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Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a\neq 20, 22$. Find the minimum possible positive value of $P(0)$.
[i]Proposed by Andrew Wu[/i]
(Note: wording changed from original to specify that $a \neq 20, 22$.) | 1. Given that \( P(x) \) is a monic quadratic polynomial, we can write it in the form:
\[
P(x) = x^2 + dx + e
\]
where \( d \) and \( e \) are constants.
2. Define \( q(x) = xP(x) \). Then \( q(x) \) is a monic cubic polynomial:
\[
q(x) = x(x^2 + dx + e) = x^3 + dx^2 + ex
\]
3. We are given that ... | 20 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to choose distinct positive integers $a, b, c, d$ dividing $15^6$ such that none of $a, b, c,$ or $d$ divide each other? (Order does not matter.)
[i]Proposed by Miles Yamner and Andrew Wu[/i]
(Note: wording changed from original to clarify) | 1. **Prime Factorization and Representation**:
We start by noting that \( 15^6 = 3^6 \cdot 5^6 \). Each divisor of \( 15^6 \) can be written in the form \( 3^x \cdot 5^y \) where \( 0 \leq x \leq 6 \) and \( 0 \leq y \leq 6 \).
2. **Distinct Exponents**:
We need to choose four distinct positive integers \( a, b,... | 1225 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Georgina calls a $992$-element subset $A$ of the set $S = \{1, 2, 3, \ldots , 1984\}$ a [b]halfthink set[/b] if
[list]
[*] the sum of the elements in $A$ is equal to half of the sum of the elements in $S$, and
[*] exactly one pair of elements in $A$ differs by $1$.
[/list]
She notices that for some values of $... | 1. **Define the set \( S \) and calculate its sum:**
\[
S = \{1, 2, 3, \ldots, 1984\}
\]
The sum of the elements in \( S \) is given by the formula for the sum of an arithmetic series:
\[
\sum_{k=1}^{1984} k = \frac{1984 \cdot 1985}{2} = 1970720
\]
Therefore, half of the sum of the elements in \... | 465 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB = 5$, $BC = 7$, and $CA = 8$, and let $D$ be a point on arc $\widehat{BC}$ of its circumcircle $\Omega$. Suppose that the angle bisectors of $\angle ADB$ and $\angle ADC$ meet $AB$ and $AC$ at $E$ and $F$, respectively, and that $EF$ and $BC$ meet at $G$. Line $GD$ meets $\Omega$ at $T$... | 1. **Identify the key elements and setup the problem:**
- Given triangle \(ABC\) with sides \(AB = 5\), \(BC = 7\), and \(CA = 8\).
- Point \(D\) is on the arc \(\widehat{BC}\) of the circumcircle \(\Omega\).
- Angle bisectors of \(\angle ADB\) and \(\angle ADC\) meet \(AB\) and \(AC\) at \(E\) and \(F\) respe... | 172 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1[/b] How many two-digit positive integers with distinct digits satisfy the conditions that
1) neither digit is $0$, and
2) the units digit is a multiple of the tens digit?
[b]p2[/b] Mirabel has $47$ candies to pass out to a class with $n$ students, where $10\le n < 20$. After distributing the candy as evenly as ... | To solve the problem, we need to find the smallest integer \( k \) such that when Mirabel distributes \( 47 - k \) candies to \( n \) students (where \( 10 \leq n < 20 \)), the candies can be distributed evenly.
1. **Identify the total number of candies after removing \( k \) candies:**
\[
47 - k
\]
We nee... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p4[/b] Define the sequence ${a_n}$ as follows:
1) $a_1 = -1$, and
2) for all $n \ge 2$, $a_n = 1 + 2 + . . . + n - (n + 1)$.
For example, $a_3 = 1+2+3-4 = 2$. Find the largest possible value of $k$ such that $a_k+a_{k+1} = a_{k+2}$.
[b]p5[/b] The taxicab distance between two points $(a, b)$ and $(c, d)$ on the coo... | Let's solve each problem step-by-step.
### Problem 1:
Define the sequence \( \{a_n\} \) as follows:
1. \( a_1 = -1 \)
2. For all \( n \ge 2 \), \( a_n = 1 + 2 + \cdots + n - (n + 1) \)
We need to find the largest possible value of \( k \) such that \( a_k + a_{k+1} = a_{k+2} \).
1. **Calculate the general form of \(... | 66 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold:
(i) The sum of every $40$ consecutive numbers is equal to $0$.
(ii) The sum of every $42$ consecutive numbers is not equal to $0$.
We denote by $S_n$ the... | 1. **Understanding the sequence and conditions:**
- We have a sequence of $n$ numbers where $n \geq 40$.
- Each number in the sequence is either $1$ or $-1$.
- Condition (i): The sum of every $40$ consecutive numbers is $0$.
- Condition (ii): The sum of every $42$ consecutive numbers is not $0$.
2. **Analy... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n>4$ be a positive integer, which is divisible by $4$. We denote by $A_n$ the sum of the odd positive divisors of $n$. We also denote $B_n$ the sum of the even positive divisors of $n$, excluding the number $n$ itself. Find the least possible value of the expression $$f(n)=B_n-2A_n,$$
for all possible values of $n... | 1. **Define the problem and variables:**
Let \( n > 4 \) be a positive integer divisible by \( 4 \). We denote by \( A_n \) the sum of the odd positive divisors of \( n \) and by \( B_n \) the sum of the even positive divisors of \( n \), excluding \( n \) itself. We aim to find the least possible value of the expre... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2-bx+c=0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$. | **
For a quadratic equation \(ax^2 + bx + c = 0\) to have exactly one real solution, the discriminant must be zero. The discriminant \(\Delta\) of the quadratic equation \(ax^2 - bx + c = 0\) is given by:
\[
\Delta = b^2 - 4ac
\]
Since the equation has exactly one real solution, we have:
\[
b^2 - 4... | 64 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1, x_2, . . . , x_{2022}$ be nonzero real numbers. Suppose that $x_k + \frac{1}{x_{k+1}} < 0$ for each $1 \leq k \leq 2022$, where $x_{2023}=x_1$. Compute the maximum possible number of integers $1 \leq n \leq 2022$ such that $x_n > 0$. | 1. **Understanding the Problem:**
We are given a sequence of nonzero real numbers \( x_1, x_2, \ldots, x_{2022} \) such that \( x_k + \frac{1}{x_{k+1}} < 0 \) for each \( 1 \leq k \leq 2022 \), with \( x_{2023} = x_1 \). We need to find the maximum number of integers \( 1 \leq n \leq 2022 \) such that \( x_n > 0 \).... | 1011 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$. | 1. **Define the function and the problem constraints:**
Let \( f \) be a function from \( \{1, 2, \ldots, 22\} \) to the positive integers such that \( mn \mid f(m) + f(n) \) for all \( m, n \in \{1, 2, \ldots, 22\} \).
2. **Propose a candidate function:**
We claim that the function \( f(n) = n \cdot \mathrm{lcm... | 2016 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$, and $(x_5, y_5)$ be the vertices of a regular pentagon centered at $(0, 0)$. Compute the product of all positive integers k such that the equality $x_1^k+x_2^k+x_3^k+x_4^k+x_5^k=y_1^k+y_2^k+y_3^k+y_4^k+y_5^k$ must hold for all possible choices of the pentagon. | 1. **Vertices on the Unit Circle:**
Since the vertices of the pentagon lie on the unit circle centered at \((0,0)\), we can represent them using complex numbers. Let the vertices be \(z_1, z_2, z_3, z_4, z_5\) where \(z_i = \exp\left(\frac{2\pi i (i-1)}{5}\right)\) for \(i = 1, 2, 3, 4, 5\). Here, \(\exp\left(\frac{... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$ satisfy $2 \leq a_i \leq 166$ and $a_i^{b_i} \cong a_{i+1}^2$ (mod 167) for each $1 \leq i \leq 7$ (where $a_8=a_1$). Compute the minimum possible value of $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$. | 1. **Understanding the problem and initial conditions:**
- We are given positive integers \(a_1, a_2, \ldots, a_7, b_1, b_2, \ldots, b_7\) such that \(2 \leq a_i \leq 166\).
- The condition \(a_i^{b_i} \equiv a_{i+1}^2 \pmod{167}\) holds for each \(1 \leq i \leq 7\) with \(a_8 = a_1\).
- We need to compute the... | 675 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$. | Given the monic polynomial \( P(x) \) of degree 2023 such that \( P(k) = k^{2023} P\left(1 - \frac{1}{k}\right) \) for every positive integer \( 1 \leq k \leq 2023 \), we need to find \( P(-1) \) in the form \( \frac{a}{b} \) where \( a \) and \( b \) are relatively prime integers. Then, we need to compute the unique i... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $\angle A = 60^o$. Line $\ell$ intersects segments $AB$ and $AC$ and splits triangle $ABC$ into an equilateral triangle and a quadrilateral. Let $X$ and $Y$ be on $\ell$ such that lines $BX$ and $CY$ are perpendicular to ℓ. Given that $AB = 20$ and $AC = 22$, compute $XY$ . | 1. **Identify the given information and set up the problem:**
- We have a triangle \(ABC\) with \(\angle A = 60^\circ\).
- Line \(\ell\) intersects segments \(AB\) and \(AC\) and splits triangle \(ABC\) into an equilateral triangle and a quadrilateral.
- Points \(X\) and \(Y\) are on \(\ell\) such that lines \... | 21 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let triangle $ABC$ be such that $AB = AC = 22$ and $BC = 11$. Point $D$ is chosen in the interior of the triangle such that $AD = 19$ and $\angle ABD + \angle ACD = 90^o$ . The value of $BD^2 + CD^2$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$. | 1. **Given Information and Initial Setup:**
- We have a triangle \(ABC\) with \(AB = AC = 22\) and \(BC = 11\).
- Point \(D\) is inside the triangle such that \(AD = 19\) and \(\angle ABD + \angle ACD = 90^\circ\).
2. **Rotation and Transformation:**
- Rotate \(\triangle ADC\) around point \(A\) such that \(A... | 36104 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on minor arc $AB$ of $\Gamma$. Suppose that $P A \cdot P B = 2$, $P C \cdot P D = 18$, and $P B \cdot P C = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integ... | 1. **Given Information and Initial Observations:**
- Rectangle \(ABCD\) is inscribed in circle \(\Gamma\).
- Point \(P\) is on the minor arc \(AB\) of \(\Gamma\).
- \(PA \cdot PB = 2\), \(PC \cdot PD = 18\), and \(PB \cdot PC = 9\).
- We need to find the area of rectangle \(ABCD\) expressed as \(\frac{a\sqr... | 21055 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4... | 1. **Understanding the Problem:**
- We are given a square \(ABCD\) with side length \(10\).
- Point \(P\) is inside the square.
- \(O_1, O_2, O_3, O_4\) are the circumcenters of triangles \(PAB, PBC, PCD, PDA\) respectively.
- We know \(PA + PB + PC + PD = 23\sqrt{2}\).
- The area of quadrilateral \(O_1O... | 5001 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that
$\bullet$ $P$ passes through $A$ and $B$,
$\bullet$ the focus $F$ of $P$ lies on $E$,
$\bullet$ the orthocenter $H$ of $\vartriangle F AB$ lies on the directrix of $P$.
If the major and minor axes of $E$ have lengths $50$ and $1... | 1. **Identify the given parameters and properties:**
- The ellipse \( E \) has major and minor axes of lengths 50 and 14, respectively.
- The foci of the ellipse are \( A \) and \( B \).
- There exists a parabola \( P \) that passes through \( A \) and \( B \).
- The focus \( F \) of the parabola \( P \) li... | 2402 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1B_1C_1$, $A_2B_2C_2$, and $A_3B_3C_3$ be three triangles in the plane. For $1 \le i \le3$, let $D_i $, $E_i$, and $F_i$ be the midpoints of $B_iC_i$, $A_iC_i$, and $A_iB_i$, respectively. Furthermore, for $1 \le i \le 3$ let $G_i$ be the centroid of $A_iB_iC_i$.
Suppose that the areas of the triangles $A_1A_2A_... | 1. **Define the points and vectors:**
Let \( a_i, b_i, c_i, d_i, e_i, f_i, g_i \) be the position vectors of the points \( A_i, B_i, C_i, D_i, E_i, F_i, G_i \) respectively, for \( i = 1, 2, 3 \).
2. **Express the centroid in terms of the vertices:**
The centroid \( G_i \) of triangle \( A_iB_iC_i \) is given by... | 917 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $\omega$ is a circle centered at $O$ with radius $8$. Let $AC$ and $BD$ be perpendicular chords of $\omega$. Let $P$ be a point inside quadrilateral $ABCD$ such that the circumcircles of triangles $ABP$ and $CDP$ are tangent, and the circumcircles of triangles $ADP$ and $BCP$ are tangent. If $AC = 2\sqrt{61}$ a... | 1. **Identify the given information and setup the problem:**
- Circle $\omega$ is centered at $O$ with radius $8$.
- Chords $AC$ and $BD$ are perpendicular.
- $P$ is a point inside quadrilateral $ABCD$ such that the circumcircles of triangles $ABP$ and $CDP$ are tangent, and the circumcircles of triangles $ADP... | 103360 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that:
$\bullet$ all four rectangles share a common vertex $P$,
$\bullet$ for each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$,
$\bullet$ for each $n = 1, 2, 3$, the opposite side of $R_n$ passes through ... | 1. **Calculate the area of the initial rectangle \( R_0 \):**
The sides of \( R_0 \) are 3 and 4. Therefore, the area \( A_0 \) of \( R_0 \) is:
\[
A_0 = 3 \times 4 = 12
\]
2. **Determine the side lengths of \( R_1 \):**
One side of \( R_1 \) is the diagonal of \( R_0 \). The length of the diagonal \( d... | 30 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Sets $A, B$, and $C$ satisfy $|A| = 92$, $|B| = 35$, $|C| = 63$, $|A\cap B| = 16$, $|A\cap C| = 51$, $|B\cap C| = 19$. Compute the number of possible values of$ |A \cap B \cap C|$. | 1. Let \( |A \cap B \cap C| = x \). We need to find the possible values of \( x \).
2. Using the principle of inclusion-exclusion for three sets, we have:
\[
|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|
\]
Substituting the given values:
\[
|A \cup B \... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of ways to color $3$ cells in a $3\times 3$ grid so that no two colored cells share an edge. | 1. **Identify the constraints**: We need to color 3 cells in a \(3 \times 3\) grid such that no two colored cells share an edge. This means that no two colored cells can be adjacent horizontally or vertically.
2. **Case 1: All three columns have at least one colored cell**:
- In this case, each column must have ex... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Mich... | 1. **Initial String and Operations**:
Michel starts with the string "HMMT". Each operation consists of:
- Replacing an occurrence of "H" with "HM".
- Replacing an occurrence of "MM" with "MOM".
- Replacing an occurrence of "T" with "MT".
2. **Length Analysis**:
Each operation increases the length of the... | 370 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Compute the number of nonempty subsets $S \subseteq\{-10,-9,-8, . . . , 8, 9, 10\}$ that satisfy $$|S| +\ min(S) \cdot \max (S) = 0.$$ | 1. We start by noting that the set \( S \) must contain both negative and positive elements because \(\min(S) \cdot \max(S)\) must be negative. Let \(\min(S) = -a\) and \(\max(S) = b\) where \(a > 0\) and \(b > 0\).
2. The given condition is:
\[
|S| + \min(S) \cdot \max(S) = 0
\]
Substituting \(\min(S) = -... | 335 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k < 10$, there exists an integer $k' > k$ such that there is at most one number between $k$ and $k'$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integer... | To solve the problem, we need to find the probability \( p \) that for every positive integer \( k < 10 \), there exists an integer \( k' > k \) such that there is at most one number between \( k \) and \( k' \) in the circle. We will use a recursive approach to determine this probability.
1. **Define the Recursive Fu... | 1390 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set of size $11$. A random $12$-tuple $(s_1, s_2, . . . , s_{12})$ of elements of $S$ is chosen uniformly at random. Moreover, let $\pi : S \to S$ be a permutation of $S$ chosen uniformly at random. The probability that $s_{i+1}\ne \pi (s_i)$ for all $1 \le i \le 12$ (where $s_{13} = s_1$) can be written a... | 1. **Define the problem and notation:**
- Let \( S \) be a set of size 11.
- A random 12-tuple \( (s_1, s_2, \ldots, s_{12}) \) of elements of \( S \) is chosen uniformly at random.
- Let \( \pi : S \to S \) be a permutation of \( S \) chosen uniformly at random.
- We need to find the probability that \( s_... | 1000000000004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$. | 1. **Define the problem and constraints:**
We need to find the maximum number of elements in a subset \( A \) of \(\{1, 2, \ldots, 2020\}\) such that the difference between any two distinct elements in \( A \) is not a prime number.
2. **Construct a potential solution:**
Consider the set \( A \) formed by taking... | 505 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three parallel lines $L_1, L_2, L_2$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$ and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$ and $C$ lies on $L_2$. Find the area of ... | 1. **Set up the problem using complex numbers:**
- Let \( B \) be the origin, i.e., \( B = 0 \).
- Since \( C \) lies on \( L_2 \) and \( L_2 \) is 3 units above \( B \), we can write \( C = m + 3i \) for some real number \( m \).
2. **Determine the coordinates of \( A \):**
- \( A \) is a \(90^\circ\) anti-c... | 45 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Ria writes down the numbers $1,2,\cdots, 101$ in red and blue pens. The largest blue number is equal to the number of numbers written in blue and the smallest red number is equal to half the number of numbers in red. How many numbers did Ria write with red pen? | 1. Let \( x \) be the largest blue number. According to the problem, the number of numbers written in blue is also \( x \). Therefore, the numbers written in blue are \( 1, 2, \ldots, x \).
2. The smallest red number is \( x+1 \) because all numbers before \( x+1 \) are written in blue.
3. The number of numbers writ... | 68 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider the set $\mathcal{T}$ of all triangles whose sides are distinct prime numbers which are also in arithmetic progression. Let $\triangle \in \mathcal{T}$ be the triangle with least perimeter. If $a^{\circ}$ is the largest angle of $\triangle$ and $L$ is its perimeter, determine the value of $\frac{a}{L}$. | 1. **Identify the triangle with the least perimeter:**
- We need to find a triangle with sides that are distinct prime numbers in arithmetic progression.
- The smallest such set of primes is \(3, 5, 7\).
- Therefore, the triangle with sides \(3, 5, 7\) has the least perimeter.
2. **Calculate the perimeter \(L... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$ | 1. **Define the sides and angles of the parallelogram:**
Let $\overline{AB}$ be the longer side of the parallelogram $ABCD$, and let $\overline{AD}$ be the shorter side. Given that the longer side is twice the shorter side, we have:
\[
\overline{AB} = 2\overline{AD}
\]
Let $\angle DAB = 2\theta$.
2. **D... | 40 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $x,y,z$ be positive real numbers such that $x^2 + y^2 = 49, y^2 + yz + z^2 = 36$ and $x^2 + \sqrt{3}xz + z^2 = 25$. If the value of $2xy + \sqrt{3}yz + zx$ can be written as $p \sqrt{q}$ where $p,q \in \mathbb{Z}$ and $q$ is squarefree, find $p+q$. | 1. **Interpret the given equations geometrically**:
We are given three equations involving \(x, y, z\):
\[
x^2 + y^2 = 49
\]
\[
y^2 + yz + z^2 = 36
\]
\[
x^2 + \sqrt{3}xz + z^2 = 25
\]
We need to find the value of \(2xy + \sqrt{3}yz + zx\).
2. **Construct a geometric interpretation**:
... | 30 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of maps $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$ such that $f(i) \le f(j)$ whenever $i < j$. | To find the number of maps \( f: \{1,2,3\} \rightarrow \{1,2,3,4,5\} \) such that \( f(i) \le f(j) \) whenever \( i < j \), we need to count the number of non-decreasing sequences of length 3 from the set \(\{1,2,3,4,5\}\).
1. **Understanding the problem**:
- We need to count the number of ways to assign values to ... | 35 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any real number $t$, let $\lfloor t \rfloor$ denote the largest integer $\le t$. Suppose that $N$ is the greatest integer such that $$\left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4$$Find the sum of digits of $N$. | 1. We start with the given equation:
\[
\left \lfloor \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor}\right \rfloor = 4
\]
This implies that:
\[
4 \leq \sqrt{\left \lfloor \sqrt{\left \lfloor \sqrt{N} \right \rfloor}\right \rfloor} < 5
\]
2. Squaring both sides, we ... | 24 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by $$x_{n+1} = - \frac{3x_n - y_n}{2}, y_{n+1} = - \frac{x_n + y_n}{2}$$Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$. | 1. **Define the sequence and transformation matrix:**
Let \( P_0 = (3,1) \) and define \( P_{n+1} = (x_n, y_n) \) for \( n \ge 0 \) by:
\[
x_{n+1} = - \frac{3x_n - y_n}{2}, \quad y_{n+1} = - \frac{x_n + y_n}{2}
\]
We can represent this transformation using a matrix \( M \):
\[
\vec{v}_n = \begin{pm... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can four married couples sit in a merry-go-round with identical seats such that men and women occupy alternate seats and no husband seats next to his wife? | 1. **Fixing the first person**: Since the merry-go-round has identical seats, we can fix one person in one seat to avoid counting rotations as different arrangements. Let's fix one man in one seat. This leaves us with 3 men and 4 women to arrange.
2. **Arranging the remaining men**: The remaining 3 men can be arranged... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $a,b,c,x,y,z$ are pairwisely different real numbers. How many terms in the following can be $1$ at most:
$$\begin{aligned}
&ax+by+cz,&&&&ax+bz+cy,&&&&ay+bx+cz,\\
&ay+bz+cx,&&&&az+bx+cy,&&&&az+by+cx?
\end{aligned}$$ | 1. **Claim and Example:**
We claim that the maximum number of terms that can be equal to 1 is 2. To illustrate this, consider the example where \(a = 1\), \(b = 2\), \(c = 3\), \(x = 4\), \(y = 27\), and \(z = -19\). In this case:
\[
ax + by + cz = 1 \quad \text{and} \quad ay + bz + cx = 1
\]
Therefore, ... | 2 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $2022$ grids in a row. Two people A and B play a game with these grids. At first, they mark each odd-numbered grid from the left with A's name, and each even-numbered grid from the left with B's name. Then, starting with the player A, they take turns performing the following action:
$\bullet$ One should sele... | 1. **Generalization and Definition:**
Let \( f(n) \) be the largest number of grids that A can guarantee to be marked with A's name for any even number \( n \). We need to prove that \( f(4n+2) = f(4n) = 2n+1 \).
2. **Base Cases:**
- For \( n = 1 \):
\[
f(2) = 1
\]
This is because there are o... | 1011 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $m$ satisfying the following proposition:
There are $999$ grids on a circle. Fill a real number in each grid, such that for any grid $A$ and any positive integer $k\leq m$, at least one of the following two propositions will be true:
$\bullet$ The difference between the numbers in... | To solve this problem, we need to find the smallest positive integer \( m \) such that for any grid \( A \) and any positive integer \( k \leq m \), at least one of the following conditions holds:
1. The difference between the numbers in grid \( A \) and the \( k \)-th grid after \( A \) in the clockwise direction is \... | 251 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest $k$ for which there exists a permutation $(a_1, a_2, \ldots, a_{2022})$ of integers from $1$ to $2022$ such that for at least $k$ distinct $i$ with $1 \le i \le 2022$ the number $\frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i}$ is an integer larger than $1$.
[i](Proposed by Oleksii Masalitin)[/i] | 1. Define \( b_i = \frac{a_1 + a_2 + \ldots + a_i}{1 + 2 + \ldots + i} \).
2. We need to find the largest \( k \) such that \( b_i \) is an integer greater than 1 for at least \( k \) distinct \( i \) with \( 1 \le i \le 2022 \).
### Achievability of \( k = 1011 \)
3. Consider the permutation where \( a_i = 2i \) for ... | 1011 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The set $S = \{1, 2, \dots, 2022\}$ is to be partitioned into $n$ disjoint subsets $S_1, S_2, \dots, S_n$ such that for each $i \in \{1, 2, \dots, n\}$, exactly one of the following statements is true:
(a) For all $x, y \in S_i$, with $x \neq y, \gcd(x, y) > 1.$
(b) For all $x, y \in S_i$, with $x \neq y, \gcd(x, y) =... | To solve this problem, we need to partition the set \( S = \{1, 2, \dots, 2022\} \) into \( n \) disjoint subsets \( S_1, S_2, \dots, S_n \) such that each subset satisfies one of the following conditions:
- (a) For all \( x, y \in S_i \) with \( x \neq y \), \(\gcd(x, y) > 1\).
- (b) For all \( x, y \in S_i \) with \(... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A finite set $M$ of real numbers has the following properties: $M$ has at least $4$ elements, and there exists a bijective function $f:M\to M$, different from the identity, such that $ab\leq f(a)f(b)$ for all $a\neq b\in M.$ Prove that the sum of the elements of $M$ is $0.$ | 1. Let \( M = \{a_1, a_2, \ldots, a_n\} \) be the set of real numbers, where \( n \geq 4 \). There exists a bijective function \( f: M \to M \) such that \( ab \leq f(a)f(b) \) for all \( a \neq b \in M \).
2. Since \( f \) is bijective, we can denote \( f(a_i) = b_i \) for \( i = 1, 2, \ldots, n \). Thus, \( M = \{b_... | 0 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
At first, on a board, the number $1$ is written $100$ times. Every minute, we pick a number $a$ from the board, erase it, and write $a/3$ thrice instead. We say that a positive integer $n$ is [i]persistent[/i] if after any amount of time, regardless of the numbers we pick, we can find at least $n$ equal numbers on the ... | 1. **Claim**: The greatest persistent number \( n \) is \( 67 \).
2. **Proof**: We need to show two things:
- We can always find at least \( 67 \) equal numbers on the board.
- There exists a sequence of choices that leads to no more than \( 67 \) equal numbers.
### Part 1: At least 67 equal numbers
3. **Assum... | 67 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For every positive integer $N\geq 2$ with prime factorisation $N=p_1^{a_1}p_2^{a_2}\cdots p_k^{a_k}$ we define \[f(N):=1+p_1a_1+p_2a_2+\cdots+p_ka_k.\] Let $x_0\geq 2$ be a positive integer. We define the sequence $x_{n+1}=f(x_n)$ for all $n\geq 0.$ Prove that this sequence is eventually periodic and determine its fund... | 1. **Define the function \( f(N) \):**
Given a positive integer \( N \geq 2 \) with prime factorization \( N = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k} \), the function \( f(N) \) is defined as:
\[
f(N) = 1 + p_1 a_1 + p_2 a_2 + \cdots + p_k a_k
\]
2. **Behavior of \( f(N) \) for composite numbers:**
For a ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
We are given some three element subsets of $\{1,2, \dots ,n\}$ for which any two of them have at most one common element. We call a subset of $\{1,2, \dots ,n\}$ [i]nice [/i] if it doesn't include any of the given subsets. If no matter how the three element subsets are selected in the beginning, we can add one more ele... | 1. **Assume otherwise**: Suppose there is a 29-element nice subset \( S \) such that we cannot add any element to \( S \) while keeping it nice. This implies that for each potential added element, there is a violating subset.
2. **Counting violating subsets**: Each violating subset must include two elements from the ... | 436 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a table there are $100$ red and $k$ white buckets for which all of them are initially empty. In each move, a red and a white bucket is selected and an equal amount of water is added to both of them. After some number of moves, there is no empty bucket and for every pair of buckets that are selected together at least... | 1. **Generalize the problem to \(a\) red and \(b\) white buckets:**
- We have \(a\) red buckets and \(b\) white buckets.
- In each move, we select one red bucket and one white bucket and add an equal amount of water to both.
2. **Analyze the condition for no empty buckets:**
- After some number of moves, ther... | 100 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a circle, 2022 points are chosen such that distance between two adjacent points is always the same. There are $k$ arcs, each having endpoints on chosen points, with different lengths. Arcs do not contain each other. What is the maximum possible number of $k$? | 1. **Numbering the Points:**
Let's number the points on the circle as \(1, 2, \ldots, 2022\). The distance between any two adjacent points is the same.
2. **Choosing Arcs:**
We need to choose arcs such that each arc has a different length and no arc contains another. We can choose arcs starting from point \(1\) ... | 1011 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ be a non-negative real number and a sequence $(u_n)$ defined as:
$u_1=6,u_{n+1} = \frac{2n+a}{n} + \sqrt{\frac{n+a}{n}u_n+4}, \forall n \ge 1$
a) With $a=0$, prove that there exist a finite limit of $(u_n)$ and find that limit
b) With $a \ge 0$, prove that there exist a finite limit of $(u_n)$ | ### Part (a)
Given \( a = 0 \), we need to prove that there exists a finite limit of \( (u_n) \) and find that limit.
1. **Initial Condition and Recurrence Relation:**
\[
u_1 = 6, \quad u_{n+1} = \frac{2n}{n} + \sqrt{\frac{n}{n}u_n + 4} = 2 + \sqrt{u_n + 4}
\]
2. **Establishing a Lower Bound:**
\[
u_n ... | 5 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the greatest integer $k\leq 2023$ for which the following holds: whenever Alice colours exactly $k$ numbers of the set $\{1,2,\dots, 2023\}$ in red, Bob can colour some of the remaining uncoloured numbers in blue, such that the sum of the red numbers is the same as the sum of the blue numbers.
Romania | 1. **Define the problem and variables:**
Let \( n = 2023 \) and \( t = k_{\max} \) be the required maximum. We need to find the greatest integer \( k \leq 2023 \) such that whenever Alice colors exactly \( k \) numbers of the set \(\{1, 2, \dots, 2023\}\) in red, Bob can color some of the remaining uncolored numbers... | 673 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given a positive integer $N$ (written in base $10$), define its [i]integer substrings[/i] to be integers that are equal to strings of one or more consecutive digits from $N$, including $N$ itself. For example, the integer substrings of $3208$ are $3$, $2$, $0$, $8$, $32$, $20$, $320$, $208$, $3208$. (The substring $08$... | 1. **Define the problem and notation:**
Given a positive integer \( N \) written in base 10, we need to find the greatest integer \( N \) such that no integer substring of \( N \) is a multiple of 9.
2. **Define integer substrings:**
Integer substrings of \( N \) are integers formed by consecutive digits of \( ... | 88888888 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider an integrable function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $af(a)+bf(b)=0$ when $ab=1$. Find the value of the following integration:
$$ \int_{0}^{\infty} f(x) \,dx $$ | 1. Given the condition \( af(a) + bf(b) = 0 \) when \( ab = 1 \), we can rewrite this as:
\[
af(a) + \frac{1}{a} f\left(\frac{1}{a}\right) = 0
\]
This implies:
\[
a f(a) = -\frac{1}{a} f\left(\frac{1}{a}\right)
\]
Multiplying both sides by \( a \), we get:
\[
a^2 f(a) = -f\left(\frac{1}{a}... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In a school there are $1200$ students. Each student is part of exactly $k$ clubs. For any $23$ students, they are part of a common club. Finally, there is no club to which all students belong. Find the smallest possible value of $k$. | 1. **Understanding the Problem:**
- There are 1200 students.
- Each student is part of exactly \( k \) clubs.
- Any group of 23 students shares at least one common club.
- No club includes all 1200 students.
- We need to find the smallest possible value of \( k \).
2. **Constructing an Example for \( k ... | 23 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the greatest common divisor of the numbers $p^6-7p^2+6$ where $p$ runs through the prime numbers $p \ge 11$. | To determine the greatest common divisor (GCD) of the numbers \( f(p) = p^6 - 7p^2 + 6 \) where \( p \) runs through the prime numbers \( p \ge 11 \), we will follow these steps:
1. **Factorize the polynomial \( f(p) \):**
\[
f(p) = p^6 - 7p^2 + 6
\]
Notice that \( f(p) \) can be rewritten as:
\[
f(p... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: "does $m$ divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Vic... | To determine the minimum number of questions Victor needs to guarantee finding William's number, we need to consider the prime factorization of numbers between 1 and 50. The key idea is to use the smallest number of questions to uniquely identify any number in this range.
1. **Prime Numbers Between 1 and 50**:
The ... | 15 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the least positive integer $n{}$ for which the following statement is true: the product of any $n{}$ odd consecutive positive integers is divisible by $45$. | To determine the least positive integer \( n \) for which the product of any \( n \) odd consecutive positive integers is divisible by \( 45 \), we need to ensure that the product is divisible by both \( 3^2 = 9 \) and \( 5 \).
1. **Divisibility by \( 3^2 = 9 \):**
- For any set of \( n \) consecutive odd integers... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$. He writes the number $0$ on the board and invites Dima to play a game. Dima tells the integer $0 < p < 10, p$ can vary from round to round. Sasha chooses $p$ cards before which he puts a “$+$” sign, and before the other cards he puts a “$-$" sign. The obtained... | 1. **Understanding the Problem:**
Sasha has 10 cards with numbers \(1, 2, 4, 8, \ldots, 512\). Initially, the number on the board is 0. In each round, Dima chooses an integer \(0 < p < 10\), and Sasha chooses \(p\) cards to put a "+" sign before and the remaining \(10-p\) cards to put a "-" sign before. The sum of t... | 1023 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $n \leq 100$ be an integer. Hare puts real numbers in the cells of a $100 \times 100$ table. By asking Hare one question, Wolf can find out the sum of all numbers of a square $n \times n$, or the sum of all numbers of a rectangle $1 \times (n - 1)$ (or $(n - 1) \times 1$). Find the greatest $n{}$ such that, after s... | 1. **Understanding the Problem:**
- We have a \(100 \times 100\) table filled with real numbers.
- Wolf can ask for the sum of numbers in an \(n \times n\) square or a \(1 \times (n-1)\) or \((n-1) \times 1\) rectangle.
- We need to find the largest \(n\) such that Wolf can determine the numbers in all cells o... | 51 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A class consists of 26 students with two students sitting on each desk. Suddenly, the students decide to change seats, such that every two students that were previously sitting together are now apart. Find the maximum value of positive integer $N$ such that, regardless of the students' sitting positions, at the end the... | 1. **Model the Problem as a Graph:**
- Consider each student as a vertex in a graph \( G \).
- Each desk with two students sitting together can be represented as an edge between two vertices.
- Since there are 26 students and each desk has 2 students, we have 13 edges in the graph.
2. **Graph Properties:**
... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the period of the repetend of the fraction $\frac{39}{1428}$ by using [i]binary[/i] numbers, i.e. its binary decimal representation.
(Note: When a proper fraction is expressed as a decimal number (of any base), either the decimal number terminates after finite steps, or it is of the form $0.b_1b_2\cdots b_sa_1a_2... | 1. **Reduce the fraction**: First, we reduce the fraction $\frac{39}{1428}$. We find the greatest common divisor (GCD) of 39 and 1428, which is 3. Thus, we can simplify the fraction as follows:
\[
\frac{39}{1428} = \frac{39 \div 3}{1428 \div 3} = \frac{13}{476}
\]
2. **Factorize the denominator**: Next, we fa... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a $2023 \times 2023$ square grid, there are beetles on some of the unit squares, with at most one beetle on each unit square. In the first minute, every beetle will move one step to its right or left adjacent square. In the second minute, every beetle will move again, only this time, in case the beetle moved righ... | 1. **Color the grid in a checkerboard pattern:**
- Consider a $2023 \times 2023$ grid. Color the grid in a checkerboard pattern such that no two adjacent cells have the same color. This means each cell is either black or white, and each row and column alternates between black and white cells.
2. **Define happy and ... | 4088485 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
3. We have a $n \times n$ board. We color the unit square $(i,j)$ black if $i=j$, red if $i<j$ and green if $i>j$. Let $a_{i,j}$ be the color of the unit square $(i,j)$. In each move we switch two rows and write down the $n$-tuple $(a_{1,1},a_{2,2},\cdots,a_{n,n})$. How many $n$-tuples can we obtain by repeating this p... | 1. **Understanding the Problem:**
We have an \( n \times n \) board where each unit square \((i,j)\) is colored based on the following rules:
- Black if \( i = j \)
- Red if \( i < j \)
- Green if \( i > j \)
We are interested in the \( n \)-tuple \((a_{1,1}, a_{2,2}, \ldots, a_{n,n})\) obtained by swit... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
2000 people are sitting around a round table. Each one of them is either a truth-sayer (who always tells the truth) or a liar (who always lies). Each person said: "At least two of the three people next to me to the right are liars". How many truth-sayers are there in the circle? | 1. **Understanding the Statements:**
- Each person at the table makes the statement: "At least two of the three people next to me to the right are liars."
- We need to determine the number of truth-sayers (T) and liars (L) based on this statement.
2. **Analyzing the Statements:**
- If a person is a truth-saye... | 666 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A regular polygon with $20$ vertices is given. Alice colors each vertex in one of two colors. Bob then draws a diagonal connecting two opposite vertices. Now Bob draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints. He gets a fish from Alice for each such segment he... | 1. **Initial Setup and Assumptions:**
- We have a regular polygon with 20 vertices.
- Alice colors each vertex in one of two colors.
- Bob draws a diagonal connecting two opposite vertices and then draws perpendicular segments to this diagonal, each segment having vertices of the same color as endpoints.
- ... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer, such that the sum and the product of it and $10$ are both square numbers. | To find the smallest positive integer \( a \) such that both the sum \( 10 + a \) and the product \( 10a \) are perfect squares, we can proceed as follows:
1. **Set up the equations:**
Let \( 10 + a = b^2 \) and \( 10a = c^2 \), where \( b \) and \( c \) are integers.
2. **Express \( a \) in terms of \( b \):**
... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer, if the number of $2$'s in its digits is greater than the number of $3$'s in its digits, we call that is a [b]good[/b] number. And if the number of $3$'s in its digits is more than the number of $2$'s in its digits, we call that is a [b]bad[/b] number. For example, there are two $2$'s and one $... | 1. **Define the sets and functions:**
- Let \( G(m,n) = \{ x \in \mathbb{N} \mid m \le x \le n; \; x \text{ is a good number} \} \)
- Let \( g(m,n) = |G(m,n)| \)
- Let \( B(m,n) = \{ y \in \mathbb{N} \mid m \le y \le n; \; y \text{ is a bad number} \} \)
- Let \( b(m,n) = |B(m,n)| \)
- We need to find \(... | 22 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $\{a_1,a_2,\ldots,a_7\}$ is a set of pair-wisely different positive integers. If $a_1,2a_2,\ldots,7a_7$ can form an arithmetic series (in this order), find the smallest positive value of $|a_7-a_1|$. | Given that $\{a_1, a_2, \ldots, a_7\}$ is a set of pairwise different positive integers, and $a_1, 2a_2, \ldots, 7a_7$ can form an arithmetic series, we need to find the smallest positive value of $|a_7 - a_1|$.
1. **Define the Arithmetic Series:**
Let the common difference of the arithmetic series be $d$. Then the... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On $5\times 5$ squares, we cover the area with several S-Tetrominos (=Z-Tetrominos) along the square so that in every square, there are two or fewer tiles covering that (tiles can be overlap). Find the maximum possible number of squares covered by at least one tile. | 1. **Understanding the Problem:**
We need to cover a $5 \times 5$ grid using S-Tetrominos (Z-Tetrominos) such that each square is covered by at most two tiles. We aim to find the maximum number of squares that can be covered by at least one tile.
2. **Coloring the Grid:**
Consider a $5 \times 5$ grid. We can col... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S=\{1,2,\dots,3000\}$. Determine the maximum possible integer $X$ that satisfies the condition:
For all bijective function $f:S\rightarrow S$, there exists bijective function $g:S\rightarrow S$ such that
$$\displaystyle\sum_{k=1}^{3000}\left(\max\{f(f(k)),f(g(k)),g(f(k)),g(g(k))\}-\min\{f(f(k)),f(g(k)),g(f(k)),g(g... | To determine the maximum possible integer \( X \) that satisfies the given condition, we need to analyze the sum involving the maximum and minimum values of the functions \( f \) and \( g \).
1. **Upper Bound Analysis:**
Consider the identity function \( f(x) = x \). We need to evaluate the sum:
\[
\sum_{k=1... | 6000000 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A marble is placed on each $33$ unit square of a $10*10$ chessboard. After that, the number of marbles in the same row or column with that square is written on each of the remaining empty unit squares. What is the maximum sum of the numbers written on the board?
| 1. **Define Variables and Initial Setup:**
Let \( x_i \) and \( y_i \) be the number of marbles in the \( i \)-th row and column of the board, respectively. Note that since there are 33 marbles on the board, we have:
\[
\sum_{i=1}^{10} x_i = \sum_{i=1}^{10} y_i = 33
\]
The total sum of the numbers writte... | 438 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$. | Given the equations:
\[ a^2(b+c) = b^2(c+a) = 2023 \]
1. **Subtract the two given equations:**
\[ a^2(b+c) - b^2(c+a) = 0 \]
This simplifies to:
\[ a^2b + a^2c - b^2c - b^2a = 0 \]
Factor out common terms:
\[ a^2b + a^2c - b^2c - b^2a = a^2b - b^2a + a^2c - b^2c = a(a-b)(b+c) + c(a^2 - b^2) = 0 \]
Si... | 2023 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$, such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube. | To determine the smallest natural number \( n \) for which there exist distinct nonzero naturals \( a, b, c \) such that \( n = a + b + c \) and \((a + b)(b + c)(c + a)\) is a perfect cube, we can proceed as follows:
1. **Assume \( \gcd(a, b, c, n) = d \)**:
Let \( a = dx \), \( b = dy \), \( c = dz \), and \( n = ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a cube $3 \times 3 \times 3$ with $27$ unit cubes. In each such cube a positive integer is written. Call a $\textit {strip}$ a block $1 \times 1 \times 3$ of $3$ cubes. The numbers are written so that for each cube, its number is the sum of three other numbers, one from each of the three strips it is in. Prove... | 1. **Initial Assumptions and Definitions**:
- We are given a $3 \times 3 \times 3$ cube with $27$ unit cubes.
- Each unit cube contains a positive integer.
- A strip is defined as a $1 \times 1 \times 3$ block of 3 cubes.
- The number in each cube is the sum of three other numbers, one from each of the thre... | 16 | Logic and Puzzles | proof | Yes | Yes | aops_forum | false |
There are $n$ students in a class, and some pairs of these students are friends. Among any six students, there are two of them that are not friends, and for any pair of students that are not friends there is a student among the remaining four that is friends with both of them. Find the maximum value of $n$. | 1. **Understanding the problem**: We need to find the maximum number of students \( n \) in a class such that among any six students, there are two who are not friends, and for any pair of students that are not friends, there is a student among the remaining four who is friends with both of them.
2. **Example construc... | 25 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Manzi has $n$ stamps and an album with $10$ pages. He distributes the $n$ stamps in the album such that each page has a distinct number of stamps. He finds that, no matter how he does this, there is always a set of $4$ pages such that the total number of stamps in these $4$ pages is at least $\frac{n}{2}$. Determine th... | To solve this problem, we need to determine the maximum possible value of \( n \) such that no matter how Manzi distributes the \( n \) stamps across the 10 pages, there is always a set of 4 pages whose total number of stamps is at least \( \frac{n}{2} \).
1. **Understanding the Distribution**:
Each page must have ... | 135 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A set of positive integers is said to be [i]pilak[/i] if it can be partitioned into 2 disjoint subsets $F$ and $T$, each with at least $2$ elements, such that the elements of $F$ are consecutive Fibonacci numbers, and the elements of $T$ are consecutive triangular numbers. Find all positive integers $n$ such that the s... | To solve the problem, we need to find all positive integers \( n \) such that the set of all positive divisors of \( n \) except \( n \) itself can be partitioned into two disjoint subsets \( F \) and \( T \), where \( F \) consists of consecutive Fibonacci numbers and \( T \) consists of consecutive triangular numbers... | 30 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all positive integers $b$ with the following property: there exists positive integers $a,k,l$ such that $a^k + b^l$ and $a^l + b^k$ are divisible by $b^{k+l}$ where $k \neq l$. | 1. We need to find all positive integers \( b \) such that there exist positive integers \( a, k, l \) with \( k \neq l \) and both \( a^k + b^l \) and \( a^l + b^k \) are divisible by \( b^{k+l} \).
2. Consider the \( p \)-adic valuation \( \nu_p \) for a prime \( p \). Recall that for any integers \( x \) and \( y \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given is a rectangle with perimeter $x$ cm and side lengths in a $1:2$ ratio. Suppose that the area of the rectangle is also $x$ $\text{cm}^2$. Determine all possible values of $x$. | 1. Let \( a \) and \( b \) be the side lengths of the rectangle, such that \( a = 2b \).
2. The perimeter of the rectangle is given by:
\[
P = 2(a + b)
\]
Since the perimeter is \( x \) cm, we have:
\[
x = 2(a + b)
\]
3. The area of the rectangle is given by:
\[
A = ab
\]
Since the ar... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A natural number $n$ is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number $n$. Find all numbers $n$ with this property. | 1. Let \( n \) be a two-digit number, which can be expressed as \( n = 10a + b \), where \( a \) and \( b \) are digits, i.e., \( 1 \leq a \leq 9 \) and \( 0 \leq b \leq 9 \).
2. According to the problem, if we insert a certain digit \( c \) between the tens digit \( a \) and the units digit \( b \) of \( n \), we obt... | 18 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given the sequence $(u_n)$ satisfying:$$\left\{ \begin{array}{l}
1 \le {u_1} \le 3\\
{u_{n + 1}} = 4 - \dfrac{{2({u_n} + 1)}}{{{2^{{u_n}}}}},\forall n \in \mathbb{Z^+}.
\end{array} \right.$$
Prove that: $1\le u_n\le 3,\forall n\in \mathbb{Z^+}$ and find the limit of $(u_n).$ | 1. **Define the functions and initial conditions:**
Given the sequence \((u_n)\) defined by:
\[
\begin{cases}
1 \le u_1 \le 3 \\
u_{n+1} = 4 - \frac{2(u_n + 1)}{2^{u_n}}, \forall n \in \mathbb{Z^+}
\end{cases}
\]
We need to prove that \(1 \le u_n \le 3\) for all \(n \in \mathbb{Z^+}\) and find t... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We color all vertexs of a convex polygon with $10$ vertexs by $2$ colors: red and blue $($each vertex is colored by $1$ color$).$
How many ways to color all the vertexs such that there are no $2$ adjacent vertex that are both colored red? | To solve the problem of counting the number of ways to color the vertices of a convex polygon with 10 vertices such that no two adjacent vertices are both colored red, we can use a combinatorial approach involving binary sequences.
1. **Define the problem in terms of binary sequences:**
- Let \( a_n \) be the numbe... | 123 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We say that a number $n \ge 2$ has the property $(P)$ if, in its prime factorization, at least one of the factors has an exponent $3$.
a) Determine the smallest number $N$ with the property that, no matter how we choose $N$ consecutive natural numbers, at least one of them has the property $(P).$
b) Determine the sm... | ### Part (a)
1. **Define the Property (P):**
A number \( n \geq 2 \) has the property \( (P) \) if, in its prime factorization, at least one of the factors has an exponent of 3.
2. **Determine the smallest \( N \):**
We need to find the smallest \( N \) such that in any set of \( N \) consecutive natural number... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Determine the largest natural number $k$ such that there exists a natural number $n$ satisfying:
\[
\sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k).
\] | 1. We start with the inequality given in the problem:
\[
\sin(n + 1) < \sin(n + 2) < \sin(n + 3) < \ldots < \sin(n + k).
\]
2. To analyze this, we need to understand the behavior of the sine function. The sine function is periodic with period \(2\pi\) and oscillates between -1 and 1. The function \(\sin(x)... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the plane, $2022$ points are chosen such that no three points lie on the same line. Each of the points is coloured red or blue such that each triangle formed by three distinct red points contains at least one blue point.
What is the largest possible number of red points?
[i]Proposed by Art Waeterschoot, Belgium[/i] | 1. **Understanding the Problem:**
We are given 2022 points in the plane, no three of which are collinear. Each point is colored either red or blue. The condition is that any triangle formed by three red points must contain at least one blue point inside it. We need to find the maximum number of red points possible u... | 1012 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given is a cube of side length $2021$. In how many different ways is it possible to add somewhere on the boundary of this cube a $1\times 1\times 1$ cube in such a way that the new shape can be filled in with $1\times 1\times k$ shapes, for some natural number $k$, $k\geq 2$? | 1. **Define the Problem and Initial Setup:**
We are given a cube of side length \(2021\). We need to determine the number of ways to add a \(1 \times 1 \times 1\) cube to the boundary of this cube such that the resulting shape can be filled with \(1 \times 1 \times k\) cuboids for some natural number \(k \geq 2\).
... | 13612182 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo).
What is the maximal possible number of edges in such a graph? | 1. **Understanding the Problem:**
We are given a simple graph with 300 vertices where no two vertices of the same degree are adjacent. We need to find the maximal possible number of edges in such a graph.
2. **Claim:**
The maximal possible number of edges is given by:
\[
\frac{1}{2} (299 + 298 \cdot 2 + \c... | 42550 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest possible value of the expression $$\frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a}$$ where $a,b,c \in \mathbb{R}$ satisfy $a+b+c = -1$ and $abc \leqslant -3$
| To determine the smallest possible value of the expression
\[ S = \frac{ab+1}{a+b} + \frac{bc+1}{b+c} + \frac{ca+1}{c+a} \]
where \(a, b, c \in \mathbb{R}\) satisfy \(a+b+c = -1\) and \(abc \leq -3\), we proceed as follows:
1. **Substitute \(1\) with \((a+b+c)^2\):**
Since \(a+b+c = -1\), we have:
\[ 1 = (a+b+c... | 3 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There is an equilateral triangle $ABC$ on the plane. Three straight lines pass through $A$, $B$ and $C$, respectively, such that the intersections of these lines form an equilateral triangle inside $ABC$. On each turn, Ming chooses a two-line intersection inside $ABC$, and draws the straight line determined by the inte... | 1. **Understanding the Problem:**
- We start with an equilateral triangle \(ABC\).
- Three lines pass through vertices \(A\), \(B\), and \(C\) respectively, forming an equilateral triangle inside \(ABC\).
- On each turn, Ming chooses an intersection point of two lines inside \(ABC\) and draws a new line throug... | 45853 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $10$ cups, each having $10$ pebbles in them. Two players $A$ and $B$ play a game, repeating the following in order each move:
$\bullet$ $B$ takes one pebble from each cup and redistributes them as $A$ wishes.
$\bullet$ After $B$ distributes the pebbles, he tells how many pebbles are in each cup to $A$. The... | 1. **Initial Setup:**
- There are 10 cups, each containing 10 pebbles.
- Players A and B take turns according to the rules specified.
2. **Strategy of Player A:**
- Player A instructs Player B to distribute the pebbles such that the cup with the most pebbles gets $\left\lfloor \frac{k}{2} \right\rfloor$ pebbl... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$?
[i]Proposed by Oleksiy Masalitin[/i] | 1. **Assume the largest value:**
Let \( a_{101} \) be the largest of the \( a_i \), and denote \( a_{101} = k \). We aim to find the maximum possible value of \( k \).
2. **Claim:**
We claim that \( a_{101-i} = k - i \) for all \( 0 \leq i \leq 100 \).
3. **Base case:**
For \( i = 0 \), we have \( a_{101-0} ... | 201 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
On a rectangular board $100 \times 300$, two people take turns coloring the cells that have not yet been colored. The first one colors cells in yellow, and the second one in blue. Coloring is completed when every cell of the board is colored. A [i]connected sequence[/i] of cells is a sequence of cells in which every tw... | 1. **Initial Setup and Strategy of the Second Player**:
- Consider the $100 \times 300$ rectangular board.
- The second player can split the board into horizontal $1 \times 2$ domino rectangles. This means each domino covers two adjacent cells in the same row.
2. **Second Player's Response**:
- Whenever the f... | 200 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Set $M$ contains $n \ge 2$ positive integers. It's known that for any two different $a, b \in M$, $a^2+1$ is divisible by $b$. What is the largest possible value of $n$?
[i]Proposed by Oleksiy Masalitin[/i] | 1. **Assume $M$ contains $n$ positive integers $a_1, a_2, \ldots, a_n$.**
We are given that for any two different $a, b \in M$, $a^2 + 1$ is divisible by $b$. This implies that for any $a_i, a_j \in M$ with $i \neq j$, $a_i^2 + 1$ is divisible by $a_j$.
2. **Consider the smallest element $a_1$ in $M$.**
Withou... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $2024$ cities in a country, every two of which are bidirectionally connected by exactly one of three modes of transportation - rail, air, or road. A tourist has arrived in this country and has the entire transportation scheme. He chooses a travel ticket for one of the modes of transportation and the city from... | 1. **Understanding the Problem:**
We have a complete graph \( K_{2024} \) where each edge is colored with one of three colors (representing rail, air, or road). The goal is to determine the largest number \( k \) such that the tourist can always visit at least \( k \) cities using only one type of transportation.
2... | 1012 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider pairs of functions $(f, g)$ from the set of nonnegative integers to itself such that
[list]
[*] $f(0) + f(1) + f(2) + \cdots + f(42) \le 2022$;
[*] for any integers $a \ge b \ge 0$, we have $g(a+b) \le f(a) + f(b)$.
[/list]
Determine the maximum possible value of $g(0) + g(1) + g(2) + \cdots + g(84)$ over all ... | To determine the maximum possible value of \( g(0) + g(1) + g(2) + \cdots + g(84) \) over all pairs of functions \((f, g)\) that satisfy the given conditions, we need to carefully analyze the constraints and derive the optimal functions \(f\) and \(g\).
1. **Constraints Analysis**:
- The first constraint is:
\... | 7993 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an equilateral triangle with side length $1$. Points $A_1$ and $A_2$ are chosen on side $BC$, points $B_1$ and $B_2$ are chosen on side $CA$, and points $C_1$ and $C_2$ are chosen on side $AB$ such that $BA_1<BA_2$, $CB_1<CB_2$, and $AC_1<AC_2$.
Suppose that the three line segments $B_1C_2$, $C_1A_2$, $A_... | 1. **Setup and Initial Conditions:**
- Let $ABC$ be an equilateral triangle with side length $1$.
- Points $A_1$ and $A_2$ are chosen on side $BC$ such that $BA_1 < BA_2$.
- Points $B_1$ and $B_2$ are chosen on side $CA$ such that $CB_1 < CB_2$.
- Points $C_1$ and $C_2$ are chosen on side $AB$ such that $AC... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The Bank of Pittsburgh issues coins that have a heads side and a tails side. Vera has a row of 2023 such coins alternately tails-up and heads-up, with the leftmost coin tails-up.
In a [i]move[/i], Vera may flip over one of the coins in the row, subject to the following rules:
[list=disc]
[*] On the first move, Vera ma... | 1. **Initial Setup**: Vera has a row of 2023 coins, alternately tails-up and heads-up, starting with the leftmost coin tails-up. This means the sequence is T, H, T, H, ..., T, H.
2. **First Move**: On the first move, Vera can flip any coin. Let's denote the coins as \( C_1, C_2, \ldots, C_{2023} \). The first move can... | 4044 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
There are $n \geq 2$ classes organized $m \geq 1$ extracurricular groups for students. Every class has students participating in at least one extracurricular group. Every extracurricular group has exactly $a$ classes that the students in this group participate in. For any two extracurricular groups, there are no more ... | ### Part (a)
Given:
- \( n = 8 \)
- \( a = 4 \)
- \( b = 1 \)
We need to find \( m \).
1. **Define the 3-tuple and calculate \( N \)**:
- Let the classes be \( c_1, c_2, \ldots, c_n \).
- Let the extracurricular groups be \( G_1, G_2, \ldots, G_m \).
- Define the 3-tuple \((G_i, G_j, c_k)\) if class \( c_k \... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1<x_2< \ldots <x_{2024}$ be positive integers and let $p_i=\prod_{k=1}^{i}(x_k-\frac{1}{x_k})$ for $i=1,2, \ldots, 2024$. What is the maximal number of positive integers among the $p_i$? | 1. **Define the sequence and initial conditions:**
Let \( x_1, x_2, \ldots, x_{2024} \) be positive integers such that \( x_1 < x_2 < \ldots < x_{2024} \). Define \( p_i = \prod_{k=1}^{i} \left( x_k - \frac{1}{x_k} \right) \) for \( i = 1, 2, \ldots, 2024 \).
2. **Consider the sequence \( x_i = x_{i-1} + 1 \) with ... | 1012 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$. Every knight answered truthfully, while every liar changed the real answer by exactly $1$. What is the minimal number of liars? | 1. **Restate the problem in graph theory terms**:
- We have a simple graph \( G \) with 2024 vertices.
- Each vertex represents a person, and the degree of each vertex represents the number of friends that person has.
- Knights tell the truth about their number of friends, while liars change their answer by ex... | 1012 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
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