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VI OM - II - Task 3
What should be the angle at the vertex of an isosceles triangle so that a triangle can be constructed with sides equal to the height, base, and one of the remaining sides of this isosceles triangle? | We will adopt the notations indicated in Fig. 9. A triangle with sides equal to $a$, $c$, $h$ can be constructed if and only if the following inequalities are satisfied:
Since in triangle $ADC$ we have $a > h$, $\frac{c}{2} + h > a$, the first two of the above inequalities always hold, so the necessary and suffic... | 106 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
IX OM - II - Task 2
Six equal disks are placed on a plane in such a way that their centers lie at the vertices of a regular hexagon with a side equal to the diameter of the disks. How many rotations will a seventh disk of the same size make while rolling externally on the same plane along the disks until it returns to... | Let circle $K$ with center $O$ and radius $r$ roll without slipping on a circle with center $S$ and radius $R$ (Fig. 16). The rolling without slipping means that different points of one circle successively coincide with different points of the other circle, and in this correspondence, the length of the arc between two ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXXVI OM - III - Problem 1
Determine the largest number $ k $ such that for every natural number $ n $ there are at least $ k $ natural numbers greater than $ n $, less than $ n+17 $, and coprime with the product $ n(n+17) $. | We will first prove that for every natural number $n$, there exists at least one natural number between $n$ and $n+17$ that is coprime with $n(n+17)$.
In the case where $n$ is an even number, the required property is satisfied by the number $n+1$. Of course, the numbers $n$ and $n+1$ are coprime. If a number $d > 1$ we... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXXVIII OM - III - Zadanie 5
Wyznaczyć najmniejszą liczbę naturalną $ n $, dla której liczba $ n^2-n+11 $ jest iloczynem czterech liczb pierwszych (niekoniecznie różnych).
|
Niech $ f(x) = x^2-x+11 $. Wartości przyjmowane przez funkcję $ f $ dla argumentów całkowitych są liczbami całkowitymi niepodzielnymi przez $ 2 $, $ 3 $, $ 5 $, $ 7 $. Przekonujemy się o tym badając reszty z dzielenia $ n $ i $ f(n) $ przez te cztery początkowe liczby pierwsze:
\begin{tabular}{lllll}
&\multicolum... | 132 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XII OM - II - Task 4
Find the last four digits of the number $ 5^{5555} $. | \spos{1} We will calculate a few consecutive powers of the number $ 5 $ starting from $ 5^4 $:
It turned out that $ 5^8 $ has the same last four digits as $ 5^4 $, and therefore the same applies to the numbers $ 5^9 $ and $ 5^5 $, etc., i.e., starting from $ 5^4 $, two powers of the number $ 5 $, whose exponents ... | 8125 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XVIII OM - II - Problem 4
Solve the equation in natural numbers
Note: The equation itself is not provided in the original text, so it is not included in the translation. | Suppose that the triple $(x, y, z)$ of natural numbers satisfies equation (1). After dividing both sides of equation (1) by $xyz$ we get
One of the numbers $x$, $y$, $z$ is less than $3$; for if $x \geq 3$, $y \geq 3$, $z \geq 3$, then the left side of equation (2) is $\leq 1$, while the right side is $> 1$.
If $x < 3... | 7 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
III OM - I - Task 4
a) Given points $ A $, $ B $, $ C $ not lying on a straight line. Determine three mutually parallel lines passing through points $ A $, $ B $, $ C $, respectively, so that the distances between adjacent parallel lines are equal.
b) Given points $ A $, $ B $, $ C $, $ D $ not lying on a plane. Deter... | a) Suppose that the lines $a$, $b$, $c$ passing through points $A$, $B$, $C$ respectively and being mutually parallel satisfy the condition of the problem, that is, the distances between adjacent parallel lines are equal. Then the line among $a$, $b$, $c$ that lies between the other two is equidistant from them. Let th... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
L OM - I - Task 3
In an isosceles triangle $ ABC $, angle $ BAC $ is a right angle. Point $ D $ lies on side $ BC $, such that $ BD = 2 \cdot CD $. Point $ E $ is the orthogonal projection of point $ B $ onto line $ AD $. Determine the measure of angle $ CED $. | Let's complete the triangle $ABC$ to a square $ABFC$. Assume that line $AD$ intersects side $CF$ at point $P$, and line $BE$ intersects side $AC$ at point $Q$. Since
$ CP= \frac{1}{2} CF $. Using the perpendicularity of lines $AP$ and $BQ$ and the above equality, we get $ CQ= \frac{1}{2} AC $, and consequently $... | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LX OM - III - Zadanie 2
Let $ S $ be the set of all points in the plane with both coordinates being integers. Find
the smallest positive integer $ k $ for which there exists a 60-element subset of the set $ S $
with the following property: For any two distinct elements $ A $ and $ B $ of this subset, there exists a po... | Let $ K $ be a subset of the set $ S $ having for a given number $ k $ the property given in the problem statement.
Let us fix any two different points $ (a, b), (c, d) \in K $. Then for some integers
$ x, y $ the area of the triangle with vertices $ (a, b) $, $ (c, d) $, $ (x, y) $ is $ k $, i.e., the equality
$ \frac... | 210 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
LV OM - III - Task 5
Determine the maximum number of lines in space passing through a fixed point and such that any two intersect at the same angle. | Let $ \ell_1,\ldots,\ell_n $ be lines passing through a common point $ O $. A pair of intersecting lines determines four angles on the plane containing them: two vertical angles with measure $ \alpha \leq 90^\circ $ and the other two angles with measure $ 180^\circ - \alpha $. According to the assumption, the value of ... | 6 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXVI - I - Problem 9
Calculate the limit
| For any natural numbers $ k $ and $ n $, where $ n \leq k $, we have
In particular, $ \displaystyle \binom{2^n}{n} \leq (2^n)^n = 2^{n^2} $. By the binomial formula, we have
and hence $ \displaystyle 2^n > \frac{n^2}{24} $. Therefore,
Since $ \displaystyle \lim_{n \to \infty} \frac{1}{n}= 0 $, it follows that $ \dis... | 0 | Calculus | math-word-problem | Incomplete | Yes | olympiads | false |
XVI OM - II - Task 4
Find all prime numbers $ p $ such that $ 4p^2 +1 $ and $ 6p^2 + 1 $ are also prime numbers. | To solve the problem, we will investigate the divisibility of the numbers \( u = 4p^2 + 1 \) and \( v = 6p^2 + 1 \) by \( 5 \). It is known that the remainder of the division of the product of two integers by a natural number is equal to the remainder of the division of the product of their remainders by that number. B... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXXVI OM - I - Zadanie 9
W urnie jest 1985 kartek z napisanymi liczbami 1,2,3,..., 1985, każda lczba na innej kartce. Losujemy bez zwracania 100 kartek. Znaleźć wartość oczekiwaną sumy liczb napisanych na wylosowanych kartkach.
|
Losowanie $ 100 $ kartek z urny zawierającej $ 1985 $ kartek można interpretować jako wybieranie $ 100 $-elementowego podzbioru zbioru $ 1985 $-elementowego. Zamiast danych liczb $ 1985 $ i $ 100 $ weźmy dowolne liczby naturalne $ n $ i $ k $, $ n \geq k $. Dla dowolnego $ k $-elementowego zbioru $ X $ będącego podzbi... | 99300 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
LII OM - I - Task 4
Determine whether 65 balls with a diameter of 1 can fit into a cubic box with an edge of 4. | Answer: It is possible.
The way to place the balls is as follows.
At the bottom of the box, we place a layer consisting of 16 balls. Then we place a layer consisting of 9 balls, each of which is tangent to four balls of the first layer (Fig. 1 and 2). The third layer consists of 16 balls that are tangent to the balls o... | 66 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXI OM - III - Task 6
Find the smallest real number $ A $ such that for every quadratic trinomial $ f(x) $ satisfying the condition
the inequality $ f.
holds. | Let the quadratic trinomial $ f(x) = ax^2 + bx + c $ satisfy condition (1). Then, in particular, $ |f(0)| \leq 1 $, $ \left| f \left( \frac{1}{2} \right) \right| \leq 1 $, and $ |f(1)| \leq 1 $.
Since
and
thus
Therefore, $ A \leq 8 $.
On the other hand, the quadratic trinomial $ f(x) = -8x^2 + 8x - 1 ... | 8 | Inequalities | math-word-problem | Incomplete | Yes | olympiads | false |
L OM - I - Task 5
Find all pairs of positive integers $ x $, $ y $ satisfying the equation $ y^x = x^{50} $. | We write the given equation in the form $ y = x^{50/x} $. Since for every $ x $ being a divisor of $ 50 $, the number on the right side is an integer, we obtain solutions of the equation for $ x \in \{1,2,5,10,25,50\} $. Other solutions of this equation will only be obtained when $ x \geq 2 $ and for some $ k \geq 2 $,... | 8 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
VI OM - II - Task 2
Find the natural number $ n $ knowing that the sum
is a three-digit number with identical digits. | A three-digit number with identical digits has the form $ 111 \cdot c = 3 \cdot 37 \cdot c $, where $ c $ is one of the numbers $ 1, 2, \ldots, 9 $, and the sum of the first $ n $ natural numbers is $ \frac{1}{2} n (n + 1) $, so the number $ n $ must satisfy the condition
Since $ 37 $ is a prime number, one of the n... | 36 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XXXIX OM - I - Problem 1
For each positive number $ a $, determine the number of roots of the polynomial $ x^3+(a+2)x^2-x-3a $. | Let's denote the considered polynomial by $ F(x) $. A polynomial of the third degree has at most three real roots. We will show that the polynomial $ F $ has at least three real roots - and thus has exactly three real roots (for any value of the parameter $ a > 0 $).
It is enough to notice that
If a continuous func... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
XXIV OM - II - Problem 3
Let $ f:\mathbb{R} \to \mathbb{R} $ be an increasing function satisfying the conditions:
1. $ f(x+1) = f(x) + 1 $ for every $ x \in \mathbb{R} $,
2. there exists an integer $ p $ such that $ f(f(f(0))) = p $. Prove that for every real number $ x $
where $ x_1 = x $ and $ x_n = f(x_{n-1}) $ ... | We will first provide several properties of the function $f$ satisfying condition $1^\circ$ of the problem. Let $f_n(x)$ be the $n$-fold composition of the function $f$, i.e., let $f_1(x) = f(x)$ and $f_{n+1}(x) = f_n(f_1(x))$ for $n = 1, 2, \ldots$. It follows that if $n = k + m$, where $k$ and $m$ are natural numbers... | 3 | Algebra | proof | Incomplete | Yes | olympiads | false |
XLIV OM - III - Problem 3
Let $ g(k) $ denote the greatest odd divisor of the positive integer $ k $, and let us assume
The sequence $ (x_n) $ is defined by the relations $ x_1 = 1 $, $ x_{n+1} = f(x_n) $. Prove that the number 800 appears exactly once among the terms of this sequence. Determine $ n $ for which ... | Let's list the first fifteen terms of the sequence $ (x_n) $, grouping them into blocks consisting of one, two, three, four, and five terms respectively:
We have obtained five rows of the infinite system (U), which can be continued according to the following rules: the $ j $-th row consists of $ j $ numbers, the ... | 166 | Number Theory | proof | Incomplete | Yes | olympiads | false |
XX OM - II - Task 2
Find all four-digit numbers in which the thousands digit is equal to the hundreds digit, and the tens digit is equal to the units digit, and which are squares of integers. | Suppose the number $ x $ satisfies the conditions of the problem and denote its consecutive digits by the letters $ a, a, b, b $. Then
The number $ x $ is divisible by $ 11 $, so as a square of an integer, it is divisible by $ 11^2 $, i.e., $ x = 11^2 \cdot k^2 $ ($ k $ - an integer), hence
Therefore,
... | 7744 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
V OM - I - Task 2
Investigate when the sum of the cubes of three consecutive natural numbers is divisible by $18$. | Let $ a - 1 $, $ a $, $ a + 1 $, be three consecutive natural numbers; the sum of their cubes
can be transformed in the following way:
Since one of the numbers $ a - 1 $, $ a $, $ a + 1 $ is divisible by $ 3 $, then one of the numbers $ a $ and $ (a + 1) (a - 1) + 3 $ is also divisible by $ 3 $. Therefore, the sum $ ... | 3 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XXXV OM - II - Task 5
Calculate the lower bound of the areas of convex hexagons, all of whose vertices have integer coordinates. | We will use the following lemma, which was the content of a competition problem in the previous Mathematical Olympiad.
Lemma. Twice the area of a triangle, whose all vertices have integer coordinates, is an integer.
Proof. The area of triangle $ABC$ is equal to half the absolute value of the determinant formed from the... | 3 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
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 |
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 |
XXVIII - II - Task 3
In a hat, there are 7 slips of paper. On the $ n $-th slip, the number $ 2^n-1 $ is written ($ n = 1, 2, \ldots, 7 $). We draw slips randomly until the sum exceeds 124. What is the most likely value of this sum? | The sum of the numbers $2^0, 2^1, \ldots, 2^6$ is $127$. The sum of any five of these numbers does not exceed $2^2 + 2^3 + 2^4 + 2^5 + 2^6 = 124$. Therefore, we must draw at least six slips from the hat.
Each of the events where we draw six slips from the hat, and the seventh slip with the number $2^{n-1}$ ($n = 1, 2, ... | 127 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
XLII OM - I - Problem 8
Determine the largest natural number $ n $ for which there exist in space $ n+1 $ polyhedra $ W_0, W_1, \ldots, W_n $ with the following properties:
(1) $ W_0 $ is a convex polyhedron with a center of symmetry,
(2) each of the polyhedra $ W_i $ ($ i = 1,\ldots, n $) is obtained from $ W_0 $ by ... | Suppose that polyhedra $W_0, W_1, \ldots, W_n$ satisfy the given conditions. Polyhedron $W_1$ is the image of $W_0$ under a translation by a certain vector $\overrightarrow{\mathbf{v}}$ (condition (2)). Let $O_0$ be the center of symmetry of polyhedron $W_0$ (condition (1)); the point $O_1$, which is the image of $O_0$... | 26 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XXIV OM - III - Task 2
Let $ p_n $ be the probability that a series of 100 consecutive heads will appear in $ n $ coin tosses. Prove that the sequence of numbers $ p_n $ is convergent and calculate its limit. | The number of elementary events is equal to the number of $n$-element sequences with two values: heads and tails, i.e., the number $2^n$. A favorable event is a sequence containing 100 consecutive heads. We estimate the number of unfavorable events from above, i.e., the number of sequences not containing 100 consecutiv... | 1 | Combinatorics | proof | Yes | Yes | olympiads | false |
XXVIII - I - Problem 11
From the numbers $ 1, 2, \ldots, n $, we choose one, with each of them being equally likely. Let $ p_n $ be the probability of the event that in the decimal representation of the chosen number, all digits: $ 0, 1, \ldots, 9 $ appear. Calculate $ \lim_{n\to \infty} p_n $. | Let the number $ n $ have $ k $ digits in its decimal representation, i.e., let $ 10^{k-1} \leq n < 10^k $, where $ k $ is some natural number. Then each of the numbers $ 1, 2, \ldots, n $ has no more than $ k $ digits. We estimate from above the number $ A_0 $ of such numbers with at most $ k $ digits that do not cont... | 0 | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false |
XXII OM - III - Problem 5
Find the largest integer $ A $ such that for every permutation of the set of natural numbers not greater than 100, the sum of some 10 consecutive terms is at least $ A $. | The sum of all natural numbers not greater than $100$ is equal to $1 + 2 + \ldots + 100 = \frac{1 + 100}{2} \cdot 100 = 5050$. If $a_1, a_2, \ldots, a_{100}$ is some permutation of the set of natural numbers not greater than $100$ and the sum of any $10$ terms of this permutation is less than some number $B$, then in p... | 505 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XV OM - II - Task 3
Prove that if three prime numbers form an arithmetic progression with a difference not divisible by 6, then the smallest of these numbers is $3$. | Suppose that the prime numbers $ p_1 $, $ p_2 $, $ p_3 $ form an arithmetic progression with a difference $ r > 0 $ not divisible by $ 6 $, and the smallest of them is $ p_1 $. Then
Therefore, $ p_1 \geq 3 $, for if $ p_1 = 2 $, the number $ p_3 $ would be an even number greater than $ 2 $, and thus would not be a pri... | 3 | Number Theory | proof | Yes | Yes | olympiads | false |
LII OM - III - Task 4
Given such integers $ a $ and $ b $ that for every non-negative integer $ n $ the number $ 2^na + b $ is a square of an integer. Prove that $ a = 0 $.
| If $ b = 0 $, then $ a = 0 $, because for $ a \ne 0 $, the numbers $ a $ and $ 2a $ cannot both be squares of integers.
If the number $ a $ were negative, then for some large natural number $ n $, the number $ 2^n a + b $ would also be negative, and thus could not be a square of an integer.
The only case left to consid... | 0 | Number Theory | proof | Yes | Incomplete | olympiads | false |
XLIII OM - I - Problem 2
In square $ABCD$ with side length $1$, point $E$ lies on side $BC$, point $F$ lies on side $CD$, the measures of angles $EAB$ and $EAF$ are $20^{\circ}$ and $45^{\circ}$, respectively. Calculate the height of triangle $AEF$ drawn from vertex $A$. | The measure of angle $ FAD $ is $ 90^\circ - (20^\circ + 45^\circ) = 25^\circ $. From point $ A $, we draw a ray forming angles of $ 20^\circ $ and $ 25^\circ $ with rays $ AE $ and $ AF $, respectively, and we place a segment $ AG $ of length $ 1 $ on it (figure 2).
From the equality $ |AG| =|AB| = 1 $, $ | \measureda... | 1 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
XIV OM - I - Task 5
How many digits does the number
have when written in the decimal system? | The task can be easily solved using 5-digit decimal logarithm tables. According to the tables, $\log 2$ is approximately $0.30103$, with the error of this approximation being less than $0.00001$, hence
therefore
after multiplying by $2^{16} = 65536$, we obtain the inequality
From this, it follows that the numb... | 19729 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
XLIV OM - I - Problem 6
The sequence $ (x_n) $ is defined as follows:
Calculate the sum $ \sum_{n=0}^{1992} 2^n x_n $. | Let's replace the parameter $1992$ with any arbitrarily chosen natural number $N \geq 1$ and consider the sequence $(x_n)$ defined by the formulas
when $N = 1992$, this is the sequence given in the problem.
Let's experiment a bit. If $N = 3$, the initial terms of the sequence $(x_n)$ are the numbers $3$, $-9$, $9$, $-... | 1992 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XLVIII OM - I - Problem 8
Let $ a_n $ be the number of all non-empty subsets of the set $ \{1,2,\ldots,6n\} $, the sum of whose elements gives a remainder of 5 when divided by 6, and let $ b_n $ be the number of all non-empty subsets of the set $ \{1,2,\ldots,7n\} $, the product of whose elements gives a remainder of ... | For a finite set of numbers $A$, the symbols $s(A)$ and $p(A)$ will denote the sum and the product of all numbers in this set, respectively. Let $n$ be a fixed natural number. According to the problem statement,
Notice that in the definition of the number $b_n$, we can remove all multiples of the number $7$ from the s... | 1 | Combinatorics | math-word-problem | Yes | Incomplete | olympiads | false |
LI OM - I - Task 10
In space, there are three mutually perpendicular unit vectors $ \overrightarrow{OA} $, $ \overrightarrow{OB} $, $ \overrightarrow{OC} $. Let $ \omega $ be a plane passing through the point $ O $, and let $ A', $ B', $ C' $ be the projections of points $ A $, $ B $, $ C $ onto the plane $ \omega $, ... | We will show that the value of expression (1) is equal to $ 2 $, regardless of the choice of plane $ \omega $.
Let $ \omega $ be any plane passing through point $ O $, and $ \pi $ be a plane containing points $ O $, $ A $, $ B $. Denote by $ \ell $ the common line of planes $ \pi $ and $ \omega $. Let $ X $, $ Y $ be t... | 2 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
L OM - I - Problem 11
In an urn, there are two balls: a white one and a black one. Additionally, we have 50 white balls and 50 black balls at our disposal. We perform the following action 50 times: we draw a ball from the urn, and then return it to the urn along with one more ball of the same color as the drawn ball. ... | Let $ P(k,n) $, where $ 1 \leq k\leq n-1 $, denote the probability of the event that when there are $ n $ balls in the urn, exactly $ k $ of them are white. Then
Using the above relationships, we prove by induction (with respect to $ n $) that $ P(k,n) = 1/(n-1) $ for $ k = 1,2,\ldots,n-1 $. In particular
T... | 51 | Combinatorics | 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 |
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 |
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 |
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 |
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 |
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 |
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 |
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 |
XXVI - I - Task 1
At the ball, there were 42 people. Lady $ A_1 $ danced with 7 gentlemen, Lady $ A_2 $ danced with 8 gentlemen, ..., Lady $ A_n $ danced with all the gentlemen. How many gentlemen were at the ball? | The number of ladies at the ball is $ n $, so the number of gentlemen is $ 42-n $. The lady with number $ k $, where $ 1 \leq k \leq n $, danced with $ k+6 $ gentlemen. Therefore, the lady with number $ n $ danced with $ n+ 6 $ gentlemen. These were all the gentlemen present at the ball. Thus, $ 42-n = n + 6 $. Solving... | 24 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXXVI OM - I - Problem 11
Provide an example of a convex polyhedron having 1985 faces, among which there are 993 faces such that no two of them share a common edge. | It is easy to obtain a number of faces significantly greater than $993$, and not only without common edges, but completely disjoint. For this purpose, consider any convex polyhedron $W$ having $5$ faces and vertices; we assume that no two faces lie in the same plane, i.e., that all dihedral angles are less than a strai... | 993 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
L OM - II - Task 5
Let $ S = \{1, 2,3,4, 5\} $. Determine the number of functions $ f: S \to S $ satisfying the equation $ f^{50} (x) = x $ for all $ x \in S $.
Note: $ f^{50}(x) = \underbrace{f \circ f \circ \ldots \circ f}_{50} (x) $. | Let $ f $ be a function satisfying the conditions of the problem. For numbers $ x \neq y $, we get $ f^{49}(f(x)) = x \neq y = f^{49}(f(y)) $, hence $ f(x) \neq f(y) $. Therefore, $ f $ is a permutation of the set $ S $. Denote by $ r(x) $ ($ x \in S $) the smallest positive integer such that $ f^{r(x)}(x) = x $. Then ... | 50 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
V OM - I - Problem 9
Points $ A, B, C, D, \ldots $ are consecutive vertices of a certain regular polygon, and the following relationship holds
How many sides does this polygon have? | \spos{1} Let $ r $ denote the radius of the circle circumscribed around a regular polygon $ ABCD\ldots $, and $ 2x $ - the central (convex) angle of this circle corresponding to the chord $ AB $. Then (Fig. 25)
Substituting these expressions into equation (1), we get
To solve equation (2), we multiply it by... | 7 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
XII OM - III - Task 6
Someone wrote six letters to six people and addressed six envelopes to them. In how many ways can the letters be placed into the envelopes so that no letter ends up in the correct envelope? | Let $ F (n) $ denote the number of all ways of placing $ n $ letters $ L_1, L_2, \ldots, L_n $ into $ n $ envelopes $ K_1, K_2, \ldots, K_n $ so that no letter $ L_i $ ends up in the correct envelope $ K_i $, or more simply, so that there is no "hit". We need to calculate $ F (6) $.
Suppose we incorrectly place all let... | 265 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
XXII OM - I - Problem 6
Let $ f(x,y,z) = \max(x^2 - yz, y^2 - xz, z^2 - xy) $. Find the set of values of the function $ f(x,y,z) $ considered for numbers $ x,y,z $ satisfying the conditions:
Note: $ \max (a, b, c) $ is the greatest of the numbers $ a, b, c $. | From (1) it follows that
thus $ x + y + z = 3 $. It follows from this that the conditions (1) added in the problem are equivalent to the conditions:
If we perform any permutation of the letters $ x $, $ y $, $ z $, the conditions of the problem will not change. Therefore, without loss of generality, we can l... | 4 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
XXII OM - III - Task 3
How many locks at least need to be placed on the treasury so that with a certain distribution of keys among the 11-member committee authorized to open the treasury, any 6 members can open it, but no 5 can? Determine the distribution of keys among the committee members with the minimum number of ... | Suppose that for some natural number $ n $ there exists a key distribution to $ n $ locks among an 11-member committee such that the conditions of the problem are satisfied. Let $ A_i $ denote the set of locks that the $ i $-th member of the committee can open, where $ i = 1, 2, \ldots, 11 $, and let $ A $ denote the s... | 462 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. Consider the set $M$ of integers $n \in[-100 ; 500]$, for which the expression $A=n^{3}+2 n^{2}-5 n-6$ is divisible by 11. How many integers are contained in $M$? Find the largest and smallest of them? | Answer: 1) 164 numbers; 2) $n_{\text {min }}=-100, n_{\text {max }}=497$. | 164 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. A crew of four pirates docked at an island to divide a treasure of gold coins left there. It was late, so they decided to postpone the division until morning. The first pirate woke up in the middle of the night and decided to take his share. He couldn't divide the coins into four equal parts, so he took two coins fi... | Answer: the first pirate 1178 coins, the second pirate 954 coins, the third pirate 786 coins, the fourth - 660 coins. | 1178 | Logic and Puzzles | math-word-problem | Yes | Problem not solved | olympiads | false |
1. Solution. According to the problem, the sum of the original numbers is represented by the expression:
$$
\begin{aligned}
& \left(a_{1}+2\right)^{2}+\left(a_{2}+2\right)^{2}+\ldots+\left(a_{50}+2\right)^{2}=a_{1}^{2}+a_{2}^{2}+\ldots+a_{50}^{2} \rightarrow \\
& {\left[\left(a_{1}+2\right)^{2}-a_{1}^{2}\right]+\left[... | Answer: will increase by 150. | 150 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Solution. In the figure, equal angles are marked with the same numbers. Triangle $A P M$ is similar to triangle $C P B$ with a similarity coefficient $k_{1}=\lambda$
$$
\begin{aligned}
& A P=\lambda x, P C=x, P Q+\lambda x=x \rightarrow P Q=x(1-\lambda) \\
& \lambda x+x=a \rightarrow x=\frac{a}{1+\lambda} \rightarr... | Answer: $M N=a(1-\lambda)=2$. | 2 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
1. A set of 60 numbers is such that adding 3 to each of them does not change the value of the sum of their squares. By how much will the sum of the squares of these numbers change if 4 is added to each number? | Answer: will increase by 240. | 240 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 4. Points $M$ and $N$ on sides $DA$ and $DC$ are the feet of the altitudes of the rhombus dropped from vertex $B$.
The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP: PC=1: 4$. Find the length of the segm... | Answer: $M N=a(1-\lambda)=3$.
# | 3 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
1. A set of 70 numbers is such that adding 4 to each of them does not change the magnitude of the sum of their squares. By how much will the sum of the squares of these numbers change if 5 is added to each number? | Answer: will increase by 350. | 350 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 12. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP: PC = 2: 3$. Find the length of the seg... | Answer: $M N=a(1-\lambda)=4$.
# | 4 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
1. A set of 80 numbers is such that adding 5 to each of them does not change the magnitude of the sum of their squares. By how much will the sum of the squares of these numbers change if 6 is added to each number? | Answer: will increase by 480. | 480 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. The length of the diagonal $AC$ of the rhombus $ABCD$ with an acute angle at vertex $A$ is 20. Points $M$ and $N$ on sides $DA$ and $DC$ are the bases of the heights of the rhombus dropped from vertex $B$. The height $BM$ intersects the diagonal $AC$ at point $P$ such that $AP: PC=3: 4$. Find the length of the segme... | Answer: $M N=a(1-\lambda)=5$.
Grading criteria, 8th grade
Preliminary round of the ROSATOM industry physics and mathematics school olympiad, mathematics
# | 5 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
3. The sum of two natural numbers is 2013. If you erase the last two digits of one of them, add one to the resulting number, and then multiply the result by five, you get the other number. Find these numbers. Enter the largest of them in the provided field. | 3. The sum of two natural numbers is 2013. If you erase the last two digits of one of them, add one to the resulting number, and then multiply the result by five, you get the other number. Find these numbers. Enter the largest of them in the provided field.
Answer: 1913 | 1913 | Number Theory | math-word-problem | Yes | Problem not solved | olympiads | false |
4. The sum of two natural numbers is 2014. If you strike out the last two digits of one of them, multiply the resulting number by three, you get a number that is six more than the other number. Find these numbers. Enter the smallest of them in the provided field. | 4. The sum of two natural numbers is 2014. If you strike out the last two digits of one of them, multiply the resulting number by three, you get a number that is six more than the other number. Find these numbers. Enter the smallest of them in the provided field.
Answer: 51 | 51 | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
5. Find the fraction $\frac{p}{q}$ with the smallest possible natural denominator, for which $\frac{1}{2014}<\frac{p}{q}<\frac{1}{2013}$. Enter the denominator of this fraction in the provided field | 5. Find the fraction $\frac{p}{q}$ with the smallest possible natural denominator, for which
$\frac{1}{2014}<\frac{p}{q}<\frac{1}{2013}$. Enter the denominator of this fraction in the provided field
Answer: 4027 | 4027 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. How many pairs of natural numbers $(a ; b)$ exist such that the number $5 a-3$ is divisible by $b$, and the number $5 b-1$ is divisible by $a$? Enter the number of such pairs of numbers in the provided field. | 6. How many pairs of natural numbers $(a ; b)$ exist such that the number $5 a-3$ is divisible by $b$, and the number $5 b-1$ is divisible by $a$? Enter the number of pairs of the specified numbers in the provided field. Answer: 18 | 18 | Number Theory | math-word-problem | Yes | Incomplete | olympiads | false |
7. The coordinates $(x ; y ; z)$ of point $M$ are consecutive terms of a geometric progression, and the numbers $x y, y z, x z$ in the given order are terms of an arithmetic progression, with $z \geq 1$ and $x \neq y \neq z$. Find the smallest possible value of the square of the distance from point $M$ to point $N(1 ; ... | 7. The coordinates $(x ; y ; z)$ of point $M$ are consecutive terms of a geometric progression, and the numbers $x y, y z, x z$ in the given order are terms of an arithmetic progression, with $z \geq 1$ and $x \neq y \neq z$. Find the smallest possible value of the square of the distance from point $M$ to point $N(1 ; ... | 18 | Algebra | math-word-problem | Yes | Incomplete | olympiads | false |
9. Find the last two digits of the number $14^{14^{14}}$. Enter your answer in the provided field. | 9. Find the last two digits of the number $14^{14^{14}}$. Enter your answer in the provided field.
Answer: 36 | 36 | Number Theory | math-word-problem | Yes | Problem not solved | olympiads | false |
10. Find the number of twos in the prime factorization of the number $2011 \cdot 2012 \cdot 2013 \cdot \ldots .4020$. Enter your answer in the provided field. | 10. Find the number of twos in the prime factorization of the number $2011 \cdot 2012 \cdot 2013 \cdot \ldots .4020$. Enter your answer in the provided field.
Answer: 2010 | 2010 | Number Theory | math-word-problem | Yes | Problem not solved | olympiads | false |
11. For what values of $a$ does the equation $|x|=a x-2$ have no solutions? Enter the length of the interval of values of the parameter $a$ in the provided field. | 11. For what values of $a$ does the equation $|x|=a x-2$ have no solutions? Enter the length of the interval of parameter $a$ values in the provided field.
Answer: 2 | 2 | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
13. For what value of $a$ does the equation $|x-2|=a x-2$ have an infinite number of solutions? Enter the answer in the provided field | 13. For what value of $a$ does the equation $|x-2|=a x-2$ have an infinite number of solutions? Enter the answer in the provided field
Answer: 1 | 1 | Algebra | math-word-problem | Yes | Problem not solved | olympiads | false |
# 2.4. Final round of the "Rosatom" Olympiad, 11th grade, set 4
## Answers and solutions
Problem 1 Answer: $x=2$ | Solution.
Transform the equation $f\left(\log _{2}\left(x\left(x^{2}-1\right)\right)-0.5 \log _{2}(x-1)^{2}\right)=f\left(\log _{2} 6\right)$. From the monotonicity of the function $f$, it follows that the equality is possible if $\log _{2}\left(x\left(x^{2}-1\right)\right)-0.5 \log _{2}(x-1)^{2}=\log _{2} 6\left(^{*}... | 2 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
4. If $\quad a=\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}, \quad$ then $\quad P(a)=\overline{a_{6} a_{1} a_{2} a_{3} a_{4} a_{5}}$, $P(P(a))=\overline{a_{5} a_{6} a_{1} a_{2} a_{3} a_{4}} \quad$ with $\quad a_{5} \neq 0, a_{6} \neq 0, a_{1} \neq 0 . \quad$ From the equality $P(P(a))=a$ it follows that $a_{1}=a_{5},... | Answer: 1) 81 is the number; 2) $a=\overline{t u t u t u}, t, u$, where $t, u$ - are any digits, not equal to zero. | 81 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Square the numbers $a=101, b=10101$. Extract the square root of the number $c=102030405060504030201$. | Answer: 1) $a^{2}=10201 ; 2$) $b^{2}=102030201 ;$ 3) $\sqrt{c}=10101010101$. | 10101010101 | Number Theory | math-word-problem | More than one problem | Yes | olympiads | false |
3. Square the numbers $a=1001, b=1001001$. Extract the square root of the number $c=1002003004005004003002001$. | 1) $a^{2}=1002001$; 2) $b^{2}=1002003002001$; 3) $\sqrt{c}=1001001001001$. | 1001001001001 | Algebra | math-word-problem | More than one problem | Yes | olympiads | false |
3. Square the numbers $a=10001, b=100010001$. Extract the square root of the number $c=1000200030004000300020001$. | 1) $a^{2}=100020001$; 2) $b^{2}=10002000300020001$; 3) $\sqrt{c}=1000100010001$. | 1000100010001 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
Task 1. Answer: 10 boys, 13 girls. | Solution. Notation: $n$ - number of boys, $m$ - number of girls, $k$ - number of laps run by each boy, $r$ - number of candies received by each boy.
Problem statement:
$$
\left\{\begin{array}{l}
n k+m(k-1)=286 \\
n r+m(r+1)=243
\end{array}, k \geq 2, r \geq 1\right.
$$
Add the equations of the system:
$$
n(k+r)+m(k... | 10 | Other | math-word-problem | Incomplete | Yes | olympiads | false |
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{5 n+3}{7 n+8}$ be reduced? For which integers $n$ can this occur? | Answer: it can be reduced by 19 when $n=19k+7, k \in Z$. | 19 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. By what natural number can the numerator and denominator of the ordinary fraction of the form $\frac{4 n+3}{5 n+2}$ be reduced? For which integers $n$ can this occur? | Answer: can be reduced by 7 when $n=7 k+1, k \in Z$. | 7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. The angle at vertex $B$ of triangle $A B C$ is $130^{\circ}$. Through points $A$ and $C$, lines perpendicular to line $A C$ are drawn and intersect the circumcircle of triangle $A B C$ at points $E$ and $D$. Find the acute angle between the diagonals of the quadrilateral with vertices at points $A, C, D$ and $E$.
P... | Solution. Let $a$ be the number of students in the first category, $c$ be the number of students in the third category, and $b$ be the part of students from the second category who will definitely lie in response to the first question (and say "YES" to all three questions), while the rest of the students from this cate... | 80 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Answer: 1) 44 integer solutions 2 ) $a=k \in Z$ | Solution.
Rewrite the system as $\left\{\begin{array}{c}3 \leq x \leq 12 \\ -12-3 a \leq x+3 y \leq-3 a\end{array}\right.$. The set of integer solutions of this system is the union over the parameter $b$ of the integer solutions of the systems $\left\{\begin{array}{c}3 \leq x \leq 12, \\ x+3 y=b, \quad b \in[-12-3 a ;... | 44 | Algebra | math-word-problem | Incomplete | Yes | olympiads | false |
3. Answer: $\operatorname{GCD}\left(x_{1}, x_{2}, x_{3}\right)=2, \operatorname{LCM}\left(x_{1}, x_{2}, x_{3}\right)=210$ | Solution.
Let's factorize the free term of the equation into prime factors: $4200=2^{3} \cdot 3 \cdot 5^{2} \cdot 7$.
If $p$ is a prime common divisor of the roots, $4200=x_{1} \cdot x_{2} \cdot x_{3}$ is divisible by $p^{3}$. The factorization of the number 4200 into factors shows that such a common divisor can only... | 210 | Number Theory | math-word-problem | Incomplete | Yes | olympiads | false |
1. Answer: 15 weights, 23 weights | Solution.
$x$ - the number of used weights of 3 kg, $y$ - the number of used weights of 5 kg.
Condition: $3 x+5 y=71, x \geq 0, y \geq 0$. The general solution of the equation:
$\left\{\begin{array}{l}x=7+5 t \\ y=10-3 t\end{array}, t \in Z \rightarrow 7+5 t \geq 0,10-3 t \geq 0 \rightarrow t \in[-1 ; 3]\right.$
Th... | 15 | Logic and Puzzles | math-word-problem | Incomplete | Yes | olympiads | false |
2. Masha chose five digits: $2,3,5,8$ and 9 and used only them to write down all possible four-digit numbers. For example, 2358, 8888, 9235, etc. Then, for each number, she multiplied the digits in its decimal representation, and then added up all the results. What number did Masha get? | Answer: $27^{4}=531441$. | 531441 | Combinatorics | math-word-problem | Yes | Problem not solved | olympiads | false |
3. In the line, integers are recorded one after another, starting with 3, and each subsequent number, except the first, is the sum of the two numbers adjacent to it. The number 7 appeared at the 1947th position. | Answer: $\Sigma_{851}=7$. | 7 | Number Theory | math-word-problem | Incomplete | Problem not solved | olympiads | false |
5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 2 and $2 \sqrt{2}$ respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 1. Find the angle between the lines $B C$ and $A D$. | Answer: $\alpha=45^{\circ}$. | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. In a convex quadrilateral $A B C D$, the lengths of sides $B C$ and $A D$ are 6 and 8, respectively. The distance between the midpoints of diagonals $B D$ and $A C$ is 5. Find the angle between the lines $B C$ and $A D$. | Answer: $\alpha=90^{\circ}$. | 90 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
5. Let's introduce the notation: $A B=2 c, A C=2 b, \measuredangle B A C=\alpha$. The feet of the perpendicular bisectors are denoted by points $P$ and $Q$. Then, in the right triangle $\triangle A M Q$, the hypotenuse $A M=\frac{b}{\cos \alpha}$. And in the right triangle $\triangle A N P$, the hypotenuse $A N=\frac{c... | Answer: $60^{\circ}$ or $120^{\circ}$. | 60 | Geometry | proof | Yes | Yes | olympiads | false |
5. In triangle $A B C$, the perpendicular bisectors of sides $A B$ and $A C$ intersect lines $A C$ and $A B$ at points $N$ and $M$ respectively. The length of segment $N M$ is equal to the length of side $B C$ of the triangle. The angle at vertex $C$ of the triangle is $40^{\circ}$. Find the angle at vertex $B$ of the ... | Answer: $80^{\circ}$ or $20^{\circ}$.
## Final round of the "Rosatom" Olympiad, 9th grade, CIS, February 2020
# | 80 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. The sum $b_{6}+b_{7}+\ldots+b_{2018}$ of the terms of the geometric progression $\left\{b_{n}\right\}, b_{n}>0$ is 6. The sum of the same terms taken with alternating signs $b_{6}-b_{7}+b_{8}-\ldots-b_{2017}+b_{2018}$ is 3. Find the sum of the squares of the same terms $b_{6}^{2}+b_{7}^{2}+\ldots+b_{2018}^{2}$. | Answer: $b_{6}^{2}+b_{7}^{2}+\ldots+b_{2018}^{2}=18$. | 18 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Let $A=\overline{a b c b a}$ be a five-digit symmetric number, $a \neq 0$. If $1 \leq a \leq 8$, then the last digit of the number $A+11$ will be $a+1$, and therefore the first digit in the representation of $A+11$ should also be $a+1$. This is possible only with a carry-over from the digit, i.e., when $b=c=9$. Then... | Answer: eight numbers of the form $\overline{a 999 a}$, where $a=1,2, \ldots, 8$. | 8 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Nine horizontal and fifteen vertical streets form a total of $\quad(9-1)(15-1)=112 \quad$ square blocks in the city "N". Let $A$ and $C$ be the nearest points of blocks $(2 ; 3)$ and $(5 ; 12)$, then the path between them (according to the rules) has
![](https://cdn.mathpix.com/cropped/2024_05_06_b772e12e98e8648fe8... | Answer: 112 blocks; $c_{\min }=10$ coins, $c_{\max }=14$ coins. | 112 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
5. Let the distance between points $A$ and $B$ be $L$ mm, and the length of the flea's jump be $d$ mm. If $L$ were divisible by $d$ exactly, the flea's strategy of jumping in a straight line connecting points $A$ and $B$ would be optimal. It would take $\frac{L}{d}=n, n \in \mathbb{N}$ jumps to reach the goal. Given $L... | Answer: $\quad n_{\min }=1554$ jumps. | 1554 | Geometry | math-word-problem | Incomplete | Yes | olympiads | false |
2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 513315 is symmetric, while 513325 is not. How many six-digit symmetric numbers exist such that adding 110 to them leaves them symmetric? | Answer: 81 numbers of the form $\overline{a b 99 b a}$, where $a=1,2, \ldots, 9, b=0,1,2, \ldots, 8$. | 81 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. In city "N", there are 12 horizontal and 16 vertical streets, of which a pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks that are squares with a side length of 100m. Each block has an address consisting of two integers $(i ; j), i=1,2,... | Answer: 165 blocks; $c_{\min }=4$ coins, $c_{\max }=8$ coins. | 165 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
5. Kuzya the flea can make a jump in any direction on a plane for exactly 15 mm. Her task is to get from point $A$ to point $B$ on the plane, the distance between which is 2020 cm. What is the minimum number of jumps she must make to do this? | Answer: $n_{\min }=1347$ jumps.
# | 1347 | Geometry | math-word-problem | Yes | Incomplete | olympiads | false |
2. Integers, the decimal representation of which reads the same from left to right and from right to left, we will call symmetric. For example, the number 5134315 is symmetric, while 5134415 is not. How many seven-digit symmetric numbers exist such that adding 1100 to them leaves them symmetric? | Answer: 810 numbers of the form $\overline{a b c 9 c b a}$, where $a=1,2, \ldots, 9$, $b=0,1,2, \ldots, 9, c=0,1,2, \ldots, 8$. | 810 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. In city "N", there are 7 horizontal and 13 vertical streets, of which a pair of horizontal and a pair of vertical streets form the rectangular boundary of the city, while the rest divide it into blocks that are squares with a side length of 100 m. Each block has an address consisting of two integers $(i ; j), i=1,2,... | # Answer: 72 blocks; $c_{\min }=8$ coins, $c_{\max }=12$ coins. | 72 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
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