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Problem 4. A $5 \times 100$ table is divided into 500 unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called adjacent if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible... |
Solution. If we colour all the cells along all edges of the board together with the entire middle row except the second and the last-but-one cell, the condition is satisfied and there are 302 black cells. The figure below exhibits this colouring for the $5 \times 8$ case.
 in a 5x100 grid where each black cell has at most two adjacent black cells. Let me think through this step by step.
First, the problem states that each cell, whether black or white, has at most two adjacent black cells. Wait, actually, let ... |
Problem 3. Find all the triples of integers $(a, b, c)$ such that the number
$$
N=\frac{(a-b)(b-c)(c-a)}{2}+2
$$
is a power of 2016 .
(A power of 2016 is an integer of the form $2016^{n}$, where $n$ is a non-negative integer.)
|
Solution. Let $a, b, c$ be integers and $n$ be a positive integer such that
$$
(a-b)(b-c)(c-a)+4=2 \cdot 2016^{n}
$$
We set $a-b=-x, b-c=-y$ and we rewrite the equation as
$$
x y(x+y)+4=2 \cdot 2016^{n}
$$
If $n>0$, then the right hand side is divisible by 7 , so we have that
$$
x y(x+y)+4 \equiv 0 \quad(\bmod 7)... | (,b,)=(k+2,k+1,k),k\in\mathbb{Z} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2 | [
false,
false
] | 0 | <think>
Okay, so I need to find all integer triples (a, b, c) such that N, defined as [(a - b)(b - c)(c - a)/2] + 2, is a power of 2016. A power of 2016 means it can be written as 2016^n where n is a non-negative integer. Let me break this down step by step.
First, let me understand the structure of N. It's given by [... |
Problem 3. Find all the triples of integers $(a, b, c)$ such that the number
$$
N=\frac{(a-b)(b-c)(c-a)}{2}+2
$$
is a power of 2016 .
(A power of 2016 is an integer of the form $2016^{n}$, where $n$ is a non-negative integer.)
|
Solution. Let $a, b, c$ be integers and $n$ be a positive integer such that
$$
(a-b)(b-c)(c-a)+4=2 \cdot 2016^{n}
$$
We set $a-b=-x, b-c=-y$ and we rewrite the equation as
$$
x y(x+y)+4=2 \cdot 2016^{n}
$$
If $n>0$, then the right hand side is divisible by 7 , so we have that
$$
x y(x+y)+4 \equiv 0 \quad(\bmod 7)... | (,b,)=(k+2,k+1,k),k\in\mathbb{Z} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2 | [
false,
false
] | 0 | <think>
Okay, let's see. The problem is asking for all integer triples (a, b, c) such that the number N, defined as [(a - b)(b - c)(c - a)]/2 + 2, is a power of 2016. A power of 2016 means it's like 2016^n where n is a non-negative integer. So, we need to find all triples (a, b, c) where this expression equals 2016^n f... |
Problem 2. Let $n$ three-digit numbers satisfy the following properties:
(1) No number contains the digit 0 .
(2) The sum of the digits of each number is 9 .
(3) The units digits of any two numbers are different.
(4) The tens digits of any two numbers are different.
(5) The hundreds digits of any two numbers are ... |
Solution. Let $S$ denote the set of three-digit numbers that have digit sum equal to 9 and no digit equal to 0 . We will first find the cardinality of $S$. We start from the number 111 and each element of $S$ can be obtained from 111 by a string of 6 A's (which means that we add 1 to the current digit) and $2 G$ 's (w... | 5 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2 | [
true,
true
] | 2 | <think>
Okay, so I need to find the largest possible number of three-digit numbers, let's call this number n, that satisfy all the given conditions. Let me read the problem again and make sure I understand each part.
First, the numbers are three-digit, and they have to meet five conditions. Let me list them again:
1.... |
"\nProblem 2. Let $n$ three-digit numbers satisfy the following properties:\n\n(1) No number contain(...TRUNCATED) | "\nSolution. Let $S$ denote the set of three-digit numbers that have digit sum equal to 9 and no dig(...TRUNCATED) | 5 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2 | [
true,
true
] | 2 | "<think>\nOkay, let's see. I need to find the largest possible value of n, which is the number of th(...TRUNCATED) |
"# Problem 1 \n\nLet $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that\n\n$$\n(...TRUNCATED) | "## Solution\n\nReplacing $1-a, 1-b, 1-c$ with $b+c, c+a, a+b$ respectively on the right hand side, (...TRUNCATED) | =b==\frac{1}{3} | Inequalities | proof | Yes | Yes | olympiads | false | 2 | [
false,
false
] | 0 | "<think>\nAlright, let me try to tackle this problem. So, we have three positive real numbers a, b, (...TRUNCATED) |
"# Problem 1 \n\nLet $a, b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove that\n\n$$\n(...TRUNCATED) | "## Solution\n\nReplacing $1-a, 1-b, 1-c$ with $b+c, c+a, a+b$ respectively on the right hand side, (...TRUNCATED) | =b==\frac{1}{3} | Inequalities | proof | Yes | Yes | olympiads | false | 2 | [
false,
false
] | 0 | "<think>\nOkay, so I need to prove this inequality:\n\n$$\n\\frac{a}{b}+\\frac{b}{a}+\\frac{b}{c}+\\(...TRUNCATED) |
"## Problem 2\n\nLet the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, a(...TRUNCATED) | "## Solution 1\n\nLet $P$ be the symmetric of $A$ with respect to $M$ (Figure 1). Then $A M=M P$ and(...TRUNCATED) | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 6 | [
true,
true,
true,
true,
true,
true
] | 6 | "<think>\nOkay, let me try to figure out this geometry problem. So, we have two circles, k₁ and k(...TRUNCATED) |
"## Problem 2\n\nLet the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, a(...TRUNCATED) | "## Solution 1\n\nLet $P$ be the symmetric of $A$ with respect to $M$ (Figure 1). Then $A M=M P$ and(...TRUNCATED) | 45 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 6 | [
true,
true,
true,
true,
true,
true
] | 6 | "<think>\nOkay, so I need to find the measure of angle NMB given the conditions in the problem. Let (...TRUNCATED) |
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