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p1. The sides of a triangle $ABC$ have lengths $AB = 26$ cm, $BC = 17$ cm, and $CA = 19$ cm. The bisectors of the angles $B$ and $C$ intersect at the point $I$. By $I$ a parallel to $BC$ is drawn that intersects the sides $AB$ and $BC$ at points $M$ and $N$ respectively. Calculate the perimeter of the triangle $AMN$.
... | 1. Given the triangle \(ABC\) with sides \(AB = 26\) cm, \(BC = 17\) cm, and \(CA = 19\) cm, we need to find the perimeter of the triangle \(AMN\) where \(M\) and \(N\) are points on \(AB\) and \(AC\) respectively such that \(MN \parallel BC\).
2. Since \(MN \parallel BC\), triangles \(BMN\) and \(BIC\) are similar. T... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The escalator of the department store, which at any given time can be seen at $75$ steps section, moves up one step in $2$ seconds. At time $0$, Juku is standing on an escalator step equidistant from each end, facing the direction of travel. He goes by a certain rule: one step forward, two steps back, then again one st... | 1. **Initial Setup**:
- The escalator has 75 steps.
- The escalator moves up one step every 2 seconds.
- Juku starts at the middle step, which is the 38th step (since \( \frac{75+1}{2} = 38 \)).
- Juku's movement pattern is: one step forward, two steps back, taking one step per second.
2. **Juku's Movemen... | 23 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
In a school tennis tournament with $ m \ge 2$ participants, each match consists of 4 sets. A player who wins more than half of all sets during a match gets 2 points for this match. A player who wins exactly half of all sets during the match gets 1 point, and a player who wins less than half of all sets gets 0 points. D... | To solve this problem, we need to find the minimum number of participants \( m \) such that one participant wins more sets than any other participant but obtains fewer points than any other participant. Let's break down the solution step by step.
1. **Define the Problem:**
- Each match consists of 4 sets.
- A pl... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $1,000,000$ piles of $1996$ coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as des... | 1. **Understanding the Problem:**
- We have $1,000,000$ piles of coins.
- Each pile contains $1996$ coins.
- One pile contains counterfeit coins, each weighing $9$ grams.
- All other piles contain real coins, each weighing $10$ grams.
- We need to determine the minimum number of weighings required to ide... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For real numbers $x, y$ and $z$ it is known that $x + y = 2$ and $xy = z^2 + 1$.
Find the value of the expression $x^2 + y^2+ z^2$. | 1. Given the equations:
\[
x + y = 2
\]
\[
xy = z^2 + 1
\]
2. Consider the quadratic equation \( t^2 - (x+y)t + xy = 0 \) with roots \( x \) and \( y \). By Vieta's formulas, we know:
\[
x + y = 2 \quad \text{(sum of the roots)}
\]
\[
xy = z^2 + 1 \quad \text{(product of the roots)}
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$. | 1. **Identify the nature of the function:**
The given function is \( f(x) = |x - 1| + |x - 2| + \cdots + |x - 2001| \). This function is a sum of absolute values, which is piecewise linear and changes slope at each of the points \( x = 1, 2, \ldots, 2001 \).
2. **Determine the median:**
The function \( f(x) \) a... | 1001000 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the least number of squares in a f... | To solve this problem, we need to determine the minimum number of ships in a fleet on a \(10 \times 10\) grid such that no new ship can be added. We will use the given formulas and reasoning to derive the solution.
1. **Understanding the Problem:**
- A ship is a figure made up of unit squares connected by common ed... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a rectangular grid of $ 10 \times 10$ unit squares. We call a [i]ship[/i] a figure made up of unit squares connected by common edges. We call a [i]fleet[/i] a set of ships where no two ships contain squares that share a common vertex (i.e. all ships are vertex-disjoint). Find the greatest natural number that, ... | 1. **Understanding the Problem:**
We need to find the greatest natural number \( n \) such that for any partition of \( n \) into positive integers, there exists a fleet of ships on a \( 10 \times 10 \) grid where the summands are exactly the numbers of squares contained in individual ships. The ships must be vertex... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$ | To find all prime numbers \( p \) for which one can find a positive integer \( m \) and nonnegative integers \( a_0, a_1, \ldots, a_m \) less than \( p \) such that
\[
\begin{cases}
a_0 + a_1 p + \cdots + a_{m-1} p^{m-1} + a_m p^m = 2013 \\
a_0 + a_1 + \cdots + a_{m-1} + a_m = 11
\end{cases}
\]
we proceed as follows... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$.
Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.
| 1. We need to find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial $P(x)$ such that all values at integer places are divisible by $n$. A simple polynomial has coefficients in $\{-1, 0, 1\}$.
2. We start by noting that a polynomial with only one non-zero term canno... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given a circle $\omega$ with radius $1$. Let $T$ be a set of triangles good, if the following conditions apply:
(a) the circumcircle of each triangle in the set $T$ is $\omega$;
(b) The interior of any two triangles in the set $T$ has no common point.
Find all positive real numbers $t$, for which for each positive inte... | 1. **Understanding the Problem:**
We are given a circle $\omega$ with radius $1$. We need to find all positive real numbers $t$ such that for any positive integer $n$, there exists a set of $n$ triangles, each having $\omega$ as its circumcircle, and no two triangles share any interior points. Additionally, the peri... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$12$ knights are sitting at a round table. Every knight is an enemy with two of the adjacent knights but with none of the others.
$5$ knights are to be chosen to save the princess, with no enemies in the group. How many ways are there for the choice? | 1. **Understanding the problem**: We have 12 knights sitting in a circle, each knight is an enemy to the two adjacent knights. We need to choose 5 knights such that no two chosen knights are enemies.
2. **Initial approach**: To ensure no two chosen knights are enemies, we need to leave at least one knight between any ... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $11$ members in the competetion committee. The problem set is kept in a safe having several locks.
The committee members have been provided with keys in such a way that every six members can open the safe, but no fi ve members can do that.
What is the smallest possible number of locks, and how many keys are ... | 1. **Define the problem and notation:**
- Let \( C = \{p_1, p_2, \dots, p_{11}\} \) be the set of 11 committee members.
- We need to determine the smallest number of locks \( n \) such that:
- Any subset of 6 members can open the safe.
- No subset of 5 members can open the safe.
2. **Determine the numb... | 2772 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There is a regular $17$-gon $\mathcal{P}$ and its circumcircle $\mathcal{Y}$ on the plane.
The vertices of $\mathcal{P}$ are coloured in such a way that $A,B \in \mathcal{P}$ are of diff erent colour, if the shorter arc connecting $A$ and $B$ on $\mathcal{Y}$ has $2^k+1$ vertices, for some $k \in \mathbb{N},$ includin... | 1. **Identify the relevant values of \(2^k + 1\):**
Since the minor arc \(AB\) has at most 9 vertices on it, the relevant values of \(2^k + 1\) are \(3, 5,\) and \(9\). This is because:
\[
2^1 + 1 = 3, \quad 2^2 + 1 = 5, \quad 2^3 + 1 = 9
\]
2. **Enumerate the vertices:**
Enumerate the vertices of the 1... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Finland is going to change the monetary system again and replace the Euro by the Finnish Mark.
The Mark is divided into $100$ pennies.
There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able
to pay for any purchase less than one mark should be minimal.
Dete... | 1. To be able to pay one penny, one of the denominations must be \(1\). Thus, we let the three denominations' values be \(1, a, b\), and without loss of generality (WLOG), \(1 < a < b\).
2. Consider a value \(d\) to be paid. For any \(d < a\), we must pay in only \(1\)'s, so we carry at most \(a-1\) coins of denominat... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer.
Show that permuting the order of digits one can obtain an integer divisible by $7.$ | To show that permuting the digits $1, 3, 7, 9$ can yield an integer divisible by $7$, we need to find a permutation of these digits such that the resulting number is congruent to $0 \pmod{7}$.
1. **List all permutations and their remainders modulo $7$:**
- $7931 \equiv 0 \pmod{7}$
- $3179 \equiv 1 \pmod{7}$
... | 7931 | Number Theory | proof | Yes | Yes | aops_forum | false |
Determine the number of real roots of the equation \[x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x +\frac{5}{2}= 0.\] | 1. **Rewrite the polynomial:**
The given polynomial is:
\[
P(x) = x^8 - x^7 + 2x^6 - 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}
\]
We can factor out \((x-1)\) from each term except the constant term:
\[
P(x) = x^7(x-1) + 2x^5(x-1) + 3x^3(x-1) + 4x(x-1) + \frac{5}{2}
\]
2. **Consider the inter... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Eight football teams play matches against each other in such a way that no two teams meet twice and no three teams play all of the three possible matches.
What is the largest possible number of matches? | 1. **Understanding the Problem:**
- We have 8 football teams.
- Each team plays against every other team exactly once.
- No three teams play all three possible matches among themselves.
2. **Graph Theory Representation:**
- Represent each team as a vertex in a graph.
- Each match between two teams is re... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$. | 1. Let \( n = 1999 \). We need to find the smallest number of pawns such that for each empty square, the total number of pawns in the row or column of that square is at least \( 1999 \).
2. Define \( U \) as the number of unmarked (empty) squares. Let \( r_i \) and \( c_i \) be the number of unmarked squares in row \(... | 1998001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:
- Any 3 participants speak a common language.
- No language is spoken more that by the half of the participants.
What is the least value of $n$? | To find the least value of \( n \) such that any 3 participants speak a common language and no language is spoken by more than half of the participants, we can proceed as follows:
1. **Define the problem constraints:**
- There are \( n \) participants.
- There are 14 languages.
- Any 3 participants speak a co... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $u_n$ be defined by $u_0=0$ and $u_{2n}=u_n$, $u_{2n+1}=1-u_n$ for each $n\in\mathbb N_0$.
(a) Calculate $u_{1990}$.
(b) Find the number of indices $n\le1990$ for which $u_n=0$.
(c) Let $p$ be a natural number and $N=(2^p-1)^2$. Find $u_N$. | ### Part (a)
1. **Understanding the sequence definition:**
- Given: \( u_0 = 0 \)
- For even \( n \): \( u_{2n} = u_n \)
- For odd \( n \): \( u_{2n+1} = 1 - u_n \)
2. **Binary representation insight:**
- The sequence \( u_n \) depends on the binary representation of \( n \).
- \( u_n = 0 \) if \( n \) ... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
For each $n\in\mathbb N$, the function $f_n$ is defined on real numbers $x\ge n$ by
$$f_n(x)=\sqrt{x-n}+\sqrt{x-n+1}+\ldots+\sqrt{x+n}-(2n+1)\sqrt x.$$(a) If $n$ is fixed, prove that $\lim_{x\to+\infty}f_n(x)=0$.
(b) Find the limit of $f_n(n)$ as $n\to+\infty$. | (a) To prove that $\lim_{x \to +\infty} f_n(x) = 0$ for a fixed $n$, we start by rewriting the function $f_n(x)$:
\[ f_n(x) = \sum_{k=-n}^{n} \sqrt{x+k} - (2n+1)\sqrt{x} \]
We can rewrite each term inside the summation as follows:
\[ \sqrt{x+k} = \sqrt{x} \sqrt{1 + \frac{k}{x}} \]
Using the first-order Taylor expan... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
(a) For given complex numbers $a_1,a_2,a_3,a_4$, we define a function $P:\mathbb C\to\mathbb C$ by $P(z)=z^5+a_4z^4+a_3z^3+a_2z^2+a_1z$. Let $w_k=e^{2ki\pi/5}$, where $k=0,\ldots,4$. Prove that
$$P(w_0)+P(w_1)+P(w_2)+P(w_3)+P(w_4)=5.$$(b) Let $A_1,A_2,A_3,A_4,A_5$ be five points in the plane. A pentagon is inscribed in... | ### Part (a)
1. **Define the function and roots of unity:**
Given the function \( P(z) = z^5 + a_4z^4 + a_3z^3 + a_2z^2 + a_1z \), we need to evaluate the sum \( P(w_0) + P(w_1) + P(w_2) + P(w_3) + P(w_4) \), where \( w_k = e^{2ki\pi/5} \) for \( k = 0, 1, 2, 3, 4 \). These \( w_k \) are the 5th roots of unity.
2. ... | 5 | Other | proof | Yes | Yes | aops_forum | false |
Given $u_0,u_1$ with $0<u_0,u_1<1$, define the sequence $(u_n)$ recurrently by the formula
$$u_{n+2}=\frac12\left(\sqrt{u_{n+1}}+\sqrt{u_n}\right).$$(a) Prove that the sequence $u_n$ is convergent and find its limit.
(b) Prove that, starting from some index $n_0$, the sequence $u_n$ is monotonous. | (a) To prove that the sequence \( u_n \) is convergent and find its limit, we will follow these steps:
1. **Boundedness and Monotonicity**:
- First, we show that the sequence \( u_n \) is bounded above by 1. We use induction to prove this.
- Base case: \( u_0 < 1 \) and \( u_1 < 1 \) (given).
- Inductive step... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB,AC$ and $AD$ are pairwise orthogonal, $AB=3$ and $CD=\sqrt2$. Find the minimum possible value of
$$BC^6+BD^6-AC^6-AD^6.$$ | 1. Denote \( x, y, z \) as the lengths \( AB, AC, AD \) respectively. We are given that:
\[
x = 3
\]
and since \( AB, AC, \) and \( AD \) are pairwise orthogonal, we can use the Pythagorean theorem in three dimensions to find:
\[
y^2 + z^2 = CD^2 = 2
\]
2. We want to find the minimum value of:
... | 1998 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell. Determine the number ... | 1. **Determine the coordinates of the relevant cells:**
- The bottom left cell (starting point) is at \((0,0)\).
- The upper right cell (ending point) is at \((6,6)\).
- The center cell (cheese) is at \((3,3)\).
2. **Calculate the number of ways to reach the center cell \((3,3)\) from the starting point \((0,... | 400 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule:
1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students;
2) Each student received the maximum possible points in each problem or got $... | 1. We start by noting that there are 30 students and each student is given 8 problems to solve. The scoring rule is such that if a problem is solved by exactly \( k \) students, each of those \( k \) students receives \( 30 - k \) points for that problem. If a problem is solved by any number of students other than \( k... | 60 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$512$ persons meet at a meeting[ Under every six of these people there is always at least two who know each other. Prove that there must be six people at this gathering, all mutual know. | To prove that there must be six people at this gathering who all mutually know each other, we can use the concept of Ramsey numbers. Specifically, we need to show that $R(5, 6) \leq 512$.
1. **Understanding Ramsey Numbers**:
- The Ramsey number $R(r, s)$ is the smallest number such that any graph of $R(r, s)$ verti... | 126 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer and $M=\{1,2,\ldots, n\}.$ A subset $T\subset M$ is called [i]heavy[/i] if each of its elements is greater or equal than $|T|.$ Let $f(n)$ denote the number of heavy subsets of $M.$ Describe a method for finding $f(n)$ and use it to calculate $f(32).$ | 1. **Define the problem and notation:**
Let \( n \) be a positive integer and \( M = \{1, 2, \ldots, n\} \). A subset \( T \subset M \) is called *heavy* if each of its elements is greater than or equal to \( |T| \). Let \( f(n) \) denote the number of heavy subsets of \( M \).
2. **Determine the number of heavy su... | 3524578 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A finite set $\{a_1, a_2, ... a_k\}$ of positive integers with $a_1 < a_2 < a_3 < ... < a_k$ is named [i]alternating [/i] if $i+a$ for $i = 1, 2, 3, ..., k$ is even. The empty set is also considered to be alternating. The number of alternating subsets of $\{1, 2, 3,..., n\}$ is denoted by $A(n)$.
Develop a method to d... | To determine \( A(n) \), we need to understand the properties of alternating subsets. An alternating subset of \(\{1, 2, 3, \ldots, n\}\) is defined such that the sum of the index and the element is even for all elements in the subset.
1. **Base Cases:**
- For \( n = 0 \), the only subset is the empty set, which is... | 5702887 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Below the standard representation of a positive integer $n$ is the representation understood by $n$ in the decimal system, where the first digit is different from $0$. Everyone positive integer n is now assigned a number $f(n)$ by using the standard representation of $n$ last digit is placed before the first.
Examples:... | To solve the problem, we need to find the smallest positive integer \( n \) such that \( f(n) = 2n \). The function \( f(n) \) is defined as the number obtained by moving the last digit of \( n \) to the front.
1. **Representation of \( n \)**:
Let \( n \) be a positive integer with \( \ell+1 \) digits. We can writ... | 105263157894736842 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a finite subset of the plane such that for any two different points $A,B\in M$ there is a point $C\in M$ such that $ABC$ is equilateral. What is the maximal number of points in $M?$ | 1. **Claim**: The maximum number of points in \( M \) is 3. We will show that any set \( M \) with more than 3 points cannot satisfy the given conditions.
2. **Base Case**: Consider an equilateral triangle with vertices \( A, B, \) and \( C \). Clearly, \( |M| \geq 3 \) since \( A, B, \) and \( C \) form an equilatera... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $10000$ trees in a park, arranged in a square grid with $100$ rows and $100$ columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.
| 1. **Prove that we cannot cut more than 2500 trees:**
- Consider the square grid of trees with coordinates ranging from \((0,0)\) to \((99,99)\).
- Divide this grid into \(2500\) smaller squares, each of size \(2 \times 2\). Each smaller square contains four points: \((2i, 2j)\), \((2i+1, 2j)\), \((2i, 2j+1)\), ... | 2500 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We consider the sequences strictely increasing $(a_0,a_1,...)$ of naturals which have the following property :
For every natural $n$, there is exactly one representation of $n$ as $a_i+2a_j+4a_k$, where $i,j,k$ can be equal.
Prove that there is exactly a such sequence and find $a_{2002}$ | 1. **Uniqueness of the Sequence**:
- We need to show that there can be at most one such sequence $\left(a_n\right)_n$.
- Since the sequence is strictly increasing and starts with non-negative integers, the smallest element must be $a_0 = 0$. If $a_0$ were not $0$, then $0$ could not be represented as $a_i + 2a_j ... | 1227132168 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A piece of paper with the shape of a square lies on the desk. It gets dissected step by step into smaller pieces: in every step, one piece is taken from the desk and cut into two pieces by a straight cut; these pieces are put back on the desk then.
Find the smallest number of cuts needed to get $100$ $20$-gons. | 1. **Understanding the Problem:**
We start with a square piece of paper and need to make cuts to eventually obtain 100 polygons, each with 20 sides (20-gons). Each cut increases the total number of sides by at most 4.
2. **Initial Considerations:**
- A square has 4 sides.
- Each cut can increase the number of... | 1699 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Fedja used matches to put down the equally long sides of a parallelogram whose vertices are not on a common line. He figures out that exactly 7 or 9 matches, respectively, fit into the diagonals. How many matches compose the parallelogram's perimeter? | 1. Let \( a \) and \( b \) be the lengths of the diagonals of the parallelogram, with \( a = 7 \) and \( b = 9 \). Let \( c \) and \( d \) be the lengths of the sides of the parallelogram.
2. It is well known that for a parallelogram, the sum of the squares of the sides is equal to the sum of the squares of the diagona... | 22 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine all real $ x$ satisfying the equation \[ \sqrt[5]{x^3 \plus{} 2x} \equal{} \sqrt[3]{x^5\minus{}2x}.\] Odd roots for negative radicands shall be included in the discussion. | To determine all real \( x \) satisfying the equation
\[ \sqrt[5]{x^3 + 2x} = \sqrt[3]{x^5 - 2x}, \]
we will analyze the equation step by step.
1. **Identify obvious solutions:**
We start by checking some simple values of \( x \):
- For \( x = 0 \):
\[
\sqrt[5]{0^3 + 2 \cdot 0} = \sqrt[5]{0} = 0,
... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand. | To solve this problem, we need to find all numbers that can be expressed in exactly 2010 different ways as the sum of powers of two with non-negative exponents, where each power can appear as a summand at most three times.
1. **Understanding the Problem:**
- We need to express a number \( n \) as a sum of powers of... | 2010 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression | 1. **Substitution**: Let \( u = x^2 \). The given equation \( x^4 - 40x^2 + q = 0 \) transforms into:
\[
u^2 - 40u + q = 0
\]
where \( u \geq 0 \).
2. **Vieta's Formulas**: For the quadratic equation \( u^2 - 40u + q = 0 \), let the roots be \( u_1 \) and \( u_2 \). By Vieta's formulas, we have:
\[
u... | 144 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other. | 1. **Partitioning the Rectangle:**
- The rectangle has dimensions \(14 \times 28\).
- We partition the rectangle into \(8\) squares, each of dimensions \(7 \times 7\).
- If we place more than \(8\) points in the rectangle, by the pigeonhole principle, at least two points will lie in the same \(7 \times 7\) squ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of all numbers which are represented as $x^2+y^2$ with $x, y \in \{1, 2, 3, \ldots, 1000\}$ and which are divisible by 121. | 1. **Identify the quadratic residues modulo 11:**
The quadratic residues modulo 11 are the possible remainders when squares of integers are divided by 11. These residues are:
\[
\{0, 1, 4, 9, 5, 3\}
\]
This can be verified by squaring each integer from 0 to 10 and taking the result modulo 11.
2. **Deter... | 8100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining $\frac{1}{2}$ points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players wit... | 1. **Define the Problem and Variables:**
- There are 14 players, each playing against every other player.
- Points: Win = 1, Loss = 0, Tie = $\frac{1}{2}$.
- The best three players (set \(A\)) have the same total points as the last nine players (set \(C\)).
- The number of ties is maximal.
2. **Total Numbe... | 29 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is called [i]nice[/i] if the sum of its digits in the number system with base $ 3$ is divisible by $ 3$.
Calculate the sum of the first $ 2005$ nice positive integers. | To solve the problem, we need to calculate the sum of the first 2005 nice positive integers. A positive integer is called *nice* if the sum of its digits in the number system with base 3 is divisible by 3.
1. **Grouping the Numbers:**
We group the positive integers into sets:
\[
A_1 = \{0, 1, 2, 3, 4, 5, 6, 7... | 6035050 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ with the following property:
For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that
\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\] | To find the smallest positive integer \( n \) such that for any integer \( m \) with \( 0 < m < 2004 \), there exists an integer \( k \) such that
\[
\frac{m}{2004} < \frac{k}{n} < \frac{m+1}{2005},
\]
we can use the following lemma:
**Lemma:**
Let \( a, b, c, d, p, q \) be positive integers such that \( \frac{a}{b}... | 4009 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the sum of absolute values for the complex roots of $ 20 x^8 \plus{} 7i x^7 \minus{}7ix \plus{} 20.$ | 1. Consider the polynomial equation:
\[
20x^8 + 7ix^7 - 7ix + 20 = 0
\]
2. Substitute \( y = ix \):
\[
20(ix)^8 + 7i(ix)^7 - 7i(ix) + 20 = 0
\]
Simplifying each term:
\[
20(i^8 x^8) + 7i(i^7 x^7) - 7i^2 x + 20 = 0
\]
Since \( i^2 = -1 \) and \( i^4 = 1 \):
\[
20(x^8) + 7(-x^7) + ... | 8 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of requi... | To solve this problem, we need to ensure that every possible rectangle within the $10 \times 10$ grid contains at least one strawberry. We will use a combinatorial approach to determine the minimum number of strawberries required.
1. **Understanding the Problem:**
- We have a $10 \times 10$ grid, which means there ... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and
\[{ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.
\]
Determine $ S_{1024}.$ | To determine \( S_{1024} \), we need to understand the recursive definition of the sequence \( (S_n) \). The sequence is defined as follows:
\[ S_1 = \{1\}, \]
\[ S_2 = \{2\}, \]
\[ S_{n+1} = \{ k \in \mathbb{N} \mid (k-1 \in S_n) \text{ XOR } (k \in S_{n-1}) \}. \]
We will use induction to find a pattern in the sets ... | 1024 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the least integer $k$ such that for any $2011 \times 2011$ table filled with integers Kain chooses, Abel be able to change at most $k$ cells to achieve a new table in which $4022$ sums of rows and columns are pairwise different. | 1. **Restate the problem with general \( n \)**:
We need to find the least integer \( k \) such that for any \( n \times n \) table filled with integers, Abel can change at most \( k \) cells to achieve a new table in which the \( 2n \) sums of rows and columns are pairwise different. Here, \( n = 2011 \).
2. **Con... | 2681 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$. | 1. Let \( O \) be the center of the two concentric circles. The radius of the larger circle \( \Omega \) is 13, so \( OB = 13 \). The radius of the smaller circle \( \omega \) is 8, so \( OD = 8 \).
2. Since \( AB \) is a diameter of \( \Omega \), \( A \) and \( B \) lie on \( \Omega \) and \( O \) is the midpoint of ... | 19 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find all integers $n$ for which both $4n + 1$ and $9n + 1$ are perfect squares. | 1. We start with the given conditions that both \(4n + 1\) and \(9n + 1\) are perfect squares. Let:
\[
4n + 1 = a^2 \quad \text{and} \quad 9n + 1 = b^2
\]
for some integers \(a\) and \(b\).
2. From the equation \(4n + 1 = a^2\), we can solve for \(n\):
\[
4n + 1 = a^2 \implies 4n = a^2 - 1 \implies n... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Today there are $2^n$ species on the planet Kerbin, all of which are exactly n steps from an original species. In an evolutionary step, One species split into exactly two new species and died out in the process. There were already $2^n-1$ species in the past, which are no longer present today can be found, but are only... | To determine the biodiversity of the planet Kerbin, we need to calculate the average degree of relationship between all pairs of species, including both the currently existing species and the extinct ones.
1. **Understanding the Evolutionary Tree:**
- The original species is at the root of the tree.
- Each evol... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If point $M(x,y)$ lies on the line with equation $y=x+2$ and $1<y<3$, calculate the value of
$A=\sqrt{y^2-8x}+\sqrt{y^2+2x+5}$ | 1. Given the point \( M(x, y) \) lies on the line with equation \( y = x + 2 \) and \( 1 < y < 3 \), we can substitute \( y = x + 2 \) into the inequality \( 1 < y < 3 \):
\[
1 < x + 2 < 3
\]
Subtracting 2 from all parts of the inequality:
\[
-1 < x < 1
\]
2. We need to calculate the value of \( A... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If in a $3$-digit number we replace with each other it's last two digits, and add the resulting number to the starting one, we find sum a $4$-digit number that starts with $173$. Which is the starting number? | 1. Let the original 3-digit number be represented as $\overline{abc}$, where $a$, $b$, and $c$ are its digits. This can be expressed as $100a + 10b + c$.
2. When we swap the last two digits, the number becomes $\overline{acb}$, which can be expressed as $100a + 10c + b$.
3. According to the problem, the sum of these tw... | 1732 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Solve the equation
$(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$ | 1. Consider the given equation:
\[
(x^2 + 2x + 1)^2 + (x^2 + 3x + 2)^2 + (x^2 + 4x + 3)^2 + \cdots + (x^2 + 1996x + 1995)^2 = 0
\]
2. Notice that each term in the sum is a square of a quadratic polynomial. Let's rewrite each term:
\[
(x^2 + kx + (k-1))^2 \quad \text{for} \quad k = 2, 3, \ldots, 1996
... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
On a past Mathematical Olympiad the maximum possible score on a problem was 5. The average score of boys was 4, the average score of girls was 3.25, and the overall average score was 3.60. Find the total number of participants, knowing that it was in the range from 31 to 50. | 1. Let \( b \) be the number of boys and \( g \) be the number of girls.
2. The average score of boys is 4, so the total score of boys is \( 4b \).
3. The average score of girls is 3.25, so the total score of girls is \( 3.25g \).
4. The overall average score is 3.60, so the total score of all participants is \( 3.6(b ... | 45 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find all four-digit natural numbers $\overline{xyzw}$ with the property that their sum plus the sum of their digits equals $2003$. | To find all four-digit natural numbers $\overline{xyzw}$ such that the sum of the number and the sum of its digits equals $2003$, we can follow these steps:
1. Let $\overline{xyzw}$ be a four-digit number, where $x, y, z, w$ are its digits. The number can be expressed as:
\[
1000x + 100y + 10z + w
\]
The s... | 1978 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
If $n$ is is an integer such that $4n+3$ is divisible by $11,$ find the form of $n$ and the remainder of $n^{4}$ upon division by $11$. | 1. **Determine the form of \( n \):**
Given that \( 4n + 3 \) is divisible by \( 11 \), we can write this as:
\[
4n + 3 \equiv 0 \pmod{11}
\]
Subtracting 3 from both sides, we get:
\[
4n \equiv -3 \pmod{11}
\]
Since \(-3 \equiv 8 \pmod{11}\), we can rewrite the equation as:
\[
4n \equiv... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest value of positive integer $ x$ , such that the number
$ A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700}$
is a perfect square . | 1. First, we rewrite the given expression \( A = 2^{182} + 4^x + 8^{700} \) in terms of powers of 2:
\[
A = 2^{182} + 2^{2x} + 2^{2100}
\]
Here, we used the fact that \( 4^x = (2^2)^x = 2^{2x} \) and \( 8^{700} = (2^3)^{700} = 2^{2100} \).
2. Next, we observe that \( 2^{2100} \) is the largest term among \... | 2008 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k,m \in N = \{0,1,2,...,\}$ | 1. **Identify the problem and the form of the numbers:**
We need to determine the number of positive integers that cannot be written in the form \(80k + 3m\), where \(k, m \in \mathbb{N} = \{0, 1, 2, \ldots\}\).
2. **Use the known result from the Frobenius coin problem:**
According to the Frobenius coin problem,... | 79 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit intege... | 1. **Expression for \( x - y \)**:
Given \( x = \overline{abcd} \) and \( y = \overline{dcba} \), we can express \( x \) and \( y \) in terms of their digits:
\[
x = 1000a + 100b + 10c + d
\]
\[
y = 1000d + 100c + 10b + a
\]
Therefore, the difference \( x - y \) is:
\[
x - y = (1000a + 100... | 8712 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We color the numbers $1, 2, 3,....,20$ with two colors white and black in such a way that both colors are used. Find the number of ways, we can perform this coloring if the product of white numbers and the product of black numbers have greatest common divisor equal to $1$. | 1. **Identify the prime factors of the numbers from 1 to 20:**
- The prime numbers between 1 and 20 are: 2, 3, 5, 7, 11, 13, 17, 19.
- The composite numbers between 1 and 20 are: 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20.
2. **Determine the condition for the greatest common divisor (GCD) to be 1:**
- For the GCD... | 29 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number ot 6-tuples $(x_1, x_2,...,x_6)$, where $x_i=0,1 or 2$ and $x_1+x_2+...+x_6$ is even | To find the number of 6-tuples \((x_1, x_2, \ldots, x_6)\) where \(x_i = 0, 1, \text{ or } 2\) and the sum \(x_1 + x_2 + \cdots + x_6\) is even, we need to consider the parity of the sum. Specifically, the sum will be even if and only if there is an even number of 1s in the tuple.
1. **Case 1: 0 ones**
- If there a... | 365 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A group of $n$ people play a board game with the following rules:
1) In each round of the game exactly $3$ people play
2) The game ends after exactly $n$ rounds
3) Every pair of players has played together at least at one round
Find the largest possible value of $n$ | 1. Let's denote the group of \( n \) people as \( P_1, P_2, \ldots, P_n \).
2. Each round involves exactly 3 players, and there are \( n \) rounds in total.
3. We need to ensure that every pair of players has played together at least once.
To find the largest possible value of \( n \), we need to consider the combinat... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A $8\times 8$ board is given. Seven out of $64$ unit squares are painted black. Suppose that there exists a positive $k$ such that no matter which squares are black, there exists a rectangle (with sides parallel to the sides of the board) with area $k$ containing no black squares. Find the maximum value of $k$.
| 1. **Understanding the Problem:**
We are given an $8 \times 8$ board with 7 out of the 64 unit squares painted black. We need to find the maximum area $k$ of a rectangle (with sides parallel to the sides of the board) that contains no black squares, regardless of the placement of the black squares.
2. **Initial Con... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$
If no element of $M$ is equal to the sum of any two elements of $M,$
find the least value that the greatest element of $M$ can take. | 1. Consider the set \( M = \{1, 3, 5, \ldots, 4007\} \). This set contains 2004 elements, and no element in this set is the sum of any two other elements in the set. This is because the sum of any two odd numbers is even, and all elements in the set are odd.
2. We need to show that the greatest element of \( M \) cann... | 4007 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many 5 digit positive integers are there such that each of its digits, except for the last one, is greater than or equal to the next digit? | 1. We need to find the number of 5-digit positive integers such that each of its digits, except for the last one, is greater than or equal to the next digit. Let the 5-digit number be represented as \(abcde\), where \(a, b, c, d, e\) are its digits.
2. The condition given is that each digit, except for the last one, i... | 715 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\textbf{Problem 1.}$ [b][/b]There are less than $400$ marbles.[i][/i] If they are distributed among $3$ childrens, there is one left over if they are distributed among $7$ children, there are 2 left over. Finally if they are distributed among $5$ children, there are none left over.
What is the largest number of the m... | 1. Let \( n \) be the number of marbles. According to the problem, we have the following congruences:
\[
n \equiv 1 \pmod{3}
\]
\[
n \equiv 2 \pmod{7}
\]
\[
n \equiv 0 \pmod{5}
\]
2. We need to solve this system of congruences using the Chinese Remainder Theorem (CRT). First, we solve the co... | 310 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$\textbf{Problem 5.}$ Miguel has two clocks, one clock advances $1$ minute per day and the other one goes $15/10$ minutes per day.
If you put them at the same correct time, What is the least number of days that must pass for both to give the correct time simultaneously? | 1. **Understanding the problem**: We have two clocks, one advances \(1\) minute per day and the other advances \(\frac{15}{10} = 1.5\) minutes per day. We need to find the least number of days after which both clocks will show the correct time simultaneously.
2. **Determine the period for each clock to show the correc... | 1440 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$\textbf{Problem 2.}$ The list of all the numbers of $5$ different numbers that are formed with the digits $1,2,3,4$ and $5$ is made.
In this list the numbers are ordered from least to greatest. Find the number that occupies the $100th$ position in the list. | 1. **Determine the total number of permutations:**
The problem involves forming numbers using the digits \(1, 2, 3, 4, 5\). Since all digits are used and each digit is unique, the total number of permutations is given by \(5!\):
\[
5! = 5 \times 4 \times 3 \times 2 \times 1 = 120
\]
Therefore, there are ... | 51342 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\textbf{Problem 1.}$ Alejandra is going to distribute candies to several children. He gives the first one a candy, the second gets two, the third gets twice as many candies as he gave the second and so on, if Alejandra has $2007$ candies, what is the minimum number of candies that is missing to be able to distribute t... | 1. **Determine the number of candies given to each child:**
- The first child receives \(1\) candy.
- The second child receives \(2\) candies.
- The third child receives \(2 \times 2 = 4\) candies.
- The fourth child receives \(2 \times 4 = 8\) candies.
- In general, the \(n\)-th child receives \(2^{(n-1... | 40 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\textbf{Problem 4.}$ What is the largest number such that dividing $17$ or $30$ by this number, the same remainder is obtained in both cases | 1. We need to find the largest number \( n \) such that when both 17 and 30 are divided by \( n \), the same remainder is obtained. This can be expressed as:
\[
17 \mod n = 30 \mod n
\]
2. This implies that the difference between 30 and 17 must be divisible by \( n \):
\[
30 - 17 \equiv 0 \mod n
\]
3.... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a circle, $15$ equally spaced points are drawn and arbitrary triangles are formed connecting $3$ of these points. How many non-congruent triangles can be drawn? | 1. **Understanding the Problem**: We need to find the number of non-congruent triangles that can be formed by connecting 3 out of 15 equally spaced points on a circle. Two triangles are congruent if they can be made to coincide by rotation or reflection.
2. **Counting Non-Congruent Triangles**: To count the non-congru... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$ | 1. **Transform the inequality**:
Given the inequality:
\[
(u^2 - 4vw)^2 > K(2v^2 - uw)(2w^2 - uv)
\]
we divide both sides by \(u^4\):
\[
\left(1 - 4\frac{v}{u}\frac{w}{u}\right)^2 > K \left(2\left(\frac{v}{u}\right)^2 - \frac{w}{u}\right) \left(2\left(\frac{w}{u}\right)^2 - \frac{v}{u}\right)
\... | 16 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let M be a subst of {1,2,...,2006} with the following property: For any three elements x,y and z (x<y<z) of M, x+y does not divide z. Determine the largest possible size of M. Justify your claim. | 1. **Understanding the Problem:**
We need to find the largest subset \( M \) of \(\{1, 2, \ldots, 2006\}\) such that for any three elements \( x, y, z \) in \( M \) with \( x < y < z \), the sum \( x + y \) does not divide \( z \).
2. **Rephrasing the Condition:**
The condition \( x + y \) does not divide \( z \... | 1004 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$. | 1. **Understanding the Problem:**
We have 2008 congruent circles on a plane, where no two circles are tangent to each other, and each circle intersects with at least three other circles. We need to determine the smallest possible value of \( N \), the total number of intersection points of these circles.
2. **Inter... | 3012 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that $ \{a_n\}$ is a sequence in which all the terms are integers, and $ a_2$ is odd. For any natural number $ n$, $ n(a_{n \plus{} 1} \minus{} a_n \plus{} 3) \equal{} a_{n \plus{} 1} \plus{} a_n \plus{} 3$. Furthermore, $ a_{2009}$ is divisible by $ 2010$. Find the smallest integer $ n > 1$ such that $ a_n$ is d... | 1. Given the recurrence relation for the sequence \( \{a_n\} \):
\[
n(a_{n+1} - a_n + 3) = a_{n+1} + a_n + 3
\]
We can rearrange and simplify this equation:
\[
n a_{n+1} - n a_n + 3n = a_{n+1} + a_n + 3
\]
\[
n a_{n+1} - a_{n+1} = n a_n + a_n + 3n - 3
\]
\[
(n-1) a_{n+1} = (n+1) a_n ... | 66 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Among the coordinates $(x,y)$ $(1\leq x,y\leq 101)$, choose some points such that there does not exist $4$ points which form a isoceles trapezium with its base parallel to either the $x$ or $y$ axis(including rectangles). Find the maximum number of coordinate points that can be chosen. | 1. **Define the problem and constraints:**
We need to choose points \((x, y)\) where \(1 \leq x, y \leq 101\) such that no four points form an isosceles trapezium with its base parallel to either the \(x\)-axis or \(y\)-axis. This includes avoiding rectangles.
2. **Identify dangerous rows:**
A row is considered ... | 201 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get $-1$ mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.) | To determine the number of different total marks a student can achieve on a multiple-choice test with 100 questions, where each correct answer gives 4 marks, each wrong answer deducts 1 mark, and unanswered questions give 0 marks, we need to consider all possible combinations of correct, wrong, and unanswered questions... | 501 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1,x_2,y_1,y_2$ be real numbers satisfying the equations $x^2_1+5x^2_2=10$, $x_2y_1-x_1y_2=5$, and $x_1y_1+5x_2y_2=\sqrt{105}$. Find the value of $y_1^2+5y_2^2$ | 1. Given the equations:
\[
x_1^2 + 5x_2^2 = 10
\]
\[
x_2y_1 - x_1y_2 = 5
\]
\[
x_1y_1 + 5x_2y_2 = \sqrt{105}
\]
2. We introduce complex numbers to simplify the problem. Let:
\[
x = x_1 + i\sqrt{5}x_2
\]
\[
y = y_1 - i\sqrt{5}y_2
\]
3. The first equation gives the magnitude... | 23 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A subset $M$ of $\{1, 2, . . . , 2006\}$ has the property that for any three elements $x, y, z$ of $M$ with $x < y < z$, $x+ y$ does not divide $z$. Determine the largest possible size of $M$. | 1. **Restate the problem with a stronger condition:**
Consider the stronger condition that there does not exist distinct elements \( x, y, z \in M \) such that \( x + y = z \). Suppose \( M = \{a_1, a_2, \ldots, a_k\} \) is a subset of such elements.
2. **Transform the set \( M \):**
Define \( M' = \{a_2 - a_1, ... | 1004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f: Z \to Z$ be such that $ f(1) \equal{} 1, f(2) \equal{} 20, f(\minus{}4) \equal{} \minus{}4$ and $ f(x\plus{}y) \equal{} f(x) \plus{}f(y)\plus{}axy(x\plus{}y)\plus{}bxy\plus{}c(x\plus{}y)\plus{}4 \forall x,y \in Z$, where $ a,b,c$ are constants.
(a) Find a formula for $ f(x)$, where $ x$ is any integer.
(b)... | Given the function \( f: \mathbb{Z} \to \mathbb{Z} \) with the properties:
\[ f(1) = 1, \quad f(2) = 20, \quad f(-4) = -4 \]
and the functional equation:
\[ f(x+y) = f(x) + f(y) + axy(x+y) + bxy + c(x+y) + 4 \quad \forall x, y \in \mathbb{Z} \]
We need to find a formula for \( f(x) \) and determine the greatest possib... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\] | 1. Consider the given expression:
\[
S = \sin\alpha_1\cos\alpha_2 + \sin\alpha_2\cos\alpha_3 + \cdots + \sin\alpha_{2007}\cos\alpha_{2008} + \sin\alpha_{2008}\cos\alpha_1
\]
2. We use the trigonometric identity for the product of sine and cosine:
\[
\sin x \cos y = \frac{1}{2} [\sin(x + y) + \sin(x - y)... | 1004 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of solutions to the following system of equations: \[ \begin{cases} a^2\plus{}bc\equiv a\pmod {37}\\ b(a\plus{}d)\equiv b\pmod {37}\\ c(a\plus{}d)\equiv c\pmod{37}\\ bc\plus{}d^2\equiv d\pmod{37}\\ ad\minus{}bc\equiv 1\pmod{37}\end{cases}\] | To find the total number of solutions to the given system of equations, we will analyze each equation step by step and use properties of modular arithmetic and field theory.
Given system of equations:
\[
\begin{cases}
a^2 + bc \equiv a \pmod{37} \\
b(a + d) \equiv b \pmod{37} \\
c(a + d) \equiv c \pmod{37} \\
bc + d^2... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \theta_1, \theta_2,\ldots , \theta_{2008}$ be real numbers. Find the maximum value of
$ \sin\theta_1\cos\theta_2 \plus{} \sin\theta_2\cos\theta_3 \plus{} \ldots \plus{} \sin\theta_{2007}\cos\theta_{2008} \plus{} \sin\theta_{2008}\cos\theta_1$ | To find the maximum value of the expression
\[
S = \sin\theta_1\cos\theta_2 + \sin\theta_2\cos\theta_3 + \ldots + \sin\theta_{2007}\cos\theta_{2008} + \sin\theta_{2008}\cos\theta_1,
\]
we can use the Cauchy-Schwarz inequality in the following form:
\[
\left( \sum_{i=1}^{n} a_i b_i \right)^2 \leq \left( \sum_{i=1}^{... | 1004 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Find the total number of solutions to the following system of equations:
$ \{\begin{array}{l} a^2 + bc\equiv a \pmod{37} \\
b(a + d)\equiv b \pmod{37} \\
c(a + d)\equiv c \pmod{37} \\
bc + d^2\equiv d \pmod{37} \\
ad - bc\equiv 1 \pmod{37} \end{array}$ | To find the total number of solutions to the given system of congruences, we will analyze each equation step-by-step.
Given system of equations:
\[
\begin{cases}
a^2 + bc \equiv a \pmod{37} \\
b(a + d) \equiv b \pmod{37} \\
c(a + d) \equiv c \pmod{37} \\
bc + d^2 \equiv d \pmod{37} \\
ad - bc \equiv 1 \pmod{37}
\end{c... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In one of the hotels of the wellness planet Oxys, there are $2019$ saunas. The managers have decided to accommodate $k$ couples for the upcoming long weekend. We know the following about the guests: if two women know each other then their husbands also know each other, and vice versa. There are several restrictions on ... | 1. **Understanding the Problem:**
- We have 2019 saunas.
- Each sauna can be used by either men only or women only.
- Each woman is only willing to share a sauna with women she knows.
- Each man is only willing to share a sauna with men he does not know.
- We need to find the maximum number of couples \(... | 2018 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We are given a map divided into $13\times 13$ fields. It is also known that at one of the fields a tank of the enemy is stationed, which we must destroy. To achieve this we need to hit it twice with shots aimed at the centre of some field. When the tank gets hit it gets moved to a neighbouring field out of precaution. ... | 1. **Initial Setup**: We have a $13 \times 13$ grid, which means there are $169$ fields in total. The tank is initially stationed in one of these fields.
2. **First Round of Shots**: We need to ensure that we hit the tank at least once. To do this, we can fire one shot at the center of each field. This requires $169$ ... | 254 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In duck language, only letters $q$, $a$, and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there?
In duck language, too, the letter $a$ is a vowel, while $q$ and... | To solve this problem, we need to count the number of valid four-letter words in duck language, where the letters are restricted to $q$, $a$, and $k$, and no two consonants can be adjacent. Here, $a$ is a vowel, and $q$ and $k$ are consonants.
Let's break down the problem step-by-step:
1. **Identify the possible posi... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three palaces, each rotating on a duck leg, make a full round in $30$, $50$, and $70$ days, respectively. Today, at noon, all three palaces face northwards. In how many days will they all face southwards? | 1. **Understanding the Problem:**
- Each palace makes a full rotation in 30, 50, and 70 days respectively.
- Today, all palaces face northwards.
- We need to find the smallest \( x \) such that all palaces face southwards.
2. **Determine the Southward Facing Days:**
- For the first palace (30 days cycle), ... | 525 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a grid rectangle of size $2010 \times 1340$. A grid point is called [i]fair [/i] if the $2$ axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the r... | 1. Let \( x \) and \( y \) be the side lengths of the top right rectangle. The area of the top right rectangle is \( xy \). The area of the bottom left rectangle is \((2010 - x)(1340 - y)\).
2. Since the areas of these two rectangles are equal, we have:
\[
xy = (2010 - x)(1340 - y)
\]
3. Expanding and simpli... | 669 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle?
[img]https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667ca... | 1. **Label the vertices and extend the sides:**
Let the vertices of the original triangle be \( A, B, \) and \( C \). Extend each side of the triangle in the clockwise direction by the length of the given side. This means:
- Extend \( AB \) to \( B' \) such that \( AB' = AB \).
- Extend \( BC \) to \( C' \) su... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$ABCDEF GH$ is a regular octagon with $10$ units side . The circle with center $A$ and radius $AC$ intersects the circle with center $D$ and radius $CD$ at point $ I$, different from $C$. What is the length of the segment $IF$? | 1. **Determine the circumradius of the regular octagon:**
The side length \( s \) of the regular octagon is given as \( 10 \) units. The circumradius \( r \) of a regular octagon with side length \( s \) can be calculated using the formula:
\[
r = \frac{s}{2} \sqrt{4 + 2\sqrt{2}}
\]
Substituting \( s = ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square with unit sides. Which interior point $P$ will the expression $\sqrt2 \cdot AP + BP + CP$ have a minimum value, and what is this minimum? | 1. **Reflecting Point $P$**:
- Let $P'$ be the reflection of $P$ along $BC$.
- Let $P''$ be the reflection of $P$ along $AB$.
2. **Transforming the Problem**:
- Notice that the problem of minimizing $\sqrt{2} \cdot AP + BP + CP$ can be transformed into minimizing $AP'' + BP'' + CP'$.
- This transformation ... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible? | 1. **Identify the given information and the goal:**
- We have a unit circle with points \( A \) and \( B \) such that the chord \( AB \) has length 1.
- We need to find the point \( C \) on the longer arc \( AB \) that maximizes the perimeter of triangle \( ABC \).
2. **Calculate the central angle \( \theta \) s... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider $998$ red points on the plane with no three collinear. We select $k$ blue points in such a way that inside each triangle whose vertices are red points, there is a blue point as well. Find the smallest $k$ for which the described selection of blue points is possible for any configuration of $998$ red points. | To solve this problem, we need to find the smallest number \( k \) of blue points such that every triangle formed by any three of the 998 red points contains at least one blue point inside it.
1. **Understanding the Problem**:
- We have 998 red points on the plane with no three collinear.
- We need to place \( ... | 1991 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We build a tower of $2\times 1$ dominoes in the following way. First, we place $55$ dominoes on the table such that they cover a $10\times 11$ rectangle; this is the first story of the tower. We then build every new level with $55$ domioes above the exact same $10\times 11$ rectangle. The tower is called [i]stable[/i] ... | To solve this problem, we need to determine the minimum number of stories required to ensure that every non-lattice point of the \(10 \times 11\) rectangle is covered by at least one domino. We will show that the minimum number of stories required is 5.
1. **Define Interior Segments**:
- An **interior segment** is ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A square has been divided into $2022$ rectangles with no two of them having a common interior point. What is the maximal number of distinct lines that can be determined by the sides of these rectangles? | To solve this problem, we need to determine the maximum number of distinct lines that can be formed by the sides of 2022 rectangles within a square, where no two rectangles share an interior point. We will use induction to prove that the maximum number of lines is \( n + 3 \) for \( n \) rectangles.
1. **Base Case:**
... | 2025 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]6.[/b] Let $H_{n}(x)$ be the [i]n[/i]th Hermite polynomial. Find
$ \lim_{n \to \infty } (\frac{y}{2n})^{n} H_{n}(\frac{n}{y})$
For an arbitrary real y. [b](S.5)[/b]
$H_n(x)=(-1)^n e^{x^2}\frac{d^n}{dx^n}\left(e^{{-x^2}}\right)$ | 1. **Definition and Base Case:**
The Hermite polynomial \( H_n(x) \) is defined as:
\[
H_n(x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} \left( e^{-x^2} \right)
\]
For the base case, \( H_0(x) = 1 \).
2. **Recurrence Relation:**
We use the recurrence relation for Hermite polynomials:
\[
H_{n+1}(x) = 2xH... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
For a real number $ x$ in the interval $ (0,1)$ with decimal representation
\[ 0.a_1(x)a_2(x)...a_n(x)...,\]
denote by $ n(x)$ the smallest nonnegative integer such that
\[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\]
Determine $ \int_0^1n(x)dx$. ($ \overli... | 1. **Understanding the Problem:**
We need to find the expected value of \( n(x) \), denoted as \( \mathbb{E}[n(X)] \), where \( X \) is uniformly distributed on the interval \( (0,1) \). The value \( n(x) \) is the smallest nonnegative integer such that the decimal representation of \( x \) contains the sequence "19... | 40 | Other | math-word-problem | Yes | Yes | aops_forum | false |
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\]
J. Suranyi | To solve the problem, we need to find the smallest residue (in absolute value) of the sum
\[
\sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \pmod{p}
\]
where \( c \) is a positive integer and \( p \) is an odd prime. Let's break down the steps:
1. **Understanding the Binomial Coefficient Modulo \( p \)**:
The binom... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
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