problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 6
values | synthetic bool 1
class |
|---|---|---|---|---|---|---|---|---|
Bangladesh National Mathematical Olympiad 2016 Higher Secondary
[u][b]Problem 2:[/b][/u]
(a) How many positive integer factors does $6000$ have?
(b) How many positive integer factors of $6000$ are not perfect squares? | ### Part (a)
1. **Prime Factorization of 6000:**
\[
6000 = 2^4 \times 3^1 \times 5^3
\]
2. **Number of Positive Integer Factors:**
The number of positive integer factors of a number \( n = p_1^{e_1} \times p_2^{e_2} \times \cdots \times p_k^{e_k} \) is given by:
\[
(e_1 + 1)(e_2 + 1) \cdots (e_k + 1)
... | 34 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
BdMO National 2016 Higher Secondary
[u][b]Problem 4:[/b][/u]
Consider the set of integers $ \left \{ 1, 2, ......... , 100 \right \} $. Let $ \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 100 \right \}$, where all of the $x_i$ are diffe... | To find the smallest possible value of the sum \( S = \left| x_2 - x_1 \right| + \left| x_3 - x_2 \right| + \cdots + \left| x_{100} - x_{99} \right| + \left| x_1 - x_{100} \right| \), we need to consider the arrangement of the integers \( \{1, 2, \ldots, 100\} \).
1. **Understanding the Problem:**
The sum \( S \) r... | 198 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Yukihira is counting the minimum number of lines $m$, that can be drawn on the plane so that they intersect in exactly $200$ distinct points.What is $m$? | 1. To find the minimum number of lines \( m \) that can be drawn on the plane so that they intersect in exactly 200 distinct points, we start by using the formula for the maximum number of intersections of \( m \) lines. The maximum number of intersections of \( m \) lines is given by:
\[
\frac{m(m-1)}{2}
\]
... | 21 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Lazim rolls two $24$-sided dice. From the two rolls, Lazim selects the die with the highest number. $N$ is an integer not greater than $24$. What is the largest possible value for $N$ such that there is a more than $50$% chance that the die Lazim selects is larger than or equal to $N$? | 1. **Define the problem and the probability condition:**
Lazim rolls two 24-sided dice and selects the die with the highest number. We need to find the largest integer \( N \) such that the probability of the highest number being at least \( N \) is more than 50%.
2. **Calculate the probability of the highest numbe... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $n$ are there subject to the constraint that $1 \leq n \leq 2020$ and $n^n$ is a perfect square? | 1. **Case 1: \( n \) is even**
If \( n \) is even, then \( n = 2k \) for some integer \( k \). We need to check if \( n^n \) is a perfect square.
\[
n^n = (2k)^{2k}
\]
Since \( 2k \) is even, \( (2k)^{2k} \) is always a perfect square because any even number raised to an even power is a perfect ... | 1032 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$56$ lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly $594$ points, what is the maximum number of them that could have the same slope? | 1. **Define the problem and variables:**
We are given 56 lines on a plane with no three lines being concurrent, and they intersect at exactly 594 points. We need to find the maximum number of lines that could have the same slope (i.e., be parallel).
2. **Set up the equation:**
Suppose \( x \) of the lines are pa... | 44 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $AB = 52$, $BC = 34$ and $CA = 50$. We split $BC$ into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median and angle bisector from $A$. What is the smallest possible value of $n$? | 1. **Identify the key points and segments:**
- Let \( M \) be the midpoint of \( BC \).
- Let \( D \) be the foot of the altitude from \( A \) to \( BC \).
- Let \( X \) be the point where the angle bisector of \( \angle BAC \) intersects \( BC \).
2. **Calculate the length of \( MC \):**
\[
MC = \frac{... | 102 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$f$ is a function on the set of complex numbers such that $f(z)=1/(z*)$, where $z*$ is the complex conjugate of $z$. $S$ is the set of complex numbers $z$ such that the real part of $f(z)$ lies between $1/2020$ and $1/2018$. If $S$ is treated as a subset of the complex plane, the area of $S$ can be expressed as $m× \pi... | 1. Given the function \( f(z) = \frac{1}{\overline{z}} \), where \( \overline{z} \) is the complex conjugate of \( z \). Let \( z = x + iy \) where \( x \) and \( y \) are real numbers, then \( \overline{z} = x - iy \).
2. We need to find the real part of \( f(z) \). First, compute \( f(z) \):
\[
f(z) = \frac{1}... | 2019 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $u, v$ be real numbers. The minimum value of $\sqrt{u^2+v^2} +\sqrt{(u-1)^2+v^2}+\sqrt {u^2+ (v-1)^2}+ \sqrt{(u-1)^2+(v-1)^2}$ can be written as $\sqrt{n}$. Find the value of $10n$. | 1. **Define the points and distances:**
Consider the coordinate plane with points \( A = (0, 0) \), \( B = (1, 0) \), \( C = (0, 1) \), \( D = (1, 1) \), and \( X = (u, v) \). The given expression can be interpreted as the sum of distances from \( X \) to these four points:
\[
\sqrt{u^2 + v^2} = AX, \quad \sqr... | 80 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A binary string is a word containing only $0$s and $1$s. In a binary string, a $1-$run is a non extendable substring containing only $1$s. Given a positive integer $n$, let $B(n)$ be the number of $1-$runs in the binary representation of $n$. For example, $B(107)=3$ since $107$ in binary is $1101011$ which has exactly ... | To solve the problem, we need to find the sum of the number of $1$-runs in the binary representations of all integers from $1$ to $255$. We denote this sum as:
\[ S = B(1) + B(2) + B(3) + \cdots + B(255) \]
We start by defining a function $a_n$ which represents the sum of $B(k)$ for all $k$ in the range $[2^{n-1}, 2^... | 255 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$. | 1. **Define the function \( h \):**
We are given a function \( h \) defined as:
\[
h(n) =
\begin{cases}
128 & \text{if } 1 \leq n \leq 15 \\
n^2 - 127 & \text{if } 16 \leq n \leq 30 \\
n^2 + 128 & \text{otherwise}
\end{cases}
\]
This function is designed to be adjective and to minimize t... | 498 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Pratyya and Payel have a number each, $n$ and $m$ respectively, where $n>m.$ Everyday, Pratyya multiplies his number by $2$ and then subtracts $2$ from it, and Payel multiplies his number by $2$ and then add $2$ to it. In other words, on the first day their numbers will be $(2n-2)$ and $(2m+2)$ respectively. Find minim... | 1. Define the initial numbers as \( n \) and \( m \) with \( n > m \). Let \( d_0 = n - m \) be the initial difference between Pratyya's and Payel's numbers.
2. On the first day, Pratyya's number becomes \( 2n - 2 \) and Payel's number becomes \( 2m + 2 \). The new difference \( d_1 \) between their numbers is:
\[
... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A pair of positive integers $(m,n)$ is called [b][i]'steakmaker'[/i][/b] if they maintain the equation 1 + 2$^m$ = n$^2$. For which values of m and n, the pair $(m,n)$ are steakmaker, find the sum of $mn$ | 1. We start with the given equation for a pair of positive integers \((m, n)\):
\[
1 + 2^m = n^2
\]
2. We need to find pairs \((m, n)\) that satisfy this equation. To do this, we can rearrange the equation:
\[
2^m = n^2 - 1
\]
Notice that \(n^2 - 1\) can be factored as:
\[
n^2 - 1 = (n - 1)(n... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Once in a restaurant [b][i]Dr. Strange[/i][/b] found out that there were 12 types of food items from 1 to 12 on the menu. He decided to visit the restaurant 12 days in a row and try a different food everyday. 1st day, he tries one of the items from the first two. On the 2nd day, he eats either item 3 or the item he did... | 1. **Understanding the Problem:**
Dr. Strange has 12 types of food items numbered from 1 to 12. He visits the restaurant for 12 consecutive days and tries a different food item each day. The selection process is as follows:
- On the 1st day, he tries one of the items from the first two (1 or 2).
- On the 2nd d... | 2048 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive real number $ [x] $ be its integer part. For example, $[2.711] = 2, [7] = 7, [6.9] = 6$. $z$ is the maximum real number such that [$\frac{5}{z}$] + [$\frac{6}{z}$] = 7. Find the value of$ 20z$. | 1. We are given the floor function $\lfloor x \rfloor$, which returns the greatest integer less than or equal to $x$. We need to find the maximum real number $z$ such that $\lfloor \frac{5}{z} \rfloor + \lfloor \frac{6}{z} \rfloor = 7$.
2. Let's denote $\lfloor \frac{5}{z} \rfloor$ by $a$ and $\lfloor \frac{6}{z} \rfl... | 30 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Sokal da tries to find out the largest positive integer n such that if n transforms to base-7, then it looks like twice of base-10. $156$ is such a number because $(156)_{10}$ = $(312)_7$ and 312 = 2$\times$156. Find out Sokal da's number. | 1. **Understanding the Problem:**
We need to find the largest positive integer \( n \) such that when \( n \) is transformed to base-7, it looks like twice of \( n \) in base-10. Mathematically, we need to find \( n \) such that:
\[
(2n)_{10} = (n)_{7}
\]
2. **Expressing the Condition Mathematically:**
... | 156 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A prime number$ q $is called[b][i] 'Kowai' [/i][/b]number if $ q = p^2 + 10$ where $q$, $p$, $p^2-2$, $p^2-8$, $p^3+6$ are prime numbers. WE know that, at least one [b][i]'Kowai'[/i][/b] number can be found. Find the summation of all [b][i]'Kowai'[/i][/b] numbers.
| To solve the problem, we need to find a prime number \( q \) such that \( q = p^2 + 10 \) and the numbers \( q \), \( p \), \( p^2 - 2 \), \( p^2 - 8 \), and \( p^3 + 6 \) are all prime numbers. We will use modular arithmetic to narrow down the possible values of \( p \).
1. **Consider the condition \( p^2 - 8 \) bein... | 59 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An invisible tank is on a $100 \times 100 $ table. A cannon can fire at any $k$ cells of the board after that the tank will move to one of the adjacent cells (by side). Then the progress is repeated. Find the smallest value of $k$ such that the cannon can definitely shoot the tank after some time. | 1. **Understanding the problem**: We have a $100 \times 100$ table, and a tank that occupies a $1 \times 1$ cell. The tank can move to any adjacent cell (up, down, left, or right) after each cannon shot. We need to determine the smallest number of cells, $k$, that the cannon can target in one shot such that the tank wi... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$.
(Folklore) | 1. We need to find the smallest value of the expression \( |3 \cdot 5^m - 11 \cdot 13^n| \) for all \( m, n \in \mathbb{N} \). We claim that the smallest value is 16.
2. First, we check if 16 is achievable. For \( (m, n) = (4, 2) \):
\[
3 \cdot 5^4 - 11 \cdot 13^2 = 3 \cdot 625 - 11 \cdot 169 = 1875 - 1859 = 16
... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least possible number of elements which can be deleted from the set $\{1,2,...,20\}$ so that the sum of no two different remaining numbers is not a perfect square.
N. Sedrakian , I.Voronovich | To solve this problem, we need to ensure that the sum of no two different remaining numbers in the set $\{1, 2, \ldots, 20\}$ is a perfect square. We will identify all pairs of numbers whose sums are perfect squares and then determine the minimum number of elements that need to be deleted to achieve this condition.
1.... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.) | 1. **Understanding the problem**: We need to find the maximum number of edges in a simple graph with 8 vertices that does not contain any quadrilateral (4-cycle).
2. **Applying Istvan Reiman's theorem**: According to Istvan Reiman's theorem, the maximum number of edges \( E \) in a simple graph with \( n \) vertices t... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
N numbers are marked in the set $\{1,2,...,2000\}$ so that any pair of the numbers $(1,2),(2,4),...,(1000,2000)$ contains at least one marked number. Find the least possible value of $N$.
I.Gorodnin | 1. **Partitioning the Set**: The set \( S = \{1, 2, \ldots, 2000\} \) can be partitioned into subsets \( S_m \) where \( S_m = \{(2m-1) \cdot 2^0, (2m-1) \cdot 2^1, \ldots, (2m-1) \cdot 2^k\} \) such that \( k \in \mathbb{Z} \) and \((2m-1) \cdot 2^k \leq 2000 < (2m-1) \cdot 2^{k+1} \).
2. **Calculating \( L_m \)**: T... | 666 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
All the numbers $1,2,...,9$ are written in the cells of a $3\times 3$ table (exactly one number in a cell) . Per move it is allowed to choose an arbitrary $2\times2$ square of the table and either decrease by $1$ or increase by $1$ all four numbers of the square. After some number of such moves all numbers of the table... | 1. **Labeling and Initial Setup**:
We are given a \(3 \times 3\) table with numbers \(1, 2, \ldots, 9\) placed in the cells. We can perform moves that increase or decrease all four numbers in any \(2 \times 2\) sub-square by 1. We need to determine the possible values of \(a\) such that all numbers in the table beco... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices.
What is the smallest number of the edges in the initial graph with 30 vertices? | 1. **Understanding the problem**: We are given a graph with 30 vertices. If any 26 vertices and their outgoing edges are removed, the remaining graph with 4 vertices must be connected. We need to find the smallest number of edges in the initial graph.
2. **Degree condition**: For the remaining 4 vertices to be connect... | 405 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $n$ so that both $n$ and $n + 100$ have odd numbers of divisors. | To find all positive integers \( n \) such that both \( n \) and \( n + 100 \) have an odd number of divisors, we need to understand the conditions under which a number has an odd number of divisors. A number has an odd number of divisors if and only if it is a perfect square. This is because the divisors of a number u... | 576 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$$Problem 4:$$The sum of $5$ positive numbers equals $2$. Let $S_k$ be the sum of the $k-th$ powers of
these numbers. Determine which of the numbers $2,S_2,S_3,S_4$ can be the greatest among them. | 1. **Given Information:**
- The sum of 5 positive numbers equals 2.
- Let \( S_k \) be the sum of the \( k \)-th powers of these numbers.
- We need to determine which of the numbers \( 2, S_2, S_3, S_4 \) can be the greatest among them.
2. **Sum of the First Powers:**
- Given \( S_1 = x_1 + x_2 + x_3 + x_4... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$. | To solve this problem, we need to find all composite numbers \( n \) such that for every proper divisor \( d \) of \( n \) (i.e., \( 1 < d < n \)), the inequality \( n-12 \geq d \geq n-20 \) holds.
1. **Understanding the inequality**:
- For every proper divisor \( d \) of \( n \), we have:
\[
n-12 \geq d ... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A circle is inscribed in the trapezoid [i]ABCD[/i]. Let [i]K, L, M, N[/i] be the points of tangency of this circle with the diagonals [i]AC[/i] and [i]BD[/i], respectively ([i]K[/i] is between [i]A[/i] and [i]L[/i], and [i]M[/i] is between [i]B[/i] and [i]N[/i]). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, fin... | 1. **Define the problem and given values:**
- A circle is inscribed in the trapezoid \(ABCD\).
- Points \(K, L, M, N\) are the points of tangency of this circle with the diagonals \(AC\) and \(BD\), respectively.
- Given: \(AK \cdot LC = 16\) and \(BM \cdot ND = \frac{9}{4}\).
2. **Introduce the tangency poin... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$ | 1. **Understanding the problem**: We need to find pairs \((n, q)\) where \(n\) is a positive integer and \(q\) is a non-integer rational number such that \(0 < q < 2000\) and \(\{q^2\} = \left\{\frac{n!}{2000}\right\}\). Here, \(\{x\}\) denotes the fractional part of \(x\).
2. **Analyzing the conditions**:
- Since... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pai... | 1. **Define the problem and the transformation:**
We are given a sequence \( (e_1, e_2, \ldots, e_N) \) where each \( e_i \) is either \( 1 \) or \( -1 \). We can choose any five consecutive terms and change their signs in one move. Two sequences are similar if one can be transformed into the other through a finite ... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle.
| 1. **Understanding the Problem:**
We need to find the minimum number of distinct terms in the 6-tuple \((a, b, c, \alpha, \beta, \gamma)\) for a scalene triangle. A scalene triangle has all sides and angles distinct, i.e., \(a \neq b \neq c\) and \(\alpha \neq \beta \neq \gamma\).
2. **Analyzing the Relationship Be... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$.
(I.Voronovich) | 1. **Identify the given information and set up the problem:**
- We have a convex pentagon \(ABCDE\) with the following properties:
- \(AB = BC\)
- \(CD = DE\)
- \(\angle ABC = 150^\circ\)
- \(\angle CDE = 30^\circ\)
- \(BD = 2\)
2. **Understand the transformations:**
- The notation \(D_{... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all... | To solve this problem, we need to determine the maximum score Ivan can achieve while ensuring that he has fewer points than any other contestant. We will use the given scoring system where a problem is worth \( n \) points if it was not solved by exactly \( n \) contestants.
1. **Understanding the scoring system**:
... | 58 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became indep... | 1. **Initial Setup:**
The problem states that there are 10 fractions, each with a unique number of members, and each fraction has at least 5 members. Therefore, the smallest possible sizes for these fractions are \(5, 6, 7, \ldots, 14\).
2. **Total Number of Deputies:**
Calculate the total number of deputies ini... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Function $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfies the following equation for all real $x$: $$f(f(x))=x^2f(x)-x+1$$. Find $f(1)$ | 1. Let \( f(1) = a \). We start by substituting \( x = 1 \) into the given functional equation:
\[
f(f(1)) = 1^2 f(1) - 1 + 1 \implies f(a) = a
\]
This implies that \( f(a) = a \).
2. Next, consider \( x = f(0) \). Substituting \( x = f(0) \) into the functional equation, we get:
\[
f(f(f(0))) = (f(0... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the least number $N$ of 4-digits numbers compiled from digits $1,2,3,4,5,6,7,8$ you need to choose, that for any two different digits, both of this digits are in
a) At least in one of chosen $N$ numbers?
b)At least in one, but not more than in two of chosen $N$ numbers? | To solve this problem, we need to ensure that any two different digits from the set \(\{1,2,3,4,5,6,7,8\}\) appear together in at least one of the chosen 4-digit numbers. Let's break down the solution step-by-step.
### Part (a)
1. **Define the Problem:**
We need to find the minimum number \(N\) of 4-digit numbers ... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The vertices of the convex quadrilateral $ABCD$ lie on the parabola $y=x^2$. It is known that $ABCD$ is cyclic and $AC$ is a diameter of its circumcircle. Let $M$ and $N$ be the midpoints of the diagonals of $AC$ and $BD$ respectively. Find the length of the projection of the segment $MN$ on the axis $Oy$. | 1. **Identify the coordinates of points \(A\), \(C\), \(M\), and \(N\):**
- Given \(A = (a, a^2)\) and \(C = (c, c^2)\).
- The midpoint \(M\) of \(AC\) is:
\[
M = \left( \frac{a+c}{2}, \frac{a^2 + c^2}{2} \right)
\]
2. **Determine the coordinates of points \(B\) and \(D\):**
- Since \(ABCD\) is... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Yesterday (=April 22, 2003) was Gittes birthday. She notices that her age equals the sum of the 4 digits of the year she was born in.
How old is she? | 1. Let the year Gittes was born be represented as \( \overline{abcd} \), where \( a, b, c, d \) are the digits of the year.
2. Given that Gittes' age equals the sum of the digits of the year she was born, we can set up the following equation:
\[
2003 - (1000a + 100b + 10c + d) = a + b + c + d
\]
3. Simplifying... | 25 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Through an internal point $O$ of $\Delta ABC$ one draws 3 lines, parallel to each of the sides, intersecting in the points shown on the picture.
[img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=289[/img]
Find the value of $\frac{|AF|}{|AB|}+\frac{|BE|}{|BC|}+\frac{|CN|}{|CA|}$. | 1. **Identify the Parallelograms:**
- Since \(OK \parallel AC\) and \(MN \parallel BC\), quadrilateral \(ONKC\) is a parallelogram. Therefore, \(CN = OK\).
- Similarly, since \(AF \parallel OD\) and \(OF \parallel AD\), quadrilateral \(AFOD\) is a parallelogram. Therefore, \(AF = OD\).
2. **Establish Similar Tri... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Work base 3. (so each digit is 0,1,2)
A good number of size $n$ is a number in which there are no consecutive $1$'s and no consecutive $2$'s. How many good 10-digit numbers are there? | 1. **Define the problem and notation:**
- We are working in base 3, so each digit can be 0, 1, or 2.
- A "good" number of size \( n \) is defined as a number with no consecutive 1's and no consecutive 2's.
- Let \( u_n \) denote the number of good numbers of length \( n \).
2. **Initial conditions:**
- For... | 4756 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$? | 1. We start with the given condition that \( x^2 \) has the same decimal part as \( x \). This can be written as:
\[
x^2 = a + x
\]
where \( a \) is an integer.
2. Rearrange the equation to form a quadratic equation:
\[
x^2 - x = a
\]
\[
x^2 - x - a = 0
\]
3. We need to find the values o... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Determine all 6-digit numbers $(abcdef)$ so that $(abcdef) = (def)^2$ where $\left( x_1x_2...x_n \right)$ is no multiplication but an n-digit number. | To determine all 6-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$, we need to follow these steps:
1. **Identify the range of $(def)$:**
Since $(abcdef)$ is a 6-digit number, $(def)$ must be a 3-digit number. Let $(def) = n$, where $100 \leq n \leq 999$.
2. **Identify possible values for $f$:**
The last... | 141376 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number. | Let's solve the first problem step-by-step.
1. **Understanding the problem:**
- We have a 3-digit number with distinct digits.
- The sum of the 5 other numbers formed by permuting these digits is 2003.
2. **Let the digits be \(a\), \(b\), and \(c\):**
- The 3-digit number can be represented as \(100a + 10b +... | 345 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake.
Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table? | 1. Let \( t \) be the number of trolls. Since there are 60 creatures in total, the number of elves is \( 60 - t \).
2. Each creature claims to sit between an elf and a troll. Since trolls always lie, their claims are false. Therefore, trolls must be seated next to other trolls or elves who made a mistake.
3. Since ex... | 20 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1,2,...,100$ are written in a board. We are allowed to choose any two numbers from the board $a,b$ to delete them and replace on the board the number $a+b-1$.
What are the possible numbers u can get after $99$ consecutive operations of these? | 1. Let's denote the initial set of numbers on the board as \( S = \{1, 2, 3, \ldots, 100\} \).
2. We are allowed to choose any two numbers \( a \) and \( b \) from the board, delete them, and replace them with \( a + b - 1 \).
3. We need to determine the final number on the board after 99 operations.
To solve this, we... | 4951 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]a)[/b] Among $9$ apparently identical coins, one is false and lighter than the others. How can you discover the fake coin by making $2$ weighing in a two-course balance?
[b]b)[/b] Find the least necessary number of weighing that must be done to cover a false currency between $27$ coins if all the others are true. | ### Part (a)
To find the fake coin among 9 coins using only 2 weighings, we can use the following strategy:
1. **First Weighing:**
- Divide the 9 coins into three groups of 3 coins each: \(A, B, C\).
- Weigh group \(A\) against group \(B\).
2. **Second Weighing:**
- If \(A\) and \(B\) are equal, the fake coi... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A rectangular table $ 9$ rows $ \times$ $ 2008$ columns is fulfilled with numbers $ 1$, $ 2$, ...,$ 2008$ in a such way that each number appears exactly $ 9$ times in table and difference between any two numbers from same column is not greater than $ 3$. What is maximum value of minimum sum in column (with minimal sum)... | 1. **Define the problem and notation:**
We are given a rectangular table with 9 rows and 2008 columns. Each number from 1 to 2008 appears exactly 9 times in the table. We denote by $\sigma(c)$ the sum of the entries in column $c$. We need to find the maximum value of the minimum sum in any column, denoted as $\max \... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find minimum of $x+y+z$ where $x$, $y$ and $z$ are real numbers such that $x \geq 4$, $y \geq 5$, $z \geq 6$ and $x^2+y^2+z^2 \geq 90$ | 1. We are given the constraints:
\[
x \geq 4, \quad y \geq 5, \quad z \geq 6, \quad \text{and} \quad x^2 + y^2 + z^2 \geq 90
\]
We need to find the minimum value of \(x + y + z\).
2. First, let's consider the sum of squares constraint:
\[
x^2 + y^2 + z^2 \geq 90
\]
Since \(x \geq 4\), \(y \geq ... | 16 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
At the round table there are $10$ students. Every of the students thinks of a number and says that number to its immediate neighbors (left and right) such that others do not hear him. So every student knows three numbers. After that every student publicly says arithmetic mean of two numbers he found out from his neghbo... | 1. Let \( n_i \) be the number that the \( i\text{-th} \) student thought.
2. According to the problem, each student publicly says the arithmetic mean of the numbers they heard from their immediate neighbors. Therefore, for the \( i\text{-th} \) student, the arithmetic mean is given by:
\[
a_i = \frac{n_{i-1} + n... | 7 | Logic and Puzzles | other | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$, $d$, $e$, $f$ and $g$ be seven distinct positive integers not bigger than $7$. Find all primes which can be expressed as $abcd+efg$ | 1. Given that \(a, b, c, d, e, f, g\) are seven distinct positive integers not bigger than 7, we have \(\{a, b, c, d, e, f, g\} = \{1, 2, 3, 4, 5, 6, 7\}\).
2. We need to find all primes which can be expressed as \(abcd + efg\).
3. Since the expression \(abcd + efg\) must be a prime number, it must be greater than 3 ... | 179 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Harry Potter can do any of the three tricks arbitrary number of times:
$i)$ switch $1$ plum and $1$ pear with $2$ apples
$ii)$ switch $1$ pear and $1$ apple with $3$ plums
$iii)$ switch $1$ apple and $1$ plum with $4$ pears
In the beginning, Harry had $2012$ of plums, apples and pears, each. Harry did some tricks an... | 1. Let \( x_k, y_k, z_k \) be the number of apples, pears, and plums after the \( k \)-th trick, respectively. Initially, we have:
\[
x_0 = y_0 = z_0 = 2012
\]
2. We analyze the effect of each trick on the quantities of apples, pears, and plums:
- Trick \( i \): Switch \( 1 \) plum and \( 1 \) pear with \(... | 2025 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find maximal positive integer $p$ such that $5^7$ is sum of $p$ consecutive positive integers | 1. We need to find the maximal positive integer \( p \) such that \( 5^7 \) can be expressed as the sum of \( p \) consecutive positive integers. Let these integers be \( a, a+1, a+2, \ldots, a+(p-1) \).
2. The sum of these \( p \) consecutive integers is given by:
\[
a + (a+1) + (a+2) + \cdots + (a+(p-1)) = \fr... | 125 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
At the beginning of school year in one of the first grade classes:
$i)$ every student had exatly $20$ acquaintances
$ii)$ every two students knowing each other had exactly $13$ mutual acquaintances
$iii)$ every two students not knowing each other had exactly $12$ mutual acquaintances
Find number of students in this cla... | 1. **Define Variables and Initial Observations:**
Let \( n \) be the number of students in the class. Each student has exactly 20 acquaintances, so the total number of pairs of acquaintances is \( 10n \). This is because each acquaintance relationship is counted twice (once for each student in the pair).
2. **Count... | 31 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Mary were searching attic and found scale and box with weights. When they sorted weights by mass, they found out there exist $5$ different groups of weights. Playing with the scale and weights, they discovered that if they put any two weights on the left side of scale, they can find other two weights and put ... | 1. **Define the weights and their groups:**
Let the weights be sorted into five different groups as follows:
\[
a_1 = a_2 = \cdots = a_k = a < b_1 = b_2 = \cdots = b_l = b < c_1 = c_2 = \cdots = c_n = c < d_1 = d_2 = \cdots = d_f = d < e_1 = e_2 = \cdots = e_t = e
\]
We need to find the minimal number of... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$
Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$ | 1. Given the equation:
\[
(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) = \frac{1007}{1008}
\]
We can rewrite \(\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\) as:
\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{ab + bc + ca}{abc}
\]
Substituting this into the given equation, we get:
\... | 2017 | Algebra | proof | Yes | Yes | aops_forum | false |
Board with dimesions $2018 \times 2018$ is divided in unit cells $1 \times 1$. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain b... | 1. **Initial Setup and Problem Understanding**:
- We have a \(2018 \times 2018\) board divided into \(1 \times 1\) unit cells.
- Each cell can contain either a black chip or a white chip, but not both.
- We perform two operations:
1. Remove all black chips from columns that contain white chips.
2. Re... | 1018081 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2,...,a_{2018}$ be a sequence of numbers such that all its elements are elements of a set $\{-1,1\}$. Sum
$$S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j$$ can be negative and can also be positive. Find the minimal value of this sum | 1. We start by interpreting the given sequence \(a_1, a_2, \ldots, a_{2018}\) where each \(a_i \in \{-1, 1\}\). We need to find the minimal value of the sum \( S = \sum_{1 \leq i < j \leq 2018} a_i a_j \).
2. Consider the sum \( S \) in terms of the total sum of the sequence elements. Let \( k \) be the number of \( 1... | -1009 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
It is given positive integer $n$. Let $a_1, a_2,..., a_n$ be positive integers with sum $2S$, $S \in \mathbb{N}$. Positive integer $k$ is called separator if you can pick $k$ different indices $i_1, i_2,...,i_k$ from set $\{1,2,...,n\}$ such that $a_{i_1}+a_{i_2}+...+a_{i_k}=S$. Find, in terms of $n$, maximum number o... | **
- To maximize the number of separators, we need to consider the possible values of \( k \) such that there exists a subset of \( k \) elements summing to \( S \).
- The sequence \( (1, 1, \ldots, 1, n-1) \) provides two separators: \( k = 1 \) and \( k = n-1 \).
6. **Conclusion:**
- The maximum number of s... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find maximal value of positive integer $n$ such that there exists subset of $S=\{1,2,...,2001\}$ with $n$ elements, such that equation $y=2x$ does not have solutions in set $S \times S$ | 1. **Define the set \( S \) and the problem constraints:**
We need to find the maximal value of a positive integer \( n \) such that there exists a subset \( S \subseteq \{1, 2, \ldots, 2001\} \) with \( n \) elements, and the equation \( y = 2x \) does not have solutions in \( S \times S \).
2. **Characterize the ... | 1335 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $Z$ shape be a shape such that it covers $(i,j)$, $(i,j+1)$, $(i+1,j+1)$, $(i+2,j+1)$ and $(i+2,j+2)$ where $(i,j)$ stands for cell in $i$-th row and $j$-th column on an arbitrary table. At least how many $Z$ shapes is necessary to cover one $8 \times 8$ table if every cell of a $Z$ shape is either cell of a table ... | To solve this problem, we need to determine the minimum number of $Z$ shapes required to cover an $8 \times 8$ table. Each $Z$ shape covers 5 cells. We will consider the possibility of overlapping and rotating the $Z$ shapes to achieve the minimum coverage.
1. **Understanding the $Z$ shape:**
The $Z$ shape covers t... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Internal angles of triangle are $(5x+3y)^{\circ}$, $(3x+20)^{\circ}$ and $(10y+30)^{\circ}$ where $x$ and $y$ are positive integers. Which values can $x+y$ get ? | 1. **Sum of Internal Angles**:
Since the sum of the internal angles of a triangle is \(180^\circ\), we have:
\[
(5x + 3y) + (3x + 20) + (10y + 30) = 180
\]
Simplifying the left-hand side:
\[
5x + 3y + 3x + 20 + 10y + 30 = 180
\]
Combine like terms:
\[
8x + 13y + 50 = 180
\]
Subtra... | 1289 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
It is given polygon with $2013$ sides $A_{1}A_{2}...A_{2013}$. His vertices are marked with numbers such that sum of numbers marked by any $9$ consecutive vertices is constant and its value is $300$. If we know that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine with which number is marked $A_... | 1. **Understanding the Problem:**
We are given a polygon with 2013 sides, and the vertices are marked with numbers such that the sum of the numbers marked by any 9 consecutive vertices is constant and equal to 300. We need to determine the number marked on \( A_{2013} \) given that \( A_{13} \) is marked with 13 and... | 67 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In each cell of $5 \times 5$ table there is one number from $1$ to $5$ such that every number occurs exactly once in every row and in every column. Number in one column is [i]good positioned[/i] if following holds:
- In every row, every number which is left from [i]good positoned[/i] number is smaller than him, and ev... | 1. **Understanding the Problem:**
We are given a \(5 \times 5\) table where each cell contains a number from 1 to 5, and each number appears exactly once in every row and every column. A number is considered "good positioned" if:
- In its row, all numbers to its left are either all smaller or all larger, and all ... | 5 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Students are in classroom with $n$ rows. In each row there are $m$ tables. It's given that $m,n \geq 3$. At each table there is exactly one student. We call neighbours of the student students sitting one place right, left to him, in front of him and behind him. Each student shook hands with his neighbours. In the end t... | 1. **Calculate the number of handshakes within rows:**
Each row has \(m\) tables, and each student shakes hands with the student to their immediate right. Therefore, in each row, there are \(m-1\) handshakes. Since there are \(n\) rows, the total number of handshakes within rows is:
\[
n(m-1)
\]
2. **Calcu... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine all four-digit numbers $\overline{abcd}$ which are perfect squares and for which the equality holds:
$\overline{ab}=3 \cdot \overline{cd} + 1$. | 1. We are given a four-digit number $\overline{abcd}$ which is a perfect square and satisfies the equation $\overline{ab} = 3 \cdot \overline{cd} + 1$.
2. Let $\overline{abcd} = x^2$ where $x$ is an integer in the range $[32, 99]$ (since $32^2 = 1024$ and $99^2 = 9801$ are the smallest and largest four-digit perfect sq... | 2809 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A board $n \times n$ is divided into $n^2$ unit squares and a number is written in each unit square.
Such a board is called [i] interesting[/i] if the following conditions hold:
$\circ$ In all unit squares below the main diagonal, the number $0$ is written;
$\circ$ Positive integers are written in all other unit square... | ### Part (a)
To determine the largest number that can appear in a $6 \times 6$ interesting board, we need to consider the conditions given:
1. In all unit squares below the main diagonal, the number $0$ is written.
2. Positive integers are written in all other unit squares.
3. The sums of all $n$ rows and the sums of ... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Some people know each other in a group of people, where "knowing" is a symmetric relation. For a person, we say that it is $social$ if it knows at least $20$ other persons and at least $2$ of those $20$ know each other. For a person, we say that it is $shy$ if it doesn't know at least $20$ other persons and at least $2... | 1. **Assume there are more than 40 people in the group.**
- Let \( A \) be the person who has the most friends.
- Let \( n \) be the number of friends of \( A \).
- Let \( S \) be the subgraph which includes \( A \) and its friends.
2. **Case 1: \( n \geq 20 \)**
- Let \( v_1, v_2, \ldots, v_n \) be the fr... | 40 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In one class in the school, number of abscent students is $\frac{1}{6}$ of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was $\frac{1}{5}$ of number of students who were present. How many students are in that class? | 1. Let \( x \) be the number of students who were present initially.
2. Let \( y \) be the number of students who were absent initially.
From the problem, we know:
\[ y = \frac{1}{6}x \]
3. When the teacher sent one student to bring chalk, the number of present students becomes \( x - 1 \), and the number of absent s... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Positive integer $n$ when divided with number $3$ gives remainder $a$, when divided with $5$ has remainder $b$ and when divided with $7$ gives remainder $c$. Find remainder when dividing number $n$ with $105$ if $4a+3b+2c=30$ | 1. **Identify the constraints and the given equation:**
- We are given that \( n \) when divided by 3, 5, and 7 gives remainders \( a \), \( b \), and \( c \) respectively.
- The equation provided is \( 4a + 3b + 2c = 30 \).
- The ranges for \( a \), \( b \), and \( c \) are:
\[
0 \leq a \leq 2, \qua... | 29 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find value of $$\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}$$ if $x$, $y$ and $z$ are real numbers usch that $xyz=1$ | Given the expression:
\[ \frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx} \]
and the condition \( xyz = 1 \), we need to find the value of the expression.
1. **Common Denominator Approach**:
- We start by finding a common denominator for the fractions. Let's consider the common denominator to be \( 1 + z + zx \).... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Which number we need to substract from numerator and add to denominator of $\frac{\overline{28a3}}{7276}$ such that we get fraction equal to $\frac{2}{7}$ | 1. We start with the given equation:
\[
\frac{\overline{28a3} - x}{7276 + x} = \frac{2}{7}
\]
Here, $\overline{28a3}$ represents a four-digit number where $a$ is a digit.
2. Cross-multiplying to eliminate the fraction, we get:
\[
7(\overline{28a3} - x) = 2(7276 + x)
\]
3. Expanding both sides:
... | 641 | Algebra | other | Yes | Yes | aops_forum | false |
Find the area of quadrilateral $ABCD$ if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is $10$ | 1. Let \(ABCD\) be a quadrilateral with \(\angle D = 90^\circ\) and \(\angle B = 90^\circ\). This implies that \(AD \perp DC\) and \(AB \perp BC\).
2. Given that \(AD = DC = x\) and the sum of the lengths of the other two sides is 10, we can denote \(CB = y\) and \(AB = 10 - y\).
3. By the Pythagorean theorem applied t... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Father is $42$ years old, and son has $14$ years. In how many years father will be twice as old as his son? | 1. Let the current age of the father be \( F = 42 \) years.
2. Let the current age of the son be \( S = 14 \) years.
3. We need to find the number of years, \( x \), after which the father will be twice as old as the son.
4. In \( x \) years, the father's age will be \( 42 + x \) and the son's age will be \( 14 + x \).... | 14 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Show tha value $$A=\frac{(b-c)^2}{(a-b)(a-c)}+\frac{(c-a)^2}{(b-c)(b-a)}+\frac{(a-b)^2}{(c-a)(c-b)}$$ does not depend on values of $a$, $b$ and $c$ | 1. We start with the given expression:
\[
A = \frac{(b-c)^2}{(a-b)(a-c)} + \frac{(c-a)^2}{(b-c)(b-a)} + \frac{(a-b)^2}{(c-a)(c-b)}
\]
2. To simplify this expression, we use the identity for the sum of cubes. Specifically, if \( x + y + z = 0 \), then:
\[
x^3 + y^3 + z^3 = 3xyz
\]
We will set \( x ... | 3 | Algebra | proof | Yes | Yes | aops_forum | false |
If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that? | 1. We start with the given conditions:
\[
19250 \equiv 11 \pmod{x}
\]
\[
20302 \equiv 3 \pmod{x}
\]
2. These congruences can be rewritten as:
\[
19250 = kx + 11 \quad \text{for some integer } k
\]
\[
20302 = mx + 3 \quad \text{for some integer } m
\]
3. Subtract the first equation ... | 53 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid $5\$$. How much did the ball cost? | 1. Let the total cost of the ball be \( C \).
2. The first person paid half of the ball price:
\[
\text{First person's payment} = \frac{C}{2}
\]
3. The second person gave one third of the money that the other three gave. Let \( x \) be the amount the second person paid. Then:
\[
x = \frac{1}{3} \left( ... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In some primary school there were $94$ students in $7$th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies $40$ students outside school program, german $27$ students and $60$ students do sports. Out of the students doing sports, $24$ of the... | 1. **Define the sets and their intersections:**
- Let \( S \) be the set of students studying Spanish.
- Let \( G \) be the set of students studying German.
- Let \( P \) be the set of students doing sports.
- Given:
\[
|S| = 40, \quad |G| = 27, \quad |P| = 60
\]
\[
|S \cap P| = 24,... | 9 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Price of some item has decreased by $5\%$. Then price increased by $40\%$ and now it is $1352.06\$$ cheaper than doubled original price. How much did the item originally cost? | 1. Let the original price of the item be \( x \).
2. The price decreased by \( 5\% \), so the new price after the decrease is:
\[
x - 0.05x = 0.95x
\]
3. The price then increased by \( 40\% \), so the new price after the increase is:
\[
0.95x + 0.40 \cdot 0.95x = 0.95x \cdot (1 + 0.40) = 0.95x \cdot 1.... | 2018 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$. | 1. We start by noting that each face of the tetrahedron is a triangle with sides \(a\), \(b\), and \(c\), and the tetrahedron has a circumradius of 1.
2. We inscribe the tetrahedron in a right parallelepiped with edge lengths \(p\), \(q\), and \(r\). The sides \(a\), \(b\), and \(c\) are the lengths of the diagonals of... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many coins can be placed on a $10 \times 10$ board (each at the center of its square, at most one per square) so that no four coins form a rectangle with sides parallel to the sides of the board? | 1. **Construct a Bipartite Graph:**
- Consider a bipartite graph where one set of vertices represents the rows of the $10 \times 10$ board and the other set represents the columns.
- An edge exists between a row vertex and a column vertex if there is a coin placed at the intersection of that row and column.
2. *... | 34 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let it be is a wooden unit cube. We cut along every plane which is perpendicular to the segment joining two distinct vertices and bisects it. How many pieces do we get? | 1. **Understanding the Problem:**
We start with a unit cube and cut it along every plane that is perpendicular to the segment joining two distinct vertices and bisects it. We need to determine the number of pieces obtained after all these cuts.
2. **Identifying the Planes:**
A unit cube has 8 vertices. The segme... | 96 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A calculator treats angles as radians. It initially displays 1. What is the largest value that can be achieved by pressing the buttons cos or sin a total of 2001 times? (So you might press cos five times, then sin six times and so on with a total of 2001 presses.) | 1. **Define Sequences**: We define the sequences \( M_k \) and \( m_k \) as the maximum and minimum values achievable with \( k \) presses of the buttons \( \cos \) or \( \sin \).
2. **Initial Values**: Initially, the calculator displays 1, so \( M_0 = m_0 = 1 \).
3. **Monotonic Properties**: Since \( \sin \) and \( ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Show that we cannot form more than $4096$ binary sequences of length $24$ so that any two differ in at least $8$ positions. | 1. **Understanding the Problem:**
We need to show that we cannot form more than $4096$ binary sequences of length $24$ such that any two sequences differ in at least $8$ positions.
2. **Using Combinatorial Arguments:**
Each binary sequence of length $24$ can be considered as an element of the set $S$. We need to... | 4096 | Combinatorics | proof | Yes | Yes | aops_forum | false |
We have four charged batteries, four uncharged batteries and a radio which needs two charged batteries to work.
Suppose we don't know which batteries are charged and which ones are uncharged. Find the least number of attempts sufficient to make sure the radio will work. An attempt consists in putting two batteries i... | To solve this problem, we need to ensure that we can find two charged batteries among the four charged and four uncharged batteries using the least number of attempts. Each attempt consists of putting two batteries in the radio and checking if it works.
1. **Understanding the Problem:**
- We have 8 batteries in tot... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is [i]bold[/i] iff it has $8$ positive divisors that sum up to $3240$. For example, $2006$ is bold because its $8$ positive divisors, $1$, $2$, $17$, $34$, $59$, $118$, $1003$ and $2006$, sum up to $3240$. Find the smallest positive bold number. | To find the smallest positive bold number \( B \) with exactly 8 positive divisors that sum up to 3240, we need to consider the structure of \( B \) based on its divisors.
1. **Determine the form of \( B \)**:
- A number with exactly 8 divisors can be expressed in two forms:
- \( B = p^7 \) where \( p \) is a ... | 1614 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integers $ c$ such that $ \minus{}2007 \leq c \leq 2007$ and there exists an integer $ x$ such that $ x^2 \plus{} c$ is a multiple of $ 2^{2007}$. | To solve the problem, we need to find the number of integers \( c \) such that \( -2007 \leq c \leq 2007 \) and there exists an integer \( x \) such that \( x^2 + c \) is a multiple of \( 2^{2007} \). This means we need \( x^2 + c \equiv 0 \pmod{2^{2007}} \), or equivalently, \( x^2 \equiv -c \pmod{2^{2007}} \).
1. **... | 670 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The venusian prophet Zabruberson sent to his pupils a $ 10000$-letter word, each letter being $ A$ or $ E$: the [i]Zabrubic word[/i]. Their pupils consider then that for $ 1 \leq k \leq 10000$, each word comprised of $ k$ consecutive letters of the Zabrubic word is a [i]prophetic word[/i] of length $ k$. It is known th... | 1. **Understanding the Problem:**
We are given a 10000-letter word composed of the letters 'A' and 'E'. We need to find the maximum number of distinct prophetic words of length 10, given that there are at most 7 distinct prophetic words of length 3.
2. **Analyzing the Constraints:**
Since there are at most 7 dis... | 504 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
One writes, initially, the numbers $1,2,3,\dots,10$ in a board. An operation is to delete the numbers $a, b$ and write the number $a+b+\frac{ab}{f(a,b)}$, where $f(a, b)$ is the sum of all numbers in the board excluding $a$ and $b$, one will make this until remain two numbers $x, y$ with $x\geq y$. Find the maximum val... | 1. **Initial Setup and Conjecture:**
We start with the numbers \(1, 2, 3, \ldots, 10\) on the board. The operation involves deleting two numbers \(a\) and \(b\) and writing the number \(a + b + \frac{ab}{f(a, b)}\), where \(f(a, b)\) is the sum of all numbers on the board excluding \(a\) and \(b\).
The conjectur... | 1320 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Every day from day 2, neighboring cubes (cubes with common faces) to red cubes also turn red and are numbered with the day number. | To solve this problem, we need to determine how many times a given day number appears in an \( n \times n \times n \) cube, where neighboring cubes to red cubes also turn red and are numbered with the day number.
1. **Day Number \( i \leq n \):**
- For the first \( n \) days, the number of times the day number \( i... | 3026 | Logic and Puzzles | other | Yes | Yes | aops_forum | false |
Consider an inifinte sequence $x_1, x_2,\dots$ of positive integers such that, for every integer $n\geq 1$:
[list] [*]If $x_n$ is even, $x_{n+1}=\dfrac{x_n}{2}$;
[*]If $x_n$ is odd, $x_{n+1}=\dfrac{x_n-1}{2}+2^{k-1}$, where $2^{k-1}\leq x_n<2^k$.[/list]
Determine the smaller possible value of $x_1$ for which $2020$ is ... | To determine the smallest possible value of \( x_1 \) for which \( 2020 \) is in the sequence, we need to analyze the sequence generation rules and backtrack from \( 2020 \).
1. **Understanding the sequence rules:**
- If \( x_n \) is even, then \( x_{n+1} = \frac{x_n}{2} \).
- If \( x_n \) is odd, then \( x_{n+1... | 1183 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer is [i]brazilian[/i] if the first digit and the last digit are equal. For instance, $4$ and $4104$ are brazilians, but $10$ is not brazilian. A brazilian number is [i]superbrazilian[/i] if it can be written as sum of two brazilian numbers. For instance, $101=99+2$ and $22=11+11$ are superbrazilians, b... | To solve the problem, we need to determine how many 4-digit numbers are superbrazilian. A superbrazilian number is defined as a number that can be written as the sum of two brazilian numbers. Let's break down the solution step by step.
1. **Identify Brazilian Numbers:**
A brazilian number is a number where the firs... | 822 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a football championship with $2021$ teams, each team play with another exactly once. The score of the match(es) is three points to the winner, one point to both players if the match end in draw(tie) and zero point to the loser. The final of the tournament will be played by the two highest score teams. Brazil Footbal... | To determine the least score such that Brazil Football Club (BFC) has a chance to play in the final match, we need to analyze the points distribution and the conditions under which BFC can qualify for the final.
1. **Total Number of Matches and Points Distribution**:
- There are \(2021\) teams, and each team plays ... | 2020 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $\angle ABC=90^{\circ}$. The square $BDEF$ is inscribed in $\triangle ABC$, such that $D,E,F$ are in the sides $AB,CA,BC$ respectively. The inradius of $\triangle EFC$ and $\triangle EDA$ are $c$ and $b$, respectively. Four circles $\omega_1,\omega_2,\omega_3,\omega_4$ are drawn inside the ... | 1. **Identify the centers of the circles and their tangencies:**
- Let the centers of the circles $\omega_1, \omega_2, \omega_3, \omega_4$ be $O_1, O_2, O_3, O_4$, respectively.
- Since each circle is tangent to its neighboring circles and the sides of the square, we have:
\[
O_1O_2 = O_2O_3 = O_3O_4 = ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A single player game has the following rules: initially, there are $10$ piles of stones with $1,2,...,10$ stones, respectively. A movement consists on making one of the following operations:
[b]i)[/b] to choose $2$ piles, both of them with at least $2$ stones, combine them and then add $2$ stones to the new pile;
[b]... | 1. **Define the initial conditions:**
- Initially, there are 10 piles of stones with 1, 2, ..., 10 stones respectively.
- Let \( a \) be the number of piles and \( b \) be the total number of stones.
2. **Calculate the initial values:**
- The initial number of piles \( a \) is 10.
- The total number of sto... | 23 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
Initially, a natural number $n$ is written on the blackboard. Then, at each minute, [i]Neymar[/i] chooses a divisor $d>1$ of $n$, erases $n$, and writes $n+d$. If the initial number on the board is $2022$, what is the largest composite number that [i]Neymar[/i] will never be able to write on the blackboard? | 1. **Initial Setup**: We start with the number \( n = 2022 \) on the blackboard. Neymar can choose any divisor \( d > 1 \) of \( n \), erase \( n \), and write \( n + d \) instead.
2. **Objective**: We need to determine the largest composite number that Neymar will never be able to write on the blackboard.
3. **Key I... | 2033 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Some cells of a $10 \times 10$ are colored blue. A set of six cells is called [i]gremista[/i] when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of $n$ for which it is possible to color $n$ chessboard cells blue such that ... | To solve this problem, we need to determine the maximum number of blue cells that can be placed on a \(10 \times 10\) grid such that no set of six cells forms a *gremista* set. A *gremista* set is defined as a set of six cells that are the intersection of three rows and two columns, or two rows and three columns, and a... | 46 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $m$ be a positive integer with $m \leq 2024$. Ana and Banana play a game alternately on a $1\times2024$ board, with squares initially painted white. Ana starts the game. Each move by Ana consists of choosing any $k \leq m$ white squares on the board and painting them all green. Each Banana play consists of choosing... | 1. **Finding the maximum number \( g(k) \) of greens that we can have while avoiding \( k \) consecutive greens:**
Divide \( 2024 \) by \( k \) and write \( 2024 = q \cdot k + r \), where \( 0 \leq r \leq k-1 \). Here, \( q = \left \lfloor \frac{2024}{k} \right \rfloor \). This groups the positions \( p_1, p_2, \ld... | 88 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square and $O$ is your center. Let $E,F,G,H$ points in the segments $AB,BC,CD,AD$ respectively, such that $AE = BF = CG = DH$. The line $OA$ intersects the segment $EH$ in the point $X$, $OB$ intersects $EF$ in the point $Y$, $OC$ intersects $FG$ in the point $Z$ and $OD$ intersects $HG$ in the point $W... | 1. **Identify the given information and setup the problem:**
- \(ABCD\) is a square with center \(O\).
- Points \(E, F, G, H\) are on segments \(AB, BC, CD, AD\) respectively such that \(AE = BF = CG = DH\).
- The line \(OA\) intersects segment \(EH\) at point \(X\).
- The line \(OB\) intersects segment \(E... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $XYZ$ be a right triangle of area $1$ m$^2$ . Consider the triangle $X'Y'Z'$ such that $X'$ is the symmetric of X wrt side $YZ$, $Y'$ is the symmetric of $Y$ wrt side $XZ$ and $Z' $ is the symmetric of $Z$ wrt side $XY$. Calculate the area of the triangle $X'Y'Z'$. | 1. **Define the coordinates of the vertices of $\triangle XYZ$:**
Let $X(0,0)$, $Y(2t,0)$, and $Z(0,\frac{1}{t})$ such that the area of $\triangle XYZ$ is 1 m$^2$.
The area of a right triangle is given by:
\[
\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}
\]
Here, the base is $2... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let N be a positive integer greater than 2. We number the vertices
of a regular 2n-gon clockwise with the numbers 1, 2, . . . ,N,−N,−N +
1, . . . ,−2,−1. Then we proceed to mark the vertices in the following way.
In the first step we mark the vertex 1. If ni is the vertex marked in the
i-th step, in the i+1-th step... | 1. **Relabeling and Modifying the Game:**
We start by relabeling the vertices. For a vertex labeled $-t$ where $t \in \{1, 2, \ldots, N\}$, we relabel it as $2N+1-t$. Additionally, we insert a $0$ between $1$ and $2N$. This results in a circle with residues modulo $2N+1$ arranged starting from $1$ in a clockwise dir... | 3810 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.