_2, A_2A_3, A_3A_4, A_4A_1 $, respectively, then the sum
is minimized when the point $ P $ is the center of the circumscrib... | To simplify the notation, we introduce the following segment lengths: $A_iA_{i+1} = a_i$, $PA_i = x_i$, $P_iA_i = y_i$ ($i = 1, 2, 3, 4$), where the index $i+1$ for $i = 4$ is replaced by $1$.
For each quadruple of points $A_i$, $A_{i+1}$, $P$, $P_i$, the following relationship holds:
Indeed, if the points $A_i$, $A_{... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
VI OM - II - Task 4
Inside the triangle $ ABC $, a point $ P $ is given; find a point $ Q $ on the circumference of this triangle such that the broken line $ APQ $ divides the triangle into two parts of equal area. | Let's consider triangles $ APB $ and $ APC $. There are three possible cases:
a) Each of the triangles $ APB $ and $ APC $ has an area less than half the area of triangle $ ABC $. This case occurs when the distances from point $ P $ to the sides $ AB $ and $ BC $ are less than half the corresponding heights of the tria... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXIII OM - II - Problem 3
The coordinates of the vertices of a triangle in the Cartesian coordinate system $ XOY $ are integers. Prove that the diameter of the circumcircle of this triangle is not greater than the product of the lengths of the sides of the triangle. | If the lengths of the sides of a triangle are $a$, $b$, $c$, and the length of the radius of the circle circumscribed around the triangle is $R$, then the area of the triangle is given by the formula
If the coordinates of the vertices of the triangle are $A = (a_1, a_2)$, $B = (b_1, b_2)$, $C = (c_1, c_2)$, then its a... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIX OM - III - Problem 6
We consider unit squares on the plane, whose vertices have both integer coordinates. Let $ S $ be a chessboard whose fields are all unit squares contained in the circle defined by the inequality $ x^2+y^2 \leq 1998^2 $. On all the fields of the chessboard, we write the number $ +1 $. We perfo... | Answer: It is not possible.
Suppose it is possible to obtain a configuration with exactly one field $ P_1 $, on which the number $ -1 $ is written. Due to the standard symmetries of the chessboard $ S $, we can assume that the field $ P_1 $ has its center at $ (a-\frac{1}{2},b-\frac{1}{2}) $, where $ a \geq b \geq 1 $ ... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XLVIII OM - I - Problem 7
Calculate the upper limit of the volume of tetrahedra contained within a sphere of a given radius $ R $, one of whose edges is the diameter of the sphere. | Let $ABCD$ be one of the considered tetrahedra. Assume that the edge $AB$ is the diameter of the given sphere. Denote the distance from point $C$ to the line $AB$ by $w$, and the distance from point $D$ to the plane $ABC$ by $h$. The volume of the tetrahedron $ABCD$ is given by
Of course, $w \leq R$, $h \leq R$, and t... | \frac{1}{3}R^3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
IX OM - I - Problem 1
Prove that if each of the functions
[ (1) \qquad (a_1x + b_1)^2 + (a_2x + b_2)^2, \]
[ (2) \qquad (b_1x + c_1)^2 + (b_2x + c_2)^2 \]
is the square of a linear function that is not constant, then the function
[ (3) \qquad (c_1x + a_1)^2 + (c_2x + a_2)^2 \]
is the square of a linear ... | First, let us note that if the function $ Ax^2 + 2Bx + C $ is the square of a linear function $ mx + n $, then its discriminant vanishes, i.e., $ B^2 - AC = 0 $. Indeed, from the identity $ Ax^2 + 2Bx + C = m^2x^2 + 2mnx + n^2 $, it follows that $ B^2 - AC = (mn)^2 - m^2n^2 = 0 $.
Conversely, if $ B^2 - AC = 0 $ and $ ... | proof | Algebra | proof | Yes | Yes | olympiads | false |
LIV OM - III - Task 4
Given a prime number p and such integers $ x $, $ y $, $ z $, that
Prove that if the numbers $ x^3 $, $ y^3 $, $ z^3 $ give the same remainders when divided by $ p $, then the number $ x^2 + y^2 + z^2 $ is divisible by $ x + y + z $. | Since the number $ y^3 - x^3 = (y - x)(x^2 + xy + y^2) $ is divisible by $ p $ and $ 0 < y - x < p $, the number $ x^2 + xy + y^2 $ is divisible by $ p $. Similarly, the numbers $ y^2 + yz + z^2 $ and $ x^2 + xz + z^2 $ are also divisible by $ p $. Therefore, the number
is also divisible by $ p $, which implies that... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXI OM - I - Problem 10
Find the largest natural number $ k $ with the following property: there exist $ k $ different subsets of an $ n $-element set, such that any two of them have a non-empty intersection. | Let the sought number be denoted by $ k(n) $. We will prove that $ k(n) = 2^{n-1} $. To each $ n $-term sequence with values $ 0 $ or $ 1 $, we assign the set of indices such that the corresponding terms of the sequence are equal to $ 1 $. This assignment establishes a one-to-one correspondence between the set of all $... | 2^{n-1} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
LIX OM - II - Task 1
Determine the maximum possible length of a sequence of consecutive integers, each of which can be expressed in the form $ x^3 + 2y^2 $ for some integers $ x, y $. | A sequence of five consecutive integers -1, 0, 1, 2, 3 satisfies the conditions of the problem: indeed, we have
On the other hand, among any six consecutive integers, there exists a number, say
$ m $, which gives a remainder of 4 or 6 when divided by 8. The number $ m $ is even; if there were a representation
in... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XV OM - I - Problem 11
In triangle $ ABC $, angle $ A $ is $ 20^\circ $, $ AB = AC $. On sides $ AB $ and $ AC $, points $ D $ and $ E $ are chosen such that $ \measuredangle DCB = 60^\circ $ and $ \measuredangle EBC = 50^\circ $. Calculate the angle $ EDC $. | Let $ \measuredangle EDC = x $ (Fig. 9). Notice that $ \measuredangle ACB = \measuredangle $ABC$ = 80^\circ $, $ \measuredangle CDB = 180^\circ-80^\circ-60^\circ = 40^\circ $, $ \measuredangle CEB = 180^\circ - 80^\circ-50^\circ = \measuredangle EBC $, hence $ EC = CB $. The ratio $ \frac{DC}{CE} $ of the sides of tr... | 30 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLVI OM - II - Problem 6
A square with side length $ n $ is divided into $ n^2 $ unit squares. Determine all natural numbers $ n $ for which such a square can be cut along the lines of this division into squares, each of which has a side length of 2 or 3. | If $ n $ is an even number, then a square of dimensions $ n \times n $ can be cut into $ 2 \times 2 $ squares. Let's assume, then, that $ n $ is an odd number. The square $ n \times n $ consists of $ n $ rows, with $ n $ cells (unit squares) in each row. We paint these rows alternately black and white (in a "zebra" pat... | nisdivisible2or3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXVI - III - Task 2
On the surface of a regular tetrahedron with edge length 1, a finite set of segments is chosen in such a way that any two vertices of the tetrahedron can be connected by a broken line composed of some of these segments. Can this set of segments be chosen so that their total length is less than $1 +... | On the surface of a regular tetrahedron with edge length $1$, one can choose a finite set of segments such that any two vertices of the tetrahedron can be connected by a broken line composed of some of these segments, and the total length of the segments belonging to this set is less than $1 + \sqrt{3}$.
Consider, in p... | \sqrt{7} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XVI OM - I - Problem 7
On the side $ AB $ of triangle $ ABC $, a point $ K $ is chosen, and points $ M $ and $ N $ are determined on the lines $ AC $ and $ BC $ such that $ KM \perp AC $, $ KN \perp BC $. For what position of point $ K $ is the area of triangle $ MKN $ the largest? | Let $ BC = a $, $ AC = b $, $ \measuredangle ACB = \gamma $, $ KM = p $, $ KN = q $ (Fig. 2).
Notice that $ \measuredangle MKN = 180^\circ - \gamma $.
Indeed, if from a point $ K_1 $ chosen on the bisector of angle $ ACB $ we drop perpendiculars $ K_1M_1 $ and $ K_1N_1 $ to the sides of this angle, then the rays $ KM $... | KisthemidpointofsideAB | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XL OM - II - Task 1
Solve the equation
Note: The equation to solve is not provided in the original text. | We rewrite the given equation in the form
where
both $ f $ and $ g $ are periodic functions with a period of $ \pi $. It is sufficient to seek solutions in the set $ (-\pi/2; 0) \cup (0; \pi/2) $, as the point $ 0 $ and the endpoints of the interval $ (-\pi/2; \pi/2) $ have been removed as they do not belon... | \frac{\pi}{4}+k\pi | Other | math-word-problem | Incomplete | Yes | olympiads | false |
I OM - I - Task 3
Construct an equilateral triangle whose vertices lie on three given parallel lines. | Analyze. Let $ABC$ be an equilateral triangle, whose vertices $A, B, C$ lie on the given parallel lines $a, b, c$ (Fig. 13a).
om1_Br_img_13a.jpg
Rotate the entire figure around point $A$ by the angle $BAC = 60^{\circ}$. Point $B$ will end up at point $C$ after the rotation, and line $b$ will take the position $b_l$. Si... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLII OM - I - Zadanie 7
Dane są liczby rzeczywiste dodatnie $ a $, $ b $. Wyznaczyć wszystkie pary liczb rzeczywistych dodatnich $ x $, $ y $ spełniające równanie
|
Stosując trzykrotnie nierówność między średnią arytmetyczną trójki liczb oraz średnią geometryczną tych liczb stwierdzamy, że
oraz
Mianownik ostatniego ułamka - to dokładnie iloczyn liczb znajdujących się po prawych stronach dwóch poprzednich równości. Wobec tego
(dla dowolnych liczb dodatnich $ a $,... | x=y=\frac{1}{} | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
IV OM - I - Task 2
Prove that the heights $ h_1 $, $ h_2 $, $ h_3 $ of a triangle and the radii $ r_1 $, $ r_2 $, $ r_3 $ of the excircles satisfy the equality
保留了源文本的换行和格式，但最后一句是中文，我将其翻译为英文如下：
Preserve the line breaks and format of the source text. | We will justify the given equality by showing that each of its sides equals the same function of the sides of the triangle. To this end, we need to use the formulas that relate the heights $h_1$, $h_2$, $h_3$ and the radii $r_1$, $r_2$, $r_3$ to the sides of the triangle.
Using the known notations: $S = \sqrt{p (p - a)... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
III OM - III - Task 1
Solve the system of equations
| Let's multiply both sides of the first equation by $3$, and then add the equations side by side; we will obtain an equation
equivalent (in the domain of real numbers) to the equation
The system of equations (1) is equivalent to the system of equations
or to a simpler system
We need to distinguish two cases:
I. $a -... | \begin{aligned}& | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XV OM - I - Problem 4
In the plane, there are 5 points, no three of which are collinear. Prove that among them, there are four points that are the vertices of a convex quadrilateral. | Let $A$, $B$, $C$, $D$, $E$ denote given points. Since no three of them are collinear, points $A$, $B$, $C$ are the vertices of a triangle, and each of the points $D$ and $E$ lies either inside or outside this triangle. Therefore, one of the following cases occurs:
a) Points $D$ and $E$ lie inside the triangle $ABC$. T... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIV OM - III - Task 6
Determine whether the volume of a tetrahedron can be calculated knowing the areas of its four faces and the radius of the circumscribed sphere (i.e., whether the volume of the tetrahedron is a function of the areas of its faces and the radius of the circumscribed sphere). | A tetrahedron is determined, up to isometry, by six parameters (for example, the lengths of the six edges). The areas of the faces and the radius of the circumscribed sphere - this is only five pieces of information (equations). Hence the conjecture that the answer to the posed question should be negative. (We emphasiz... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXVII OM - III - Problem 3
Prove that for any tetrahedron, the products of the lengths of its opposite edges are the lengths of the sides of some triangle. | Let $ABCD$ be a given tetrahedron. On the rays $AB^\to$, $AC^\to$, and $AD^\to$, we find points $B$, $C$, and $D$ respectively such that (1) $AB$, $AC$, $AD$, (Fig. 15). We will show that the triangle $B$ satisfies the conditions of the problem.
First of all, from (1) it follows that
Therefore, triangles $ABC$ and $AC... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXXIII OM - III - Problem 6
Prove that in any tetrahedron, the sum of all dihedral angles is greater than $2\pi$. | We will first prove the following lemma.
Lemma. The sum of the measures of the planar angles at the vertices of a trihedral angle is less than $2\pi$.
Proof. Let $O$ be the vertex of the trihedral angle, $A$, $B$, $C$ - points lying on the respective edges at equal distance from $O$.
om33_3r_img_8.jpg
Let $O'$ be the o... | |\alpha|+|\beta|+|\gamma|+|\delta|+|\varepsilon|+|\zeta|>2\pi | Geometry | proof | Yes | Yes | olympiads | false |
LVIII OM - I - Problem 7
Given is a tetrahedron $ABCD$. The angle bisector of $\angle ABC$ intersects the edge $AC$ at point $Q$. Point $P$ is symmetric to $D$ with respect to point $Q$. Point $R$ lies on the edge $AB$, such that $BR = \frac{1}{2}BC$. Prove that a triangle can be formed with segments of lengths $BP$, ... | Let $ S $ be the midpoint of edge $ BC $, and $ T $ - the point symmetric to point $ C $ with respect to point $ Q $ (Fig. 3).
om58_1r_img_3.jpg
Since $ BS={1\over 2}BC=BR $, points $ R $ and $ S $ are symmetric with respect to the angle bisector $ BQ $ of angle $ ABC $, and thus $ QR=QS $. Furthermore, triangle $ CTB... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXI - II - Problem 1
Students $ A $ and $ B $ play according to the following rules: student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length 1 on the plane, then student $ B $ gives a number $ s_1 $, equal to $ 1 $ or $ -1 $; then student $ A $ chooses a vector $ \overrightarrow{a_1} $ of length $ 1 $, and ... | We will prove by induction that student $ B $ can choose the numbers $ \varepsilon_1, \varepsilon_2, \ldots, \varepsilon_n $ such that the length of the vector $ \overrightarrow{\omega_n} = \sum_{k=1}^n \varepsilon_k \overrightarrow{a_k} $ is not less than $ \sqrt{n} $.
For $ n = 1 $, the vector has a length of $ 1 $ r... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
Given an integer $ n \geq 5 $. Determine the number of solutions in real numbers $ x_1, x_2, x_3, \ldots, x_n $ of the system of equations
where $ x_{-1}=x_{n-1}, $ $ x_{0}=x_{n}, $ $ x_{1}=x_{n+1}, $ $ x_{2}=x_{n+2} $. | Adding equations side by side, we obtain
Thus, the numbers $ x_1, x_2, \ldots, x_n $ take values 0 or 1. The task reduces to finding the number of solutions to the system of equations
in numbers belonging to the set $ \{0,1\} $.
Notice that the system of equations (1) is satisfied when $ x_i = x_2 = \dots =... | 2 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
II OM - III - Problem 6
Given a circle and a segment $ MN $. Find a point $ C $ on the circle such that the triangle $ ABC $, where $ A $ and $ B $ are the points of intersection of the lines $ MC $ and $ NC $ with the circle, is similar to the triangle $ MNC $. | First, let us note that if points $M$ and $N$ lie on a given circle $k$, then any point $C$ on the circle, except for points $M$ and $N$, satisfies the condition of the problem; points $A$ and $B$ then coincide with points $M$ and $N$, and triangle $ABC$ coincides with triangle $MNC$. We will henceforth omit this case ... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLIII OM - III - Zadanie 2
Wyznaczyć wszystkie funkcje $ f $ określone na zbiorze liczb wymiernych dodatnich, o wartościach w tym samym zbiorze, spełniające dla wszystkich dodatnich liczb wymiernych warunki
|
Przypuśćmy, że funkcja $ f $ spełnia podane warunki. Wykażemy, że dla każdej liczby wymiernej $ u>0 $ oraz każdej liczby naturalnej $ k $ zachodzi równość (1)
Traktujemy $ u $ jako ustalone. Dla $ k=1 $ równość (1) zachodzi na mocy pierwszego z warunków danych w założeniach. Przyjmijmy słuszność (1) dla pewne... | f(x)=x | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXIX OM - III - Task 2
On a plane, there are points with integer coordinates, at least one of which is not divisible by A. Prove that these points cannot be paired in such a way that the distance between the points of each pair is equal to 1; this means that an infinite chessboard with fields cut out at coordinates di... | Consider the set $ A $ of points on the plane with integer coordinates $ (i, j) $, where $ 0 \leq i, j \leq 4k $. Notice that if we pair the points according to the conditions of the problem, the sum of the coordinates of one point in each pair will be odd, and the other will be even.
The set $ A $ has $ (4k + 1)^2 $ p... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXVII OM - I - Problem 8
Prove that on the surface of a regular tetrahedron with edge length 1, there exist points $ P $ and $ Q $ such that any point on the surface of this tetrahedron can be connected to one of the points $ P $, $ Q $ by a broken line of length $ \leq \frac{\sqrt{3}}{3} $ lying on the surface of the... | We will prove that as points $P$ and $O$ we can take the midpoints of edges $\overline{AB}$ and $\overline{CD}$ of a given regular tetrahedron $ABCD$. Consider the planar development of two faces $ABC$ and $BCD$ of this tetrahedron. The resulting planar figure $ABCD$ is a rhombus with side length $1$ and acute angle $\... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XII OM - I - Problem 10
Prove that if the equation
has three real roots, then
Is the converse statement true? | If equation (1) has 3 roots $x_1$, $x_2$, $x_3$ (not necessarily distinct), then $a \neq 0$ and according to Vieta's formulas
Hence
If the numbers $x_1$, $x_2$, $x_3$ are real, it follows from the above equality that
Next
where $u_1 = x_2x_3$, $u_2 = x_3x_1$, $u_3 = x_1x_2$; relying on the result of the previous ca... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLV OM - I - Problem 12
Prove that the sums of opposite dihedral angles of a tetrahedron are equal if and only if the sums of opposite edges of this tetrahedron are equal. | For a quadruple of non-coplanar points $ U $, $ V $, $ X $, $ Y $, we will denote by $ |\measuredangle X(UV)Y| $ the measure of the convex dihedral angle formed by the half-planes with common edge $ UV $, passing through points $ X $ and $ Y $.
Let us consider a tetrahedron $ ABCD $. We will prove the equivalence of th... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XII OM - I - Problem 5
Solve in integers the equation
Note: The equation is not provided in the original text, so it is not included in the translation. | If the pair of integers $ (x, y) $ satisfies equation (1), then $ x > 0 $ (since for an integer $ y $, the number $ y^2 - 1 $ is an integer and different from $ 1 $); simultaneously, the pair $ (x, - y) $ also satisfies this equation. The task thus reduces to solving equation (1) in natural numbers. We will write it in... | (3,3)(3,-3) | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XLVI OM - I - Problem 5
Given are positive numbers $ a $, $ b $. Prove the equivalence of the statements:
保留了源文本的换行和格式，但最后一句是中文，翻译成英文应为: "The statements are to be proven equivalent while retaining the original text's line breaks and format." 但为了保持格式的一致性，这里不添加额外的句子。 | Assume the truth of statement (2). Substituting $ x = 1+\frac{1}{\sqrt{a}} $, we obtain the inequality
thus $ a + \sqrt{a} + \sqrt{a} + 1 > b $, or equivalently,
Since $ b $ is a positive number, it follows that condition (1) holds.
Conversely, from condition (1), we derive (2) as follows. Let $ x $ be any number gre... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XVIII OM - I - Problem 6
Prove that if the real numbers $ a, b, c, d $ satisfy the equations
then
(Note: The equations are missing in the provided text, so they are also missing in the translation.) | From the given equalities, it follows that
therefore also
from which it follows that
Each term on the left side of this equality is $ \geq 0 $, the sum of these numbers is thus equal to zero only when each of them is zero, i.e., when $ a = b = c = d = 0 $. | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XVII OM - III - Problem 2
Prove that when $ k, m, n $ are any non-negative integers, the polynomial
is divisible by the polynomial $ x^2 + x + 1 $. | For any non-negative integer $ p $, the difference $ x^{3p} - 1 = (x^3)p - 1 $ is divisible by $ x^3 - 1 $, and thus by $ x^2 + x + 1 $. Therefore, the difference
is divisible by $ x^2 + x + 1 $. This implies that the polynomial $ P(x) $ is divisible by $ x^2 + x + 1 $. | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
IX OM - I - Zadanie 6
Dowieść, że jeżeli liczby $ a $, $ b $, $ c $, $ d $ spełniają warunki
to funkcja
przybiera wszystkie wartości rzeczywiste.
|
Z nierówności, które spełniają $ a $, $ b $, $ c $, $ d $ wynika, że równanie (1) jest równoważne równaniu
skąd po uporządkowaniu otrzymujemy równanie
Należy udowodnić, że równanie (2) ma dla każdej wartości $ y $ przynajmniej jeden pierwiastek $ x $.
Zależnie od obranej wartości $ y $ rozróżnimy $ 2 $ prz... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XI OM - III - Task 1
Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $. | Since $ 2^5 > 5^2 $, the inequality $ 2^n > n^2 $ is true for $ n = 5 $. Suppose this inequality is true when $ n $ equals some integer $ k > 4 $, i.e., that $ 2^k > k^2 $. Then
the inequality $ 2^n > n^2 $ is also true when $ n = k + 1 $. By the principle of mathematical induction, the inequality is true for every in... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
L OM - I - Problem 9
Points $ D $, $ E $, $ F $ lie on the sides $ BC $, $ CA $, $ AB $ of triangle $ ABC $, respectively. The incircles of triangles $ AEF $, $ BFD $, $ CDE $ are tangent to the incircle of triangle $ DEF $. Prove that the lines $ AD $, $ BE $, $ CF $ intersect at a single point. | Let $ c_1 $ be the incircle of triangle $ DEF $, and $ c_2 $ be the incircle of triangle $ ABC $. Since the incircles of triangles $ AEF $ and $ DEF $ are tangent, we have $ AF + DE = AE + DF $. This means that a circle can be inscribed in quadrilateral $ AEDF $; let this circle be denoted by $ c_A $. Point $ D $ is th... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LII OM - II - Problem 4
Determine all natural numbers $ n \geq 3 $ for which the following statement is true:
In any $ n $-term arithmetic sequence $ a_1, a_2, \ldots, a_n $, if the number $ 1 \cdot a_1 + 2 \cdot a_2 + \ldots + n \cdot a_n $ is rational, there exists a term that is a rational number. | The considered statement is true only for numbers $ n \geq 3 $ that give a remainder of $ 1 $ when divided by $ 3 $.
Let $ a_i = a + ir $ ($ i = 1,2,\ldots,n $) be an arithmetic sequence. Using the formulas
we determine that the sum given in the problem is equal to
From this, we obtain
First, assume t... | n\geq3thatgiveremainderof1whendivided3 | Number Theory | proof | Yes | Yes | olympiads | false |
VI OM - III - Task 5
In the plane, a line $ m $ and points $ A $ and $ B $ lying on opposite sides of the line $ m $ are given. Find a point $ M $ on the line $ m $ such that the difference in distances from this point to points $ A $ and $ B $ is as large as possible. | Let $ B' $ be the point symmetric to point $ B $ with respect to line $ m $ (Fig. 18). If point $ P $ is any point on line $ m $, then
The difference $ |AP - BP| $ thus reaches its maximum value, equal to the length of segment $ AB $, when $ |AP - BP| $, i.e., when points $ A $, $ B $, and $ P $ lie on the same straig... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXIX OM - I - Problem 3
For a given point $ P $ lying inside the triangle $ ABC $, we consider
points $ A' $ and $ A'' $ lying on the ray $ AP^{\rightarrow} $ such that $ A' \in AP $, $ |A'A''| = |AP| $;
points $ B' $ and $ B'' $ lying on the ray $ BP^{\rightarrow} $ such that $ B' \in BP $, $ |B'B''| = |BP| $;
point... | Let's denote: $ |PA| = a $, $ |PB| = b $, $ |PA| < |PB|. From the conditions of the problem, it follows that
and similarly $ |PB| (figure 2).
om39_1r_img_2.jpg
We calculate the areas of the triangles:
Notice now that the measures of any two angles appearing in these relations are either equal or supplemen... | notfound | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
VII OM - I - Zadanie 6
Rozwiązać równanie
|
Przypuśćmy, że pewna liczba $ x $ spełnia równanie (1); wówczas $ x \geq 0 $ i $ a + \sqrt{x} \geq 0 $. Niech $ a + \sqrt{x}=y $, zatem $ a = y - \sqrt{x} $; podstawienie tej wartości do równania (1) daje równanie $ x = y - \sqrt{x} + \sqrt{x} $ Przekształcając to równanie, otrzymujemy kolejno
Poni... | Algebra | math-word-problem | Incomplete | Yes | olympiads | false | |
XXXIV OM - II - Problem 5
The angle bisectors of $ CAB, ABC, BCA $ of triangle $ ABC $ intersect the circumcircle of this triangle at points $ K, L, M $, respectively. Prove that | Let's denote the angles of triangle $ABC$ by $A$, $B$, and $C$ respectively. The angle $AKB$ is subtended by the same arc as angle $C$, so $\measuredangle AKB = C$. In triangle $AKB$, the sum of the angles is $180^\circ$, so $\measuredangle ABK = 180^\circ - \measuredangle AKB - \measuredangle BAK = 180^\circ - C - A/2... | notfound | Geometry | proof | Incomplete | Incomplete | olympiads | false |
VIII OM - II - Task 1
Prove that if $ n $ is an integer, then
is also an integer. | Since
it is enough to prove that the number $ N = n (n^2 - 1) (n^2 - 4) $ is divisible by $ 120 $. Indeed, the number $ N $ equal to $ (n - 2) (n - 1) n (n + 1) (n+2) $ is the product of five consecutive natural numbers. Among such numbers, at least two are even, at least one is divisible by $ 4 $, at least one i... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
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