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1. To each vertex of a regular 63-gon, we assign one of the numbers 1 or -1. To each of its sides, we append the product of the numbers at its vertices and sum all the numbers next to the individual sides. Find the smallest possible non-negative value of such a sum. | SOLUTION. Let $S$ be the value we are examining, i.e., the sum of the numbers on the sides of the 63-gon. If we assign the number 1 to each vertex of the 63-gon, we get $S=63$, because each of its sides will be assigned the number 1. It is clear that for any chosen numbering of the vertices, it can be reached by gradua... | 3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. Dana wrote a three-digit number on paper, which when divided by seven gives a remainder of 2. By swapping the first two digits, a three-digit number was formed that when divided by seven gives a remainder of 3. The number formed by swapping the last two digits of the original number gives a remainder of 5 when divid... | SOLUTION. Let's denote the digits of Dana's number as $a, b, c$. The information about the remainders when divided by seven from the problem can be rewritten into equations
$$
\begin{aligned}
& 100 a+10 b+c=7 x+2, \\
& 100 b+10 a+c=7 y+3, \\
& 100 a+10 c+b=7 z+5 .
\end{aligned}
$$
The digits $a$ and $b$ cannot be zer... | 6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. We have a certain number of boxes and a certain number of balls. If we put exactly one ball in each box, we will have n balls left. However, if we set aside exactly n boxes, we can distribute all the balls so that there are exactly $n$ balls in each of the remaining boxes. How many boxes and how many balls do we hav... | SOLUTION. If we denote $x$ as the number of boxes and $y$ as the number of balls, the problem leads to the system of equations
$$
x+n=y \quad \text { and } \quad(x-n) \cdot n=y
$$
with unknowns $x$, $y$, and $n$ from the set of natural numbers. By eliminating the unknown $y$, we get the equation $x+n=(x-n) \cdot n$, ... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
6. Klárka had a three-digit number written on a piece of paper. When she correctly multiplied it by nine, she got a four-digit number that started with the same digit as the original number, the middle two digits were the same, and the last digit was the sum of the digits of the original number. What four-digit number ... | SOLUTION. Let us look for the original number $x=100a+10b+c$, whose digits are $a, b, c$. Let the digit that appears in the middle two places of the resulting product be denoted by $d$. From the problem statement, we have:
$$
9(100a+10b+c)=1000a+100d+10d+(a+b+c),
$$
where the expression in the last parentheses repres... | 2007 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Construct a trapezoid, given the lengths of its diagonals $9 \mathrm{~cm}$ and $12 \mathrm{~cm}$, the length of the midline $8 \mathrm{~cm}$, and the distance between the midpoints of the diagonals $2 \mathrm{~cm}$. | SOLUTION. Let's choose the notation according to Fig. $1, K P$ is the midline in triangle $A C D$,
Fig. 1
thus $|K P|=\frac{1}{2}|D C|$, similarly $|Q L|=\frac{1}{2}|D C|,|P L|=\frac{1}{2}|A... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. Find all pairs of natural numbers $a, b$ for which
$$
n(a, b)+D(a, b)=63
$$
where $n(a, b)$ denotes the least common multiple and $D(a, b)$ the greatest common divisor of the numbers $a, b$. | SOLUTION. We will use what we stated in the 1st problem. Let $a=D p, b=D q, n=D p q$, where $D$ is the greatest common divisor, $n$ is the least common multiple of the numbers $a, b$, and the numbers $p, q$ are coprime. According to the problem statement, $D(1+p q)=63$ must hold, so we have the following options (witho... | 8 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. On the board, one or several different two-digit natural numbers are written. We will call a digit c on the board good if the sum of those numbers on the board that contain the digit c is equal to the number 71. (Sometimes it may be a "sum" of a single number with the given digit)
a) Which of the digits 0 to 9 can ... | SOLUTION. a) The following examples of one or two numbers written on the board show that the digits 1, 2, 3, 4, 5, 7 can be good:
$$
\begin{array}{ll}
\text { 1: } & \{71\}, \\
\text { 2: } & \{29,42\}, \\
\text { 3: } & \{32,39\}, \\
\text { 4: } & \{24,47\}, \\
\text { 5: } & \{15,56\}, \\
\text { 7: } & \{71\} .
\e... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. A $10 \times 10$ table is filled with numbers 1 and -1 such that the sum of the numbers in each row, except one, is equal to 0, and the sum of the numbers in each column, except one, is equal to the same number s. Determine the largest possible value of s and show that it cannot be larger. Also, provide an example o... | SOLUTION. In the first part of the solution, we will show that the value of $s$ never exceeds the number 2. For the value $s=2$, we will provide an example of a suitable table in the second part.
Let us consider any $10 \times 10$ table filled according to the problem statement and, in addition to the number $s$, let ... | 2 | Combinatorics | proof | Yes | Yes | olympiads | false |
2. Determine the largest four-digit number $n$, for which the sum $n^{19}+99^{n}$ is divisible by ten. | 2. The given sum is odd for even $n$. It is therefore divisible by ten only when $n$ is odd. For $n=2k+1$, the term $99^{n}=(100-1)^{2k+1}=10A-1$ gives a remainder of 9 when divided by ten, and thus the second term $n^{19}$ must give a remainder of 1 when divided by ten.
The decimal representation of the number $3^{19... | 9991 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Determine the smallest positive integer that can be inserted between the digits 20 and 16 so that the resulting number $20 \ldots 16$ is a multiple of 2016. | 1. The number 2016 is a multiple of 9, so the resulting number must have a digit sum divisible by 9. This will occur precisely when the inserted number also has a digit sum divisible by 9, i.e., it must be a multiple of nine. We will try the positive multiples of the number 9 starting from the smallest: the numbers 9, ... | 36 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Let $\mathbb{N}$ denote the set of all natural numbers, and consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any $x, y \in \mathbb{N}$, the following holds:
$$
f(x f(y))=y f(x) .
$$
Determine the smallest possible value of $f(2007)$.
(P. Calábek) | Solution. Consider any function $f$ with the required properties. First, we will show that it is injective. If $f\left(y_{1}\right)=f\left(y_{2}\right)$, then for all natural numbers $x$ we have
$$
y_{1} f(x)=f\left(x f\left(y_{1}\right)\right)=f\left(x f\left(y_{2}\right)\right)=y_{2} f(x) \text {, }
$$
and since $f... | 18 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Find the largest three-digit number from which, after erasing any digit, we get a prime number. | 1. We are looking for the largest number $\overline{a b c}$, for which all three numbers $\overline{a b}, \overline{a c}$, and $\overline{b c}$ are prime numbers. The two-digit prime numbers $\overline{a b}$ and $\overline{a c}$ are odd, so their last digits must be odd. Additionally, none of them can contain the digit... | 731 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Let's examine whether it is possible to fill an $n \times n$ square table with natural numbers from 1 to $n^{2}$ such that in every $2 \times 2$ square part, at least one multiple of five is written.
a) Prove that for no even $n$ is this possible.
b) Find the largest odd $n$ for which this is possible. | 2. For even $n=2k$, divide the table into non-overlapping $2 \times 2$ squares. There will be exactly $k^2$ of these, and in each of them, a different multiple of five must be written. For this, we need $k^2$ multiples of five, the smallest of which are the numbers $5, 10, \ldots, 5k^2$. However, the last of these numb... | 9 | Combinatorics | proof | Yes | Yes | olympiads | false |
5. We have cards with numbers 5, 6, 7, ..., 55 (each card has one number). What is the maximum number of cards we can select so that the sum of the numbers on any two selected cards is not a palindrome? (A palindrome is a number that reads the same backward as forward.) | SOLUTION. To be able to express ourselves more concisely, we will choose numbers directly, not cards.
First, note that for the sum $s$ of any two given numbers, $11=$ $=5+6 \leqq s \leqq 55+54=109$. Among the numbers from 11 to 109, the palindromes are precisely all multiples of 11 and the number 101. Now, realize tha... | 25 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. Find the bases $z$ of all number systems in which the four-digit number $(1001)_{z}$ is divisible by the two-digit number $(41)_{z}$. | 1. Since the digit 4 appears in the notation of a two-digit number, it necessarily follows that $z \geq 5$. From the expanded notations $(1001)_z = z^3 + 1$ and $(41)_z = 4z + 1$, we are looking for exactly those natural numbers $z \geq 5$ for which the number $z^3 + 1$ is a multiple of the number $4z + 1$. Using the E... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Prove that there exists a unique digit $c$, for which there is a unique natural number $n$ ending in the digit $c$ and having the property that the number $2n+1$ is the square of a prime number. | SOLUTION. Let the (odd) number $2n+1$ be the square of a prime number $p$, then $p$ is also an odd number. From the relation $p^2 = 2n + 1$, it follows that $n = \frac{1}{2}(p^2 - 1) = \frac{1}{2}(p - 1)(p + 1)$. Let's create a table of the first few odd prime numbers $p$ and their corresponding numbers $n$:
| $p$ | 3... | 2 | Number Theory | proof | Yes | Yes | olympiads | false |
1. Each cell of a $68 \times 68$ table is to be colored with one of three colors (red, blue, white). In how many ways can this be done so that every triplet of adjacent cells in each row and each column contains cells of all three colors?
(Josef Tkadlec) | Solution. Once we determine the colors of some two adjacent fields in a row or column of the table, the coloring of all its other fields is uniquely determined by the requirements of the problem. The colors in each row and column thus alternate regularly with a period of 3.
Let's say that $a b c a b c .$. is the color... | 12 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. What is the smallest possible sum of four natural numbers such that pairs formed from these numbers have the greatest common divisors of $2, 3, 4, 5, 6$, and 9? Provide an example of a quadruple $s$ with such a sum and explain why there does not exist a quadruple $s$ with a smaller sum.
(Tomáš Jurík) | Solution. Suppose we have four numbers with the required properties. Since exactly three pairs have an even greatest common divisor (GCD), exactly three of them are even and one is odd (they cannot all be even, and two even numbers are insufficient for three even GCDs). Let the three even numbers be $a, b,$ and $c$, su... | 95 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. A $10 \times 10$ table is filled with numbers $1$ and $-1$ such that the sum of the numbers in each row $i$ and each column is divisible by three. Determine the largest possible sum of the numbers in the table and show that it cannot be greater. Also, provide an example of a table with the determined largest sum.
(... | SOLUTION. Consider any $10 \times 10$ table filled according to the problem statement and estimate the sums of its numbers in individual rows, knowing that these are multiples of the number 3. The same will likely apply to the sums of numbers in columns.
In one row, there are ten numbers $\pm 1$, so for their sum, we ... | 60 | Combinatorics | proof | Yes | Yes | olympiads | false |
1. In how many ways can a $3 \times 3$ square table be filled with the numbers $2,2,3,3,3,4,4,4,4$ so that the sum of the numbers in each $2 \times 2$ square of this table is equal to 14? | ANOTHER SOLUTION. Just like in the previous solution, we will show that the number 14 can be obtained from the given numbers as a sum of four numbers only as $4+4+4+2$ or $4+4+3+3$. It follows from this that in any $2 \times 2$ square that contains the number 2 (or 3), no other 2 can be present. In the central square (... | 4 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. Consider any arithmetic sequences of real numbers $\left(x_{i}\right)_{i=1}^{\infty}$ and $\left(y_{i}\right)_{i=1}^{\infty}$, which have the same first term and satisfy for some $k>1$ the equalities
$$
x_{k-1} y_{k-1}=42, \quad x_{k} y_{k}=30 \quad \text { and } \quad x_{k+1} y_{k+1}=16 .
$$
Find all such sequenc... | Solution. Let $c$, resp. $d$ be the differences of the sought sequences, so that from the expressions $x_{i}=$ $=x_{1}+(i-1) c$ and $y_{i}=x_{1}+(i-1) d$ we then get for each $i$ the equality
$$
x_{i} y_{i}=x_{1}^{2}+(i-1) x_{1}(c+d)+(i-1)^{2} c d \text {. }
$$
We will thus address the question of when for some index... | 15 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Find the smallest natural number $n$ such that in the decimal representation of $\sqrt{n}$, two nines immediately follow the decimal point. | SOLUTION. Let $a$ be the nearest larger natural number to the irrational number $\sqrt{n}$. According to the problem, we then have $a-0.01 \leqq \sqrt{n}$. Since $a^2$ is a natural number greater than $n$, it must hold that
$$
(a-0.01)^2 \leqq n \leqq a^2-1
$$
After rearranging the inequality between the extreme term... | 2600 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Let $a, b, c$ be positive real numbers such that $a b + b c + c a = 1$. Determine the values that the expression
$$
\frac{a\left(b^{2}+1\right)}{a+b}+\frac{b\left(c^{2}+1\right)}{b+c}+\frac{c\left(a^{2}+1\right)}{c+a}
$$
can take.
(Josef Tkadlec \& Patrik Bak) | SOLUTION. Given the condition $a b+b c+c a=1$, we first modify the first of the three addends of the expression under consideration, similarly to problem 4 from the homework round,
$$
\frac{a\left(b^{2}+1\right)}{a+b}=\frac{a\left(b^{2}+a b+b c+c a\right)}{a+b}=\frac{a(b+a)(b+c)}{a+b}=a(b+c)=a b+c a .
$$
Similarly, t... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Find the greatest three-digit number $n$, for which the sum
$$
1^{2}+2^{3}+3^{4}+4^{5}+\ldots+n^{n+1}
$$
is divisible by three. | 2. First, we notice that when dividing by three, the numbers $k$ and $k^{3}$ always give the same remainder (it is sufficient to verify this for $k \in \{0,1,2\}$). This means that all numbers $k, k^{3}, k^{5}, k^{7}, \ldots$ give the same remainder, which depends only on the remainder when the number $k$ is divided by... | 997 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Determine the number of pairs $(a, b)$ of natural numbers $(1 \leqq a<b \leqq 86)$ for which the product $a b$ is divisible by three. | 1. First, we calculate the total number of pairs of numbers such that \(1 \leqq ab\). The number of all pairs \((a, b)\) of natural numbers such that \(1 \leqq a < b \leqq 86\) is therefore equal to \(\frac{1}{2} \cdot 86 \cdot 85 = 3655\). (This is also the number of all unordered pairs of natural numbers from the set... | 2002 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Find the smallest natural number ending with the four-digit number 2018, which is a multiple of the number 2017. | 1. We will take advantage of the fact that each multiple of seven by a single-digit number ends with a different digit. We will gradually add the digits of the sought factor (to the given factor 2017) from the right, so that the result ends with the four-digit number 2018.
In the first step, we look for a digit whose ... | 8782018 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. For integers $x, y, z$ it holds that $x^{2}+y-z=10, x^{2}-y+z=22$. Find the smallest possible value of the expression $x^{2}+y^{2}+z^{2}$. | 2. By adding the given equations, we get
$$
\left(x^{2}+y-z\right)+\left(x^{2}-y+z\right)=2 x^{2}=10+22=32,
$$
thus $x^{2}=16$ or $x= \pm 4$. If we substitute this result back into the first or second of the given equations, in both cases we get $z=y+6$.
Now let's substitute both obtained equations into the expressi... | 34 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Determine the largest natural number $k$ for which it is possible to place $k$ rooks and $k$ bishops on an $8 \times 8$ chessboard such that no piece threatens another. (A bishop threatens any square on the same diagonal, and a rook threatens any square on the same row or column.)
(Josef Tkadlec) | Solution. Consider any valid placement of $k$ rooks and $k$ archers. We will prove that the inequality $k \leqq 5$ holds.
If a rook is placed in any row (or column) of the chessboard, it must be the only piece in that row (or column). Therefore, for the number $k$ of placed rooks, it must be true that $k \leqq 8$, and... | 5 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. How many numbers at most can be chosen from the set $M=\{1,2, \ldots, 2018\}$ such that the difference of any two chosen numbers is not equal to a prime number?
The regional round of category B takes place
## on Tuesday, April 10, 2018
so that it starts no later than 10:00 AM and the contestants have 4 hours of p... | 4. Selecting 505 numbers $1,5,9,13,17, \ldots, 2017 \in \mathrm{M}$, which give a remainder of 1 when divided by four, clearly has the required property, because the difference between any two selected numbers is a multiple of four, thus a composite number. (Since 4 is the smallest composite number, it is not possible ... | 505 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Let $n$ be a positive integer. Denote all its positive divisors as $d_{1}, d_{2}, \ldots, d_{k}$ such that $d_{1}<d_{2}<\ldots<d_{k}$ (thus $d_{1}=1$ and $d_{k}=n$). Determine all values of $n$ for which $d_{5}-d_{3}=50$ and $11 d_{5}+8 d_{7}=3 n$. (Matúš Harminc) | Solution. We distinguish whether the sought $n$ is odd or even.
(i) Let $n$ be odd, then all $d_{i}$ are odd. From the equality $11 d_{5}+8 d_{7}=3 n$ it follows that $d_{7} \mid 11 d_{5}$ and also $d_{5} \mid 8 d_{7}$, or $d_{5} \mid d_{7}$. From $d_{5}\left|d_{7}\right| 11 d_{5}$, considering $d_{7}>d_{5}$, we have ... | 2013 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Let's have an $8 \times 8$ chessboard and for each "edge" that separates two of its squares, write a natural number that indicates the number of ways the entire chessboard can be cut into $2 \times 1$ rectangles such that the given edge is part of the cut. Determine the last digit of the sum of all such written numb... | Solution. There are $7 \cdot 8=56$ vertical edges and the same number of horizontal edges, making a total of $56 \cdot 2=112$. When the chessboard is cut in any way, 32 rectangles of $2 \times 1$ are formed, so each such cut touches exactly $112-32=80$ edges, contributing the number 80 to the total sum. Therefore, the ... | 0 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. A cinema received 234 visitors. Determine for which $n \geqq 4$ it could have happened that the visitors could be seated in $n$ rows such that each visitor in the $i$-th row knew exactly $s_j$ visitors in the $j$-th row for any $i, j \in\{1,2, \ldots, n\}, i \neq j$. (The relationship of knowing is symmetric.)
(Tom... | Solution. For each $k \in\{1,2, \ldots, n\}$, let $p_{k}$ be the number of viewers in the $k$-th row. The condition of the problem for given $i$ and $j$ means that the number of acquaintances between viewers in the $i$-th and $j$-th rows is equal to $j p_{i}$. By swapping the roles of the numbers $i$ and $j$, we find t... | 12 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Determine the maximum possible area of triangle $ABC$, whose medians satisfy the inequalities $t_{a} \leqq 2, t_{b} \leqq 3, t_{c} \leqq 4$.
(Pavel Novotný) | Solution. Let $T$ be the centroid of triangle $ABC$ and $K, L, M$ the midpoints of sides $BC, CA, AB$. The medians divide triangle $ABC$ into six smaller triangles of equal area: For example, triangle $AMT$ has side $|AM|=\frac{1}{2} c$ and its height to side $AM$ has a length of $\frac{1}{3} v_{c}$, so $S_{AMT}=\frac{... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Do there exist ten consecutive natural numbers that are successively divisible by the numbers $9,7,5,3,1,1,3,5,7,9$?
(Jaroslav Zhouf) | SOLUTION. Yes, such ten numbers do exist. An example is the following ten consecutive numbers, under which we immediately write their required divisors:
Indeed, divisibility by 1 is trivial,... | 153 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Let's consider 20 statements:
| "I have exactly one sister." | "I have exactly one brother." |
| :--- | :--- |
| "I have exactly two sisters." | "I have exactly two brothers." |
| "I have exactly ten sisters." | "I have exactly ten brothers." |
a) Each of the four siblings made a different one of these 20 statemen... | SOLUTION. a) Yes, it is possible. In the case where the four siblings are two brothers and two sisters, they can truthfully make four mutually different statements. One brother says: "I have exactly one brother" and the other: "I have exactly two sisters," one sister says: "I have exactly one sister" and the other: "I ... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
4. How many ordered quadruples of natural numbers $(a, b, c, d)$ with a sum of 100 satisfy the equations
$$
(a+b)(c+d)=(b+c)(a+d)=(a+c)(b+d) ?
$$
(Patrik Bak) | SOLUTION. First, we will equivalently transform the first equation from the problem:
$$
\begin{aligned}
(a+b)(c+d) & =(b+c)(a+d), \\
a c+a d+b c+b d & =b a+b d+c a+c d, \\
a d+b c-a b-c d & =0, \\
a(d-b)-c(d-b) & =0, \\
(a-c)(d-b) & =0 .
\end{aligned}
$$
Similarly, we find that the second equation is equivalent to an... | 129 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
## 4. Find the smallest value of the fraction
$$
V(n)=\frac{n^{3}-10 n^{2}+17 n-4}{n^{2}-10 n+18}
$$
$where\ n$ is any natural number greater than 2. | SOLUTION. First, let's calculate the values of the expression $V(n)$ for several natural numbers $n \geq 3:$
| $n$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $V(n)$ | $5 \frac{1}{3}$ | $5 \frac{1}{3}... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Find the largest five-digit natural number that is divisible by 101 and reads the same forwards and backwards. | 1. Any of the considered five-digit numbers has a decimal representation of the form $\overline{a b c b a}$. By expanding and rearranging it, we obtain the equality
$$
\overline{a b c b a}=10001 a+1010 b+100 c=101(99 a+10 b+c)+2 a-c .
$$
From this, it follows that the number in question is divisible by 101 if and onl... | 49894 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. The number n is the product of three (not necessarily distinct) prime numbers. If we increase each of them by 1, the product of the increased numbers will be 963 more than the original product. Determine the original number $n$. | SOLUTION. We are looking for $n=p \cdot q \cdot r$ with prime numbers $p \leqq q \leqq r$, which satisfy the equation
$$
(p+1)(q+1)(r+1)=p q r+963
$$
The right-hand side of this equation is an odd number in the case of the smallest prime $p=2$, so then the factors $q+1$ and $r+1$ from the product on the left side mus... | 2013 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Find the largest integer d for which it is possible to fill a $43 \times 47$ table with ones and twos such that the sum of the numbers in each row and each column is divisible by d. (Also prove that no larger number d satisfies the condition of the problem.)
(Tomáš Bárta) | SOLUTION. The answer is $d=47$.
In the first part of the solution, we prove that for $d=47$, the required filling of the table actually exists: Write the same 47 numbers in each row, with ones in 39 rows and twos in the remaining 4 rows (see figure). Then the sum of the numbers in each row is 47 or $2 \cdot 47$, and t... | 47 | Number Theory | proof | Yes | Yes | olympiads | false |
1. Four teams participated in a hockey tournament, with each team playing exactly one match against each of the others. The number of goals scored in each match divides the total number of goals scored in the tournament, and no two matches had the same number of goals. What is the minimum number of goals that could hav... | 1. If each team plays one match against each other, each team will play a total of three matches in the tournament, and the total number of matches will be $\frac{1}{2} \cdot 4 \cdot 3=6$. Therefore, we need to find six different natural numbers (zero does not divide any number) with the smallest possible sum such that... | 24 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Find all eight-digit numbers with this property: if we strike out the first two and the last two digits of the number, we get a four-digit number that is 2019 times smaller than the original number.
(Pavel Calábek) | Solution. In any suitable eight-digit number $N$, let $A$ be its first two-digit number, $B$ the following four-digit number, and $C$ the last two-digit number. If we interpret $A, B, C$ as numbers written in the decimal system, then $10 \leq A \leq 99$, $1000 \leq B \leq 9999$ (according to the problem, the number $B$... | 19990119 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. On a game board of size $9 \times 9$ squares, there is a ship consisting of eight squares around the perimeter of a $3 \times 3$ square (the ship is colored in the image).
a) How many squares do we need to shoot at to be certain that we will hit the ship at least in two different places? We do not learn about the f... | SOLUTION. a) A $9 \times 9$ square can be divided into 9 smaller $3 \times 3$ squares. We must shoot at least twice into each of these smaller squares, meaning we need at least 18 shots in total. The example in the figure shows that 18 shots are sufficient.
| | | | | | |
| :---: | :---: | :---: | :---: | :---: |... | 18 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1. Find the smallest four-digit number $n$ for which the system
$$
\begin{aligned}
x^{3}+y^{3}+y^{2} x+x^{2} y & =n, \\
x^{2}+y^{2}+x+y & =n+1
\end{aligned}
$$
has only integer solutions. | 1. Assume that the parameter $n$ is a natural number, and solve the given system in the domain of real numbers. The left side of the first equation is equal to $\left(x^{2}+y^{2}\right)(x+y)$, and thus for the numbers $s=x+y$ and $t=x^{2}+y^{2}$, we have $t \cdot s=n$ and $t+s=n+1$. The numbers $s, t$ are therefore the... | 1013 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. In an urn, there are only white and black balls, the number of which, rounded to the nearest hundred, is 1000. The probability of drawing two black balls is $\frac{17}{43}$ greater than the probability of drawing two white balls. How many white and how many black balls are in the urn? (The probability of drawing any... | SOLUTION. Let there be $n$ balls in the urn, of which $b$ are white (and $n-b$ are black). Then the probability of drawing two white balls is equal to the ratio
$$
\frac{\left(\begin{array}{l}
b \\
2
\end{array}\right)}{\left(\begin{array}{l}
n \\
2
\end{array}\right)}=\frac{b(b-1)}{n(n-1)}
$$
while the probability o... | 299 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. Let $a, b$ be any positive real numbers such that $a^{2}+b^{2}=1$. Find the smallest possible value of the expression
$$
\frac{a^{2}\left(a+b^{3}\right)}{b-b^{3}}+\frac{b^{2}\left(b+a^{3}\right)}{a-a^{3}}
$$
and determine for which pairs $(a, b)$ this value is achieved. (Tomáš Bárta) | SOLUTION. Using the condition $a^{2}+b^{2}=1$, we first simplify the first fraction of the given expression:
$$
\frac{a^{2}\left(a+b^{3}\right)}{b-b^{3}}=\frac{a^{2}\left(a+b^{3}\right)}{b\left(1-b^{2}\right)}=\frac{a^{2}\left(a+b^{3}\right)}{b \cdot a^{2}}=\frac{a+b^{3}}{b}=\frac{a}{b}+b^{2} .
$$
Similarly, for the ... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. A $3 \times 3$ table is filled with distinct natural numbers such that in each row and column, the sum of the outer numbers equals the number written between them. Determine the smallest number that can be written in the center of the table.
(Tomáš Jurík) | Solution. Let's denote the numbers in the corners of the table as $a, b, c, d$ (from left to right, top to bottom). These four numbers uniquely determine all other numbers in the table, as we can sequentially calculate the numbers between them and finally the number $a+b+c+d$ in the center of the table.
| $a$ | | $b$... | 12 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
4. Let $p, q (p<q)$ be given natural numbers. Determine the smallest natural number $m$ with the property: The sum of all fractions in simplest form, which have denominator $m$ and whose values lie in the open interval $(p, q)$, is at least $56\left(q^{2}-p^{2}\right)$.
The regional round of category A takes place
##... | 4. We will show that the smallest $m$ is 113 (independent of the values of $p, q$). Let's assume $m>1$. For any natural numbers $c, d$, let $S_{m}(c, d)$ denote the sum of all fractions in simplest form that lie in the open interval $(c, d)$ and whose denominator is $m$. Then the inequality holds:
$$
S_{m}(c, c+1) \le... | 113 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. The function $f: \mathbb{N} \rightarrow \mathbb{N}$ is given such that $f(n)=1$ if $n$ is odd, and $f(n)=k$ for every even number $n=2^{k} l$, where $k$ is a natural number and $l$ is an odd number. Determine the largest natural number $n$ for which
$$
f(1)+f(2)+\ldots+f(n) \leqq 123456 .
$$ | SOLUTION. Let
$$
S(n)=f(1)+f(2)+\ldots+f(n)
$$
From the problem statement, it follows that $S(1)=1$. Since $f(n) \geq 1$ for all natural numbers $n$, $S: \mathbb{N} \rightarrow \mathbb{N}$ is an increasing function. If $n$ is a natural number of the form $n=2^{k}$, where $k$ is a natural number, we determine the sum ... | 82307 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. Find all tetrahedra that have a net in the shape of a deltoid and exactly four edges of a given length a. (A deltoid is understood to be a convex quadrilateral symmetric with respect to only one of its diagonals; thus, neither a square nor a rhombus is considered a deltoid.) | SOLUTION. In the first (more significant) part of the solution, we will find all tetrahedra that have a net in the shape of a kite; then it will be relatively easy to determine which of the found tetrahedra have exactly four equal edges.
Consider, therefore, an arbitrary tetrahedron $ABCD$ and describe the lengths of ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. Assume that for positive real numbers $a, b, c, d$ the following holds:
$$
a b+c d=a c+b d=4 \quad \text { and } \quad a d+b c=5 .
$$
Find the smallest possible value of the sum $a+b+c+d$ and determine which quadruples $a, b, c, d$ achieve it.
The regional round of category A takes place
## on Tuesday, January 1... | 4. Since the given equations contain mixed products of variables, it will be advantageous to examine the square of the sum $a+b+c+d$. By rearrangement and substitution of the given equations, we get
$$
\begin{aligned}
(a+b+c+d)^{2} & =a^{2}+b^{2}+c^{2}+d^{2}+2(ab+cd+ac+bd+ad+bc)= \\
& =a^{2}+b^{2}+c^{2}+d^{2}+2(4+4+5)... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Let's say that three mutually distinct natural numbers form a sum triplet if the sum of the first two of them equals the third number. Determine the maximum number of sum triplets that can be found in a set of twenty natural numbers. | 1. For any twenty natural numbers
$$
x_{1}<x_{2}<\ldots<x_{20}
$$
we estimate how many sum triples, i.e., triples $\left\{x_{i}, x_{j}, x_{k}\right\}$ satisfying the conditions $1 \leqq i<j<k \leqq 20$ and $x_{i}+x_{j}=x_{k}$, can exist, first for a fixed index $k \in \{3,4, \ldots, 20\}$. Let these triples be $\left... | 90 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Provide an example of a set $M$ of two-digit numbers that has the maximum number of elements and at the same time satisfies both of the following conditions:
(i) Any two numbers from M are coprime.
(ii) If we change the order of the digits of any number from M, we get another number from the set M. | 2. Due to condition (i), the set M can contain at most one of the numbers $11,22,33, \ldots, 99$ written with two identical digits, which are all divisible by eleven. Due to condition (ii) and divisibility by two, there cannot be any number in M written with two different even digits; with one even digit, M can contain... | 13 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. On the board, we will write ten different natural numbers. In each step, we first underline each number that is not the sum of any two different numbers written on the board, and then erase all underlined numbers. For example:
 According to the problem statement, all numbers on the board are positive and distinct. In each step, we will certainly underline the smallest number on the board, and - if it is not the only one - also the second smallest number, as it cannot equal the sum of two different numbers on the board. Since we e... | 4 | Number Theory | proof | Yes | Yes | olympiads | false |
5. Anna and Bedřich are playing a card game. Each of them has five cards with values from 1 to 5 (one of each). In each of the five rounds, both lay down one card, and the one with the higher number gets a point. If the cards have the same number, no one gets a point. Used cards are not returned to the game. Whoever ha... | SOLUTION. The described game is apparently fair in the sense that both players have an equal number of ways to win. To find the required number, it is enough to determine how many ways a draw can occur, i.e., one of the results $0: 0, 1: 1$, and $2: 2$.
The case $0: 0$ occurs if both players lay down the same cards in... | 31 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Let $ABCD$ be a rhombus with the shorter diagonal $BD$ and $E$ an internal point on its side $CD$, which lies on the circumcircle of triangle $ABD$. Determine the size of its interior angle at vertex $A$, given that the circumcircles of triangles $ACD$ and $BCE$ have exactly one common point. | Let's denote the sought internal angle at vertex $A$ of the considered rhombus as $\alpha$. Further, let $k$ be the circumcircle of triangle $A C D$ and $l$ the circumcircle of triangle $B C E$.
From the conditions of the problem, it follows that $A B E D$ is an isosceles trapezoid with the shorter base $E D$. Therefo... | 72 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. Determine the number of all triples of natural numbers $a, b, c$, for which
$$
a + ab + abc + ac + c = 2017 \text{.}
$$ | SOLUTION. First, we will rearrange the left side of the given equation
$$
\begin{aligned}
a+a b+a b c+a c+c & =a(1+b)+a c(1+b)+c=a(1+b)(1+c)+c= \\
& =a(1+b)(1+c)+(1+c)-1=(1+c)(a(1+b)+1)-1,
\end{aligned}
$$
thanks to which we obtain the equivalent equation
$$
(1+c)(a(1+b)+1)=2018 .
$$
The number 2018 can be written ... | 29 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Find the largest possible number of integers that can be selected from the set $\{1,2,3, \ldots, 100\}$ such that no two of them differ by 2 or 5. | SOLUTION. First, we will show that from any seven consecutive numbers, denoted as
$$
a, a+1, a+2, a+3, a+4, a+5, a+6,
$$
we can select at most three numbers in the required way. To do this, we divide all seven numbers into three sets
$$
A=\{a, a+2, a+5\}, \quad B=\{a+1, a+3\}, \quad C=\{a+4, a+6\}.
$$
Since we can ... | 44 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Find all integers $n>2$ such that the number $n^{n-2}$ is the $n$-th power of an integer.
(Patrik Bak) | SOLUTION. We will show that only $n=4$ satisfies the condition. For brevity, we will use "n-th power" instead of "n-th power of an integer" throughout the solution.
A positive integer is an $n$-th power if and only if each prime number in its prime factorization appears with an exponent that is a multiple of $n$. Ther... | 4 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. If a, b, c are distinct positive real numbers, what is the smallest possible number of distinct numbers among the numbers $a+b, b+c, c+a, ab, bc, ca, abc$?
(Patrik Bak) | SOLUTION. Since $a, b, c$ are distinct positive numbers, so are the numbers $a b, b c, c a$, because, for example, from $a b=b c$ it follows that $a=c$ (thanks to $b \neq 0$). We see, then, that in the examined septet of numbers $a+b, b+c, c+a, a b, b c, c a, a b c$ there are at least 3 different values. We will first ... | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Determine the number of ten-digit numbers in which two adjacent digits can be crossed out to obtain a number that is 99 times smaller. | SOLUTION. Let $n$ be the number satisfying the conditions of the problem. By erasing the last two digits, we reduce $n$ by at least a hundred times, so we can limit ourselves to erasing digits that are not the last. After erasing two adjacent digits, two parts of the number $n$ remain, with the first part possibly bein... | 50 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. On the meadow, there are children and adults. The percentage of boys among all children is equal to the percentage of girls among all present people and also the number of all adults. How many boys, girls, and adults are on the meadow? | SOLUTION. Let's denote $c, d$ and $v$ as the number of boys, girls, and adults on the meadow, respectively. We have
$$
\frac{100 c}{c+d}=\frac{100 d}{c+d+v}=v
$$
From the first equality, we get $d^{2}=c^{2}+v c$, which we substitute into the equality between the first and third expression, which we first rearrange to... | 32 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
3. For any four-element subset $\mathrm{P}$ of the set $\{1,2,3, \ldots, 12\}$, denote
$$
\mathrm{Q}=\{3 x: x \in \mathrm{P}\} \quad \text { and } \quad \mathrm{R}=\{4 x: x \in \mathrm{P}\} \text {. }
$$
Determine the number of such sets $P$ for which the numbers from $P, Q, R$ give all possible non-zero remainders w... | Solution. We will calculate with all numbers modulo 13, i.e., as remainders upon division by 13. Furthermore, it is clear that not only the set $P$, but also the derived sets $Q$ and $R$ are four-element sets. Therefore, if the three mentioned sets are to contain all non-zero remainders modulo 13, of which there are tw... | 9 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. Determine the number of all triples of two-digit natural numbers $a, b, c$, whose product $abc$ has a representation in which all digits are the same. Triples differing only in the order of the numbers are considered the same, i.e., they are counted only once. | SOLUTION. For two-digit numbers $a, b, c$, the product $abc$ is a four-digit, five-digit, or six-digit number. If all the digits of the number $abc$ are equal to the same digit $k$, one of the equalities $abc = k \cdot 1111, abc = k \cdot 11111$, or $abc = k \cdot 111111, k \in \{1,2, \ldots, 9\}$ holds.
The numbers $... | 10 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Given a trapezoid $A B C D$ with bases $A B$ and $C D$. Let $E$ be the midpoint of side $A B$, $F$ the midpoint of segment $D E$, and $G$ the intersection of segments $B D$ and $C E$. Express the area of trapezoid $A B C D$ in terms of its height $v$ and the length of segment $F G$, assuming that points $A, F, C$ li... | SOLUTION. According to the problem, angles $E F D$ and $A F C$ are right angles, so we have (Fig. 3)
$$
\begin{aligned}
& |\Varangle C D F|=|\Varangle A E F| \quad \text { (alternate interior angles), } \\
& |\Varangle C F D|=|\Varangle A F E| \quad \text { (vertical angles). }
\end{aligned}
$$
Furthermore, point $F$... | 3 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
2. For any natural number $n$, let us form all possible "words" of length $n$ from the letters $A$ and $B$, and let $p_{n}$ denote the number of those words that do not contain the quadruple $AAAA$ of consecutive letters $A$, nor the triple $BBB$ of consecutive letters $B$. Determine the value of the expression
$$
\fr... | Solution. Let the number of suitable words of length $n$ ending with the letter $A$, respectively $B$, be denoted by $a_{n}$, respectively $b_{n}$. Then
$$
p_{n}=a_{n}+b_{n} \text {. }
$$
Let $n \geqq 4$. A suitable word ending with the letter $A$ has one of the endings $B A, B A A$, or $B A A A$. The number of words... | 2 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. Given a segment $A B$. Construct a point $C$ such that the area of triangle $A B C$ equals $1 / 8$ of the area $S$ of a square with side $A B$, and the sum of the areas of squares with sides $A C$ and $B C$ equals $S$. How many solutions does the problem have for a given placement of segment $A B$ in the plane?
The... | 3. The condition that the area of triangle $ABC$ is to equal $\frac{1}{8}$ of the area $S$ of the square with side $AB$ means that the height of triangle $ABC$ to side $AB$ has a length of $\frac{1}{4}|AB|$, so point $C$ must lie on one of the two lines parallel to line $AB$ and at a distance of $\frac{1}{4}|AB|$ from ... | 4 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. The product of all positive divisors of a natural number $n$ is $20^{15}$. Determine $n$. | 1. The number $20^{15}=2^{30} 5^{15}$ is divisible only by the prime numbers 2 and 5, so the sought number $n$ must be of the form $n=2^{a} 5^{b}$, where $a, b$ are natural numbers. Each of its positive divisors is thus of the form $2^{\alpha} 5^{\beta}$, where $\alpha \in\{0,1, \ldots, a\}$ and $\beta \in\{0,1, \ldots... | 400 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. Let $\tau(k)$ denote the number of all positive divisors of a natural number $k$, and suppose the number $n$ is a solution to the equation $\tau(1.6 n)=1.6 \tau(n)$. Determine the value of the ratio $\tau(0.16 n): \tau(n)$. | SOLUTION. If the prime factorization of the number $n$ is $n=\prod_{i=1}^{k} p_{i}^{s_{i}},$ where $p_{1}, \ldots, p_{k}$ are distinct primes and $s_{1}, \ldots, s_{k}$ are non-negative integers, the number of its positive divisors is $\tau(n)=\prod_{i=1}^{k}\left(s_{i}+1\right)$. If $n=2^{\alpha} 5^{\beta} n^{\prime}$... | 1 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. We will write down all five-digit numbers in which each of the digits $4,5,6,7,8$ appears exactly once. Then we will strike out one (any of them) and sum all the remaining ones. What are the possible values of the digital sum of such a result?
The regional round of category B takes place
## on Tuesday, April 5, 20... | 4. The final digit sum is uniquely determined and is the number 33.
To solve the problem, it will be advantageous to first determine the sum $S$ of all five-digit numbers containing each of the digits $4,5,6,7,8$. There are clearly as many such numbers as there are different permutations of the five digits, i.e., $5!=... | 33 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Let's say a positive integer $k$ is fair if the number of 2021-digit palindromes that are multiples of $k$ is the same as the number of 2022-digit palindromes that are multiples of $k$. Does the set $M=\{1,2, \ldots, 35\}$ contain more numbers that are fair or more numbers that are not fair?
(A palindrome is a natu... | SOLUTION. We will show that the given 35-element set $M$ contains at least 18 fair numbers, which is more than the number of numbers in it that are not fair.
It suffices to prove that every positive divisor of the number $2^{1010} \cdot 3^{2} \cdot 5^{1010}$ is a fair number. From this, it follows that all 18 numbers ... | 18 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. Let a regular n-gon be denoted as $A_{1} A_{2} \ldots A_{n}$. The point $A_{3}$ is reflected over the axis $A_{2} A_{4}$, resulting in the point $A_{3}^{\prime}$. Then, the point $A_{3}^{\prime}$ is reflected over the axis $A_{1} A_{3}$, resulting in the point $A_{3}^{\prime \prime}$. For which $n \geqq 4$ is the po... | SOLUTION. Let $S$ be the center of the circle circumscribed around the $n$-gon $A_{1} A_{2} \ldots A_{n}$. Since the central angle $A_{i} S A_{i+1}$ is equal to $\frac{360^{\circ}}{n}$ for each $i=1, \ldots, n-1$, all the acute inscribed angles over the chords $A_{i} A_{i+1}$ have a size of $\frac{180^{\circ}}{n}$. Fou... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. For the sequence $\left(a_{n}\right)_{n=1}^{\infty}$, we know that for all natural numbers $n$,
$$
a_{n+1}=\frac{a_{n}^{2}}{a_{n}^{2}-4 a_{n}+6} .
$$
a) Find all values of $a_{1}$ for which this sequence is constant.
b) Let $a_{1}=5$. Determine the greatest integer not exceeding $a_{2018}$. (Vojtech Bálint) | SOLUTION. a) Suppose the sequence $\left(a_{n}\right)_{n=1}^{\infty}$ is constant. Then it must hold that $a_{2}=a_{1}$, which we can write using the relation from the problem as
$$
a_{1}=\frac{a_{1}^{2}}{a_{1}^{2}-4 a_{1}+6}
$$
We can easily rearrange this equation to $a_{1}\left(a_{1}-2\right)\left(a_{1}-3\right)=0... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. For any three-digit number, we determine its remainders when divided by the numbers 2, 3, 4, ..., 10 and then sum the nine resulting numbers. Find the smallest possible value of such a sum. | SOLUTION. Let $S(n)$ denote the sum of the mentioned remainders of the three-digit number $n$. We will explain why $S(n) \geqq 3$.
- For odd $n$, $S(n) \geqq 5$ (consider the remainders when dividing by even numbers $2,4,6,8$, 10). Hence, let $n$ be even from now on.
- If $4 \nmid n$, then $S(n) \geqq 4$ ( $n$ gives a... | 3 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. Find the smallest natural number $k$, for which the following holds: If we select any $k$ different numbers from the set $\{1,4,7,10,13, \ldots, 1999\}$, then there exist two different numbers among the selected ones whose sum equals 2000. | 1. Let $\mathrm{M}=\{1,4,7,10,13, \ldots, 1999\}$ and list all sums of two different (we will not emphasize this further) numbers from M that equal the number 2000:
$$
2000=1+1999=4+1996=7+1993=\ldots=997+1003 \text {. }
$$
With the exception of the single number 1000, each number in M appears in exactly one sum (the... | 335 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. On a grid of size $12 \times 12$ squares, there is a ship consisting of eight cells along the perimeter of a $3 \times 3$ square (it is marked in gray in the figure). What is the minimum number of cells that need to be shot at to ensure that the ship is hit at least once?
(Jozef Rajník)
, according to Vièta's formulas, the following equalities hold:
$$
x_{1}+x_{2}=-4 p \quad \text { and } \quad x_{1} x_{2}=5 p^{2}+6 p-16
$$
from which we can compute the desired sum
$$
\begin{aligned}
x_{1}^{2}+x_{2}^{2} & =\left(x_{1}+x_... | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Determine the number of paths of length 14 that lead along the edges of the grid from point $A$ to point $B$. The length of each edge is one.
 | Another solution. For easier expression, let's denote some points of the network as in Fig. 2. Each path from point $A$ to point $B$ of length 14 consists of seven edges to the right and seven edges upwards. The diagonal $X_{1} X_{6}$ must be crossed exactly once, and that is at one of the six points $X_{1}, \ldots, X_... | 678 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. Determine the smallest values that the expression $V=(a-b)^{2}+(b-c)^{2}+(c-a)^{2}$ can take, given that the real numbers $a$, $b$, $c$ satisfy the pair of conditions
$$
\begin{aligned}
a+3 b+c & =6 \\
-a+b-c & =2
\end{aligned}
$$ | SOLUTION. By adding both equations, we find that $b=2$. Substituting $b$ into one of them yields $c=-a$. Therefore, $V=(a-2)^{2}+(2+a)^{2}+(-2 a)^{2}$. After squaring and adding, we find that $V=6 a^{2}+8 \geqq 8$. Equality occurs precisely when $a=0, b=2$ and $c=0$.
The sought minimum value of the expression $V$ is t... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The number n is the product of three different prime numbers. If we increase the two smaller ones by 1 and leave the largest one unchanged, the product of the three will increase by 915. Determine the number n. | SOLUTION. Let $n=p q r, p<q<r$. The equation $(p+1)(q+1) r=p q r+915$ can be equivalently rearranged to the form $(p+q+1) \cdot r=915=3 \cdot 5 \cdot 61$, from which it follows that the prime number $r$ can only take some of the values 3, 5, and 61. For $r=3$, however, from the last equation we get $(p+q+1) \cdot 3=3 \... | 2013 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. A rectangle $2005 \times 2007$ is divided into black and white unit squares. Prove that for one of the colors (black or white), there are more than 95800 rectangles (composed of unit squares) that do not overlap each other and whose corner squares are all of the chosen color, with each side containing at least two s... | SOLUTION. We will look for a rectangle of the smallest area in which there must be a rectangle with all corner cells of the same color. A width of 2 is not enough (for any length, for example, one entire row could be black and the other white). Let's consider a rectangle of width 3. Its columns can be colored in eight ... | 95810 | Combinatorics | proof | Yes | Yes | olympiads | false |
1. A natural number $m>1$ is called a $k$-fold divisor of a natural number $n$ if the equality $n=m^{k} q$ holds, where $q$ is an integer that is not a multiple of the number $m$. Determine how many sevenfold divisors the number $100 !=1 \cdot 2 \cdot 3 \cdot \ldots \cdot 100$ has. | Solution. First, we derive a general formula for the number $P_{k}(n)$ of all $k$-fold divisors of a number $n$ with the factorization $n=p_{1}^{a_{1}} p_{2}^{a_{2}} \ldots p_{N}^{a_{N}}$, where $p_{i}$ are distinct primes and the exponents $a_{i}$ are non-negative integers. It holds that $m^{k} \mid n$ if and only if ... | 2520 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. The angle $A C B$ of triangle $A B C$ has a size of $140^{\circ}$. The bisector of angle $A B C$ intersects side $A C$ at point $X$. Point $Y$ lies on side $A B$ such that angle $Y C B$ has a size of $100^{\circ}$. Determine the size of angle $Y X B$. | Solution. Let $P$ denote the orthogonal projection of point $X$ onto line $BC$ (Fig. 3). Since $|\varangle XCY| = 40^{\circ} = |\varangle XCP|$, point $X$ lies not only on the bisector of angle $ABC$ but also on the bisector of angle $YCP$. Therefore, point $X$ has the same distance from the three lines $AB$, $BC$, and... | 50 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
2. On one website, there is a poll to determine the best hockey player of the last decade. The number of votes for each player is given as a percentage, rounded to the nearest whole number. After Mirek voted for Jaromír Jágr, his share of $7 \%$ did not change. What is the minimum number of people, including Mirek, who... | 2. Let $p$ be the number of participants in the survey, including Miro, and $j$ be the number of votes for Jágar. The $7\%$ can be rounded from numbers in the interval $\langle 6.5\% ; 7.5\%)$ or $\langle 0.065 ; 0.075)$. Before Miro's vote, Jágar had $j-1$ votes and after, he had $j$ votes. Therefore, it must hold tha... | 107 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. On the board, there is a positive integer $n$. In one step, we can erase the number on the board and write either its double, or its double increased by 1. For how many initial numbers $n$ different from 2019 can we achieve that the number 2019 appears on the board after a finite number of steps?
(Josef Tkadlec) | 1. In one step, the current natural number $k$ is increased either to the even number $m=2k$, or to the odd number $m=2k+1$. Depending on the parity of the new number $m$, we can reconstruct the previous number $k$: either $k=m/2$, or $k=(m-1)/2$ — depending on whether $m$ is even or odd.
The odd number 2019 will thus... | 10 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Find all three-digit numbers with this property: if we strike out the middle digit of the number and multiply the resulting two-digit number by the square of the struck-out digit, we get the original three-digit number again.
(Tomáš Jurík) | 2. A three-digit number $n=\overline{a b c}$ has the desired property if and only if its digits $a$, $b$, $c$ satisfy the equation
$$
100 a+10 b+c=(10 a+c) b^{2} .
$$
Since the digit $b$ has the most complex representation in it, we will examine its possible values one by one.
First, we notice that $b \in\{0,1\}$ ca... | 135 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. Determine all natural numbers $n>1$ such that in some number system with $b a s e z \geqq 5$ the following divisibility criterion holds: a three-digit number $(a b c)_{z}$ is divisible by the number n if and only if the number $c+3 b-4 a$ is divisible by $n$. | SOLUTION. Since $(a b c)_{z}$ is the number $a z^{2}+b z+c$, we need to determine when the equivalence generally holds: $n \mid c+3 b-4 a$ if and only if $n \mid a z^{2}+b z+c$. Here, $a, b, c$ are arbitrary digits in base $z$, i.e., numbers from the set $\{0,1, \ldots, z-1\}$. Note that $z-1 \geq 4$, since we assume t... | 13 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
4. A $10 \times 10$ table is filled with numbers 1 and -1 such that the sum of the numbers in each row, except one, is equal to zero, and the sum of the numbers in each column, except one, is also equal to zero. Determine the maximum possible sum of all the numbers in the table.
(Patrik Bak) | SOLUTION. If we add up all the numbers in the considered table row by row, we will conclude that the total sum is equal to the sum of the numbers in the exceptional row where it is not zero. This sum is at most 10, and it equals 10 if all the numbers in the given row are ones. (We reach the same conclusion if we add up... | 10 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. Determine all possible values of the sum $a+b+c+d$, where $a, b, c, d$ are natural numbers satisfying the equation
$$
\left(a^{2}-b^{2}\right)\left(c^{2}-d^{2}\right)+\left(b^{2}-d^{2}\right)\left(c^{2}-a^{2}\right)=2021 .
$$
(Mária Dományová, Patrik Bak) | SOLUTION. Let's expand the left side of the given equation and further simplify:
$$
\begin{aligned}
\left(a^{2}-b^{2}\right)\left(c^{2}-d^{2}\right) & +\left(b^{2}-d^{2}\right)\left(c^{2}-a^{2}\right)= \\
& =\left(a^{2} c^{2}-a^{2} d^{2}-b^{2} c^{2}+b^{2} d^{2}\right)+\left(b^{2} c^{2}-a^{2} b^{2}-c^{2} d^{2}+a^{2} d^... | 90 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. For natural numbers $k, l, m$ it holds that
$$
\frac{k+m+k l m}{l m+1}=\frac{2051}{404}
$$
Determine all possible values of the product klm. | SOLUTION. Although the equation in the problem contains three unknowns, we will be able to solve it uniquely thanks to the fact that we are looking for a solution only in the set of natural numbers. We will try to separate the integer part from both fractions (which is easy in the case of a numerical fraction):
$$
\fr... | 2015 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. In a square table $11 \times 11$, we have written the natural numbers $1,2, \ldots, 121$ sequentially from left to right and from top to bottom. Using a square tile $4 \times 4$, we covered exactly 16 cells in all possible ways. How many times was the sum of the 16 covered numbers a perfect square? | SOLUTION. Let the number covered by the top left corner of the tile be denoted as $z$. The entire tile must lie within the given table, so the values of $z$ can only be the numbers written in the first eight rows and the first eight columns of the table (for example, if $z=10$, the tile would protrude, and thus could n... | 5 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. In a $4 \times 4$ table, positive real numbers are written such that the product in each pentad of the shape 母 is equal to 1. Determine the maximum number of different numbers written in the table.
Note: The shape 母 refers to a specific arrangement of five cells in the table, resembling the Chinese character 母. For... | SOLUTION. Let $a, b, c, d, e, f, g, h, i$ be the numbers inscribed in the left upper $3 \times 3$ square of the table (Fig. 1). By comparing the products for pentads of the form $\mathbb{\text { a }}$ and placed in this part of the table, we must have $a b c d e=b d e f g$, or $a c=f g$. Similarly, for pentads $\mathrm... | 3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Determine how many numbers we can select $z$ from the set $\{1,2,3, \ldots, 75599,75600\}$ such that the set includes the number 75600 and for any two selected numbers $a$, $b$, it holds that $a$ is a divisor of $b$ or $b$ is a divisor of $a$. (List all possibilities.) | SOLUTION. Let's consider a set M that satisfies the conditions given in the problem. Since M contains the number 75600, it must be at least a one-element set. Further, notice that if we remove some number $a \neq 75600$ from the set M, we get a set $\mathrm{M}^{\prime} \subset \mathrm{M}$, which also satisfies the give... | 11 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Given an acute triangle $A B C$ that is not isosceles. Let $K$ be the intersection of the angle bisector of $\angle A C B$ with the perpendicular bisector of side $A B$. The line $C K$ intersects the altitudes from vertices $A$ and $B$ at points, which we will denote as $P$ and $Q$ respectively. Assume that triangle... | 3. Let's denote the internal angles of triangle $ABC$ in the usual way. From the congruence of the peripheral angles $ACK$ and $BCK$ in the circumcircle of triangle $ABC$, it follows that the corresponding chords $AK$ and $BK$ are congruent, so point $K$ bisects the arc $AB$ that lies opposite vertex $C$ (Fig. 3). Acco... | 60 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. On the board, the numbers 1, 2, .., 33 are written. In one step, we choose two numbers on the board, one of which is a divisor of the other, erase both, and write their (integer) quotient on the board. We continue this process until only numbers remain on the board, none of which is a divisor of another. (In one ste... | SOLUTION. The numbers on the board will likely remain only from the set $M=\{1,2, \ldots, 33\}$. The prime numbers 17, 19, 23, 29, and 31 will always be written, each exactly once, because they have no divisors other than 1, and the set $M$ does not contain any of their multiples (so they can never disappear from the b... | 7 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. The quadratic equation $x^{2}-35 x+334=0$, whose coefficients are written in a number system with base $z(z \geqq 6)$, has two distinct real roots. Determine $z$ and both roots.
School - the written part of the 1st round of category B takes place
## on Tuesday, January 26, 1999
so that it starts in the morning an... | ## 3. Given equation
$$
x^{2}-(3 z+5) x+\left(3 z^{2}+3 z+4\right)=0
$$
has two distinct real roots if and only if its discriminant $D$ is positive,
$$
\begin{aligned}
D & =(3 z+5)^{2}-4\left(3 z^{2}+3 z+4\right)=-3 z^{2}+18 z+9= \\
& =-3\left(z^{2}-6 z-3\right)=-3\left((z-3)^{2}-12\right)>0
\end{aligned}
$$
from w... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Determine the smallest natural number $n$ with the following property: If we choose $n$ different natural numbers less than 2006, there are two among them such that the ratio of the sum and the difference of their squares is greater than three. | SOLUTION. First, we will determine for which natural numbers $a, b$ the mentioned inequality holds
$$
\frac{a^{2}+b^{2}}{a^{2}-b^{2}}>3
$$
For the fraction on the left side to be positive, it must be that $a^{2}>b^{2}$, or $a>b$. If this necessary condition is met, we multiply both sides of the inequality by the posi... | 21 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Find the smallest natural number $k$ such that every $k$-element set of pairwise coprime three-digit numbers contains at least one prime number. | SOLUTION. For constructing a set of two-by-two coprime three-digit composite numbers with a large number of elements, we can use the fact that powers of two different prime numbers are coprime. The set
$$
\left\{2^{7}, 3^{5}, 5^{3}, 7^{3}, 11^{2}, 13^{2}, 17^{2}, 19^{2}, 23^{2}, 29^{2}, 31^{2}\right\}
$$
contains 11 ... | 12 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. Let $P$ be the intersection of the diagonals of a convex quadrilateral $A B C D$. Calculate its area if the areas of triangles $A B C, B C D$ and $D A P$ are $8 \mathrm{~cm}^{2}, 9 \mathrm{~cm}^{2}$, $10 \mathrm{~cm}^{2}$, respectively. | 2. Let $S_{XYZ}$ denote the area of triangle $XYZ$ in $\mathrm{cm}^{2}$, and let $S=S_{ABP}$. According to the problem, we have $S_{ADP}=10$, $S+S_{BCP}=8$, and $S_{BCP}+S_{CDP}=9$. From the second equation, it follows that $S_{BCP}=8-S$. Substituting this into the third equation, we get $S_{CDP}=1+S$ (Fig. 1).
 Determine the smallest possible integer value of the resulting ... | Solution. a) The resulting expression can always be written (without reducing) as the ratio $A: B$ of two products $A$ and $B$ of natural numbers, for which
$$
A \cdot B=2 \cdot 3 \cdot 4 \cdot \ldots \cdot 29=29 !=2^{25} \cdot 3^{13} \cdot 5^{6} \cdot 7^{4} \cdot 11^{2} \cdot 13^{2} \cdot 17 \cdot 19 \cdot 23 \cdot 2... | 1292646 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
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