problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | difficulty float64 0.75 2.02k | difficulty_raw listlengths 3 8 |
|---|---|---|---|---|
An economist and a statistician play a game on a calculator which does only one
operation. The calculator displays only positive integers and it is used in the following
way: Denote by $n$ an integer that is shown on the calculator. A person types an integer,
$m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the fir... | 951 |
To solve this problem, we need to understand the specific condition under which the current displayed number \( n \) on the calculator can be transformed to another integer through the operation described, where \( m \) is chosen from the set \(\{1, 2, \ldots, 99\}\).
The process involves finding \( m\% \) of \( n \)... | 7 | [
7,
7,
7,
7,
7,
7,
6,
8
] |
One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210. | -1 | Given the problem, we have to find the number in position 210 under the constraints provided. We have 268 numbers written in a circle, denoted as \( a_1, a_2, \ldots, a_{268} \), and we know that the sum of any 20 consecutive numbers is 75.
This implies:
\[
a_i + a_{i+1} + \cdots + a_{i+19} = 75
\]
for all \( i \). G... | 5.875 | [
6,
6,
6,
6,
6,
6,
6,
5
] |
Find all triples $(a, b, c)$ of real numbers such that
$$ a^2 + ab + c = 0, $$
$$b^2 + bc + a = 0, $$
$$c^2 + ca + b = 0.$$ | (0, 0, 0)\left(-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}\right) |
To find all triples \((a, b, c)\) of real numbers satisfying the system of equations:
\[
\begin{align*}
1. \quad & a^2 + ab + c = 0, \\
2. \quad & b^2 + bc + a = 0, \\
3. \quad & c^2 + ca + b = 0,
\end{align*}
\]
we proceed by considering possible symmetric solutions first and also evaluate the possibility of all v... | 5.75 | [
5,
5,
6,
7,
7,
5,
5,
6
] |
A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$? | 100 |
Consider a \(100 \times 100\) square grid. We are tasked to tile this grid in two ways:
1. **Tiling with Dominoes:** Dominoes are \(1 \times 2\) or \(2 \times 1\) tiles. This means that each domino covers exactly two squares in the grid.
2. **Tiling with \(2 \times 2\) Squares:** Each square tile covers four grid sq... | 5.75 | [
6,
6,
5,
6,
6,
6,
6,
5
] |
Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$. | 90^\circ |
Consider triangle \(\triangle ABC\) with altitudes \(AD\), \(BF\), and \(CE\). The orthocenter of the triangle is denoted by \(H\). A line through \(D\) that is parallel to \(AB\) intersects line \(EF\) at point \(G\).
To find the angle \(\angle CGH\), follow these steps:
1. **Identify the orthocenter \(H\):**
Si... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$. | (3, 2, 6) |
To find all triples \((p, q, r)\) of positive integers, where \(p, q\) are also primes, such that:
\[
\frac{r^2 - 5q^2}{p^2 - 1} = 2,
\]
we start by rearranging the equation:
\[
r^2 - 5q^2 = 2(p^2 - 1).
\]
This can be further rewritten as:
\[
r^2 = 5q^2 + 2(p^2 - 1).
\]
Since \(p\) and \(q\) are primes, we will ... | 6.125 | [
6,
7,
6,
6,
6,
6,
6,
6
] |
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\] | \frac{1}{2n} |
To determine the largest real constant \( C_n \) such that for all positive real numbers \( a_1, a_2, \ldots, a_n \), the inequality
\[
\frac{a_1^2 + a_2^2 + \ldots + a_n^2}{n} \geq \left( \frac{a_1 + a_2 + \ldots + a_n}{n} \right)^2 + C_n \cdot (a_1 - a_n)^2
\]
holds, we start by rewriting the inequality:
\[
\frac... | 7 | [
7,
7,
7,
7,
6,
8,
7,
7
] |
Let $n \geq 3$ be an integer. Rowan and Colin play a game on an $n \times n$ grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid and Colin is allowed to permute the columns. A grid coloring is [i]orderly[/i] if: [list] [*]no matter how Rowan permutes the ro... | 2(n! + 1) |
To determine the number of orderly colorings on an \( n \times n \) grid where each square is either red or blue, we must first understand the conditions of the game as described.
An orderly coloring must satisfy two main conditions:
1. No matter how the rows are permuted by Rowan, Colin can permute the columns to re... | 6.75 | [
6,
7,
7,
6,
7,
7,
8,
6
] |
It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4? | 32 |
To determine how many numbers \( 2^k \), for \( 1 \leq k \leq 332 \), have the first digit as 4, we can approach the problem using logarithms to examine the leading digits.
### Step 1: Understanding the Leading Digit
For a number \( 2^k \) to have a first digit of 4, it must satisfy:
\[
4 \times 10^m \leq 2^k < 5 \t... | 6.125 | [
7,
6,
6,
6,
6,
6,
6,
6
] |
Find all positive integers $n$ such that the inequality $$\left( \sum\limits_{i=1}^n a_i^2\right) \left(\sum\limits_{i=1}^n a_i \right) -\sum\limits_{i=1}^n a_i^3 \geq 6 \prod\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \dots, a_n$. | 3 |
To find all positive integers \( n \) such that the given inequality:
\[
\left( \sum_{i=1}^n a_i^2\right) \left(\sum_{i=1}^n a_i \right) -\sum_{i=1}^n a_i^3 \geq 6 \prod_{i=1}^n a_i
\]
holds for any \( n \) positive numbers \( a_1, \dots, a_n \), we proceed as follows:
1. **Case \( n = 1 \):**
- Substitute into ... | 5.625 | [
5,
6,
6,
6,
5,
6,
5,
6
] |
Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$.
Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers. | f(x) = 1 \text{ or } f(x) = 2x - 1. |
We need to find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that \( f(a) + f(b) \) divides \( 2(a + b - 1) \) for all \( a, b \in \mathbb{N} \).
Begin by analyzing the given divisibility condition:
\[
f(a) + f(b) \mid 2(a + b - 1).
\]
### Step 1: Simplify for Special Cases
Consider the case where \( a = ... | 6.5 | [
6,
6,
7,
7,
6,
7,
7,
6
] |
Determine all quadruples $(x,y,z,t)$ of positive integers such that
\[ 20^x + 14^{2y} = (x + 2y + z)^{zt}.\] | (1, 1, 3, 1) |
To solve the problem of determining all quadruples \((x, y, z, t)\) of positive integers such that:
\[
20^x + 14^{2y} = (x + 2y + z)^{zt},
\]
we begin by analyzing the constraints in the given equation with small values for \(x\), \(y\), \(z\), and \(t\). Our task is to find values that satisfy the equation when sub... | 7.125 | [
7,
7,
7,
7,
7,
8,
7,
7
] |
The point $P$ is inside of an equilateral triangle with side length $10$ so that the distance from $P$ to two of the sides are $1$ and $3$. Find the distance from $P$ to the third side. | 5\sqrt{3} - 4 |
Given an equilateral triangle with side length \(10\) and a point \(P\) inside the triangle, we are required to find the distance from \(P\) to the third side, knowing the distances from \(P\) to the other two sides are \(1\) and \(3\).
First, recall the area formula of a triangle in terms of its base and correspondi... | 4 | [
4,
4,
4,
4,
4,
4,
4,
4
] |
The quadrilateral $ABCD$ has the following equality $\angle ABC=\angle BCD=150^{\circ}$. Moreover, $AB=18$ and $BC=24$, the equilateral triangles $\triangle APB,\triangle BQC,\triangle CRD$ are drawn outside the quadrilateral. If $P(X)$ is the perimeter of the polygon $X$, then the following equality is true $P(APQRD)=... | 10 |
Given that the quadrilateral \(ABCD\) satisfies \(\angle ABC = \angle BCD = 150^\circ\), and that equilateral triangles \(\triangle APB\), \(\triangle BQC\), and \(\triangle CRD\) are drawn outside the quadrilateral. We are provided with the lengths \(AB = 18\) and \(BC = 24\), and the equality for the perimeters:
\... | 6.125 | [
6,
6,
6,
7,
6,
7,
6,
5
] |
Find all pairs of positive integers $m, n$ such that $9^{|m-n|}+3^{|m-n|}+1$ is divisible by $m$ and $n$ simultaneously. | (1, 1) \text{ and } (3, 3) |
To solve the problem, we need to find all pairs of positive integers \( (m, n) \) such that \( 9^{|m-n|} + 3^{|m-n|} + 1 \) is divisible by both \( m \) and \( n \) simultaneously.
Let's denote \( d = |m-n| \). The expression becomes \( f(d) = 9^d + 3^d + 1 \).
### Step-by-step Analysis:
1. **Case \( d = 0 \):**
... | 5.375 | [
6,
5,
5,
5,
6,
5,
6,
5
] |
We are given $n$ coins of different weights and $n$ balances, $n>2$. On each turn one can choose one balance, put one coin on the right pan and one on the left pan, and then delete these coins out of the balance. It's known that one balance is wrong (but it's not known ehich exactly), and it shows an arbitrary result o... | 2n - 1 |
Given are \( n \) coins of different weights and \( n \) balances, where \( n > 2 \). One of these balances is faulty and provides arbitrary results on each turn. Our goal is to determine the smallest number of turns required to find the heaviest coin.
### Strategy
1. **Initial Understanding**: We need to find which... | 6.75 | [
6,
7,
7,
6,
7,
7,
7,
7
] |
What is the largest possible rational root of the equation $ax^2 + bx + c = 0{}$ where $a, b$ and $c{}$ are positive integers that do not exceed $100{}$? | \frac{1}{99} |
To determine the largest possible rational root of the quadratic equation \( ax^2 + bx + c = 0 \), where \( a, b, \) and \( c \) are positive integers not exceeding 100, we use the Rational Root Theorem. This theorem states that any rational root, expressed as \(\frac{p}{q}\), must have \( p \) as a divisor of the con... | 5 | [
4,
4,
6,
4,
6,
4,
6,
6
] |
Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors.
(M Levin) | 11250, 4050, 7500, 1620, 1200, 720 |
To find all natural numbers divisible by 30 with exactly 30 different divisors, we first explore the properties of divisors and the structure of such numbers.
A natural number \( n \) has exactly 30 divisors if its prime factorization can be expressed to fit the divisors' formula:
\[
(n = p_1^{a_1} \cdot p_2^{a_2} \c... | 6.125 | [
6,
6,
6,
6,
6,
6,
6,
7
] |
Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.
[i]Greece[/i] | 1 |
To find the area of the pentagon \(ABCDE\), we will use the given conditions:
1. \(AB = AE = CD = 1\),
2. \(\angle ABC = \angle DEA = 90^\circ\),
3. \(BC + DE = 1\).
We start by placing the pentagon in the coordinate plane to simplify calculations:
- Let \(A\) be at the origin \((0, 0)\).
- Since \(AB = 1\) and \(\... | 5.125 | [
5,
5,
4,
5,
6,
5,
5,
6
] |
Find all pairs $(n, p)$ of positive integers such that $p$ is prime and
\[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \] | (5, 2) |
To find all pairs \((n, p)\) of positive integers such that \( p \) is prime and:
\[
1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdots + p^2)
\]
we first express these sums using known formulas:
1. The sum of the first \( n \) positive integers is given by:
\[
\frac{n(n + 1)}{2}
\]
2. The sum of the square... | 5.625 | [
5,
6,
6,
6,
6,
5,
6,
5
] |
Each side of square $ABCD$ with side length of $4$ is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Just write the numbers without proof.
[asy]
import graph; si... | {6,7,7.5,8,8.5,9,10} |
Consider square \(ABCD\) with a side length of \(4\). Each side of the square is divided into equal parts by three points, creating four sections of equal length, each measuring \( \frac{4}{4} = 1 \) unit. Let's label these points for each side as follows:
- On side \(AB\) (from \(A\) to \(B\)): \(P_1\), \(P_2\), and ... | 5.125 | [
5,
5,
4,
5,
6,
5,
5,
6
] |
Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$). What is the maximal possible number of different integers on the blackboard? | 2 |
To determine the maximal possible number of different positive integers on the blackboard, given the condition that the sum of any two of them must be a power of two, we proceed as follows:
First, recall that a power of two can be expressed as \(2^k\) for some integer \(k\). The integers on the blackboard must sum to... | 5.125 | [
5,
4,
6,
5,
5,
5,
5,
6
] |
For each positive integer $n$, find the number of $n$-digit positive integers that satisfy both of the following conditions:
[list]
[*] no two consecutive digits are equal, and
[*] the last digit is a prime.
[/list] | \frac{2}{5} \cdot 9^n - \frac{2}{5} \cdot (-1)^n |
To solve this problem, we need to determine the number of \( n \)-digit positive integers that meet two criteria:
1. No two consecutive digits are equal.
2. The last digit is a prime number.
### Step 1: Count All \( n \)-Digit Numbers
The total number of \( n \)-digit numbers is \( 9 \times 10^{n-1} \). The first di... | 5.5 | [
6,
6,
6,
6,
4,
6,
5,
5
] |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$ | f(x) = x |
Let \( f: \mathbb{R} \to \mathbb{R} \) be a function satisfying the following conditions:
1. \( f(x+1) = f(x) + 1 \).
2. \( f(x^2) = f(x)^2 \).
We aim to find all such functions.
### Step 1: Analyze the First Condition
The first condition \( f(x+1) = f(x) + 1 \) implies that \( f \) is a function that increases lin... | 6.5 | [
6,
7,
6,
7,
7,
6,
7,
6
] |
Let $A$ be a $n\times n$ matrix such that $A_{ij} = i+j$. Find the rank of $A$.
[hide="Remark"]Not asked in the contest: $A$ is diagonalisable since real symetric matrix it is not difficult to find its eigenvalues.[/hide] | 2 |
Let \( A \) be an \( n \times n \) matrix where each entry \( A_{ij} = i + j \). We aim to find the rank of this matrix.
**Step 1: Analyze the Structure of Matrix \( A \)**
The entry \( A_{ij} \) depends linearly on the indices \( i \) and \( j \):
\[
A = \begin{bmatrix}
2 & 3 & 4 & \cdots & n+1 \\
3 & 4 & 5 & \cdot... | 4.125 | [
4,
3,
4,
4,
6,
4,
4,
4
] |
Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \} $ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$. If all of these sums have different r... | 2^{n(n-1)/2} |
Let \( n \) be a positive integer, and consider the set of integers \( \{ a_1, a_2, \ldots, a_n \} \) where each \( a_i \) is an element of the set \( \{ 0, 1, 2, \ldots, 2^n - 1 \} \). We associate each subset of this set with the sum of its elements, and the empty subset is associated with the sum \( 0 \).
A set \(... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity
\[ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. \]
Determine all possible values of $Q \left( - \frac{1}{2} \right)$. | 2^n |
Given the problem, we have two polynomials \( P(x) \) and \( Q(x) \) with real coefficients such that their degrees do not exceed \( n \). They satisfy the identity:
\[
P(x) x^{n+1} + Q(x)(x+1)^{n+1} = 1.
\]
We want to determine all possible values of \( Q\left(-\frac{1}{2}\right) \).
### Step 1: Analyzing the Degr... | 6.375 | [
7,
6,
6,
6,
7,
7,
6,
6
] |
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$, such that $$f(xy+f(x^2))=xf(x+y)$$ for all reals $x, y$. | f(x) = 0 \text{ and } f(x) = x | To find the functions \( f : \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the functional equation:
\[
f(xy + f(x^2)) = x f(x + y),
\]
for all real numbers \( x \) and \( y \), we will proceed with the following steps:
### Step 1: Explore Simple Solutions
First, test simple function solutions like \( f(x) = 0 \) a... | 7.5 | [
8,
7,
7,
8,
9,
7,
7,
7
] |
There is a city with $n$ citizens. The city wants to buy [i]sceptervirus[/i] tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following:
[list]
[*][i]Virus positive[/i]: there is at least one currently infected person among the people whose samp... | n |
To determine the smallest number of tests required to ascertain if the sceptervirus is currently present or has been present in the city, let's analyze the given conditions for the test results:
1. **Virus positive**: Indicates there is at least one currently infected individual among the tested samples, and none of ... | 5.5 | [
5,
5,
5,
7,
5,
5,
7,
5
] |
Positive integers are put into the following table.
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|}
\hline
1 & 3 & 6 & 10 & 15 & 21 & 28 & 36 & & \\ \hline
2 & 5 & 9 & 14 & 20 & 27 & 35 & 44 & & \\ \hline
4 & 8 & 13 & 19 & 26 & 34 & 43 & 53 & & \\ \hline
7 & 12 & 18 & 25 & 33 & 42 & & & & \\ \hline
1... | (62, 2) |
We will analyze the pattern in the table to determine where the number \(2015\) is located. The arrangement of numbers in the table has a specific formulation in terms of line and column indices. We will first observe the pattern, derive the general formula for numbers at position \((i, j)\), and then use it to find t... | 6.25 | [
6,
6,
6,
7,
7,
6,
6,
6
] |
All letters in the word $VUQAR$ are different and chosen from the set $\{1,2,3,4,5\}$. Find all solutions to the equation \[\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R}=V^{{{U^Q}^A}^R}.\] | (5, 2, 1, 3, 4) \text{ and } (5, 2, 1, 4, 3) |
Let's consider the given problem: we need to find all solutions for the letters \( V, U, Q, A, R \) in the equation:
\[
\frac{(V+U+Q+A+R)^2}{V-U-Q+A+R} = V^{{{U^Q}^A}^R},
\]
where each letter is from the set \(\{1,2,3,4,5\}\) and all letters are different.
### Step-by-step Strategy:
1. **Analyze the Equation**:
... | 6.75 | [
7,
6,
7,
6,
7,
7,
7,
7
] |
As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is... | 65 |
We are given a \( 40 \times 30 \) rectangle (the paper) with a filled \( 10 \times 5 \) rectangle inside it. The objective is to cut out the filled rectangle using four straight cuts with the aim of minimizing the total length of the cuts. Each cut divides the remaining paper into two pieces, and we keep the piece con... | 5.75 | [
6,
6,
6,
5,
5,
6,
6,
6
] |
In the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$. | 75^\circ |
We are given a parallelogram \(ABCD\) with \(\angle D = 60^\circ\), \(AD = 2\), and \(AB = \sqrt{3} + 1\). Point \(M\) is the midpoint of \(AD\), and segment \(CK\) is the angle bisector of \(\angle C\). We need to find \(\angle CKB\).
### Step 1: Analyzing the Parallelogram Properties
In a parallelogram, opposite si... | 5 | [
5,
5,
4,
6,
6,
4,
5,
5
] |
The cells of a $8 \times 8$ table are initially white. Alice and Bob play a game. First Alice paints $n$ of the fields in red. Then Bob chooses $4$ rows and $4$ columns from the table and paints all fields in them in black. Alice wins if there is at least one red field left. Find the least value of $n$ such that Alice ... | 13 |
Consider a \( 8 \times 8 \) table where Alice and Bob play a game. Initially, all cells in this table are white. Alice begins by painting \( n \) of the cells red. After that, Bob selects 4 rows and 4 columns and paints all cells in these rows and columns black. Alice wins if at least one red cell remains unpainted by... | 4.875 | [
5,
5,
4,
6,
4,
4,
5,
6
] |
We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]? | 5 |
To solve this problem, we need to find the greatest number of consecutive natural numbers greater than 25 that can each be represented as the sum of two different prime numbers, which we refer to as semi-prime.
First, let's identify a few prime numbers as potential candidates: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ... | 4 | [
4,
5,
4,
4,
4,
3,
4,
4
] |
The following construction is used for training astronauts:
A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a... | n - 1 |
Consider the problem of determining the number of revolutions an astronaut performs relative to the ground while being fastened to a circle \( C_3 \) of radius \( R \). This circle rolls inside another circle \( C_2 \) of radius \( 2R \), which in turn rolls along the inside of a fixed circle \( C_1 \) of radius \( nR... | 7.875 | [
8,
7,
8,
8,
8,
8,
8,
8
] |
A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly.
Alice and Bob can discuss a strategy before the game with the ... | k+1 |
Let a sequence of length \(2^k\) consisting of only 0s and 1s be given. Alice selects a member from this sequence, revealing both its position and value to Bob. Our goal is to determine the largest number \(s\) such that Bob, by following an optimal strategy agreed upon with Alice prior to the game, can always correct... | 7.125 | [
7,
7,
7,
7,
7,
7,
8,
7
] |
Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that
\[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \]
holds for all $ x, y \in \mathbb{R}$. | f(x) = x + 1 |
To solve the functional equation
\[
f(xf(x) + 2y) = f(x^2) + f(y) + x + y - 1
\]
for all real numbers \( x \) and \( y \), we need to find all possible functions \( f : \mathbb{R} \rightarrow \mathbb{R} \) that satisfy this condition.
**Step 1: Initial Evaluation**
Let's substitute \( y = 0 \) into the equation:
... | 7.5 | [
8,
8,
8,
8,
8,
7,
7,
6
] |
Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$. | (1, 1) |
Let us find all positive integers \( a, b \) such that the following conditions are satisfied:
1. \( b^{619} \mid a^{1000} + 1 \)
2. \( a^{619} \mid b^{1000} + 1 \)
To find these solutions, we start by analyzing the given divisibility conditions.
First, consider the condition \( b^{619} \mid a^{1000} + 1 \). This i... | 7.75 | [
9,
7,
8,
7,
7,
8,
8,
8
] |
Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$. | \left(\left(\frac{u}{v}\right)^{\frac{u}{v-u}}, \left(\frac{u}{v}\right)^{\frac{v}{v-u}}\right) |
To find all pairs of distinct rational numbers \((a, b)\) such that \(a^a = b^b\), we start by setting up the equation:
\[
a^a = b^b.
\]
This can be rewritten using logarithms as:
\[
a \ln a = b \ln b.
\]
Consider \(a = \left(\frac{u}{v}\right)^{\frac{u}{v-u}}\) and \(b = \left(\frac{u}{v}\right)^{\frac{v}{v-u}}\)... | 7.25 | [
7,
7,
7,
7,
8,
7,
8,
7
] |
Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] | (3, 3, 2, 3), (3, 37, 3, 13), (37, 3, 3, 13), (3, 17, 3, 7), (17, 3, 3, 7) |
To find all prime numbers \(a, b, c\) and positive integers \(k\) that satisfy the equation
\[
a^2 + b^2 + 16c^2 = 9k^2 + 1,
\]
we proceed as follows:
1. **Modulo Consideration**: Observe the equation modulo 3. We have:
\[
a^2 \equiv 0 \text{ or } 1 \pmod{3}, \quad b^2 \equiv 0 \text{ or } 1 \pmod{3}, \quad 16... | 6.875 | [
7,
7,
7,
7,
7,
7,
7,
6
] |
$M$ is the midpoint of the side $AB$ in an equilateral triangle $\triangle ABC.$ The point $D$ on the side $BC$ is such that $BD : DC = 3 : 1.$ On the line passing through $C$ and parallel to $MD$ there is a point $T$ inside the triangle $\triangle ABC$ such that $\angle CTA = 150.$ Find the $\angle MT D.$
[i](K. Ivano... | 120^\circ |
Given an equilateral triangle \(\triangle ABC\), where \(M\) is the midpoint of side \(AB\). We have a point \(D\) on side \(BC\) such that the segment \(BD : DC = 3 : 1\). We need to find \(\angle MTD\) given that there is a point \(T\) on a line passing through \(C\) and parallel to \(MD\) inside \(\triangle ABC\) s... | 6.375 | [
7,
6,
6,
6,
6,
7,
7,
6
] |
Define the polynomials $P_0, P_1, P_2 \cdots$ by:
\[ P_0(x)=x^3+213x^2-67x-2000 \]
\[ P_n(x)=P_{n-1}(x-n), n \in N \]
Find the coefficient of $x$ in $P_{21}(x)$. | 61610 |
To find the coefficient of \( x \) in \( P_{21}(x) \), we need to evaluate the transformation of the polynomial \( P_0(x) \) through a series of substitutions as defined by the recurrence relation \( P_n(x) = P_{n-1}(x-n) \).
Initially, we have:
\[ P_0(x) = x^3 + 213x^2 - 67x - 2000. \]
### Step-by-Step Transformati... | 6 | [
6,
5,
6,
6,
6,
7,
6,
6
] |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$. | f(x) = x - 1 \text{ or } f(x) = -x - 1. |
We are tasked with finding all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following functional equation holds:
\[
f(x)f(y) + f(x+y) = xy.
\]
**Step 1: Plug-in Specific Values**
First, let's test for simple values of \( x \) and \( y \).
1. Set \( y = 0 \):
... | 8 | [
8,
9,
8,
7,
8,
9,
8,
7
] |
Turbo the snail plays a game on a board with $2024$ rows and $2023$ columns. There are hidden monsters in $2022$ of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at ... | 3 |
To solve this problem, we will analyze the board's structure and derive a strategy for Turbo to ensure he reaches the last row in a guaranteed number of attempts. We'll consider the distribution of monsters and Turbo's possible paths.
Given:
- The board has 2024 rows and 2023 columns.
- There is exactly one monster i... | 7.125 | [
8,
7,
6,
7,
8,
6,
7,
8
] |
A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if ... | 2013 |
To solve this problem, we need to determine the least number of lines, \( k \), required to ensure that any Colombian configuration of 4027 points (where 2013 are red and 2014 are blue, with no three points collinear) can be separated such that no region contains points of both colors.
The steps to find the solution ... | 6.75 | [
7,
6,
6,
6,
6,
7,
9,
7
] |
Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that
$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$. | (2, 1, 4, 1) \text{ and } (2, 1, 1, 4) |
To find all positive integers \( a \), \( b \), \( c \), and \( p \), where \( p \) is a prime number, satisfying the equation:
\[
73p^2 + 6 = 9a^2 + 17b^2 + 17c^2,
\]
we proceed as follows:
### Step 1: Investigate the Equation
The equation is balanced on both sides, with terms involving squares of integers and a ... | 6.5 | [
6,
7,
7,
6,
6,
6,
7,
7
] |
Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:
[list]
[*]If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the car... | 2^{n-2} |
Fiona starts with \( n \) cards labeled from 1 to \( n \) lined up in increasing order. Her task is to perform a series of pile rearrangements until she consolidates them into a single pile. Given the specific operation she can perform, we explore how many distinct final single pile sequences she can achieve.
**Under... | 6.25 | [
6,
6,
6,
7,
6,
6,
6,
7
] |
Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$ | 3 |
To find all prime numbers \( p \) such that there exists a unique \( a \in \mathbb{Z}_p \) for which the equation \( a^3 - 3a + 1 = 0 \) holds, we proceed as follows:
The equation in question is \( a^3 - 3a + 1 = 0 \). We need to determine under what conditions (i.e., for which primes \( p \)) this polynomial has exa... | 5.875 | [
6,
6,
6,
6,
5,
6,
6,
6
] |
What is the smallest value that the sum of the digits of the number $3n^2+n+1,$ $n\in\mathbb{N}$ can take? | 3 |
We need to find the smallest value of the sum of the digits of the expression \( 3n^2 + n + 1 \) for \( n \in \mathbb{N} \).
To determine this, we will explore values of \( n \) and calculate the resulting expression \( 3n^2 + n + 1 \), then find the sum of its digits. Our goal is to find the smallest possible sum.
... | 3.625 | [
4,
3,
3,
4,
3,
3,
4,
5
] |
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after t... | 0, 2, 4, 6, \ldots, 1022 |
We begin with the numbers \(1, 2, 3, \ldots, 1024\) written on a blackboard. The task involves repeatedly dividing these numbers into pairs, taking the non-negative difference of each pair, and then continuing this process until only one number remains after ten such operations. We aim to determine all possible final ... | 6.25 | [
6,
8,
6,
6,
6,
6,
6,
6
] |
Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square. | 3 |
Let \( n \) be an integer such that \( n \geq 1 \), and we want to find all values of \( n \) for which \( n \cdot 2^{n+1} + 1 \) is a perfect square. We set:
\[
n \cdot 2^{n+1} + 1 = k^2
\]
for some integer \( k \). Rearranging gives:
\[
n \cdot 2^{n+1} = k^2 - 1
\]
which can be factored as:
\[
(k - 1)(k + 1) = ... | 5.5 | [
5,
6,
5,
6,
6,
5,
5,
6
] |
Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \neq b$) have equal areas. Find $x$. | \frac{\sqrt{a^2 + b^2}}{2} |
We are given two isosceles triangles with side lengths \( x, x, a \) and \( x, x, b \), where \( a \neq b \), and they have equal areas. We need to find the value of \( x \).
### Step-by-Step Solution
1. **Area of an Isosceles Triangle:**
For an isosceles triangle with sides \( x, x, a \), the area \( A_1 \) can... | 5.625 | [
6,
6,
6,
6,
5,
5,
5,
6
] |
Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$. | f(x) = x \text{ or } f(x) = -x |
To find all functions \( f: \mathbb{R} \to \mathbb{R} \) satisfying the given functional equation:
\[
f(xf(x) + f(y)) = f^2(x) + y \quad \text{for all } x, y \in \mathbb{R},
\]
we proceed with the following steps.
### Step 1: Analyzing the Functional Equation
First, we substitute \( y = 0 \) into the equation:
\[... | 8 | [
8,
8,
9,
7,
8,
8,
8,
8
] |
Find all functions $f: R \to R$ such that, for any $x, y \in R$:
$f\left( f\left( x \right)-y \right)\cdot f\left( x+f\left( y \right) \right)={{x}^{2}}-{{y}^{2}}$ | f(x) = x |
Consider the functional equation that must be satisfied for all functions \( f: \mathbb{R} \to \mathbb{R} \):
\[
f\left( f(x) - y \right) \cdot f\left( x + f(y) \right) = x^2 - y^2
\]
### Step 1: Analyze the structure
1. **Initial Observation**: The equation is symmetric in a way that resembles the difference of sq... | 7.75 | [
8,
7,
8,
8,
8,
6,
8,
9
] |
What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?
(A Shapovalov) | 16 |
To find the maximum number of colors that can be used to paint an \(8 \times 8\) chessboard such that each square is adjacent (horizontally or vertically) to at least two other squares of its own color, we need to carefully analyze and construct a feasible coloring pattern under the given constraints.
### Step-by-Ste... | 5.5 | [
6,
6,
6,
6,
5,
5,
6,
4
] |
Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist? | 2^{n-1} |
To solve this problem, we need to determine how many distinct ways the child can build a wall with \( n \) identical cubes. Each cube can be placed in such a way that it shares a face with the previous cube. This can be done either by placing the new cube on the ground or on top of the previously placed cube.
Let's ... | 3.125 | [
3,
3,
3,
3,
3,
3,
4,
3
] |
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
[i] | f(x) = 0f(x) = 1 - xf(x) = x - 1 |
Let \(\mathbb{R}\) be the set of real numbers. We are tasked with finding all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for any real numbers \( x \) and \( y \), the functional equation:
\[
f(f(x)f(y)) + f(x+y) = f(xy)
\]
is satisfied.
### Step 1: Checking Simple Functions
#### Case 1: Constan... | 6.875 | [
7,
7,
7,
7,
6,
7,
7,
7
] |
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived. | -\frac{1}{8} |
Let \(a, b, c, d\) be real numbers such that \(a^2 + b^2 + c^2 + d^2 = 1\). We want to determine the minimum value of the expression \((a-b)(b-c)(c-d)(d-a)\).
To find the minimum value of \((a-b)(b-c)(c-d)(d-a)\), we first recognize the symmetry and potential simplifications. The key is to find a particular symmetric... | 6.625 | [
7,
6,
7,
7,
6,
6,
7,
7
] |
A new website registered $2000$ people. Each of them invited $1000$ other registered people to be their friends. Two people are considered to be friends if and only if they have invited each other. What is the minimum number of pairs of friends on this website? | 1000 |
Consider a new website with \(2000\) registered people. Each person invites \(1000\) other people from the group to be their friends. According to the rules, two people are actually considered friends if and only if they have invited each other.
We need to determine the minimum number of pairs of friends on this webs... | 4.625 | [
4,
5,
5,
4,
4,
4,
6,
5
] |
If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$. | 2015 |
Given the inequality \( s = a^3 + b^3 - 60ab(a + b) \geq 2012 \), we are tasked with finding the least possible value of \( s \).
To solve this, we start by rewriting \( s \) in terms of a simplified expression:
\[
s = a^3 + b^3 - 60ab(a + b).
\]
Notice that we can factor and simplify the expression using the identi... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Find all pairs $(x,y)$ of nonnegative integers that satisfy \[x^3y+x+y=xy+2xy^2.\] | (0, 0), (1, 1), (2, 2) |
We need to find all pairs \((x, y)\) of nonnegative integers that satisfy the equation:
\[
x^3 y + x + y = xy + 2xy^2
\]
Let's start by simplifying the given equation. We rewrite the equation as follows:
\[
x^3 y + x + y = xy + 2xy^2
\]
Rearranging the terms, we get:
\[
x^3 y + x + y - xy - 2xy^2 = 0
\]
Collect ... | 5.375 | [
5,
5,
6,
6,
5,
5,
6,
5
] |
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$? | 31 |
To solve the problem, we are tasked with finding the number of integers \( n > 1 \) such that \( n \) divides \( x^{13} - x \) for every positive integer \( x \).
First, observe that if \( n \mid x^{13} - x \) for every integer \( x \), then \( n \mid x^{13} - x \) for each \( x \) in particular values, such as \( x ... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
Let $\,{\mathbb{R}}\,$ denote the set of all real numbers. Find all functions $\,f: {\mathbb{R}}\rightarrow {\mathbb{R}}\,$ such that \[ f\left( x^{2}+f(y)\right) =y+\left( f(x)\right) ^{2}\hspace{0.2in}\text{for all}\,x,y\in \mathbb{R}. \] | f(x) = x |
Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function satisfying the functional equation:
\[
f(x^2 + f(y)) = y + (f(x))^2 \quad \text{for all } x, y \in \mathbb{R}.
\]
**Step 1**: Evaluate the functional equation at specific points.
First, let's substitute \( y = 0 \) into the functional equation:
\[
f(x^2 ... | 6.875 | [
8,
7,
7,
6,
7,
6,
7,
7
] |
A set of positive integers is called [i]fragrant[/i] if it contains at least two elements and each of its elements has a prime factor in common with at least one of the other elements. Let $P(n)=n^2+n+1$. What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for wh... | 6 |
To solve this problem, we need to find the smallest positive integer \( b \) such that there exists a non-negative integer \( a \) for which the set
\[
\{P(a+1), P(a+2), \ldots, P(a+b)\}
\]
is fragrant. The polynomial \( P(n) = n^2 + n + 1 \).
A set is considered fragrant if it contains at least two elements and eac... | 6 | [
6,
6,
6,
6,
6,
6,
6,
6
] |
A magician has one hundred cards numbered 1 to 100. He puts them into three boxes, a red one, a white one and a blue one, so that each box contains at least one card. A member of the audience draws two cards from two different boxes and announces the sum of numbers on those cards. Given this information, the magician l... | 12 |
Given the problem, let's denote the three boxes as \( R \) (red), \( W \) (white), and \( B \) (blue). Each box must contain at least one card, and the numbers on the cards range from 1 to 100. The magician must be able to determine the box from which no card has been drawn using only the sum of the numbers on the two... | 6.25 | [
5,
7,
6,
6,
7,
7,
7,
5
] |
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$. | f(x) = 0, \quad f(x) = \frac{1}{2}, \quad f(x) = x^2. |
To solve the given functional equation for all functions \( f: \mathbb{R} \to \mathbb{R} \):
\[
(f(x) + f(z))(f(y) + f(t)) = f(xy - zt) + f(xt + yz),
\]
we start by analyzing specific cases to deduce possible forms for \( f(x) \).
1. **Testing the Zero Function:**
Substitute \( f(x) = 0 \) for all \( x \). The ... | 8 | [
8,
9,
7,
8,
8,
8,
7,
9
] |
For each integer $a_0 > 1$, define the sequence $a_0, a_1, a_2, \ldots$ for $n \geq 0$ as
$$a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer,} \\
a_n + 3 & \text{otherwise.}
\end{cases}
$$
Determine all values of $a_0$ such that there exists a number $A$ such that $a_n = A$ for infinitel... | 3 \mid a_0 |
We are given a sequence defined by \( a_0, a_1, a_2, \ldots \) where the recurrence relation for \( n \geq 0 \) is:
\[
a_{n+1} =
\begin{cases}
\sqrt{a_n} & \text{if } \sqrt{a_n} \text{ is an integer}, \\
a_n + 3 & \text{otherwise}.
\end{cases}
\]
The goal is to determine all starting values \( a_0 \) such that the se... | 6.5 | [
8,
7,
6,
6,
7,
6,
6,
6
] |
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. $f(2)=0,f(3)>0,f(9999)=3333$ and for all $m,n:$ \[ f(m+n)-f(m)-f(n)=0 \text{ or } 1. \] Determine $f(1982)$. | 660 |
We are given that the function \( f(n) \) is defined on positive integers and it takes non-negative integer values. It satisfies:
\[ f(2) = 0, \]
\[ f(3) > 0, \]
\[ f(9999) = 3333, \]
and for all \( m, n \):
\[ f(m+n) - f(m) - f(n) = 0 \text{ or } 1. \]
We need to determine \( f(1982) \).
### Analysis of the Func... | 6 | [
6,
6,
6,
5,
6,
6,
6,
7
] |
Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\ne0$ for $0\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$. | f(x) = \begin{cases}
\frac{2}{2 - x}, & 0 \leq x < 2, \\
0, & x \geq 2.
\end{cases} |
We need to find all functions \( f: [0, \infty) \to [0, \infty) \) that satisfy the following conditions:
1. \( f(2) = 0 \).
2. \( f(x) \neq 0 \) for \( 0 \leq x < 2 \).
3. \( f(xf(y))f(y) = f(x+y) \) for all \( x, y \geq 0 \).
Let's begin by analyzing these conditions:
1. **Condition \( f(2) = 0 \):** According to ... | 7.125 | [
7,
7,
7,
8,
7,
6,
8,
7
] |
We call a positive integer $N$ [i]contagious[/i] if there are $1000$ consecutive non-negative integers such that the sum of all their digits is $N$. Find all contagious positive integers. | \{13500, 13501, 13502, \ldots\} |
To determine which positive integers \( N \) are contagious, we consider 1000 consecutive non-negative integers and the sum of all their digits equating to \( N \).
Let the consecutive integers be \( x, x+1, x+2, \ldots, x+999 \). We need to calculate the sum of the digits of these 1000 numbers.
Let's start by consi... | 5.875 | [
6,
5,
7,
5,
5,
7,
6,
6
] |
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following equality for all $x,y\in\mathbb{R}$ \[f(x)f(y)-f(x-1)-f(y+1)=f(xy)+2x-2y-4.\][i] | f(x) = x^2 + 1 |
To determine all functions \( f: \mathbb{R} \to \mathbb{R} \) that satisfy the given functional equation for all \( x, y \in \mathbb{R} \):
\[
f(x)f(y) - f(x-1) - f(y+1) = f(xy) + 2x - 2y - 4,
\]
we proceed as follows.
### Step 1: Substitute Special Values
1. **Substitute \( x = 0 \) and \( y = 0 \):**
\[
f... | 7 | [
8,
7,
7,
7,
7,
6,
6,
8
] |
Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like... | 12 |
To solve this problem, we will analyze the movement of Esmeralda's special knight on a \(5 \times 5\) board and determine the longest possible path without revisiting any square. The knight moves in an "L" shape spanning across a \(2 \times 4\) rectangle, rather than the traditional \(2 \times 3\) from chess. These mo... | 6 | [
6,
6,
6,
6,
7,
5,
6,
6
] |
Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer.
[i]Laurentiu Panaitopol, Romania[/i] | (5, 3) |
We are tasked with finding all pairs of positive integers \( m, n \geq 3 \) such that there exist infinitely many positive integers \( a \) making the expression
\[
\frac{a^m + a - 1}{a^n + a^2 - 1}
\]
an integer. To solve this problem, we aim to explore potential values of \( m \) and \( n \) and identify condition... | 7.75 | [
7,
8,
9,
7,
7,
8,
8,
8
] |
The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$. | 120^\circ |
Given triangle \( \triangle ABC \), where \( AB = AC \) and \( \angle BAC < 60^\circ \), points \( D \) and \( E \) are chosen on side \( AC \) such that \( EB = ED \) and \( \angle ABD \equiv \angle CBE \). We are tasked with finding \(\angle COD\), where \( O \) is the intersection of the internal bisectors of \(\an... | 6.25 | [
6,
6,
6,
7,
6,
6,
6,
7
] |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(xy) = f(x)f(y) + f(f(x + y))$$
holds for all $x, y \in \mathbb{R}$. | f(x) = 0 \text{ and } f(x) = x - 1. |
We are tasked with finding all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) satisfying the equation:
\[
f(xy) = f(x)f(y) + f(f(x+y))
\]
for all \( x, y \in \mathbb{R} \).
### Step 1: Consideration of Simple Cases
First, let us consider the case where \( x = 0 \):
\[
f(0) = f(0)f(y) + f(f(y))
\]
for all \... | 7.875 | [
8,
7,
8,
7,
9,
8,
8,
8
] |
Determine all positive integers relatively prime to all the terms of the infinite sequence \[ a_n=2^n+3^n+6^n -1,\ n\geq 1. \] | 1 |
To solve the problem, we need to determine all positive integers that are relatively prime to every term of the sequence defined by:
\[
a_n = 2^n + 3^n + 6^n - 1, \quad n \geq 1.
\]
**Step 1: Understanding the sequence properties**
To determine an integer relatively prime to all \( a_n \), we first investigate the ... | 6.75 | [
6,
7,
8,
6,
7,
7,
7,
6
] |
A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20.
Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance be... | 100 |
Let us consider the problem where Amy and Ben take turns placing stones on a 20x20 grid consisting of sites \((x, y)\) where \(x\) and \(y\) are integers between 1 and 20 inclusive. Amy's condition for placing a red stone is that the distance between any two red stones is not equal to \(\sqrt{5}\). This occurs specifi... | 6.625 | [
6,
7,
7,
6,
7,
7,
7,
6
] |
Let $ ABC$ be a triangle with $ AB \equal{} AC$ . The angle bisectors of $ \angle C AB$ and $ \angle AB C$ meet the sides $ B C$ and $ C A$ at $ D$ and $ E$ , respectively. Let $ K$ be the incentre of triangle $ ADC$. Suppose that $ \angle B E K \equal{} 45^\circ$ . Find all possible values of $ \angle C AB$ .
[i]Jan ... | 60^\circ \text{ and } 90^\circ |
Given a triangle \( ABC \) with \( AB = AC \) (isosceles triangle), we are tasked with finding all possible values of \( \angle CAB \) given the specific geometric conditions.
### Problem Setup
1. **Notation and Known Values:**
- Let \( ABC \) be an isosceles triangle with \( AB = AC \).
- The angle bisector o... | 6.5 | [
6,
7,
6,
7,
5,
7,
7,
7
] |
$100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number?
(D. Fomin, Leningrad) | 101 |
Initially, we have $100$ numbers written on the blackboard: \(1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{100}\). In each step, two numbers \(a\) and \(b\) are selected and replaced with the number \(a + b + ab\). This process is repeated until only one number remains.
To solve this, a useful observation involves r... | 6.25 | [
8,
6,
6,
5,
6,
6,
6,
7
] |
Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$. | (m, n) = (0, 0) \text{ and } (m, n) = (-2, 2) |
We are tasked with finding all integer pairs \((m, n)\) such that:
\[
m^5 - n^5 = 16mn
\]
**Step 1: Algebraic Manipulation**
We begin by rewriting the given equation as:
\[
m^5 - n^5 - 16mn = 0
\]
**Step 2: Factorization**
Using the identity for the difference of powers, we have:
\[
m^5 - n^5 = (m - n)(m^4 + m^... | 6.125 | [
6,
6,
6,
7,
6,
6,
5,
7
] |
An $8\times8$ array consists of the numbers $1,2,...,64$. Consecutive numbers are adjacent along a row or a column. What is the minimum value of the sum of the numbers along the diagonal? | 88 |
We have an \(8 \times 8\) array filled with the numbers from 1 to 64, where consecutive numbers are adjacent either along a row or along a column. Our task is to find the minimum possible value of the sum of the numbers along a diagonal of this array.
### Analysis
Let's denote the elements of the array by \( a_{ij} ... | 6.125 | [
6,
7,
6,
6,
6,
6,
6,
6
] |
There are $100$ piles of $400$ stones each. At every move, Pete chooses two piles, removes one stone from each of them, and is awarded the number of points, equal to the non- negative difference between the numbers of stones in two new piles. Pete has to remove all stones. What is the greatest total score Pete can get,... | 3920000 | To solve this problem, we need to find the greatest total score Pete can get by removing all stones. Initially, we have 100 piles, each containing 400 stones.
### Strategy
To maximize the total score, Pete should aim to keep the piles as balanced as possible until they are empty. This involves making the difference be... | 6.5 | [
7,
6,
6,
6,
7,
6,
6,
8
] |
Let $n$ be a positive integer. A sequence of $n$ positive integers (not necessarily distinct) is called [b]full[/b] if it satisfies the following condition: for each positive integer $k\geq2$, if the number $k$ appears in the sequence then so does the number $k-1$, and moreover the first occurrence of $k-1$ comes befor... | n! |
To solve this problem, we need to determine how many sequences of length \( n \) consisting of positive integers are considered "full" according to the defined condition. The condition implies a hierarchical appearance of integers in the sequence, such that if an integer \( k \) appears, then \( k-1 \) must also appea... | 5.5 | [
6,
6,
5,
6,
4,
6,
6,
5
] |
For any positive integer $n$, we define the integer $P(n)$ by :
$P(n)=n(n+1)(2n+1)(3n+1)...(16n+1)$.
Find the greatest common divisor of the integers $P(1)$, $P(2)$, $P(3),...,P(2016)$. | 510510 |
To find the greatest common divisor (GCD) of the integers \( P(1), P(2), P(3), \ldots, P(2016) \), where \( P(n) = n(n+1)(2n+1)(3n+1)\cdots(16n+1) \), we will first consider each part of the product and determine if there is a consistent factor across all \( P(n) \).
### Step 1: Analyze the Form of \( P(n) \)
The ex... | 6.875 | [
7,
6,
7,
7,
7,
7,
7,
7
] |
Let $ S \equal{} \{1,2,3,\cdots ,280\}$. Find the smallest integer $ n$ such that each $ n$-element subset of $ S$ contains five numbers which are pairwise relatively prime. | 217 |
Let \( S = \{1, 2, 3, \ldots, 280\} \). We are tasked with determining the smallest integer \( n \) such that every \( n \)-element subset of \( S \) contains at least five numbers that are pairwise relatively prime.
To solve this problem, we need to understand the prime factorization properties of the numbers within... | 6.125 | [
7,
5,
6,
7,
6,
6,
7,
5
] |
Find all triples $(a, b, c)$ of positive integers such that $a^3 + b^3 + c^3 = (abc)^2$. | (3, 2, 1), (3, 1, 2), (2, 3, 1), (2, 1, 3), (1, 3, 2), (1, 2, 3) |
We are required to find all triples \((a, b, c)\) of positive integers such that:
\[
a^3 + b^3 + c^3 = (abc)^2
\]
### Step 1: Analyze the Equation
We start by analyzing the given equation:
\[
a^3 + b^3 + c^3 = (abc)^2
\]
We will assume, without loss of generality, that \(a \leq b \leq c\). This implies that \(a\)... | 6.375 | [
7,
7,
5,
7,
6,
6,
6,
7
] |
At a meeting of $ 12k$ people, each person exchanges greetings with exactly $ 3k\plus{}6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting? | 36 |
Let the total number of people at the meeting be \( n = 12k \). Each person exchanges greetings with exactly \( 3k + 6 \) others. We need to determine the value of \( n \).
Given that for any two people, the number of people who exchange greetings with both is the same, this problem is essentially about identifying a... | 6.25 | [
6,
6,
6,
6,
7,
6,
6,
7
] |
Let $n$ be a positive integer. Compute the number of words $w$ that satisfy the following three properties.
1. $w$ consists of $n$ letters from the alphabet $\{a,b,c,d\}.$
2. $w$ contains an even number of $a$'s
3. $w$ contains an even number of $b$'s.
For example, for $n=2$ there are $6$ such words: $aa, bb, cc, d... | 2^{n-1}(2^{n-1} + 1) |
We are tasked with determining the number of words \( w \), consisting of \( n \) letters from the alphabet \(\{a, b, c, d\}\), that satisfy the following properties:
1. The word \( w \) contains an even number of \( a \)'s.
2. The word \( w \) contains an even number of \( b \)'s.
Let's approach the problem by consi... | 5.625 | [
6,
4,
6,
6,
7,
6,
5,
5
] |
The equation
$$(x-1)(x-2)\cdots(x-2016)=(x-1)(x-2)\cdots (x-2016)$$
is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting eq... | 2016 |
Given the equation:
\[
(x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016)
\]
This equation has 2016 linear factors on each side of the equation. Our goal is to find the smallest number \( k \) such that removing \( k \) factors from these \( 4032 \) factors still leaves at least one factor on each side and results ... | 6.125 | [
6,
6,
6,
6,
6,
6,
7,
6
] |
Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple:
\begi... | 3 |
To solve the problem, we need to determine the minimum number \( s \) of initial integer-valued \( 2022 \)-tuples that Lucy has to write on the blackboard such that any other integer-valued \( 2022 \)-tuple can be formed using the operations defined.
### Step-by-Step Analysis:
1. **Operations Description**:
- Ad... | 5.625 | [
6,
6,
6,
6,
5,
5,
5,
6
] |
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i =... | 506 |
To solve the given problem, we first need to understand the requirements for a \(\pm 1\)-sequence. We are looking for the largest integer \( C \) such that, for any sequence of numbers \( a_1, a_2, \ldots, a_{2022} \) where each \( a_i \) is either \( +1 \) or \( -1 \), there exists a subsequence satisfying certain co... | 6.125 | [
7,
6,
6,
6,
6,
6,
6,
6
] |
Determine all integers $ k\ge 2$ such that for all pairs $ (m$, $ n)$ of different positive integers not greater than $ k$, the number $ n^{n\minus{}1}\minus{}m^{m\minus{}1}$ is not divisible by $ k$. | 2 \text{ and } 3 | Let us analyze the problem, which requires us to determine all integers \( k \ge 2 \) such that for all pairs \( (m, n) \) of different positive integers not greater than \( k \), the expression \( n^{n-1} - m^{m-1} \) is not divisible by \( k \).
### Step 1: Understand the condition
The condition states:
- For \( n,... | 6.875 | [
6,
7,
7,
7,
7,
7,
7,
7
] |
Sir Alex plays the following game on a row of 9 cells. Initially, all cells are empty. In each move, Sir Alex is allowed to perform exactly one of the following two operations:
[list=1]
[*] Choose any number of the form $2^j$, where $j$ is a non-negative integer, and put it into an empty cell.
[*] Choose two (not neces... | 2 \sum_{i=0}^{8} \binom{n}{i} - 1 |
To determine the maximum number of moves that Sir Alex could have made in this game, we need to analyze the operations and how each affects the game state.
Sir Alex has 9 cells initially empty. The objective is to have one cell contain the number \(2^n\) at the end, while all others are empty. During the game, Sir Al... | 6 | [
6,
7,
5,
6,
7,
6,
6,
5
] |
$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take?
A. Gribalko | 2019! |
Given the problem, we have a sequence of \(2020\) positive integers, say \(a_1, a_2, \ldots, a_{2020}\). Each term in the sequence starting with the third term (\(a_i\) for \(i \geq 3\)) is divisible by its preceding term and the sum of its two immediate predecessors. Formally, this can be expressed as:
\[
a_i \text{... | 6.75 | [
7,
6,
7,
7,
7,
7,
6,
7
] |
The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$. | 4018021 |
We are given the sequence \( \{a_n\}_{n \geq 0} \) which satisfies the relation:
\[
a_{m+n} + a_{m-n} - m + n - 1 = \frac{1}{2} (a_{2m} + a_{2n})
\]
for all non-negative integers \( m \) and \( n \) with \( m \ge n \). We are also given that \( a_1 = 3 \), and we need to find \( a_{2004} \).
We start by plugging sp... | 6.875 | [
7,
7,
7,
7,
6,
7,
8,
6
] |
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.) | 59 |
Given an \(8 \times 8\) chessboard where each square initially contains a rook, we need to determine the maximal number of rooks that can be removed such that each removed rook initially attacked an odd number of other rooks. A rook attacks another rook if they are positioned in the same row or column and there are no... | 6.75 | [
6,
7,
7,
7,
7,
7,
7,
6
] |
There is an integer $n > 1$. There are $n^2$ stations on a slope of a mountain, all at different altitudes. Each of two cable car companies, $A$ and $B$, operates $k$ cable cars; each cable car provides a transfer from one of the stations to a higher one (with no intermediate stops). The $k$ cable cars of $A$ have $k$ ... | n^2 - n + 1 |
To solve the problem involving cable car companies \( A \) and \( B \), we must determine the smallest integer \( k \) such that there are always two stations linked by both companies.
### Understanding the Problem
1. **Stations and Connections:**
- We have \( n^2 \) stations on a mountain, each at different alti... | 6.25 | [
7,
6,
7,
6,
6,
6,
7,
5
] |
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i] | f(x) = 2 |
To solve the problem, we need to find all functions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) such that for all positive real numbers \( x \) and \( y \), the functional equation holds:
\[
f(x)f(y) = 2f(x + yf(x)).
\]
### Step 1: Analyze the Functional Equation for Simplicity
Firstly, let's test if a const... | 7.25 | [
7,
7,
8,
8,
8,
7,
6,
7
] |
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \] | f(x) = -1 \text{ and } f(x) = x - 1. |
To find all functions \( f:\mathbb{R}\rightarrow\mathbb{R} \) satisfying the given functional equation, we start with the condition:
\[ f(0)\neq 0, \]
and the functional equation for all \( x, y \in \mathbb{R} \):
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]
### Step 1: Test Simple Fu... | 7.375 | [
7,
7,
7,
9,
8,
7,
7,
7
] |
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