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The integer lattice in the plane is colored with 3 colors. Find the least positive real $S$ with the property: for any such coloring it is possible to find a monochromatic lattice points $A,B,C$ with $S_{\triangle ABC}=S$.
[i]Proposed by Nikolay Beluhov[/i]
EDIT: It was the problem 3 (not 2), corrected the source tit... | To solve this problem, we need to find the smallest positive real number \( S \) such that for any 3-coloring of the integer lattice points in the plane, there exists a monochromatic triangle with area \( S \).
1. **Initial Considerations**:
- Suppose such \( S \) exists. Since the area of a triangle formed by latt... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ i... | 1. **Define the problem and setup:**
Let \( I \) be the center of the incircle of a non-isosceles triangle \( ABC \). Define \( A_1 = AI \cap BC \) and \( B_1 = BI \cap AC \). Let \( l_a \) be the line through \( A_1 \) parallel to \( AC \) and \( l_b \) be the line through \( B_1 \) parallel to \( BC \). Define \( ... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In every cell of a board $9 \times 9$ is written an integer. For any $k$ numbers in the same row (column), their sum is also in the same row (column). Find the smallest possible number of zeroes in the board for
$a)$ $k=5;$
$b)$ $k=8.$ | ### Part (a): \( k = 5 \)
1. **Assume there are at least two non-zero numbers with the same sign in a row:**
- Without loss of generality (WLOG), assume these numbers are positive.
- Let these numbers be \( a_1, a_2, \ldots, a_5 \) in a row, where \( a_1 \leq a_2 \leq \ldots \leq a_5 \).
2. **Consider the sum o... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the set of all sequences from 0’s or 1’s with length 4. What’s the minimal number of sequences that can be chosen, so that an arbitrary sequence from $A$ differs at most in 1 position from one of the chosen? | 1. **Define the set \( A \)**:
The set \( A \) consists of all sequences of length 4 composed of 0's and 1's. Therefore, the total number of sequences in \( A \) is \( 2^4 = 16 \).
2. **Determine the coverage of each sequence**:
Each sequence can differ from another sequence in at most 1 position. This means tha... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $c_0,c_1>0$. And suppose the sequence $\{c_n\}_{n\ge 0}$ satisfies
\[ c_{n+1}=\sqrt{c_n}+\sqrt{c_{n-1}}\quad \text{for} \;n\ge 1 \]
Prove that $\lim_{n\to \infty}c_n$ exists and find its value.
[i]Proposed by Sadovnichy-Grigorian-Konyagin[/i] | 1. **Define the sequence and initial conditions:**
Given the sequence $\{c_n\}_{n \ge 0}$ with $c_0, c_1 > 0$ and the recurrence relation:
\[
c_{n+1} = \sqrt{c_n} + \sqrt{c_{n-1}} \quad \text{for} \; n \ge 1
\]
2. **Assume the sequence converges:**
Suppose $\lim_{n \to \infty} c_n = L$. If the limit exi... | 4 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers? | 1. **Understanding the definition of "heavy" numbers:**
A natural number \( n > 1 \) is called "heavy" if it is coprime with the sum of its divisors, denoted by \( \sigma(n) \). This means \( \gcd(n, \sigma(n)) = 1 \).
2. **Analyzing powers of 2:**
For any natural number \( n \), if \( n = 2^k \) (a power of 2),... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of solutions to the equation:
$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $
With {x} we denote the fractional part of the number x. | 1. **Understanding the problem**: We need to find the number of solutions to the equation \(6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018\), where \(\{x\}\) denotes the fractional part of \(x\). Recall that \(0 \leq \{x\} < 1\).
2. **Isolate the integer part**: Let \(x = [x] + \{x\}\), where \([x]\) is the integer part of \(... | 5 | Other | math-word-problem | Yes | Yes | aops_forum | false |
The towns in one country are connected with bidirectional airlines, which are paid in at least one of the two directions. In a trip from town A to town B there are exactly 22 routes that are free. Find the least possible number of towns in the country. | To solve this problem, we need to determine the minimum number of towns \( n \) such that there are exactly 22 free routes from town \( A \) to town \( B \). We will analyze the problem step-by-step.
1. **Initial Analysis for Small \( n \)**:
- For \( n = 2 \), there is only 1 route from \( A \) to \( B \).
- Fo... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The number 1 is a solution of the equation
$(x + a)(x + b)(x + c)(x + d) = 16$,
where $a, b, c, d$ are positive real numbers. Find the largest value of $abcd$. | 1. Given the equation \((x + a)(x + b)(x + c)(x + d) = 16\) and knowing that \(x = 1\) is a solution, we substitute \(x = 1\) into the equation:
\[
(1 + a)(1 + b)(1 + c)(1 + d) = 16
\]
2. We aim to find the maximum value of \(abcd\). To do this, we will use the Arithmetic Mean-Geometric Mean Inequality (AM-GM... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $p_1,p_2,\dots ,p_n$ be all prime numbers lesser than $2^{100}$. Prove that
$\frac{1}{p_1} +\frac{1}{p_2} +\dots +\frac{1}{p_n} <10$. | 1. **Lemma Proof:**
We need to prove the lemma: \(\sum_{k=1}^{2^n-1} \frac{1}{k} < n\) for \(n \ge 2\).
**Base Case:**
For \(n = 2\):
\[
\sum_{k=1}^{2^2-1} \frac{1}{k} = 1 + \frac{1}{2} + \frac{1}{3} = \frac{11}{6} < 2
\]
The base case holds.
**Inductive Step:**
Assume the lemma holds for s... | 8 | Number Theory | proof | Yes | Yes | aops_forum | false |
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive? | 1. **Identify the range and common ratio:**
We are given that the terms of the geometric progression (GP) must lie between 100 and 1000 inclusive, and the common ratio \( r \) is greater than 1. We need to find the maximum number of terms in such a GP.
2. **Assume a specific GP:**
Let's consider a specific examp... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be label... | 1. **Identify the fixed triangles:**
- The vertices \( P_2 \) and \( P_5 \) are each part of only one triangle. Therefore, the triangles containing these vertices must be labeled as \( \triangle_2 \) and \( \triangle_5 \) respectively.
- Thus, \( \triangle_{P_1P_2P_3} \) is \( \triangle_2 \) and \( \triangle_{P_4... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$? | 1. **Identify the possible units digits of perfect squares:**
The units digit of a perfect square can only be one of the following: \(0, 1, 4, 5, 6, 9\). This is because:
- \(0^2 = 0\)
- \(1^2 = 1\)
- \(2^2 = 4\)
- \(3^2 = 9\)
- \(4^2 = 16\) (units digit is 6)
- \(5^2 = 25\) (units digit is 5)
-... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the fes... | To determine the minimum number of days the festival must last, we can use a combinatorial approach based on Sperner's lemma. Here is the detailed solution:
1. **Define the Problem in Terms of Sets:**
- Let \( n = 11 \) be the number of theatrical groups.
- Each day, a subset of these groups performs, and the re... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Alice and Bob are in a hardware store. The store sells coloured sleeves that fit over keys to distinguish them. The following conversation takes place:
[color=#0000FF]Alice:[/color] Are you going to cover your keys?
[color=#FF0000]Bob:[/color] I would like to, but there are only $7$ colours and I have $8$ keys.
[color=... | To determine the smallest number of colors needed to distinguish \( n \) keys arranged in a circle, we need to consider the symmetries of the circle, which include rotations and reflections. Let's analyze the problem step-by-step.
1. **Case \( n \leq 2 \):**
- For \( n = 1 \), only one color is needed.
- For \( ... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ ABC$ be a right angled triangle of area 1. Let $ A'B'C'$ be the points obtained by reflecting $ A,B,C$ respectively, in their opposite sides. Find the area of $ \triangle A'B'C'.$ | 1. Given that \( \triangle ABC \) is a right-angled triangle with area 1. Let \( A', B', C' \) be the reflections of \( A, B, C \) in their opposite sides respectively.
2. Since \( A' \) is the reflection of \( A \) across \( BC \), the line segment \( AA' \) is perpendicular to \( BC \) and \( AA' = 2AH \), where \( H... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define $ \{ a_n \}_{n\equal{}1}$ as follows: $ a_1 \equal{} 1989^{1989}; \ a_n, n > 1,$ is the sum of the digits of $ a_{n\minus{}1}$. What is the value of $ a_5$? | 1. **Establishing the congruence relationship:**
- Given \( a_1 = 1989^{1989} \), we need to find \( a_5 \).
- Note that \( a_n \) is the sum of the digits of \( a_{n-1} \) for \( n > 1 \).
- By properties of digit sums, \( a_{n-1} \equiv a_n \pmod{9} \). Therefore, \( a_1 \equiv a_2 \equiv a_3 \equiv a_4 \equ... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Anumber of schools took part in a tennis tournament. No two players from the same school played against each other. Every two players from different schools played exactly one match against each other. A match between two boys or between two girls was called a [i]single[/i] and that between a boy and a girl was called ... | 1. Let there be \( n \) schools. Suppose the \( i^{th} \) school sends \( B_i \) boys and \( G_i \) girls. Let \( B = \sum B_i \) and \( G = \sum G_i \). We are given that \( |B - G| \leq 1 \).
2. The number of same-sex matches (singles) is given by:
\[
\frac{1}{2} \sum B_i(B - B_i) + \frac{1}{2} \sum G_i(G - G_... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$, compute $\frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} + \frac{1+\gamma}{1-\gamma}$. | 1. Let $\alpha, \beta, \gamma$ be the roots of the polynomial equation \(x^3 - x - 1 = 0\).
2. Define \(r = \frac{1+\alpha}{1-\alpha}\), \(s = \frac{1+\beta}{1-\beta}\), and \(t = \frac{1+\gamma}{1-\gamma}\).
3. We need to find the value of \(r + s + t\).
4. Express \(\alpha, \beta, \gamma\) in terms of \(r, s, t\):
... | -7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of real solutions $a$ to the equation:
\[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \]
Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$. | 1. We start with the given equation:
\[
\left\lfloor \frac{a}{2} \right\rfloor + \left\lfloor \frac{a}{3} \right\rfloor + \left\lfloor \frac{a}{5} \right\rfloor = a
\]
where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \(x\).
2. Since \(\left\lfloor x \right\rfloor\) ... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There is a board numbered $-10$ to $10$. Each square is coloured either red or white, and the sum of the numbers on the red squares is $n$. Maureen starts with a token on the square labeled $0$. She then tosses a fair coin ten times. Every time she flips heads, she moves the token one square to the right. Every time she... | 1. **Determine the total number of possible outcomes:**
Since Maureen flips a fair coin 10 times, each flip has 2 possible outcomes (heads or tails). Therefore, the total number of possible outcomes is:
\[
2^{10} = 1024
\]
2. **Identify the number of favorable outcomes:**
We are given that the probabili... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]Randy:[/b] "Hi Rachel, that's an interesting quadratic equation you have written down. What are its roots?''
[b]Rachel:[/b] "The roots are two positive integers. One of the roots is my age, and the other root is the age of my younger brother, Jimmy.''
[b]Randy:[/b] "That is very neat! Let me see if I can fi... | 1. **Prove that Jimmy is two years old.**
Let's denote the quadratic equation as \( P(x) = ax^2 + bx + c \). According to the problem, the roots of this equation are Rachel's age and Jimmy's age, which are both positive integers. Let these roots be \( r_1 \) and \( r_2 \).
By Vieta's formulas, we know:
\[
... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersectio... | 1. **Identify the key points and properties:**
- Let $ABC$ be a triangle with $AC > AB$.
- $P$ is the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle A$.
- Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX \perp AB$ and $PY \perp AC$.
-... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For two real numbers $ a$, $ b$, with $ ab\neq 1$, define the $ \ast$ operation by
\[ a\ast b=\frac{a+b-2ab}{1-ab}.\] Start with a list of $ n\geq 2$ real numbers whose entries $ x$ all satisfy $ 0<x<1$. Select any two numbers $ a$ and $ b$ in the list; remove them and put the number $ a\ast b$ at the end of the... | ### Part (a)
1. **Commutativity and Associativity for \( n = 3 \)**:
- First, we show that the operation \( \ast \) is commutative, i.e., \( a \ast b = b \ast a \).
\[
a \ast b = \frac{a + b - 2ab}{1 - ab} = b \ast a
\]
- Next, we prove the associativity for \( n = 3 \). We need to show that \( (a... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer. | 1. We start with the given equation:
\[
n^3 + 2n^2 = b
\]
where \( b \) is the square of an odd integer.
2. We can factor the left-hand side:
\[
n^3 + 2n^2 = n^2(n + 2)
\]
Let \( b = k^2 \) for some odd integer \( k \). Thus, we have:
\[
n^2(n + 2) = k^2
\]
3. Since \( k \) is an odd ... | 7 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 4
-----
The points $(1,y_1)$ and $(-1,y_2)$ lie on the curve $y=px^2+qx+5$. If $y_1+y_2=14$, then the value of $p$ is
$\textbf{(A) } 2 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 2-q \qquad \textbf{(E) }\text{none of these}$ | 1. Given the points \((1, y_1)\) and \((-1, y_2)\) lie on the curve \(y = px^2 + qx + 5\), we can substitute these points into the equation to find expressions for \(y_1\) and \(y_2\).
For the point \((1, y_1)\):
\[
y_1 = p(1)^2 + q(1) + 5 = p + q + 5
\]
For the point \((-1, y_2)\):
\[
y_2 = p... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 6
-----
The $y$-intercept of the graph of the function defined by $y=\frac{4(x+3)(x-2)-24}{(x+4)}$ is
$\textbf{(A) } -24 \qquad \textbf{(B) } -12 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } -4 \qquad \textbf{(E) } -48$ | 1. To find the $y$-intercept of the function \( y = \frac{4(x+3)(x-2) - 24}{x+4} \), we need to set \( x = 0 \) and solve for \( y \).
2. Substitute \( x = 0 \) into the function:
\[
y = \frac{4(0+3)(0-2) - 24}{0+4}
\]
3. Simplify the expression inside the numerator:
\[
y = \frac{4 \cdot 3 \cdot (-2) -... | -12 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Source: 1976 Euclid Part A Problem 8
-----
Given that $a$, $b$, and $c$ are the roots of the equation $x^3-3x^2+mx+24=0$, and that $-a$ and $-b$ are the roots of the equation $x^2+nx-6=0$, then the value of $n$ is
$\textbf{(A) } 1 \qquad \textbf{(B) } -1 \qquad \textbf{(C) } 7 \qquad \textbf{(D) } -7 \qquad \textbf{(E... | 1. Given the polynomial \(x^3 - 3x^2 + mx + 24 = 0\) with roots \(a\), \(b\), and \(c\), we can use Vieta's formulas to find the relationships between the coefficients and the roots:
\[
a + b + c = 3,
\]
\[
ab + bc + ca = m,
\]
\[
abc = -24.
\]
2. Given the polynomial \(x^2 + nx - 6 = 0\) wi... | -1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The value of $\frac{1998- 998}{1000}$ is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 1000 \qquad \textbf{(C)}\ 0.1 \qquad \textbf{(D)}\ 10 \qquad \textbf{(E)}\ 0.001$ | 1. Start with the given expression:
\[
\frac{1998 - 998}{1000}
\]
2. Perform the subtraction in the numerator:
\[
1998 - 998 = 1000
\]
3. Substitute the result back into the expression:
\[
\frac{1000}{1000}
\]
4. Simplify the fraction:
\[
\frac{1000}{1000} = 1
\]
Thus, the value ... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
In the multiplication question, the sum of the digits in the
four boxes is
[img]http://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy83L2NmMTU0MzczY2FhMGZhM2FjMjMwZDcwYzhmN2ViZjdmYjM4M2RmLnBuZw==&rn=U2NyZWVuc2hvdCAyMDE3LTAyLTI1IGF0IDUuMzguMjYgUE0ucG5n[/img]
$\textbf{(A)}\ 13 \qquad \textbf{(B)}\ 1... | 1. First, we need to identify the digits in the four boxes from the multiplication problem. The image shows the multiplication of two numbers, resulting in a product. The digits in the boxes are the individual digits of the product.
2. Let's assume the multiplication problem is \( 12 \times 12 \). The product of \( 12... | 12 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Tuesday’s high temperature was 4°C warmer than that of Monday’s. Wednesday’s high temperature
was 6°C cooler than that of Monday’s. If Tuesday’s high temperature was 22°C, what was
Wednesday’s high temperature?
$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 24 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 32 \qquad \textbf{(E)... | 1. Let the high temperature on Monday be denoted by \( M \).
2. According to the problem, Tuesday’s high temperature was 4°C warmer than Monday’s. Therefore, we can write:
\[
T = M + 4
\]
3. It is given that Tuesday’s high temperature was 22°C. Thus:
\[
T = 22
\]
4. Substituting \( T = 22 \) into the ... | 12 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point.
If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?
$\textbf{(A)}\ 7 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 11$ | 1. Let \( x \) be the total number of games played.
2. Juan won exactly 3 games. Each win gives him 2 points, so his points from wins are \( 3 \times 2 = 6 \).
3. Since Juan won 3 games, he lost \( x - 3 \) games. Each loss deducts 1 point, so his points from losses are \( -(x - 3) = -x + 3 \).
4. Therefore, Juan's tot... | 7 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10
(that is, $10, 100, 1000, 10 000$ , ...). If $p >q$, the last digit of $p – q$ cannot be
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$ | 1. Given two natural numbers \( p \) and \( q \) that do not end in zero, and their product \( p \cdot q \) is a power of 10, i.e., \( 10^k \) for some integer \( k \).
2. Since \( p \cdot q = 10^k \), we can express \( p \) and \( q \) in terms of their prime factors. Specifically, \( 10^k = 2^k \cdot 5^k \). Therefor... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Forty-two cubes with 1 cm edges are glued together to form a solid rectangular block. If the perimeter of the base of the block is 18 cm, then the height, in cm, is
$\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ \dfrac{7}{3} \qquad \textbf{(D)}\ 3 \qquad \textbf{(E)}\ 4$ | 1. Let the dimensions of the rectangular block be \( l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
2. The volume of the block is given by the product of its dimensions:
\[
l \times w \times h = 42 \text{ cubic centimeters}
\]
3. The perimeter of the base ... | 3 | Geometry | MCQ | Yes | Yes | aops_forum | false |
p1. Suppose the number $\sqrt[3]{2+\frac{10}{9}\sqrt3} + \sqrt[3]{2-\frac{10}{9}\sqrt3}$ is integer. Calculate it.
p2. A house A is located $300$ m from the bank of a $200$ m wide river. $600$ m above and $500$ m from the opposite bank is another house $B$. A bridge has been built over the river, that allows you to ... | 1. Denote \( \sqrt[3]{2+\frac{10}{9}\sqrt{3}} = a \) and \( \sqrt[3]{2-\frac{10}{9}\sqrt{3}} = b \). We need to find \( a + b \).
2. First, calculate \( a^3 \) and \( b^3 \):
\[
a^3 = 2 + \frac{10}{9}\sqrt{3}, \quad b^3 = 2 - \frac{10}{9}\sqrt{3}
\]
3. Adding these, we get:
\[
a^3 + b^3 = \left(2 + \fr... | 2 | Other | math-word-problem | Yes | Yes | aops_forum | false |
p1. The next game is played between two players with a pile of peanuts. The game starts with the man pile being divided into two piles (not necessarily the same size). A move consists of eating all the mana in one pile and dividing the other into two non-empty piles, not necessarily the same size. Players take turns m... | To solve problem P4, we need to determine the value of \( z \) given the conditions:
- \( N \) is a number with 2002 digits and is divisible by 9.
- \( x \) is the sum of the digits of \( N \).
- \( y \) is the sum of the digits of \( x \).
- \( z \) is the sum of the digits of \( y \).
1. **Sum of Digits and Divisibi... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
By dividing $2023$ by a natural number $m$, the remainder is $23$. How many numbers $m$ are there with this property? | 1. We start with the given condition that dividing \(2023\) by a natural number \(m\) leaves a remainder of \(23\). This can be expressed as:
\[
2023 = mn + 23
\]
where \(m\) is a natural number and \(n\) is an integer.
2. Rearranging the equation, we get:
\[
mn = 2023 - 23 = 2000
\]
Therefore,... | 12 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ be a number such that $x +\frac{1}{x}=-1$. Determine the value of $x^{1994} +\frac{1}{x^{1994}}$. | 1. Let \( x \) be a number such that \( x + \frac{1}{x} = -1 \). We need to determine the value of \( x^{1994} + \frac{1}{x^{1994}} \).
2. Define \( a_n = x^n + \frac{1}{x^n} \). We aim to find a recurrence relation for \( a_n \).
3. Using the given equation \( x + \frac{1}{x} = -1 \), we can derive the recurrence re... | -1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$.... | 1. **Define the function \( q(N) \)**: The function \( q(N) \) represents the sum of the digits of \( N \). For example, if \( N = 128 \), then \( q(128) = 1 + 2 + 8 = 11 \).
2. **Understand the problem**: We need to repeatedly apply the function \( q \) to \( 2^{2004} \) until we reach a single digit number. This pro... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A quadrilateral $ABCD$ is inscribed in a circle. Suppose that $|DA| =|BC|= 2$ and$ |AB| = 4$. Let $E $ be the intersection point of lines $BC$ and $DA$. Suppose that $\angle AEB = 60^o$ and that $|CD| <|AB|$. Calculate the radius of the circle. | 1. Given that quadrilateral $ABCD$ is inscribed in a circle, we know that $|DA| = |BC| = 2$ and $|AB| = 4$. Let $E$ be the intersection point of lines $BC$ and $DA$. We are also given that $\angle AEB = 60^\circ$ and $|CD| < |AB|$.
2. Since $ABCD$ is a cyclic quadrilateral, we can use the property that opposite angles... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are real numbers $a,b$ and $c$ and a positive number $\lambda$ such that $f(x)=x^3+ax^2+bx+c$ has three real roots $x_1, x_2$ and $x_3$ satisfying
$(1) x_2-x_1=\lambda$
$(2) x_3>\frac{1}{2}(x_1+x_2)$.
Find the maximum value of $\frac{2a^3+27c-9ab}{\lambda^3}$ | 1. Given the polynomial \( f(x) = x^3 + ax^2 + bx + c \) with roots \( x_1, x_2, x_3 \), we know from Vieta's formulas:
\[
x_1 + x_2 + x_3 = -a,
\]
\[
x_1x_2 + x_2x_3 + x_3x_1 = b,
\]
\[
x_1x_2x_3 = -c.
\]
2. We are given the conditions:
\[
x_2 - x_1 = \lambda,
\]
\[
x_3 > \fr... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In a $7\times 8$ chessboard, $56$ stones are placed in the squares. Now we have to remove some of the stones such that after the operation, there are no five adjacent stones horizontally, vertically or diagonally. Find the minimal number of stones that have to be removed. | To solve this problem, we need to ensure that no five stones are adjacent horizontally, vertically, or diagonally on a $7 \times 8$ chessboard. We start with 56 stones and need to determine the minimal number of stones to remove to meet this condition.
1. **Initial Setup and Constraints**:
- The chessboard has dime... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $k$ with the following property: if each cell of a $100\times 100$ grid is dyed with one color and the number of cells of each color is not more than $104$, then there is a $k\times1$ or $1\times k$ rectangle that contains cells of at least three different colors. | To find the smallest positive integer \( k \) such that any \( 100 \times 100 \) grid dyed with colors, where no color appears in more than 104 cells, contains a \( k \times 1 \) or \( 1 \times k \) rectangle with at least three different colors, we need to analyze the distribution of colors and the constraints given.
... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a=1+10^{-4}$. Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$. Let $x_i$ be the sum of the elements of the $i$-th row and $y_i$ be the sum of the elements of the $i$-th column for each integer $i\in [1,n]$. Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2... | 1. Let \( A \) be a \( 2023 \times 2023 \) matrix with each entry \( a_{ij} \) such that \( 1 \leq a_{ij} \leq a \), where \( a = 1 + 10^{-4} \).
2. Define \( x_i \) as the sum of the elements in the \( i \)-th row:
\[
x_i = \sum_{j=1}^{2023} a_{ij}
\]
Similarly, define \( y_i \) as the sum of the elements... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of integers between $1$ and $100$, inclusive, that have an odd number of factors. Note that $1$ and $4$ are the first two such numbers. | To determine the number of integers between \(1\) and \(100\) that have an odd number of factors, we need to understand the conditions under which a number has an odd number of factors.
1. **Understanding Factors**:
- A number \(n\) has an odd number of factors if and only if \(n\) is a perfect square. This is beca... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the remainder when $(1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)$ is divided by $43$. Your answer should be an integer between $0$ and $42$. | 1. Let \( p = 43 \). We need to find the remainder when \( (1^2+1)(2^2+1)(3^2+1)\dots(42^2+1) \) is divided by \( 43 \).
2. Consider the polynomial \( f(x) = (x^2 + 1^2)(x^2 + 2^2) \cdots (x^2 + (p-1)^2) \).
3. For any \( a \) coprime to \( p \), the set \( \{a, 2a, \dots, (p-1)a\} \) is the same as \( \{1, 2, \dots,... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $RICE$ be a quadrilateral with an inscribed circle $O$ such that every side of $RICE$ is tangent to $O$. Given taht $RI=3$, $CE=8$, and $ER=7$, compute $IC$. | 1. Let $X$, $Y$, $Z$, and $W$ be the points of tangency on sides $RI$, $IC$, $CE$, and $ER$ respectively.
2. By the property of tangents from a point to a circle, we have:
- $RX = RW$
- $IY = IX$
- $CZ = CY$
- $EW = EZ$
3. Let $RX = RW = a$, $IY = IX = b$, $CZ = CY = c$, and $EW = EZ = d$.
4. The lengths of... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$. | 1. **Identify the primes less than or equal to 17:**
The primes less than or equal to 17 are:
\[
2, 3, 5, 7, 11, 13, 17
\]
There are 7 such primes.
2. **Consider the number 1:**
The number 1 is a special case because it is not divisible by any prime number. Therefore, it can be included in our set w... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ben works quickly on his homework, but tires quickly. The first problem takes him $1$ minute to solve, and the second problem takes him $2$ minutes to solve. It takes him $N$ minutes to solve problem $N$ on his homework. If he works for an hour on his homework, compute the maximum number of problems he can solve. | 1. We need to determine the maximum number of problems \( N \) that Ben can solve in 60 minutes, given that the time taken to solve the \( i \)-th problem is \( i \) minutes.
2. The total time taken to solve the first \( N \) problems is given by the sum of the first \( N \) natural numbers:
\[
\sum_{i=1}^N i = 1... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A two-digit positive integer is $\textit{primeable}$ if one of its digits can be deleted to produce a prime number. A two-digit positive integer that is prime, yet not primeable, is $\textit{unripe}$. Compute the total number of unripe integers. | 1. **Identify the digits that are not prime:**
- The digits that are not prime are \(1, 4, 6, 8, 9\).
2. **Identify the two-digit numbers ending in non-prime digits:**
- Since we are looking for two-digit numbers that are prime but not primeable, we need to consider numbers ending in \(1\) or \(9\) (since these ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)=\sum_{i=1}^{2014}|x-i|$. Compute the length of the longest interval $[a,b]$ such that $f(x)$ is constant on that interval. | 1. Consider the function \( f(x) = \sum_{i=1}^{2014} |x - i| \). This function represents the sum of the absolute differences between \( x \) and each integer from 1 to 2014.
2. To understand where \( f(x) \) is constant, we need to analyze the behavior of \( f(x) \) as \( x \) varies. The absolute value function \( |x... | 1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $k$ is $2014$-ambiguous if the quadratics $x^2+kx+2014$ and $x^2+kx-2014$ both have two integer roots. Compute the number of integers which are $2014$-ambiguous. | 1. To determine if a positive integer \( k \) is \( 2014 \)-ambiguous, we need to check if both quadratics \( x^2 + kx + 2014 \) and \( x^2 + kx - 2014 \) have integer roots.
2. For the quadratic \( x^2 + kx + 2014 \) to have integer roots, its discriminant must be a perfect square. The discriminant of \( x^2 + kx + 2... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1,a_2,a_3,\dots,a_6$ be an arithmetic sequence with common difference $3$. Suppose that $a_1$, $a_3$, and $a_6$ also form a geometric sequence. Compute $a_1$. | 1. Let \( a = a_1 \). Since \( a_1, a_2, a_3, \dots, a_6 \) is an arithmetic sequence with common difference \( 3 \), we can express the terms as follows:
\[
a_1 = a, \quad a_2 = a + 3, \quad a_3 = a + 6, \quad a_4 = a + 9, \quad a_5 = a + 12, \quad a_6 = a + 15
\]
2. Given that \( a_1, a_3, a_6 \) form a geo... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In a rectangle $ABCD$, two segments $EG$ and $FH$ divide it into four smaller rectangles. $BH$ intersects $EG$ at $X$, $CX$ intersects $HF$ and $Y$, $DY$ intersects $EG$ at $Z$. Given that $AH=4$, $HD=6$, $AE=4$, and $EB=5$, find the area of quadrilateral $HXYZ$. | 1. **Identify the given dimensions and points:**
- Rectangle \(ABCD\) with \(AH = 4\), \(HD = 6\), \(AE = 4\), and \(EB = 5\).
- Points \(E\) and \(G\) divide \(AB\) and \(CD\) respectively.
- Points \(F\) and \(H\) divide \(AD\) and \(BC\) respectively.
- \(BH\) intersects \(EG\) at \(X\).
- \(CX\) inte... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Four men are each given a unique number from $1$ to $4$, and four women are each given a unique number from $1$ to $4$. How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to ea... | 1. **Labeling and Initial Constraints**:
- Denote the men as \( A, B, C, D \) and the women as \( a, b, c, d \).
- Each man and woman pair share the same number: \( A \) and \( a \) have number 1, \( B \) and \( b \) have number 2, and so on.
- The arrangement must alternate between men and women, and no two p... | 12 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a unit circle with center $O$. Let $P$ be a point outside the circle such that the two line segments passing through $P$ and tangent to the circle form an angle of $60^\circ$. Compute the length of $OP$. | 1. Let \( O \) be the center of the unit circle, and let \( P \) be a point outside the circle such that the two line segments passing through \( P \) and tangent to the circle form an angle of \( 60^\circ \). Let \( A \) and \( B \) be the points of tangency of the two line segments from \( P \) to the circle.
2. Sin... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A circle $A$ is circumscribed about a unit square and a circle $B$ is inscribed inside the same unit square. Compute the ratio of the area of $A$ to the area of $B$. | 1. **Determine the radius of circle \( B \):**
- Circle \( B \) is inscribed inside the unit square.
- The side length of the unit square is 1.
- The radius of circle \( B \) is half the side length of the square.
\[
r_B = \frac{1}{2}
\]
- The area of circle \( B \) is:
\[
\text{Area}_B = \pi... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to write $91$ as the sum of at least $2$ consecutive positive integers? | 1. Suppose we write \( 91 \) as the sum of the consecutive integers from \( a \) to \( b \), inclusive. This gives us the equation:
\[
\frac{(b+a)}{2}(b-a+1) = 91
\]
because the sum of a sequence is equal to the mean of the sequence multiplied by the number of terms.
2. Removing fractions by multiplying bo... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $D$ is a point on $AB$ between $A$ and $B$, $E$ is a point on $AC$ between $A$ and $C$, and $F$ is a point on $BC$ between $B$ and $C$ such that $AF$, $BE$, and $CD$ all meet inside $\triangle ABC$ at a point $G$. Given that the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and... | 1. Given the area of $\triangle ABC$ is $15$, the area of $\triangle ABE$ is $5$, and the area of $\triangle ACD$ is $10$.
2. We need to find the area of $\triangle ABF$.
First, let's analyze the given areas and their implications:
- The area of $\triangle ABE$ is $5$, so the area of $\triangle BCE$ is $15 - 5 = 10$.
... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a given positive integer $m$, the series
$$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$
evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$. | 1. We start with the given series:
\[
\sum_{k=1, k \neq m}^{\infty} \frac{1}{(k+m)(k-m)}
\]
We can use partial fraction decomposition to rewrite the summand:
\[
\frac{1}{(k+m)(k-m)} = \frac{A}{k+m} + \frac{B}{k-m}
\]
Solving for \(A\) and \(B\), we get:
\[
1 = A(k-m) + B(k+m)
\]
Sett... | 7 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop? | 1. Let \( C \) be the number of cats, \( D \) be the number of ducks, and \( P \) be the number of parrots. We are given that \( C = 56 \).
2. Let \( T \) be the total number of animals in the pet shop. We know that \( \frac{1}{12} \) of the animals are ducks:
\[
D = \frac{1}{12}T
\]
3. We are also given tha... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping? | To solve this problem, we need to cover a $5 \times 5$ square using the minimum number of smaller squares of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$, and $4 \times 4$. We must ensure that no two squares overlap.
1. **Calculate the total area to be covered:**
The area of the $5 \times 5$ square is:
\[
5... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Eric has $2$ boxes of apples, with the first box containing red and yellow apples and the second box containing green apples. Eric observes that the red apples make up $\tfrac{1}{2}$ of the apples in the first box. He then moves all of the red apples to the second box, and observes that the red apples now make up $\tfr... | 1. Let the number of red apples be \( r \), yellow apples be \( y \), and green apples be \( g \).
2. Since the red apples make up \(\frac{1}{2}\) of the apples in the first box, and the only other color in the first box is yellow, we have:
\[
r = y
\]
3. After moving all the red apples to the second box, Eric... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If $a$, $6$, and $b$, in that order, form an arithmetic sequence, compute $a+b$. | 1. Let the common difference of the arithmetic sequence be \( d \).
2. In an arithmetic sequence, the terms are given by:
- The first term: \( a \)
- The second term: \( a + d \)
- The third term: \( a + 2d \)
3. Given that the second term is 6, we have:
\[
a + d = 6
\]
4. The third term is given by:
... | 12 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The sum
$$\sum_{n=0}^{2016\cdot2017^2}2018^n$$
can be represented uniquely in the form $\sum_{i=0}^{\infty}a_i\cdot2017^i$ for nonnegative integers $a_i$ less than $2017$. Compute $a_0+a_1$. | To solve the problem, we need to represent the sum \( \sum_{n=0}^{2016 \cdot 2017^2} 2018^n \) in the form \( \sum_{i=0}^{\infty} a_i \cdot 2017^i \) where \( 0 \leq a_i < 2017 \). We are asked to compute \( a_0 + a_1 \).
1. **Understanding the Sum**:
The given sum is a geometric series with the first term \( a = 1... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given that $x\ge0$, $y\ge0$, $x+2y\le6$, and $2x+y\le6$, compute the maximum possible value of $x+y$. | To find the maximum possible value of \(x + y\) given the constraints \(x \ge 0\), \(y \ge 0\), \(x + 2y \le 6\), and \(2x + y \le 6\), we can follow these steps:
1. **Graph the inequalities**:
- The inequality \(x \ge 0\) represents the region to the right of the y-axis.
- The inequality \(y \ge 0\) represents ... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
The set $S = \{ (a,b) \mid 1 \leq a, b \leq 5, a,b \in \mathbb{Z}\}$ be a set of points in the plane with integeral coordinates. $T$ is another set of points with integeral coordinates in the plane. If for any point $P \in S$, there is always another point $Q \in T$, $P \neq Q$, such that there is no other integeral po... | To solve this problem, we need to find the minimum number of points in set \( T \) such that for any point \( P \in S \), there exists a point \( Q \in T \) (with \( P \neq Q \)) such that the line segment \( PQ \) does not contain any other points with integer coordinates.
1. **Understanding the Set \( S \)**:
The... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$. Find the least $k$ so that for any $b \in M$, there exist $a_{i}, a_{j}\in A$, satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ($a_{i}$ and $a_{j}$ do not have to be different) . | To solve the problem, we need to find the smallest subset \( A \subseteq M \) such that for any \( b \in M \), there exist \( a_i, a_j \in A \) satisfying \( b = a_i \) or \( b = a_i \pm a_j \).
1. **Initial Consideration for \( k \leq 3 \)**:
- If \( k \leq 3 \), then \( A \) has at most three elements. Let these... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given $2018 \times 4$ grids and tint them with red and blue. So that each row and each column has the same number of red and blue grids, respectively. Suppose there're $M$ ways to tint the grids with the mentioned requirement. Determine $M \pmod {2018}$. | 1. **Understanding the problem**: We need to tint a \(2018 \times 4\) grid such that each row and each column has the same number of red and blue grids. We are to determine the number of ways to do this, denoted as \(M\), and find \(M \pmod{2018}\).
2. **Column combinations**: Each column \(C_i\) can have the followin... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We are given a convex quadrilateral $ABCD$ in the plane.
([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?
([i]ii[/i]) Find (with proof) the maximum possible num... | 1. **Given a convex quadrilateral \(ABCD\) and a point \(P\) such that the areas of \(\triangle ABP\), \(\triangle BCP\), \(\triangle CDP\), and \(\triangle DAP\) are equal, we need to determine the condition that must be satisfied by \(ABCD\).**
Let's denote the area of each of these triangles as \(A\). Therefore,... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a competition there are $18$ teams and in each round $18$ teams are divided into $9$ pairs where the $9$ matches are played coincidentally. There are $17$ rounds, so that each pair of teams play each other exactly once. After $n$ rounds, there always exists $4$ teams such that there was exactly one match played betw... | 1. **Understanding the Problem:**
We have 18 teams, and in each round, they are divided into 9 pairs to play 9 matches. There are 17 rounds in total, and each pair of teams plays exactly once. We need to find the maximum value of \( n \) such that after \( n \) rounds, there always exist 4 teams with exactly one mat... | 7 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $M$ be a set consisting of $n$ points in the plane, satisfying:
i) there exist $7$ points in $M$ which constitute the vertices of a convex heptagon;
ii) if for any $5$ points in $M$ which constitute the vertices of a convex pentagon, then there is a point in $M$ which lies in the interior of the pentagon.
Find the... | To find the minimum value of \( n \) for the set \( M \) of points in the plane, we need to satisfy the given conditions:
1. There exist 7 points in \( M \) which constitute the vertices of a convex heptagon.
2. For any 5 points in \( M \) which constitute the vertices of a convex pentagon, there is a point in \( M \) ... | 11 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, \ldots , a_{11}$ be 11 pairwise distinct positive integer with sum less than 2007. Let S be the sequence of $1,2, \ldots ,2007$. Define an [b]operation[/b] to be 22 consecutive applications of the following steps on the sequence $S$: on $i$-th step, choose a number from the sequense $S$ at random, say $... | To solve this problem, we need to determine whether the number of odd operations is larger than the number of even operations, and by how many. We will use the properties of permutations and the given operations to derive the solution.
1. **Define the Operations:**
- We start with the sequence \( S = \{1, 2, \ldots... | 0 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $ f$ be a function given by $ f(x) = \lg(x+1)-\frac{1}{2}\cdot\log_{3}x$.
a) Solve the equation $ f(x) = 0$.
b) Find the number of the subsets of the set \[ \{n | f(n^{2}-214n-1998) \geq 0,\ n \in\mathbb{Z}\}.\] | ### Part (a): Solve the equation \( f(x) = 0 \)
1. Given the function \( f(x) = \lg(x+1) - \frac{1}{2} \cdot \log_{3}x \), we need to solve \( f(x) = 0 \).
2. This implies:
\[
\lg(x+1) - \frac{1}{2} \cdot \log_{3}x = 0
\]
3. Rewrite the equation in terms of common logarithms:
\[
\lg(x+1) = \frac{1}{2} \... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions | To solve this problem, we need to determine the minimum number of $+1$'s that must be placed in a $4 \times 4$ grid such that if any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative.
1. **Understanding the Condition**:
The condition states that for any 2 rows and 2 columns delete... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$1989$ equal circles are arbitrarily placed on the table without overlap. What is the least number of colors are needed such that all the circles can be painted with any two tangential circles colored differently. | To solve the problem of determining the least number of colors needed to paint 1989 equal circles placed on a table such that no two tangential circles share the same color, we can use graph theory concepts. Specifically, we can model the problem using a graph where each circle represents a vertex and an edge exists be... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$5$ points are given in the plane, any three non-collinear and any four non-concyclic. If three points determine a circle that has one of the remaining points inside it and the other one outside it, then the circle is said to be [i]good[/i]. Let the number of good circles be $n$; find all possible values of $n$. | 1. **Understanding the problem**: We are given 5 points in the plane, no three of which are collinear and no four of which are concyclic. We need to determine the number of "good" circles, where a circle is defined as good if it passes through three of the points, has one of the remaining two points inside it, and the ... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on? | **
- We need to ensure that every pair of people shares at least one correct answer.
- We can use a combinatorial design to ensure this property. One such design is a projective plane of order 4, which has 21 points (people) and 21 lines (questions), with each line containing 5 points and each point lying on 5 li... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $A\subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b\in A$($a,b$ is allowed to be same), $a+b+30k$ is always not the product of two consecutive integers. Please find $A$ with largest possible cardinality. | 1. **Define Bad Numbers:**
Let us call a nonnegative integer \( s \) *bad* if there exists an integer \( k \) such that \( s + 30k \) is the product of two consecutive integers. This means \( s \) is bad if \( s + 30k = n(n+1) \) has a solution with \( n \) being an integer.
2. **Quadratic Equation Analysis:**
... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ \left(x_{n}\right)$ be a real sequence satisfying $ x_{0}=0$, $ x_{2}=\sqrt[3]{2}x_{1}$, and $ x_{n+1}=\frac{1}{\sqrt[3]{4}}x_{n}+\sqrt[3]{4}x_{n-1}+\frac{1}{2}x_{n-2}$ for every integer $ n\geq 2$, and such that $ x_{3}$ is a positive integer. Find the minimal number of integers belonging to this sequence. | **
Using the roots of the characteristic equation, the general solution for \( x_n \) is:
\[
x_n = A \left( \frac{\sqrt[3]{2}}{2} (1 + \sqrt{3}) \right)^n + B \left( \frac{\sqrt[3]{2}}{2} (1 - \sqrt{3}) \right)^n
\]
Given \( x_0 = 0 \), we have:
\[
A + B = 0 \implies B = -A
\]
Thus,
\[
... | 5 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$. | 1. We start with the given equation:
\[
x^2 + x + 4 = \sum_{i=1}^{n} (a_i x + b_i)^2
\]
where \(a_i\) and \(b_i\) are rational numbers.
2. To find the least possible value of \(n\), we need to express \(x^2 + x + 4\) as a sum of squares of linear polynomials with rational coefficients.
3. Let's expand the... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given $ n$ points arbitrarily in the plane $ P_{1},P_{2},\ldots,P_{n},$ among them no three points are collinear. Each of $ P_{i}$ ($1\le i\le n$) is colored red or blue arbitrarily. Let $ S$ be the set of triangles having $ \{P_{1},P_{2},\ldots,P_{n}\}$ as vertices, and having the following property: for any two segme... | 1. **Define the problem and notation:**
We are given \( n \) points \( P_1, P_2, \ldots, P_n \) in the plane, with no three points collinear. Each point is colored either red or blue. We need to find the smallest \( n \) such that there exist two triangles in the set \( S \) (the set of all triangles formed by these... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value,
a) Prove that $ AB \geq 2BC$,
b) Find the value of $ AQ ... | ### Part (a): Prove that \( AB \geq 2BC \)
1. **Given**: \(ABCD\) is a rectangle with area 2, \(P\) is a point on side \(CD\), and \(Q\) is the point where the incircle of \(\triangle PAB\) touches the side \(AB\).
2. **Objective**: Prove that \(AB \geq 2BC\).
3. **Approach**: We need to find the minimum value of \(... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the complex plane, consider squares having the following property: the complex numbers its vertex correspond to are exactly the roots of integer coefficients equation $ x^4 \plus{} px^3 \plus{} qx^2 \plus{} rx \plus{} s \equal{} 0$. Find the minimum of square areas. | 1. **Identify the roots of the polynomial:**
The given polynomial is \( P(x) = x^4 + px^3 + qx^2 + rx + s \). We are told that the roots of this polynomial correspond to the vertices of a square in the complex plane. Let the roots be \( \alpha + \beta i, \alpha - \beta i, -\alpha + \beta i, -\alpha - \beta i \), whe... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$. Find the coefficients' sum of this polynomial. | 1. **Expressing \(a^{2005} + b^{2005}\) as a polynomial \(P(a+b, ab)\):**
Given that \(a^{2005} + b^{2005}\) can be expressed as a polynomial \(P(a+b, ab)\), we need to find the sum of the coefficients of this polynomial. The sum of the coefficients of a polynomial \(P(x, y)\) is given by evaluating the polynomial a... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For $n$ people, if it is known that
(a) there exist two people knowing each other among any three people, and
(b) there exist two people not knowing each other among any four people.
Find the maximum of $n$.
Here, we assume that if $A$ knows $B$, then $B$ knows $A$. | To find the maximum number of people \( n \) such that the given conditions hold, we will use the principles of graph theory and Ramsey's theorem.
1. **Restate the problem in graph theory terms:**
- Let each person be represented by a vertex in a graph.
- An edge between two vertices indicates that the correspon... | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive real $k$ satisfying the following condition: for any given four DIFFERENT real numbers $a,b,c,d$, which are not less than $k$, there exists a permutation $(p,q,r,s)$ of $(a,b,c,d)$, such that the equation $(x^{2}+px+q)(x^{2}+rx+s)=0$ has four different real roots. | To find the smallest positive real \( k \) such that for any four different real numbers \( a, b, c, d \) which are not less than \( k \), there exists a permutation \((p, q, r, s)\) of \((a, b, c, d)\) such that the equation \((x^2 + px + q)(x^2 + rx + s) = 0\) has four different real roots, we need to ensure that bot... | 4 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Three points of a triangle are among 8 vertex of a cube. So the number of such acute triangles is
$\text{(A)}0\qquad\text{(B)}6\qquad\text{(C)}8\qquad\text{(D)}24$ | 1. **Identify the possible side lengths of the triangle:**
- The side lengths of the triangle formed by the vertices of the cube can be $1$, $\sqrt{2}$, or $\sqrt{3}$.
- $1$: Edge of the cube.
- $\sqrt{2}$: Diagonal of a face of the cube.
- $\sqrt{3}$: Space diagonal of the cube.
2. **Determine the c... | 8 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer. $f(n)$ denotes the number of $n$-digit numbers $\overline{a_1a_2\cdots a_n}$(wave numbers) satisfying the following conditions :
(i) for each $a_i \in\{1,2,3,4\}$, $a_i \not= a_{i+1}$, $i=1,2,\cdots$;
(ii) for $n\ge 3$, $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$.
(1) Fin... | 1. **Define the functions and initial conditions:**
Let \( g(n) \) denote the number of \( n \)-digit wave numbers satisfying \( a_1 < a_2 \). Let \( u(n) \) denote the number of \( n \)-digit wave numbers satisfying \( a_1 > a_2 \). Clearly, \( g(n) = u(n) \), so \( f(n) = g(n) + u(n) = 2g(n) \).
Let \( v_i(k)... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a$ is real number. Sequence $a_1,a_2,a_3,....$ satisfies
$$a_1=a, a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n\ne 0 \\
0, & a_n=0
\end{cases}
(n=1,2,3,..)$$
Find all possible values of $a$ such that $|a_n|<1$ for all positive integer $n$.
| 1. **Initial Condition and Function Definition**:
Given the sequence \(a_1, a_2, a_3, \ldots\) defined by:
\[
a_1 = a, \quad a_{n+1} =
\begin{cases}
a_n - \frac{1}{a_n}, & a_n \ne 0 \\
0, & a_n = 0
\end{cases}
\]
we need to find all possible values of \(a\) such that \(|a_n| < 1\) for all po... | 0 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
If $x_i$ is an integer greater than 1, let $f(x_i)$ be the greatest prime factor of $x_i,x_{i+1} =x_i-f(x_i)$ ($i\ge 0$ and i is an integer).
(1) Prove that for any integer $x_0$ greater than 1, there exists a natural number$k(x_0)$, such that $x_{k(x_0)+1}=0$
Grade 10: (2) Let $V_{(x_0)}$ be the number of different n... | 1. **Prove that for any integer \( x_0 \) greater than 1, there exists a natural number \( k(x_0) \), such that \( x_{k(x_0)+1} = 0 \):**
Let \( x_i \) be an integer greater than 1. Define \( f(x_i) \) as the greatest prime factor of \( x_i \). We construct a sequence \( \{x_i\} \) where \( x_{i+1} = x_i - f(x_i) \... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive sequences $\{a_n\},\{b_n\}$ satisfy:$a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$.
Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$,where $m$ is a given positive integer. | 1. **Initial Setup and Sequence Definition:**
Given the sequences $\{a_n\}$ and $\{b_n\}$ with $a_1 = b_1 = 1$ and the recurrence relation for $b_n$:
\[
b_n = a_n b_{n-1} - \frac{1}{4} \quad \text{for} \quad n \geq 2
\]
2. **Expression to Minimize:**
We need to find the minimum value of:
\[
4\sqrt... | 5 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
We want to colour all the squares of an $ nxn$ board of red or black. The colorations should be such that any subsquare of $ 2x2$ of the board have exactly two squares of each color. If $ n\geq 2$ how many such colorations are possible? | 1. **Understanding the Problem:**
We need to color an \( n \times n \) board such that any \( 2 \times 2 \) subsquare has exactly two red squares and two black squares. This implies that the board must be colored in a checkerboard pattern.
2. **Checkerboard Pattern:**
A checkerboard pattern alternates colors in ... | 1 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $x \in R-\{-7\}$, determine the smallest value of the expression
$$\frac{2x^2 + 98}{(x + 7)^2}$$ | To determine the smallest value of the expression
\[ \frac{2x^2 + 98}{(x + 7)^2}, \]
we will analyze the given expression and use calculus to find its minimum value.
1. **Rewrite the expression:**
\[ f(x) = \frac{2x^2 + 98}{(x + 7)^2}. \]
2. **Simplify the expression:**
\[ f(x) = \frac{2x^2 + 98}{x^2 + 14x + 4... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The Matini company released a special album with the flags of the $ 12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope has two flags chosen randomly. Determine the minimum number of envelopes that need to be opened to that the probability of having a repeated flag is $50\%$. | 1. **Define the problem and the variables:**
- We have 12 different flags.
- Each envelope contains 2 flags chosen randomly.
- We need to determine the minimum number of envelopes to be opened such that the probability of having at least one repeated flag is at least 50%.
2. **Calculate the probability of all... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)$ be defined by $a_1=a_2=1$ and $a_n=a_{n-1}+a_{n-2}$ for $n>2$. Compute the sum $\frac{a_1}2+\frac{a_2}{2^2}+\frac{a_3}{2^3}+\ldots$.
| 1. Let \( S = \frac{a_1}{2} + \frac{a_2}{2^2} + \frac{a_3}{2^3} + \ldots \). Given the sequence \( (a_n) \) defined by \( a_1 = a_2 = 1 \) and \( a_n = a_{n-1} + a_{n-2} \) for \( n > 2 \), we recognize that \( (a_n) \) is the Fibonacci sequence.
2. We can express \( S \) as:
\[
S = \frac{1}{2} + \frac{1}{4} + \... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the least possible cardinality of a set $A$ of natural numbers, the smallest and greatest of which are $1$ and $100$, and having the property that every element of $A$ except for $1$ equals the sum of two elements of $A$. | To find the least possible cardinality of a set \( A \) of natural numbers, where the smallest and greatest elements are \( 1 \) and \( 100 \) respectively, and every element of \( A \) except for \( 1 \) equals the sum of two elements of \( A \), we need to prove that the cardinality cannot be \( \leq 8 \).
1. **Assu... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An $8\times 8$ chessboard is made of unit squares. We put a rectangular piece of paper with sides of length 1 and 2. We say that the paper and a single square overlap if they share an inner point. Determine the maximum number of black squares that can overlap the paper. | 1. **Understanding the Problem:**
We have an $8 \times 8$ chessboard with alternating black and white squares. We need to place a $1 \times 2$ rectangular piece of paper on the board such that it overlaps the maximum number of black squares.
2. **Initial Configuration:**
Consider the chessboard where each squar... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider all words containing only letters $A$ and $B$. For any positive integer $n$, $p(n)$ denotes the number of all $n$-letter words without four consecutive $A$'s or three consecutive $B$'s. Find the value of the expression
\[\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.\] | To solve the problem, we need to find a recurrence relation for \( p(n) \), the number of \( n \)-letter words without four consecutive \( A \)'s or three consecutive \( B \)'s.
1. **Define the recurrence relation:**
Let's consider the possible endings of the \( n \)-letter words:
- If the word ends in \( B \),... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$. | To find the maximum possible area of the triangle \(ABC\) whose medians have lengths satisfying the inequalities \(m_a \le 2\), \(m_b \le 3\), and \(m_c \le 4\), we use the formula for the area of a triangle in terms of its medians:
\[
E = \frac{1}{3} \sqrt{2(m_a^2 m_b^2 + m_b^2 m_c^2 + m_c^2 m_a^2) - (m_a^4 + m_b^4 +... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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