problem stringlengths 0 45.3k | solution stringlengths 0 26.8k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 9
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XXXIX OM - I - Problem 5
Prove that there are infinitely many natural numbers $ n $ for which $ [n\sqrt{2}] $ is a power of 2.
Note: $ [x] $ is the greatest integer not greater than $ x $.
| The number $ \sqrt{2} $ can be replaced by any number $ x $ from the interval $ (1; 2) $.
Let $ x \in (1; 2) $ and let $ M $ be the set of all natural numbers $ n $ for which $ [nx] $ is a power of two (with a non-negative integer exponent). We will show in several ways that the set $ M $ is infinite; for $ x = \sqrt{2... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XLVII OM - II - Zadanie 1
Rozstrzygnąć, czy każdy wielomian o współczynnikach całkowitych jest sumą trzecich potęg wielomianów o współczynnikach całkowitych.
|
Jeśli $ P(x) = a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n $ jest wielomianem o współczynnikach całkowitych, to $ (P(x))^3 = b_0 + b_1x + b_2 x^2 + b_3 x^3 +\ldots + b_{3n}x^{3n} $; przy tym $ b_0 = a_0^3 $, $ b_1= 3a_0^2 a_1 $. Tak więc trzecia potęga wielomianu o współczynnikach całkowitych ma współczynnik przy $ x $ p... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXXVI OM - II - Problem 5
Prove that for a natural number $ n $ greater than 1, the following conditions are equivalent:
a) $ n $ is an even number,
b) there exists a permutation $ (a_0, a_1, a_2, \ldots, a_{n-1}) $ of the set $ \{0,1,2,\ldots,n-1\} $ such that the sequence of remainders when dividing by $ n $ of the ... | We prove that condition a) implies condition b). We assume that $ n $ is an even number and we take
Let $ b_k=a_0+a_1+\ldots+a_k $ for $ k = 0,1,\ldots,n-1 $. Then for $ j<n $ the following equalities hold
(also for $ j = 0 $) and
Therefore, when divided by $ n $:
$ b_0 $ gives a remainder of $ 0 $
$ b_{2j} $ gi... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXV OM - III - Problem 2
For a fixed natural number $ n $ and each $ i = 1, 2, \ldots, n $, $ j = 1, 2, \ldots $ we define
while for $ i = n + 1,\ldots, 2n $ $ j = 1,2,\ldots, n $
Prove that for every permutation $ p $ of the set $ \{1, 2,\ldots, 2n\} $ the inequality is satisfied
Please note th... | If the permutation $ p $ of the numbers $ 1,2,\ldots,n $ assigns $ s $ values not greater than $ n $ and $ n-s $ values greater than $ n $, then the sum $ \displaystyle \sum_{k=1}^n a_{j,p(k)} $ equals $ 1+(n - s) \cdot \frac{-1}{n} = \frac{s}{n} $ or $ (n-s) \cdot \frac{-1}{n} $ depending on whether there exists $ k \... | \frac{}{n} | Inequalities | proof | Incomplete | Yes | olympiads | false |
VIII OM - III - Task 1
Through the midpoint $ S $ of the segment $ MN $ with endpoints lying on the legs of an isosceles triangle, a line parallel to the base of the triangle is drawn, intersecting its legs at points $ K $ and $ L $. Prove that the orthogonal projection of the segment $ MN $ onto the base of the trian... | Since the broken lines $ MNL $ and $ MKL $ have common ends $ M $ and $ L $ (Fig. 19), their projections on the base of the triangle are equal, which we will write as
Considering that the projection of a broken line equals the sum of the projections of its sides, we obtain from this
By drawing $ MP \paralle... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVII OM - I - Problem 3
Prove that for every natural number $ k $ the polynomial
divides $ x^5+1 $. | Let $ P_k $ be the considered polynomial. Let us adopt the notation
and notice that
Divisibility of the polynomial $ P_k $ by $ x^5+1 $:
Induction on $ k $. For $ k = 1 $ we have
From equations (3) and (5), it is clear that the divisibility of $ P_k $ by $ x^5 + 1 $ is equivalent to the divisibility o... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XIX OM - II - Problem 2
Given a circle $ k $ and a point $ H $ inside it. Inscribe a triangle in the circle such that point $ H $ is the orthocenter of the triangle. | Suppose the sought triangle is triangle $ABC$ (Fig. 11).
Since, by assumption, point $H$ lies inside the circle $k$, triangle $ABC$ is acute-angled and point $H$ lies inside triangle $ABC$. Let's call $H'$ the second point of intersection of line $AH$ with circle $k$; it lies on the opposite side of line $BC$ from poin... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LII OM - I - Task 1
Solve in integers the equation
| We reduce the given equation to the form $ f (x) = f (2000) $, where
Since $ f $ is an increasing function on the interval $ \langle 1,\infty ) $, the given equation has only one solution in this interval, which is $ x = 2000 $. On the set $ (-\infty,0\rangle $, the function $ f $ is decreasing, so there is at mo... | 2000 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XXI OM - I - Problem 8
On a billiard table in the shape of a square, there is a ball. The ball was hit so that it bounced off the cushion (the angle of incidence equals the angle of reflection) at an angle $ \alpha $. Prove that the ball will move along a closed broken line if and only if $ \tan \alpha $ is a rational... | Let's assume for simplicity that the billiard table is a square with vertices at points $A(0, 0)$, $B(1, 0)$, $C(1, 1)$, $D(0, 1)$ (Fig. 6). Suppose that at the initial moment, the ball is located at point $P(a, b)$, where $0 < a < 1$, $0 < b < 1$.
Let $v$ be the speed at which it moves at the initial moment, and $\alp... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LIV OM - I - Task 4
We consider the set of all $ k $-term sequences with terms from the set $ \{1, 2, 3,\ldots , m\} $. From each such sequence, we select the smallest term and sum the selected terms. Prove that the resulting sum is equal to | Let $ L_i $ $ (i =1,2,\ldots,m) $ denote the number of $ k $-term sequences with terms from the set $ \{1, 2, 3, \ldots , m\} $, where each term is not less than $ i $. Such sequences are $ k $-term sequences with terms from a $ (m-i+1) $-element set.
Hence, $ L_i = (m-i+1)^k $. Therefore, the number of sequences with... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XI OM - I - Task 2
Prove that if the sum $ a + b $ of two integers is divisible by an odd number $ n $, then $ a^n + b^n $ is divisible by $ n^2 $.
Does this theorem hold for even $ n $? | It is known that when $ n $ is an odd number, the binomial $ a^n + b^n $ is divisible by the binomial $ a + b $, because the following equality holds:
To prove that if $ a + b $ is divisible by $ n $, then $ a^n + b^n $ is divisible by $ n^2 $, it suffices to show that the number
is divisible by $ n $.
For this... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXII OM - I - Problem 2
Prove that for any positive numbers $ p $ and $ q $ satisfying the condition $ p + q = 1 $, the 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”是中文，应删除或翻译为英文：“Note that the last sentence '保留了源文本的换行和格式' is in Chinese and should be deleted or translated to English: 'Note that the l... | From the conditions of the problem, it follows that $0 < p < 1$ and $q = 1 - p$. Therefore, the problem reduces to showing that the function $f(p) = p \log p + (1 - p) \log (1 - p)$ in the interval $(0; 1)$ takes values no less than $-\log 2$. We calculate the derivative of this function:
Solving the equation $f'(p) =... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
X OM - I - Problem 11
On the same side of the line $ AB $, three semicircles were constructed with diameters $ AB = a + b $, $ AC = a $, $ CB = b $. Calculate the radius of the circle inscribed in the figure bounded by these semicircles, given $ a $ and $ b $. | Let $ O $, $ S $, $ T $ denote the centers of semicircles with diameters $ AB $, $ AC $ and $ CB $, $ U $ - the center of the circle inscribed in the figure bounded by these semicircles, and $ x $ - its diameter (Fig. 17).
The lengths of all segments in Fig. 17 can easily be expressed in terms of $ a $, $ b $, $ x $. U... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XLVIII OM - I - Problem 5
The angle bisectors of the interior angles $ A $, $ B $, $ C $ of triangle $ ABC $ intersect the opposite sides at points $ D $, $ E $, $ F $, and the circumcircle of triangle $ ABC $ at points $ K $, $ L $, $ M $. Prove that | Given the geometric inequality to be proven, we will reduce it to a simple algebraic inequality. Notice that $ |AD| = |AK|-|DK| $, $ |BE| = |BL| -|EL| $, $ |CF| = |CM|-|FM| $, and therefore
\[ \frac{|AD|}{|DK|} = \frac{|AK|}{|DK|}-1, \quad \frac{|BE|}{|EL|}=\frac{|BL|}{|EL|}-1, \quad \frac{|CF|}{|FM|}=\frac{|CM|}{|FM|... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XL OM - I - Problem 11
Points $ A_1 $, $ A_2 $, $ A_3 $, $ A_4 $ lie on the surface of a sphere, such that the distance between any two of them is less than the length of the edge of a regular tetrahedron inscribed in this sphere. Prove that all these points can be illuminated by a single point light source placed out... | Let $ C_1C_2C_3C_4 $ be any regular tetrahedron inscribed in a given sphere; let point $ O $ be its center, and $ r $ be the length of the radius. Consider the vectors
\[
\mathbf{u}_1 = \overrightarrow{OA_1}, \quad \mathbf{u}_2 = \overrightarrow{OA_2}, \quad \mathbf{u}_3 = \overrightarrow{OA_3}, \quad \mathbf{u}_4 = \... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LVI OM - II - Task 6
Numbers $ a $, $ b $, $ c $ belong to the interval $ \langle 0;1\rangle $. Prove that | We will show that for any numbers $ a $, $ b $, $ c $ belonging to the interval $ \langle 0; 1\rangle $, the following inequality holds:
Indeed, if $ a = 0 $, then the above inequality is satisfied. For $ a>0 $, inequality (1) is equivalent to the inequality $ a + b + c \leq 2bc + 2 $, which can be written as
The las... | proof | Inequalities | proof | Yes | Yes | olympiads | false |
XVI OM - I - Problem 9
Prove that if the value of the function $ y = ax^2 + bx + c $ for three consecutive integer values of the variable $ x $ is an integer, then it is also an integer for every integer value of $ x $. | Suppose that when $ x $ takes integer values $ k - 1 $, $ k $, $ k + 1 $, then $ y $ takes integer values $ m $, $ n $, $ p $. From the equality
we obtain by subtracting the equality
From (4) and (5) it follows similarly that
From (6) the number $ 2a $ is an integer, and from (4) and (5) we conclude that the numbers... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XLVI OM - III - Problem 2
The diagonals of a convex pentagon divide this pentagon into a pentagon and ten triangles. What is the maximum possible number of triangles with equal areas? | om46_3r_img_12.jpg
Let's denote the considered pentagon by $ABCDE$, and the pentagon formed by the intersection points of the diagonals by $KLMNP$ so that the following triangles are those mentioned in the problem:
$\Delta_0$: triangle $LEM$; $\quad \Delta_1$: triangle $EMA$;
$\Delta_2$: triangle $MAN$; $\quad \Delta... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XIV OM - I - Task 1
Prove that if $ a $, $ b $, $ m $ are integers such that $ a^2 + 2mb^2 $ is a perfect square, then $ a^2 + mb^2 $ is the sum of the squares of two integers. | Let $ mb^2 = n $; the theorem we need to prove takes a simpler form: Prove that if $ a^2 + 2n $, where $ a $ and $ n $ are integers, is a perfect square, then $ a^2 + n $ is the sum of the squares of two integers.
According to the assumption
therefore
from (1) it follows that $ k^2 - a^2 $ is an even number, so the n... | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
XLIV OM - I - Problem 9
Prove that for any real numbers $ a $, $ b $, $ c $, the inequality holds
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：
Preserving the line breaks and format of the source text, the last sentence in Chinese should be translated into English as above.
However, to strictly follow the instruction, h... | Let us denote:
We need to prove the inequality
Since
at most one of the numbers $ A $, $ B $, $ C $ can be negative. If there is such a number, then $ ABC \leq 0 $, so inequality (1) holds.
Now assume that $ A, B, C \geq 0 $. Notice that
Similarly, we observe that
Therefore,
... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XXVII OM - III - Task 5
A fishing vessel is fishing in the territorial waters of a foreign state without permission. Each cast of the net results in a catch of the same value. During each subsequent cast, the probability of the vessel being intercepted by the border guard is $ \frac{1}{k} $, where $ k $ is a fixed nat... | The probability of an event where a ship is not caught with a certain cast of the net is $1 - \frac{1}{k}$. Since events involving catching or not catching a ship with each subsequent cast of the net are independent, the probability of the event where the ship is not caught after $m$-times casting the net is $\left(1 -... | n=k-1n=k | Algebra | math-word-problem | Yes | Yes | olympiads | false |
XLIV OM - III - Problem 4
A convex polyhedron is given, all of whose faces are triangles. The vertices of this polyhedron are colored with three colors. Prove that the number of faces having vertices of all three colors is even. | For a tetrahedron, the following statement holds: the number of faces with vertices of three different colors is $0$ or $2$ (an easy exercise).
Now consider any polyhedron $W$ with $n$ triangular faces (and vertices colored with three colors). Choose any point $P$ inside $W$. Each face of the polyhedron $W$ serves as t... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXVIII - II - Task 4
Given a pyramid with a quadrilateral base such that each pair of circles inscribed in adjacent faces has a common point. Prove that the points of tangency of these circles with the base of the pyramid lie on one circle. | We will first prove a few auxiliary facts that we will use to solve the problem. We will first show that if in a quadrilateral $ABCD$ (not necessarily convex) the sums of the lengths of opposite sides are equal:
if
then the angle bisectors of the angles of this quadrilateral intersect at one point.
Indeed, from condi... | proof | Geometry | proof | Yes | Yes | olympiads | false |
I OM - B - Problem 11
A vertical pole of height $a$, illuminated by sunlight, casts a shadow on a horizontal plane; at one moment the shadow has a length of $a$, at another moment the shadow had a length of $2a$, and at a third moment the shadow's length was $3a$. Prove that the sum of the angles of inclination of the... | The angles mentioned in the problem are $\angle ACB$, $\angle ADB$, $\angle AEB$ in Figure 5, where $AB = a$, $BC = a$, $BD = 2a$, $BE = 3a$, and $\angle B = 90^{\circ}$. Since $\angle ACB = 45^{\circ}$, it is necessary to prove that
$\angle ADB + \angle AEB = 45^{\circ}$.
om1_Br_img_5.jpg
To do this, rotate triangle ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
IV OM - III - Task 2
Find the geometric place of the center of a rectangle, whose vertices lie on the circumference of a given triangle. | First, we will determine the geometric locus of the center $L$ of the rectangle $EFGH$, whose two vertices $E$ and $F$ lie on the side $AB$ of the triangle $ABC$, vertex $G$ on the side $BC$, and vertex $H$ on the side $AC$. Such rectangles exist only if neither of the angles $A$ and $B$ of the triangle is obtuse. In t... | proof | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XLII OM - I - Problem 9
Let $ f $ be a transformation of the plane having the following property: for any pair of points $ P $, $ Q $, whose distance $ |PQ| $ is a rational number, the distance of their images $ P', $ Q' satisfies the equality $ |P'Q'| = |PQ| $. Prove that $ f $ is an isometry. | We will first show that the transformation $ f $ is injective.
Suppose this is not the case and that there exist different points $ U $, $ V $ that are mapped to the same point $ U $. Take any rational numbers $ q, r > 0 $ satisfying the conditions $ q + r > |UV| > q - r > 0 $. We can then construct a triangle $ UVW $ ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XI OM - I - Task 4
Three consecutive segments on a straight line are given. Find the point from which these segments are visible at equal angles. | Let $AB$, $BC$, $CD$ be three consecutive segments on a line (Fig. 2). Notice that any point on this line, lying outside the segment $AD$, can be considered a solution to the problem, as from such a point each of the segments $AB$, $BC$, $CD$ is seen at an angle of zero. Ignoring this uninteresting case, we will look f... | TheproblemhassolutionifonlyiftheconditionAB<BC<CDorBC<AB<CDissatisfied | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XXII OM - I - Problem 4
Determine the angles that a plane passing through the midpoints of three skew edges of a cube makes with the faces of the cube. | Let $P$, $Q$, $R$ be the midpoints of three skew edges of a cube with edge length $a$ (Fig. 7). Let $n$ be the plane of triangle $PQR$, and $\pi$ be the plane of the base $ABCD$.
Triangle $PQR$ is equilateral because by rotating the cube $90^\circ$ around a vertical axis (passing through the centers of faces $ABCD$ and... | 5445 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
X OM - II - Task 1
What necessary and sufficient condition should the coefficients $ a $, $ b $, $ c $, $ d $ satisfy so that the equation
has two opposite roots? | Suppose that equation (1) has two opposite roots $ x_1 $ and $ x_2 = - x_1 $. In this case,
By adding and subtracting equations (2) and (3), we obtain
We multiply equations (4) and (5) by $ ax_1 $ and $ b_1 $, respectively, and then subtract them, which gives the equation
If $ x_1 \ne 0 $,... | ad-=0(\neq0ac\leq0)or(0b\neq00) | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
LII OM - I - Problem 3
Find all natural numbers $ n \geq 2 $ such that the inequality
holds for any positive real numbers $ x_1,x_2,\ldots,x_n $. | The only number satisfying the conditions of the problem is $ n = 2 $.
For $ n = 2 $, the given inequality takes the form
Thus, the number $ n = 2 $ satisfies the conditions of the problem.
If $ n \geq 3 $, then inequality (1) is not true for any positive real numbers. The numbers
serve as a counterexample.... | 2 | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false |
XXXVI OM - I - Problem 6
Calculate the lengths of the parallel sides of a trapezoid given the lengths of the other sides and the diagonals. | Let $ABCD$ be the considered trapezoid, $AB\parallel CD$, $|AB| = a$, $|BC| = b$, $|CD| = c$, $|DA| = d$, $|AC| = p$, $|BD| = q$ (Figure 2). Given $b$, $d$, $p$, $q$, we need to find $a$ and $c$.
om36_1r_img_2.jpg
First, note that if $p = q$, then it must be $b = d < p$ and in this case, the problem does not have a uni... | \frac{2(p^2-q^2)}{b^2-^2+p^2-q^2},\quad\frac{2(q^2-p^2)}{b^2-^2+q^2-p^2} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LIV OM - I - Task 6
Points $ A $, $ B $, $ C $, $ D $ lie in that order on circle $ o $. Point $ M $ is the midpoint of that arc $ AB $ of circle $ o $ which does not contain points $ C $ and $ D $; point $ N $ is the midpoint of that arc $ CD $ of circle $ o $ which does not contain points $ A $ and $ B $. Prove that | From Ptolemy's theorem (see LI Mathematical Olympiad, Report of the Main Committee, Appendix D, p. 112) applied to quadrilateral $AMBN$, we obtain the equality $BM \cdot (AN + BN) = MN \cdot AB$. Therefore,
the problem reduces to showing that
Without loss of generality, assume that $AN < BN$; then $DM < CM$. Let $A'$... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
VIII OM - III - Task 3
Prove that if the function $ ax^2 + bx + c $ takes an integer value for every integer value of the variable $ x $, then $ 2a $, $ a + b $, $ c $ are integers and vice versa. | Suppose that when the number $ x $ is an integer, the number $ f (x) = ax^2 + bx + c $ is also an integer. Then
1) $ f (0) = c $, so $ c $ is an integer
2) $ f (1) = a + b + c $, hence $ a + b = f (1) - c $, so $ a + b $ is an integer
3) $ f (2) = 4a + 2b + c $, hence $ 2a = f (2) - 2 (a + b) - c $, so $ 2a $ is an int... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XIX OM - II - Task 4
Prove that if the numbers $ a, b, c $ are the lengths of the sides of a triangle, and the sum of the numbers $ x, y, z $ is equal to zero, then
(Note: The original text ends abruptly, so the translation also ends at the same point.) | Let $ U $ denote the left-hand side of inequality (1). Since $ x + y + z = 0 $, then $ z = - (x + y) $, hence
thus
The numbers $ a $, $ b $, $ c $, satisfy the inequality
Therefore
If $ xy > 0 $, then from (3) it follows that
hence from (2) we obtain
If $ xy < 0 $, then by (3)
Thus from (2) ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLVII OM - II - Problem 3
Prove that if each of the numbers $ a $, $ b $, $ c $ is not less than $ -\frac{3}{4} $ and if $ a+b+c=1 $, then | For $ x \geq - \frac{3}{4} $, the inequality holds
(see: Note 1). We transform this inequality as follows:
Now it is enough to substitute the given numbers $ a $, $ b $, $ c $:
and we have the thesis of the problem. | proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
XXXVI OM - II - Problem 3
Let $ L $ be the set of all broken lines $ ABCDA $, where $ A, B, C, D $ are distinct vertices of a fixed regular 1985-gon. A broken line is randomly chosen from the set $ L $. Calculate the probability that it is the boundary of a convex quadrilateral. | Let $ W $ be the set of all vertices of a given regular polygon. Each broken line from the set $ L $ can be associated with a set of its vertices, i.e., a certain four-point subset of the set $ W $. Conversely, every four points from $ W $ are the set of vertices of exactly three different broken lines from the set $ L... | \frac{1}{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXV OM - II - Problem 6
The sequence $ (x_n) $ is defined by the formulas
If $ c $ is a natural number, prove that all numbers $ x_n $ are natural numbers. | The terms $ x_1 = c $, $ x_2 = 2c^2 - 1 $ are natural numbers.
Assume for some $ n $ that all terms $ x_k $ for $ k \leq n $ are natural. We will prove that $ x_{n+1} $ is a natural number. From the given formula, it follows that
thus
and similarly
From the last equality, we obtain
and by subtracting $ c^2x^2_n $ f... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XLII OM - I - Problem 11
For a given natural number $ n>10^3 $ and each $ k \in \{1,\ldots, n\} $, consider the remainder $ r_k $ from the division of $ 2^n $ by $ k $. Prove that $ r_1+\ldots r_n > 2n $. | For $ j = 0,1,2,3,4 $, let $ K_j $ denote the set of positive integers not exceeding $ n $, which are divisible by $ 2^j $ but not by $ 2^{j+1} $, and have at least one odd divisor greater than $ 1 $. Thus,
Let $ j \in \{0,1,2,3,4\} $ and consider any number $ k \in K_j $. Then $ k = 2^j m $, where $ m $ is odd a... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXXII - I - Problem 12
For a given natural number $ n > 1 $, we define the sequence $ (a_0, a_1, \ldots, a_n) $ as follows:
Prove that $ a_n < 1 $. | First, we will prove by induction that
For $ k = 0 $, we have $ \frac{1}{2} \left( 1+ \frac{1}{n^2}\right)^0 = \frac{1}{2} = a_0 $.
If $ a_k = \frac{1}{2} \left( 1+ \frac{1}{n^2}\right)^k $ for some $ k, 0 \leq k \leq n-1 $, then
Indeed, by the principle of induction, we have
In particular,
By the preparatory probl... | proof | Algebra | proof | Incomplete | Incomplete | olympiads | false |
XXVIII - II - Task 6
What is the maximum number of parts that the edges of $ n $ squares can divide a plane into? | The common part of the boundaries of two squares is either a finite set of points (and in this case, there are at most $8$ points, since each side of one square can intersect the boundary of the other square in at most two points), or it contains a segment (and in this case, it is a segment, the union of two segments -... | (2n-1)^2+1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
I OM - B - Task 9
Prove that if the sum of three natural numbers is divisible by 3, then the sum of the cubes of these numbers is
also divisible by 3. | Let $a, b, c$ be natural numbers whose sum is divisible by 3. To prove that the difference
$ (a^3 + b^3 + c^3) - (a + b + c) $ is divisible by 3. Indeed,
Each of the three terms on the right side of the equation is
divisible by 3, because it is the product of three consecutive integers. The sum of them is therefore al... | proof | Number Theory | proof | Yes | Incomplete | olympiads | false |
LIX OM - I - Task 9
Determine the smallest real number a with the following property:
For any real numbers $ x, y, z \geqslant a $ satisfying the condition $ x + y + z = 3 $
the inequality holds | Answer: $ a = -5 $.
In the solution, we will use the following identity:
Suppose that the number $ a \leqslant 1 $ has the property given in the problem statement. The numbers
$ x = a, y = z = 2\cdot (\frac{3-a}{2}) $ satisfy the conditions $ x, y, z \geqslant a $ and $ x+y+z = 3 $, thus
by virtue of (1),... | -5 | Inequalities | math-word-problem | Yes | Yes | olympiads | false |
XLVI OM - III - Problem 4
For a given natural number $ n \geq 1 $, determine the minimum value of the expression
where $ x_1, x_2, \ldots , x_n $ are positive numbers satisfying the condition
Please note that the mathematical expression is missing in the original text, so it is not included in the tran... | The key to the solution is the inequality
(1)
Here is its proof: the difference between the expression on the left side (1) and the expression on the right side (1), multiplied by the positive factor \( kx \), is equal to
\[
kx \left( x^k + \frac{k}{x} - (k+1) \right) = kx^{k+1} - k(k+1)x + k^2
\]
This sum is non-n... | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false | |
XLIV OM - I - Problem 11
In six different cells of an $ n \times n $ table, we place a cross; all arrangements of crosses are equally probable. Let $ p_n $ be the probability that in some row or column there will be at least two crosses. Calculate the limit of the sequence $ (np_n) $ as $ n \to \infty $. | Elementary events are determined by six-element subsets of the set of $n^2$ cells of the table; there are $\binom{n^2}{6}$ of them. Let $\mathcal{Z}$ be the complementary event to the event considered in the problem. The configurations favorable to event $\mathcal{Z}$ are obtained as follows: we place the first cross i... | 30 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XVI OM - I - Problem 11
Prove the theorem: A closed broken line with five sides, none of whose three vertices lie on the same straight line, can intersect itself at one, two, three, or five points, but it cannot intersect itself at four points. | Let's call a point of a closed broken line an ordinary point if it belongs only to one side and each vertex belongs only to two sides, and a double point if it belongs only to two sides of the broken line but is not a vertex. If no three vertices of a closed broken line with five sides lie on the same straight line, th... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVII OM - I - Problem 2
Prove that if a real number $ x_1 $ satisfies the equation $ x^3 + 2px + q = 0 $ ($ p, q $ - given real numbers), then $ x_1q \leq p^2 $. | When $ x_1 = 0 $, the inequality $ x_1q \leq p^2 $ is true, so let's assume that $ x_1 \ne 0 $.
According to the assumption, $ x_1^3 + 2px_1 + q = 0 $, so the number $ x_1 $ is a root of the quadratic equation
The discriminant of this equation is therefore non-negative, so $ p^2 - x_1q \geq 0 $, hence $ x_1q \leq p^2... | proof | Algebra | proof | Yes | Incomplete | olympiads | false |
LX OM - I - Task 4
Prove that for any non-negative real numbers
$ a $, $ b $, $ c $, the following inequality holds:
\[4(\sqrt{a^3b^3} + \sqrt{b^3c^3} + \sqrt{c^3a^3}) \leqslant 4c^3 + (a+b)^3.\] | Using the inequality $ x+y \geqslant 2xy $ valid for
any non-negative numbers $ x $, $ y $ we obtain
Therefore, the thesis of the problem follows from the following calculation:
| proof | Inequalities | proof | Yes | Incomplete | olympiads | false |
XIII OM - II - Problem 5
In the plane, a square $ Q $ and a point $ P $ are given. Point $ K $ runs along the perimeter of square $ Q $. Find the geometric locus of the vertex $ M $ of the equilateral triangle $ KPM $. | Let $ K $ be any point on the perimeter of a given square. On the plane of the square, there are two equilateral triangles with base $ KP $. The vertex of one of them is obtained by rotating point $ K $ around point $ P $ by an angle of $ 60^\circ $ in the direction of the clockwise; the vertex of the second one is fou... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XIV OM - I - Task 8
Two similar but unequal polygons are given; the corresponding sides of these polygons lie on parallel lines, the distance between which is the same for all pairs of sides. Must these be regular polygons with a common center? | The answer to the posed question is negative: two polygons satisfying the mentioned conditions may not be regular. We can state this as follows.
Let $W$ be any irregular polygon, the sides or extensions of whose sides are tangent to a circle $C$ with radius $r$ and center $O$. Construct a polygon $W_1$ similar to $W$ w... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
Given are non-negative integers $ k_1 < k_2 < \dots < k_m $. Let
Determine the number of odd coefficients of the polynomial
The above text is translated into English, preserving the original text's line breaks and formatting. | At the beginning, we will prove the following lemma:
Lemma
If $ 1 \leq t \leq 2^k $, then the number $ \binom{2^k}{t} $ is even.
Proof
Notice that
For any numbers $ j $ and $ p $ such that $ 1\leq j \leq k $ and $ 1\leq p < t $, the following equivalence holds:
Therefore, the number 2 appears in the same power i... | 2^ | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false |
II OM - I - Problem 8
Find the geometric locus of the midpoints of segments of a given length $ a $, whose ends lie on two mutually perpendicular lines (intersecting or skew).
Translated the text into English while preserving the original formatting and line breaks. | A. Let $ m $ and $ n $ be two perpendicular lines intersecting at point $ O $ (Fig. 40).
Let point $ S $ be the midpoint of segment $ MN $ of length $ a $ with endpoints lying on lines $ m $ and $ n $. In the rectangle $ OMPN $ (which reduces to segment $ MN $ when $ M $ or $ N $ coincides with $ O $), we have
Point $... | notfound | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
III OM - I - Task 1
Two common internal tangents and both common external tangents are drawn to two circles. Prove that the segment of the internal tangent contained between the external tangents is equal to the segment of the external tangent contained between its points of tangency. | We will obtain the proof of the theorem by applying the theorem that the segments of tangents drawn from a certain point to a circle are equal. For example, in Fig. 10, representing the considered figure, $ PK = PR $, $ PL = PS $, $ QM = QR $, $ QN = QS $.
Considering that due to the symmetry of the figure, the equalit... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXII - III - Problem 1
In space, there are two intersecting lines $ a $ and $ b $. We consider all pairs of planes $ \alpha $ and $ \beta $ that are perpendicular and such that $ a\subset \alpha $, $ b\subset \beta $. Prove that there exists a circle such that through each of its points there passes a line $ \alpha \... | We will distinguish two cases.
1. If lines $a$ and $b$ are perpendicular, consider the plane $\alpha$ perpendicular to line $b$ and containing line $a$, and take any circle contained in this plane. For any point $A$ of this circle, draw the plane $\beta$ containing line $b$ and point $A$. Of course, the line $\alpha \c... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXVIII OM - II - Problem 2
Prove that the sum of the planar angles at each vertex of a given tetrahedron is $180^{\circ}$ if and only if all its faces are congruent. | We unfold the surface of the considered tetrahedron $ABCD$ along the edges $AD$, $BD$, $CD$ and lay the faces $BCD$, $CAD$, $ABD$ on the plane of the triangle $ABC$. We obtain the net of this tetrahedron, that is, the hexagon $AMBKCL$, in which the triangles $BCK$, $CAL$, $ABM$ are respectively congruent to the mention... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXIX OM - I - Problem 7
Given a transformation of the plane onto itself such that the image of every circle is a circle. Prove that the image of every line is a line. | Let the considered transformation be denoted by $ f $. Let $ l $ be any line, and $ f(l) $ - its image under the transformation $ f $.
The proof of the given theorem will be based on the following three statements:
[(1)] \textrm{Together with any two points, the set}\ f(l) \ \textrm{contains the line passing through t... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXV OM - II - Task 1
Let $ Z $ be an $ n $-element set. Find the number of such pairs of sets $ (A, B) $, where $ A $ is a subset of $ B $ and $ B $ is a subset of $ Z $. We assume that each set includes itself and the empty set. | As is known, if $ 0 \leq r \leq s $, then the number of $ r $-element subsets of an $ s $-element set is equal to $ \binom{s}{r} $, and the number of all subsets of an $ s $-element set is equal to $ 2^s $.
A $ k $-element subset $ A $, where $ 0 \leq k \leq n $, can be chosen from a set $ Z $ in $ \binom{n}{k} $ ways.... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | |
XXXV OM - II - Task 4
In the Mathematical Olympiad, $3n$ participants take part. They have designated seats in three rows, with $n$ seats in each, and they are admitted to the hall one by one, after which they immediately take their seats. Calculate the probability that until the moment the last participant takes thei... | Elementary events are permutations of a $3n$-element set. Their number is $(3n)!$. We will calculate the number of favorable events for the event mentioned in the problem statement. The first player to enter the room can be any player (there are $3n$ possibilities). The next person can be one of the $2n$ whose seats ar... | notfound | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false |
XXIV OM - III - Problem 3
The polyhedron $ W $ has the following properties:
a) it has a center of symmetry,
b) the intersection of the polyhedron $ W $ with any plane containing the center of symmetry and any edge of the polyhedron is a parallelogram,
c) there exists a vertex of the polyhedron $ W $ that belongs to e... | A polyhedron is a bounded set of points in space that is not contained in a single plane and is the intersection of a finite number of half-spaces. Since a half-space is a convex set, and the intersection of convex sets is a convex set, every polyhedron is a convex set. (Some other definitions of a polyhedron are also ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLIX OM - I - Problem 6
In triangle $ABC$, where $|AB| > |AC|$, point $D$ is the midpoint of side $BC$, and point $E$ lies on side $AC$. Points $P$ and $Q$ are the orthogonal projections of points $B$ and $E$ onto line $AD$, respectively. Prove that $|BE| = |AE| + |AC|$ if and only if $|AD| = |PQ|$. | Note 1: In the official print of the problem texts for the first-level competition of the 49th Mathematical Olympiad, problem 6 was formulated with an error: the assumption that $ |AB| > |AC| $ was missing. (The corrected version of the problem was sent to schools as an Errata a few weeks later.) Without this assumptio... | proof | Geometry | proof | Yes | Yes | olympiads | false |
L OM - II - Task 1
Given is a function $ f: \langle 0,1 \rangle \to \mathbb{R} $ such that $ f\left(\frac{1}{n}\right) = (-1)^n $ for $ n = 1, 2,\ldots $. Prove that there do not exist increasing functions $ g: \langle 0,1 \rangle \to \mathbb{R} $, $ h: \langle 0,1 \rangle \to \mathbb{R} $, such that $ f = g - h $. | Suppose there exist functions $ g $ and $ h $ satisfying the conditions of the problem. Since these functions are increasing, for every $ k \in \mathbb{N} $ we have
From the monotonicity of the function $ g $ and the above inequalities, we get
Thus, $ g(1)-g(0) > 2k $ for any $ k \in \mathbb{N} $, which i... | proof | Algebra | proof | Yes | Yes | olympiads | false |
XXXIII OM - I - Problem 2
Let $ p $ be a fixed prime number. Prove that the number
is divisible by $ p $. | The number $ N = 11 \ldots 122 \ldots 2 \ldots 99 \ldots 9 - 123456789 $ is the difference of two numbers whose last digits are nines, so the last digit of the number $ N $ is zero, meaning the number is divisible by $ 2 $ and by $ 5 $, and thus the statement is true for $ p = 2 $ and for $ p = 5 $. For $ p = 3 $, we c... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XX OM - I - Task 4
Given points $ A, B, C, D $ that do not lie on the same plane. Determine the plane such that the distances from these points to the plane are equal. | Suppose the plane $ \pi $ is equidistant from points $ A $, $ B $, $ C $, $ D $. None of the given points lie on the plane $ \pi $, for otherwise all four would have to lie on it, contrary to the assumption. The points $ A $, $ B $, $ C $, $ D $ do not lie on the same side of the plane $ \pi $, for they would then have... | 7 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XXXI - I - Problem 10
A plane is divided into congruent squares by two families of parallel lines. $ S $ is a set consisting of $ n $ such squares. Prove that there exists a subset of the set $ S $, in which the number of squares is no less than $ \frac{n}{4} $ and no two squares have a common vertex. | A set of squares formed on a plane can be divided into two subsets in such a way that two squares sharing a side belong to different subsets (similar to the division of a chessboard into black and white squares). Such a division can be described as follows. We introduce a coordinate system on the plane, whose axes are ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
LVIII OM - I - Problem 10
Given is an acute triangle $ ABC $. Points $ P $ and $ U $ lie on side $ BC $, points $ Q $ and $ S $ lie on side $ CA $, points $ R $ and $ T $ lie on side $ AB $, such that
Prove that triangles $ PQR $ and $ STU $ are congruent. | Since $ PR\perp BC $ and $ US\perp BC $, then $ PR\parallel US $. Similarly, $ QP\parallel ST $ and $ RQ\parallel TU $. Triangles $ PQR $ and $ STU $ are therefore similar, as triangles with corresponding sides parallel (Fig. 4).
om58_1r_img_4.jpg
From the right triangles $ PUS $ and $ RPU $, we read the following rel... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XXXIII OM - III - Problem 5
Integers $ x_0, x_1, x_2, \ldots, x_n = x_0, x_{n+1} = x_1 $ satisfy for $ k = 1, 2, \ldots, n $ the inequality
Prove that the difference
is divisible by 4. | From the condition
it follows that $ x_k \ne 0 $ for $ k = 1, 2, \ldots, n $. By setting $ \varepsilon_k = \frac{x_k}{|x_k|} $, we obtain the inequality
From this, we get
Since we have $ \varepsilon_k = \pm 1 $, the equality $ \varepsilon_k = (-1)^{\frac{x_k-1}{2}} $ holds.
Substituting $ \lambda_k = ... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
III OM - III - Task 2
On the sides $ BC $, $ CA $, $ AB $ of triangle $ ABC $, points $ M $, $ N $, $ P $ have been chosen in such a way that
where $ k $ denotes a given number greater than $ 1 $; then the segments $ AM $, $ BN $, $ CP $ were drawn. Given the area $ S $ of triangle $ ABC $, calculate the area of... | Let's denote the intersection points of the lines $AM$, $BN$, and $CP$ by $X$, $Y$, and $Z$, as indicated in Fig. 36.
Since the areas of triangles with equal heights are in the ratio of their bases, we have
therefore
The ratios $\frac{BM}{BC}$ and $\frac{AX}{AM}$ are calculated based on equation (1):
... | \frac{S}{(k+1)^2} | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XV OM - I - Problem 7
Given a circle and points $ A $ and $ B $ inside it. Find a point $ P $ on this circle such that the angle $ APB $ is subtended by a chord $ MN $ equal to $ AB $. Does the problem have a solution if the given points, or only one of them, lie outside the circle? | Suppose that point $ P $ of a given circle $ C $ with radius $ r $ is a solution to the problem (Fig. 7).
Since points $ A $ and $ B $ lie inside the circle $ C $, points $ M $ and $ N $ lie on the rays $ PA $ and $ PB $ respectively, and angle $ APB $ coincides with angle $ MPN $. Triangles $ APB $ and $ MPN $ have eq... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
IV OM - I - Problem 5
In trapezoid $ABCD$, a line parallel to the bases $AB$ and $DC$ of the trapezoid is drawn through the intersection point $S$ of the diagonals, intersecting sides $AD$ and $BC$ at points $M$ and $N$. Prove that 1) $MN$ equals the harmonic mean of the bases, 2) if $\measuredangle ADC = \measuredang... | 1° First, let us note (Fig. 22) that segments $MS$ and $SN$ are equal. Indeed, triangles $MSD$ and $SNC$ are similar to triangles $ABD$ and $ABC$, respectively, and in both cases the scale of similarity is the same, since according to Thales' theorem, we have $\frac{MD}{AD} = \frac{NC}{BC}$. Therefore, each of the segm... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
LVI OM - I - Problem 3
In an acute triangle $ABC$, point $D$ is the orthogonal projection of point $C$ onto line $AB$. Point $E$ is the orthogonal projection of point $D$ onto line $BC$. Point $F$ lies on segment $DE$, such that
To prove that lines $CF$ and $AE$ are perpendicular. | Through point $ D $, draw a line parallel to line $ AE $, which intersects side $ BC $ at point $ G $ (Fig. 1). By the theorem of Thales, we obtain the proportions
from which it follows that lines $ FG $ and $ DB $ are parallel. From this and the similarity of right triangles $ CDE $ and $ DBE $, we obtain the equa... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XL OM - II - Task 2
For a randomly chosen permutation of the set $ \{1,\ldots, n\} $, denote by $ X(\mathbf{f}) $ the largest number $ k \leq n $ such that $ f_i < f_{i+1} $ for all indices $ i < k $. Prove that the expected value of the random variable $ X $ is $ \sum_{k=1}^n \frac{1}{k!} $. | The considered random variable of course depends on $ n $. Let us denote it by $ X_n $. We will find a recursive relationship between the expected values of the variables $ X_{n-1} $ and $ X_n $.
Take any permutation $ f = (f_1, \ldots , f_n) $ of the set $ \{1,\ldots, n\} $. By discarding the term $ f_n $, we obtain a... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
LIX OM - I -Zadanie 10
Dana jest liczba pierwsza p. Ciąg liczb całkowitych dodatnich $ a_1, a_2, a_3, \dots $ spełnia warunek
Wykazać, że pewien wyraz tego ciągu jest $ p $-tą potęgą liczby całkowitej.
(Uwaga: Symbol $ [x] $ oznacza największą liczbę całkowitą nie przekraczającą $ x $.)
|
Zdefiniujmy ciągi liczb całkowitych nieujemnych $ b_1, b_2, b_3, \dots $ oraz $ r_1, r_2, r_3, \dots $
w następujący sposób:
dla $ n =1,2,3,\dots $. Inaczej mówiąc, $ r_n $ jest różnicą między wyrazem an a największą nie
przekraczającą tego wyrazu $ p $-tą potęgą liczby całkowitej. Warunek (1) przepisujemy te... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXVI - III - Task 3
Determine the smallest positive number $ \alpha $, for which there exists a positive number $ \beta $ such that for every $ x $ in the interval $ \rangle 0; 1\rangle $ it is
For the determined value of $ \alpha $, find the smallest positive number $ \beta $ satisfying this inequality. | For $ x \in \langle 0;\ 1 \rangle $ we have the equality
The function $ h(x) =\displaystyle \frac{1}{2} (\sqrt{1+x} + \sqrt{1-x}+2) (\sqrt{1-x^2} + 1) $ in the interval $ \langle 0; 1 \rangle $ satisfies the inequality $ 0 < h(x) \leq 4 $, because
and $ 0 \leq \sqrt{1-x^2} \leq 1 $ for $ 0 \leq x \leq 1 $. Therefore,... | \alpha=2,\beta=4 | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false |
XIII OM - I - Problem 1
Prove that if $ x^2 + y^2 = a^2 $ and $ a \geq 0 $, then
(Note: The original text ends abruptly, so the translation also ends at the same point.) | Since
and $(x - y)^2 \geq 0$, therefore $(x + y)^2 \leq 2a^2$, from which $|x + y| \leq a \sqrt{2}$, hence
Remark. Using initial knowledge from analytic geometry, a simple geometric interpretation of the proven theorem can be provided. In a rectangular coordinate system $(x, y)$, the equation $x^2 + y^2 = a^2$ ($... | proof | Geometry | proof | Incomplete | Yes | olympiads | false |
XV OM - I - Problem 12
The tetrahedron $ABCD$ is cut into two parts by a plane passing through vertex $D$, point $M$ on edge $AB$, and point $N$ on edge $BC$. Prove that the area of triangle $DMN$ is less than the area of one of the triangles $DAB$, $DBC$, or $DCA$. | The theorem we need to prove can be considered the spatial counterpart of the following theorem from plane geometry:
The segment $CM$ connecting vertex $C$ of triangle $ABC$ with an internal point $M$ on the base $AB$ of this triangle is smaller than one of the sides $AC$ and $BC$.
A short proof of this theorem can be ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXI OM - I - Problem 11
Prove that in any division of the plane into three sets, there exist two points in at least one of them that are 1 unit apart. | Suppose this theorem is false. Consider equilateral triangles $ABC$ and $ABD$ (Fig. 7) with a common base $\overline{AB}$ and a side length of $1$. From our assumption, it follows that each vertex of triangle $ABC$ belongs to a different one of the three distinct subsets into which we have divided the plane. Similarly,... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLVIII OM - I - Problem 10
Points $ P $, $ Q $ lie inside an acute-angled triangle $ ABC $, such that $ |\measuredangle ACP| = |\measuredangle BCQ| $ and $ |\measuredangle CAP| = |\measuredangle BAQ| $. Points $ D $, $ E $, $ F $ are the orthogonal projections of point $ P $ onto the sides $ BC $, $ CA $, $ AB $, resp... | In the assumptions of equality
they are equivalent to the equalities
Angles $ ACP $ and $ BCQ $, considered as areas of the plane, can be disjoint (as in Figure 6) or they can overlap; this has no significance. A similar remark applies to each of the other three pairs of angles in relations (1) and (2).
Let $ U... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXXIV OM - I - Problem 9
Natural numbers $ m, n $ satisfy the condition $ \sqrt{7} - \frac{m}{n} $. Prove that $ \sqrt{7} - \frac{m}{n} > \frac{1}{mn} $. | Suppose that the natural numbers $ m $, $ n $ satisfy the conditions
From these conditions
we have the following inequalities
Since it cannot be that $ m = 1 $, the double inequality $ m^2 < 7n^2 < m^2+3 $ is satisfied.
Therefore, either (1) $ 7n^2 = m^2+1 $ or $ 7n^2 = m^2+2 $. We will show that this is impossible.... | proof | Number Theory | proof | Incomplete | Yes | olympiads | false |
XXIV OM - I - Problem 2
Prove that among 25 different positive numbers, one can choose two such that neither their sum nor their difference is equal to any of the remaining numbers. | Let us denote a given set of numbers by $A = \{a_1, a_2, \ldots, a_{25}\}$, where $0 < a_1 < a_2 < \ldots < a_{25}$. Suppose that for any $r$, $s$, where $1 \leq r < s \leq 25$, we have: $a_s + a_r \in A - \{a_r, a_s\}$ or $a_s - a_r \in A - \{a_r, a_s\}$. Of course, $a_r - a_s < 0$ and therefore the number $a_r - a_s$... | proof | Combinatorics | proof | Yes | Yes | olympiads | false |
XXIII OM - II - Problem 4
A cube with edge length $ n $ has been divided by planes parallel to its faces into $ n^3 $ unit cubes. How many pairs of such unit cubes exist that have no more than two vertices in common? | If two unit cubes have at least three vertices in common, then they share an entire face. Conversely, every face of a unit cube not contained in a face of the larger cube determines a pair of unit cubes with at least three vertices in common. We will determine the number of such faces. There are $n^3$ unit cubes, each ... | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false | |
XXXIV OM - II - Task 2
Given are three non-negative numbers $ a, b, c $, such that the sum of any two is not less than the remaining one. Prove that
(Note: The original text ends abruptly, and the problem statement is incomplete. The translation reflects this.) | We apply the inequality
\[
\sqrt{x} + \sqrt{y} \leq \sqrt{2(x + y)}
\]
which can be derived from the properties of the root function, or proven directly as follows. Squaring both sides, we obtain the equivalent inequality
\[
x + y + 2\sqrt{xy} \leq 2(x + y)
\]
which we then transform into the form
\[
2\sqrt{xy} \l... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
VII OM - I - Problem 9
Prove that a triangle can be constructed from segments of lengths $ a $, $ b $, $ c $ if and only if | From segments of lengths $ a $, $ b $, $ c $, a triangle can be constructed if and only if
If the numbers $ a $, $ b $, $ c $ satisfy inequality (1) and are (as measures of segments) positive, then they also satisfy inequalities (2) and vice versa. To this end, we transform inequality (1). Writing this inequality in t... | proof | Geometry | proof | Yes | Yes | olympiads | false |
VI OM - III - Task 6
Through points $ A $ and $ B $, two oblique lines $ m $ and $ n $ perpendicular to the line $ AB $ have been drawn. On line $ m $, a point $ C $ (different from $ A $) has been chosen, and on line $ n $, a point $ D $ (different from $ B $). Given the lengths of segments $ AB = d $ and $ CD = l $,... | Fig. 20 shows the considered figure in an orthogonal projection onto the plane determined by the lines $AB$ and $n$, hence on this drawing $AB \bot n$. For simplicity, we denote the projections of points and lines with the same letters as the points and lines themselves. Let us draw through point $A$ a line $p$ paralle... | \sqrt{(\frac{} | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LII OM - III - Task 1
Prove that for every natural number $ n \geq 2 $ and any non-negative real numbers $ x_1,x_2,\ldots,x_n $, the following inequality holds
The translation is provided as requested, preserving the original text's line breaks and format. | From the inequality between the geometric mean and the arithmetic mean applied to the numbers
we obtain
Therefore,
Note: The original text uses mathematical symbols and expressions which are directly translated here. If you need the mathematical expressions in a more readable format, please let me kno... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XXXIII OM - III - Problem 3
Determine all pairs of positive numbers $ x, y $ satisfying the system of equations
where $ a, b $ are given positive numbers. | If a pair of positive numbers $ (x,y) $ satisfies the given system of equations, then $ y = \sqrt{a^2 + b^2 - x^2} $,
Consider the function $ f $ defined in the interval $ (0, \sqrt{a^2 + b^2} $ by the formula
The function takes the value $ 0 $ for $ x = a $ and $ x = b $. We will show that it has no other ... | (,b)(b,) | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XI OM - III - Task 2
A plane passing through the height of a regular tetrahedron intersects the planes of the lateral faces along $ 3 $ lines forming angles $ \alpha $, $ \beta $, $ \gamma $ with the plane of the base of the tetrahedron. Prove that | We accept the notation given in Fig. 27, where $HD$ represents the height of the regular tetrahedron $ABCD$, and $MD$, $ND$, $PD$ are the lines of intersection of the plane through $HD$ with the lateral faces $BCD$, $CAD$, and $ABD$, respectively. The angles $\alpha$, $\beta$, $\gamma$ are the angles that the segments ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
IX OM - I - Problem 9
Prove that the inequality holds
where $ a $, $ b $, $ c $, $ d $ denote positive numbers. | Inequalities (1) can be given a simpler form by introducing the following notations
Then $ 3(a+b+c + d) = x + y + z + u $ and inequality (1) takes the form
where $ x > 0 $, $ y > 0 $, $ z > 0 $, $ u > 0 $. We will provide two proofs of inequality (2).
\spos{1} Notice that if $ m > 0 $ and $ n > 0 $, then
... | proof | Inequalities | proof | Incomplete | Yes | olympiads | false |
XVII OM - III - Problem 5
Given a convex hexagon $ABCDEF$, in which each of the diagonals $AD, BE, CF$ divides the hexagon into two parts of equal area. Prove that these three diagonals pass through one point. | \spos{1} The area of a polygon $ ABC\ldots $ will be denoted by the symbol $ (ABC\ldots) $. According to the assumption (Fig. 15)
From (1) and (2) we obtain
from which it follows that $ AE \parallel BD $, since the vertices $ A $ and $ E $ of triangles $ ABD $ and $ EDB $ with the common side $ BD $ lie on ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XVIII OM - I - Problem 7
On a plane, there are $2n$ points. Prove that there exists a line that does not pass through any of these points and divides the plane into half-planes, each containing $n$ of the given points. Formulate and prove an analogous theorem for space. | Let $A_1, A_2, \ldots, A_{2n}$ be given points in the plane and let $a_{ik}$ ($i, k = 1, 2, \ldots, 2n$, $i \ne k$) denote the line passing through points $A_i$ and $A_k$. The set $Z$ of all lines $a_{ik}$ contains at most $\binom{2n}{2} = n(2n-1)$ distinct lines. Therefore, through any point $P$ in the plane, one can ... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XLVI OM - I - Problem 6
Inside triangle $ABC$, a point $P$ is chosen. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at points $A'$, $B'$, $C'$ respectively. Let: $u = |AP|:|PA'|$, $v = |BP|:|PB'|$, $w = |CP|:|PC'$. Express the product $uvw$ in terms of the sum $u + v + w$. | Let $ S $ denote the area of triangle $ ABC $, and $ S_a $, $ S_b $, $ S_c $ denote the areas of triangles $ PBC $, $ PCA $, $ PAB $, respectively. Let segment $ PD $ be the altitude in triangle $ PBC $, and segment $ AE $ be the altitude in triangle $ ABC $ (Figure 3). From the similarity of triangles $ AEA $ and $ PD... | uvw=u+v+w+1 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
XXX OM - III - Task 4
Given are real numbers $ A > 1 $ and $ B > 1 $ and a sequence $ (x_n) $ of real numbers in the interval [1, A\cdot B] Prove that there exists a sequence $ (y_n) $ of numbers in the interval $ [1, A] $, such that | The inequality given in the task is equivalent to the following
Notice that if
then inequality (1) will be satisfied. Indeed, from (2) it follows that
For the solution of the problem, it is therefore sufficient to find a sequence $ (y_k) $ such that $ 1 \leq y_k \leq A $ and to satisfy condition (2), i.e., $ \frac{x... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XLV OM - II - Task 3
Prove that if a circle can be circumscribed around a hexagonal cross-section of a cube passing through its center, then the cross-section is a regular hexagon. | We place a cube in a coordinate system so that its vertices are points with all coordinates equal to $+1$ or $-1$. The center of the cube is then the point $O = (0,0,0)$, the origin of the coordinate system. The plane considered in the problem passes through $O$ and thus has an equation of the form
\[ ax + by + cz = 0... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XXVII OM - I - Problem 9
In a circle of radius 1, a triangle with side lengths $a, b, c$ is inscribed. Prove that the triangle is acute if and only if $a^2 + b^2 + c^2 > 8$, right-angled if and only if $a^2 + b^2 + c^2 = 8$, and obtuse if and only if $a^2 + b^2 + c^2 < 8$. | From the Law of Sines, we have $ \displaystyle 2 = \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} $, where $ \alpha $, $ \beta $, $ \gamma $ are the angles of the considered triangle opposite to the sides of lengths $ a $, $ b $, $ c $, respectively. Therefore,
From the Law of Cosines, we ha... | proof | Geometry | proof | Yes | Yes | olympiads | false |
XII OM - II - Task 1
Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers. | Suppose the statement made in the text of the problem is not true and that the equality holds
where $ k $, $ r $, $ n $ denote natural numbers. According to the formula for the sum of terms of an arithmetic progression, we obtain from this
Each of the natural numbers $ 2k + r $ and $ r + 1 $ is greater than... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XXIX OM - III - Problem 5
Given a real number $ a $. We define the sequence $ (a_n) $ as follows
Prove that the sequence $ (a_n) $ has infinitely many non-negative terms and infinitely many non-positive terms. | If a term of the sequence $(a_n)$ is equal to zero, then all subsequent terms of this sequence are also equal to zero, and the thesis of the problem is satisfied. Let us assume, therefore, that all terms of the sequence $(a_n)$ are different from zero.
Let $d_n = a_{n+1} - a_n$. We calculate that
Therefore, $|d_n| ... | proof | Algebra | proof | Incomplete | Yes | olympiads | false |
XXX OM - II - Task 5
Prove that among any ten consecutive natural numbers there exists one that is relatively prime to each of the other nine. | Every common divisor of two natural numbers is also a divisor of their difference. Therefore, if among ten consecutive natural numbers, some two are not relatively prime, then they have a common divisor which is a prime number less than $10$, i.e., one of the numbers $2$, $3$, $5$, $7$.
Among ten consecutive natural nu... | proof | Number Theory | proof | Yes | Yes | olympiads | false |
XLIII OM - III - Problem 5
The base of a regular pyramid is a $2n$-sided regular polygon $A_1, A_2, \ldots, A_n$. A sphere passing through the vertex $S$ of the pyramid intersects the lateral edges $SA_i$ at points $B_i$ ($i = 1,2,\ldots, 2n$). Prove that | om43_3r_img_12.jpg
Let's adopt the following notation:
(Figure 12). It is tacitly assumed in the problem that $ B_i $ is the point of intersection of the given sphere with the edge $ SA_i $, different from $ S $. Therefore, $ x_i > 0 $.
Let the center of the sphere be denoted by $ O $. The following vector equali... | proof | Geometry | proof | Yes | Yes | olympiads | false |
LV OM - II - Task 3
Determine the number of infinite sequences $ a_1,a_2,a_3,\dots $ with terms equal to $ +1 $ and $ -1 $, satisfying the equation
and the condition: in every triplet of consecutive terms $ (a_n, a_{n+1}, a_{n+2}) $, both $ +1 $ and $ -1 $ appear. | Let $ (a_n) $ be such a sequence. We will start by showing that
Suppose $ a_{3k+1} = a_{3k+2} = a $. Then it must be that $ a_{3k} = a_{3k+3} = b $ and $ b \neq a $. Multiplying the above equalities by $ a_2 $, we get $ a_{6k+2} = a_{6k+4} = c $ and $ a_{6k} = a_{6k+6} = d $, where $ c \neq d $. In the triplet $ (a_{6... | 2 | Combinatorics | math-word-problem | Incomplete | Yes | olympiads | false |
XXIV OM - I - Problem 6
Prove the following theorem: The four altitudes of a tetrahedron $ABCD$ intersect at a single point if and only if $AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2$. | For any tetrahedron $ABCD$, let $a = \overrightarrow{AB}$, $b = \overrightarrow{AC}$, $c = \overrightarrow{AD}$. Since $\overrightarrow{DC} = b - c$, $\overrightarrow{BD} = c - a$, $\overrightarrow{CB} = a - b$, the equations given in the problem take the form $a^2 + (b - c)^2 = b^2 + (c - a)^2 = c^2 + (a - b)^2$, that... | proof | Geometry | proof | Yes | Incomplete | olympiads | false |
XXIII OM - I - Problem 12
Prove that | Let $ a_n = \underbrace{\sqrt{2 + \sqrt{ 2 + \ldots + \sqrt{2}}}}_{n \ \text{square roots}} $. Then $ a_1 = \sqrt{2} $ and for $ n = 1, 2, \ldots $.
Using the formula for the sum of $ n $ terms of a geometric sequence, we get
and therefore
Thus, the formula given in the problem can be equivalently written as:
We wi... | proof | Other | proof | Incomplete | Yes | olympiads | false |
XXXVII OM - I - Problem 10
Prove that the sequence $ (\sin n) $ does not have a limit. | Let's consider sequences of open intervals
($ k = 1, 2, 3, \ldots $). The length of each of them (equal to $ \pi/2 $) is greater than $ 1 $. Therefore, in each interval $ I_k $ there is a natural number $ m_k $ (if there are two, let's choose one of them, for example the smaller one, and denote it by $ m_k $). Similar... | proof | Calculus | proof | Yes | Yes | olympiads | false |
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