problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$ | 1. **Given Conditions and Polynomial Properties:**
We are given that \( P(x) \) is a polynomial with integer coefficients such that:
\[
P(-10) = 145 \quad \text{and} \quad P(9) = 164.
\]
We need to find the smallest possible value of \( |P(0)| \).
2. **Using Polynomial Properties:**
It is well known ... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $m$ and $n$ are positive integers with $m < n$ such that the interval $[m, n)$ contains more multiples of $2021$ than multiples of $2000$. Compute the maximum possible value of $n - m$. | 1. **Define the problem in terms of multiples:**
Let \( k \) be the number of multiples of 2000 in the interval \([m, n)\). Since there are more multiples of 2021 than 2000, there must be \( k+1 \) multiples of 2021 in the same interval.
2. **Determine the length of the interval for multiples of 2000:**
The \( k... | 1999 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy
\[x+y+z = -\frac{\sqrt{3}}{2}-i\sqrt{5}\]
and
\[xyz=\sqrt{3} + i\sqrt{5}.\]
If $x=x_1+ix_2, y=y_1+iy_2,$ and $z=z_1+iz_2$ for real $x_1,x_2,y_1,y_2,z_1$ and $z_2$ then
\[(x_1x_2+y_1y_2+z_1z_2)^2\]
can be written as $\tfrac{a}{b}$ for r... | 1. **Determine the magnitude of \(x\), \(y\), and \(z\):**
Given that \(x\), \(y\), and \(z\) are complex numbers of equal magnitude, and \(xyz = \sqrt{3} + i\sqrt{5}\), we can find the magnitude of these numbers. Let \( |x| = |y| = |z| = r \). Then,
\[
|xyz| = |x| \cdot |y| \cdot |z| = r^3.
\]
Since \( ... | 1516 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Leo the fox has a $5$ by $5$ checkerboard grid with alternating red and black squares. He fills in the
grid with the numbers $1, 2, 3, \dots, 25$ such that any two consecutive numbers are in adjacent squares
(sharing a side) and each number is used exactly once. He then computes the sum of the numbers in
the $13$ squar... | 1. **Identify the color of the center square:**
- The center square of a \(5 \times 5\) checkerboard grid is the square at position \((3, 3)\).
- Since the grid alternates colors, and assuming the top-left corner is red, the center square \((3, 3)\) will also be red.
2. **Determine the color pattern:**
- The ... | 169 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be a positive integer. Brothers Michael and Kylo each select a positive integer less than or equal to $N$, independently and uniformly at random. Let $p_N$ denote the probability that the product of these two integers has a units digit of $0$. The maximum possible value of $p_N$ over all possible choices of $N$... | 1. **Understanding the Problem:**
We need to find the probability that the product of two randomly chosen integers from 1 to \( N \) has a units digit of 0. This happens if and only if at least one of the integers is divisible by 2 and at least one is divisible by 5.
2. **Probability Calculation:**
Let \( A \) b... | 2800 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1, 2, \dots, 9\}.$ Compute the number of functions $f : S \rightarrow S$ such that, for all $s \in S, f(f(f(s))) =s$ and $f(s) - s$ is not divisible by $3$. | 1. **Understanding the Problem:**
We need to find the number of functions \( f: S \rightarrow S \) such that for all \( s \in S \), \( f(f(f(s))) = s \) and \( f(s) - s \) is not divisible by 3. The set \( S \) is \( \{1, 2, \dots, 9\} \).
2. **Graph Representation:**
Consider the directed graph with vertices \(... | 1728 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1, 2, \dots , 2021\}$, and let $\mathcal{F}$ denote the set of functions $f : S \rightarrow S$. For a function $f \in \mathcal{F},$ let
\[T_f =\{f^{2021}(s) : s \in S\},\]
where $f^{2021}(s)$ denotes $f(f(\cdots(f(s))\cdots))$ with $2021$ copies of $f$. Compute the remainder when
\[\sum_{f \in \mathcal{F}} |... | 1. **Understanding the Problem:**
We need to compute the sum of the sizes of the sets \( T_f \) for all functions \( f \) from \( S \) to \( S \), where \( S = \{1, 2, \dots, 2021\} \). The set \( T_f \) is defined as \( T_f = \{ f^{2021}(s) : s \in S \} \), where \( f^{2021}(s) \) denotes applying \( f \) to \( s \... | 255 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $X_0$ be the interior of a triangle with side lengths $3, 4,$ and $5$. For all positive integers $n$, define $X_n$ to be the set of points within $1$ unit of some point in $X_{n-1}$. The area of the region outside $X_{20}$ but inside $X_{21}$ can be written as $a\pi + b$, for integers $a$ and $b$. Compute $100a + b... | 1. **Understanding the Problem:**
We are given a triangle with side lengths \(3, 4,\) and \(5\). This is a right triangle. We need to find the area of the region outside \(X_{20}\) but inside \(X_{21}\), where \(X_n\) is the set of points within \(1\) unit of some point in \(X_{n-1}\).
2. **Area Calculation for \(X... | 2062 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has a right angle at $C$, and $D$ is the foot of the altitude from $C$ to $AB$. Points $L, M,$ and $N$ are the midpoints of segments $AD, DC,$ and $CA,$ respectively. If $CL = 7$ and $BM = 12,$ compute $BN^2$. | 1. **Identify the given information and setup the problem:**
- Triangle \(ABC\) is a right triangle with \(\angle C = 90^\circ\).
- \(D\) is the foot of the altitude from \(C\) to \(AB\).
- Points \(L\), \(M\), and \(N\) are the midpoints of segments \(AD\), \(DC\), and \(CA\), respectively.
- Given \(CL = ... | 193 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let ABCD be a trapezoid with $AB \parallel CD, AB = 5, BC = 9, CD = 10,$ and $DA = 7$. Lines $BC$ and $DA$ intersect at point $E$. Let $M$ be the midpoint of $CD$, and let $N$ be the intersection of the circumcircles of $\triangle BMC$ and $\triangle DMA$ (other than $M$). If $EN^2 = \tfrac ab$ for relatively prime pos... | 1. **Identify the given elements and relationships:**
- Trapezoid \(ABCD\) with \(AB \parallel CD\), \(AB = 5\), \(BC = 9\), \(CD = 10\), and \(DA = 7\).
- Lines \(BC\) and \(DA\) intersect at point \(E\).
- \(M\) is the midpoint of \(CD\).
- \(N\) is the intersection of the circumcircles of \(\triangle BMC... | 90011 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $AEF$ be a triangle with $EF = 20$ and $AE = AF = 21$. Let $B$ and $D$ be points chosen on segments $AE$ and $AF,$ respectively, such that $BD$ is parallel to $EF.$ Point $C$ is chosen in the interior of triangle $AEF$ such that $ABCD$ is cyclic. If $BC = 3$ and $CD = 4,$ then the ratio of areas $\tfrac{[ABCD]}{[AE... | 1. **Identify Similar Triangles:**
Given that $BD \parallel EF$, we can conclude that $\triangle ABD \sim \triangle AEF$ by the AA (Angle-Angle) similarity criterion. This is because $\angle BAD = \angle BAE$ and $\angle ADB = \angle AEF$.
2. **Determine the Ratio of Similar Triangles:**
Since $\triangle ABD \si... | 5300 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $OA = ON = 11$ and $OH = 7.$ Compute $BC^2$. | 1. **Identify Key Points and Properties:**
- Let \( M \) be the midpoint of \( BC \).
- Let \( H \) be the orthocenter of \( \triangle ABC \).
- Let \( O \) be the circumcenter of \( \triangle ABC \).
- Let \( N \) be the reflection of \( M \) over \( H \).
- Given: \( OA = ON = 11 \) and \( OH = 7 \).
... | 288 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $O$ and $A$ be two points in the plane with $OA = 30$, and let $\Gamma$ be a circle with center $O$ and radius $r$. Suppose that there exist two points $B$ and $C$ on $\Gamma$ with $\angle ABC = 90^{\circ}$ and $AB = BC$. Compute the minimum possible value of $\lfloor r \rfloor.$ | 1. **Fixing the triangle $\triangle ABC$**: We start by fixing the triangle $\triangle ABC$ such that $AB = BC = 2$. This is a valid assumption because we can scale the figure later to match the given conditions.
2. **Locating point $O$**: Since $O$ is the center of the circle $\Gamma$ and $B$ and $C$ lie on $\Gamma$,... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB,$ and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC.$ Suppose that $BC = 27, CD = 25,$ and $AP = 10.$ If $MP = \tfrac {a}{b}$ for relatively prime positive integers $a$ ... | 1. Given that $ABCD$ is a trapezoid with $AB \parallel CD$ and $AD = BD$. Let $M$ be the midpoint of $AB$, and let $P \neq C$ be the second intersection of the circumcircle of $\triangle BCD$ and the diagonal $AC$. We are given $BC = 27$, $CD = 25$, and $AP = 10$. We need to find $MP$ in the form $\frac{a}{b}$ for rela... | 2705 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many integers between $2$ and $100$ inclusive [i]cannot[/i] be written as $m \cdot n,$ where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$? Note that there are $25$ primes less than $100.$ | To solve the problem, we need to find the number of integers between $2$ and $100$ inclusive that cannot be written as $m \cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$.
We start by identifying the numbers that can be written in the form $p^k$, where $p$ is a prime and $k \... | 35 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the product of all real $x$ for which \[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \] | 1. Let \( y = 2^x \). Then, the given equation \( 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0 \) can be rewritten in terms of \( y \).
2. Rewrite each term in the equation using \( y \):
\[
2^{3x+1} = 2 \cdot 2^{3x} = 2 \cdot (2^x)^3 = 2y^3,
\]
\[
17 \cdot 2^{2x} = 17 \cdot (2^x)^2 = 17y^2,
\]
\[
... | -3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Evaluate the sum \[ 11^2 - 1^1 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2. \] | 1. We start by rewriting the given sum:
\[
11^2 - 1^2 + 12^2 - 2^2 + 13^2 - 3^2 + \cdots + 20^2 - 10^2.
\]
This can be expressed as:
\[
\sum_{k=1}^{10} (10+k)^2 - k^2.
\]
2. Next, we simplify each term inside the sum:
\[
(10+k)^2 - k^2 = (10+k)(10+k) - k^2 = 100 + 20k + k^2 - k^2 = 100 + 20k... | 2100 | Algebra | other | Yes | Yes | aops_forum | false |
How many subsets $A$ of $ \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \} $ have the property that no two elements of $A$ sum to $11$? | 1. Identify the pairs of elements in the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ that sum to $11$. These pairs are:
\[
(1, 10), (2, 9), (3, 8), (4, 7), (5, 6)
\]
There are 5 such pairs.
2. For each pair $(a, b)$ where $a + b = 11$, we have three choices:
- Neither $a$ nor $b$ is in the subset $A$.
- ... | 243 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are 111 StarCraft programmers. The StarCraft team SKT starts with a given set of eleven programmers on it, and at the end of each season, it drops a progamer and adds a programmer (possibly the same one). At the start of the second season, SKT has to fi eld a team of five programmers to play the opening match. H... | 1. **Initial Setup**: We start with a team of 11 programmers. At the end of the season, one programmer is dropped and another is added. This results in a new team of 11 programmers.
2. **Choosing the Programmer to Drop**: There are 11 options for the programmer to be dropped from the team.
3. **Choosing the Replaceme... | 61593840 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
7. What is the minimum value of the product $$\prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}}$$ given that $(a_1, a_2, a_3, a_4, a_5, a_6)$ is a permutation of $(1, 2, 3, 4, 5, 6)$? (note $a_7 = a_1, a_8 = a_2 \ldots$)
8. Danielle picks a positive integer $1 \le n \le 2016$ uniformly at random. What is the probabilit... | ### Problem 7
We need to find the minimum value of the product
\[ \prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}} \]
given that \((a_1, a_2, a_3, a_4, a_5, a_6)\) is a permutation of \((1, 2, 3, 4, 5, 6)\) and \(a_7 = a_1, a_8 = a_2, \ldots\).
1. Notice that the product can be rewritten as:
\[ \prod_{i=1}^6 \frac{a_... | 299 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
17. Let $... | To solve the given problem, we need to find the value of the expression \((p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha)))\) where \(\alpha\) is a root of \(p(p(p(p(x)))) = 0\) and \(p(x) = x^2 - x + 1\).
1. **Find the roots of \(p(x) = x^2 - x + 1\):**
\[
p(x) = x^2 - x + 1
\]
The roots of this quadrat... | -1 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Each person in Cambridge drinks a (possibly different) $12$ ounce mixture of water and apple juice,
where each drink has a positive amount of both liquids. Marc McGovern, the mayor of Cambridge, drinks $\frac{1}{6}$ of the total amount of water drunk and $\frac{1}{8}$ of the total amount of apple juice drunk. How many ... | 1. Let \( p \) be the number of people in Cambridge.
2. Let \( w \) be the total amount of water consumed by all people in Cambridge.
3. Let \( a \) be the total amount of apple juice consumed by all people in Cambridge.
4. Since each person drinks a 12-ounce mixture of water and apple juice, we have:
\[
w + a = ... | 7 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
| 1. Let \( P(x) = a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + a_0 \), where \( a_4, a_3, a_2, a_1, a_0 \) are nonnegative integers each less than 100.
2. Given \( P(10) = 331633 \) and \( P(-10) = 273373 \), we can write:
\[
P(10) = a_4 \cdot 10^4 + a_3 \cdot 10^3 + a_2 \cdot 10^2 + a_1 \cdot 10 + a_0 = 331633
\]
... | 100 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Altitudes $BE$ and $CF$ of acute triangle $ABC$ intersect at $H$. Suppose that the altitudes of triangle $EHF$ concur on line $BC$. If $AB=3$ and $AC=4$, then $BC^2=\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$. | 1. **Assume the altitudes of $\Delta EHF$ intersect at $M$.** Since $EM \perp BF$, we know that:
\[
\angle BEM = 90^\circ - \angle ACF = \angle BAC
\]
Similarly, since $FM \perp CE$, we have:
\[
\angle CFM = 90^\circ - \angle ABE = \angle BAC
\]
Therefore, $\Delta EMC \sim \Delta ABC$ and $\Delt... | 1301 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$ with $AB=8$ and $AC=10$, the incenter $I$ is reflected across side $AB$ to point $X$ and across side $AC$ to point $Y$. Given that segment $XY$ bisects $AI$, compute $BC^2$. (The incenter is the center of the inscribed circle of triangle .) | 1. **Reflecting the Incenter:**
- Given that the incenter \( I \) is reflected across side \( AB \) to point \( X \) and across side \( AC \) to point \( Y \), we know that:
\[
\angle XAB = \angle BAI = \angle IAC = \angle CAY = \theta
\]
and
\[
AX = AI = AY
\]
This implies th... | 84 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a parallelogram. Let $E$ be the midpoint of $AB$ and $F$ be the midpoint of $CD$. Points $P$ and $Q$ are on segments $EF$ and $CF$, respectively, such that $A, P$, and $Q$ are collinear. Given that $EP = 5$, $P F = 3$, and $QF = 12$, find $CQ$. | 1. **Identify the given information and the relationships between the points:**
- $ABCD$ is a parallelogram.
- $E$ is the midpoint of $AB$.
- $F$ is the midpoint of $CD$.
- $P$ is on segment $EF$ such that $EP = 5$ and $PF = 3$.
- $Q$ is on segment $CF$ such that $QF = 12$.
- Points $A$, $P$, and $Q$ ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = x^3 + 3x - 1$ have roots $ a, b, c$. Given that $\frac{1}{a^3 + b^3}+\frac{1}{b^3 + c^3}+\frac{1}{c^3 + a^3}$ can be written as $\frac{m}{n}$, where $m, n$ are positive integers and $gcd(m, n) = 1$, find $100m + n$.
| 1. **Identify the roots and use Vieta's formulas:**
Given the polynomial \( f(x) = x^3 + 3x - 1 \), let the roots be \( a, b, c \). By Vieta's formulas, we have:
\[
a + b + c = 0,
\]
\[
ab + bc + ca = 3,
\]
\[
abc = 1.
\]
2. **Express the sum of the reciprocals:**
We need to find:
\... | 3989 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
$ABCDE$ is a cyclic convex pentagon, and $AC = BD = CE$. $AC$ and $BD$ intersect at $X$, and $BD$ and $CE$ intersect at $Y$ . If $AX = 6$, $XY = 4$, and $Y E = 7$, then the area of pentagon $ABCDE$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $ b$, $c$ are integers, $c$ is positive, $b$ is square-free, and gcd$... | 1. **Given Information and Setup:**
- We have a cyclic convex pentagon \(ABCDE\) with \(AC = BD = CE\).
- Points of intersection: \(AC\) and \(BD\) intersect at \(X\), and \(BD\) and \(CE\) intersect at \(Y\).
- Given lengths: \(AX = 6\), \(XY = 4\), and \(YE = 7\).
2. **Coordinate System and Points:**
- A... | 2752 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$,$C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $AJG$ is an equilateral triangle, then the area o... | 1. **Define the coordinates of the square \(ABCD\):**
- Let \(A(0,0)\), \(B(3,0)\), \(C(3,3)\), and \(D(0,3)\).
2. **Determine the coordinates of the squares \(EHGF\) and \(IHJK\):**
- Let \(H(h,h)\) be a point on the line \(AC\) (since \(A\), \(C\), and \(H\) are collinear).
- Let \(E(3,e)\) be a point on \(... | 48 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1. [/b] Evaluate $S$.
$$S =\frac{10000^2 - 1}{\sqrt{10000^2 - 19999}}$$
[b]p2. [/b] Starting on a triangular face of a right triangular prism and allowing moves to only adjacent faces, how many ways can you pass through each of the other four faces and return to the first face in five moves?
[b]p3.[/b] Given th... | 1. Assume \( p \neq 2, 3 \). Then \( p \) is odd, so \( 2^{p+1} \equiv 1 \pmod{3} \).
2. Residue testing also gives \( p^3 - p^2 - p \equiv 0, 2 \pmod{3} \).
3. If \( p^3 - p^2 - p \equiv 0 \pmod{3} \), then \( p \) cannot be prime since \( p \neq 3 \).
4. If \( p^3 - p^2 - p \equiv 2 \pmod{3} \), then the whole exp... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let circle $O$ have radius $5$ with diameter $\overline{AE}$. Point $F$ is outside circle $O$ such that lines $\overline{F A}$ and $\overline{F E}$ intersect circle $O$ at points $B$ and $D$, respectively. If $F A = 10$ and $m \angle F AE = 30^o$, then the perimeter of quadrilateral ABDE can be expressed as $a + b\sqrt... | 1. **Identify Key Elements and Given Information:**
- Circle \( O \) has a radius of \( 5 \), so the diameter \( \overline{AE} \) is \( 10 \).
- Point \( F \) is outside the circle such that lines \( \overline{FA} \) and \( \overline{FE} \) intersect the circle at points \( B \) and \( D \), respectively.
- Gi... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Sherlock and Mycroft are playing Battleship on a $4\times4$ grid. Mycroft hides a single $3\times1$ cruiser somewhere on the board. Sherlock can pick squares on the grid and fire upon them. What is the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser? | To determine the smallest number of shots Sherlock has to fire to guarantee at least one hit on the cruiser, we need to consider the possible placements of the $3 \times 1$ cruiser on the $4 \times 4$ grid.
1. **Possible Placements of the Cruiser**:
- The cruiser can be placed horizontally or vertically.
- For h... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of $99$ gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each ... | To solve the problem, we need to show that the total length of the cuts in a \(25 \times 25\) checkerboard, when cut into smaller square boards, is divisible by 4.
1. **Understanding the Problem:**
- A \(25 \times 25\) checkerboard has 25 rows and 25 columns.
- We need to cut this checkerboard along the gridline... | 1200 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] How many real roots does the equation $2x^7 + x^5 + 4x^3 + x + 2 = 0$ have?
[b]p2.[/b] Given that $f(n) = 1 +\sum^n_{j=1}(1 +\sum^j_{i=1}(2i + 1))$, find the value of $f(99)-\sum^{99}_{i=1} i^2$.
[b]p3.[/b] A rectangular prism with dimensions $1\times a \times b$, where $1 < a < b < 2$, is bisected by a ... | To solve the problem, we need to find the value of \( q \) in the polynomial \( x^4 + px^3 + qx^2 + rx + s \) given the conditions on the roots.
1. **Identify the roots and their properties:**
- The polynomial has four non-real roots.
- The sum of two of the roots is \( 4 + 7i \).
- The product of the other t... | 71 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 1[/u]
[b]p1.[/b] If this mathathon has $7$ rounds of $3$ problems each, how many problems does it have in total? (Not a trick!)
[b]p2.[/b] Five people, named $A, B, C, D,$ and $E$, are standing in line. If they randomly rearrange themselves, what’s the probability that nobody is more than one spot away fro... | ### Problem 1
1. There are 7 rounds, and each round has 3 problems.
2. To find the total number of problems, we multiply the number of rounds by the number of problems per round:
\[
7 \times 3 = 21
\]
The final answer is \(\boxed{1365}\) | 1365 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] What is the smallest positive prime divisor of $101^2 -99^2$?
[b]p2.[/b] The product $81 \cdot 13579$ equals $1099A99$ for some digit $A$. What is $A$?
[b]p3.[/b] On a MMATHS team of $6$ students, all students forget which of the $5$ MMATHS sites (Yale, UVA, UF, UM, and Columbia) they are supposed to att... | 1. We start with the given equation:
\[
\frac{1}{a} + \frac{1}{b} = \frac{1}{24}
\]
2. Multiply both sides by \(24ab\) to clear the denominators:
\[
24b + 24a = ab
\]
3. Rearrange the equation to facilitate factoring:
\[
ab - 24a - 24b = 0
\]
4. Add \(576\) (which is \(24^2\)) to both sides t... | 41 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 5 [/u]
[b]p13.[/b] Circles $\omega_1$, $\omega_2$, and $\omega_3$ have radii $8$, $5$, and $5$, respectively, and each is externally tangent to the other two. Circle $\omega_4$ is internally tangent to $\omega_1$, $\omega_2$, and $\omega_3$, and circle $\omega_5$ is externally tangent to the same three circl... | Given the equations:
\[
2(\sqrt[3]{2})^2 + \sqrt[3]{2}a + 2b + (\sqrt[3]{2})^2c + \sqrt[3]{2}d + e = 0
\]
\[
25\sqrt{5}i + 25a - 5\sqrt{5}ib - 5c + \sqrt{5}id + e = 0
\]
Let's introduce the substitutions:
\[
m = (\sqrt[3]{2})^2, \quad n = \sqrt[3]{2}, \quad p = \sqrt{5}i
\]
This transforms the system into:
\[
2m + na... | 7 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Suppose $\frac{x}{y} = 0.\overline{ab}$ where $x$ and $y$ are relatively prime positive integers and $ab + a + b + 1$ is a multiple of $12$. Find the sum of all possible values of $y$.
[b]p2.[/b] Let $A$ be the set of points $\{(0, 0), (2, 0), (0, 2),(2, 2),(3, 1),(1, 3)\}$. How many distinct circles pass ... | To determine the number of distinct circles that pass through at least three points in the set \( A = \{(0, 0), (2, 0), (0, 2), (2, 2), (3, 1), (1, 3)\} \), we need to consider the geometric properties and configurations of these points.
1. **Identify Collinear Points**:
- First, we check if any three points are co... | 13 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] When Charles traveled from Hawaii to Chicago, he moved his watch $5$ hours backwards instead of $5$ hours forwards. He plans to wake up at $7:00$ the next morning (Chicago time). When he wakes up during the night and sees that his watch says $6:00$, how many more hours should he sleep? (He has a $12$-hour wa... | 1. Partition the given set into the ordered sets:
\[
S_1 = \{1, 3, 5, 7, 9, 11\}, \quad S_2 = \{2, 4, 6, 8, 10\}
\]
Note that any subset of the given set corresponds to a unique pair of subsets, one of \( S_1 \) and \( S_2 \), and vice versa.
2. The given condition is equivalent to not having any two conse... | 273 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A nine-digit number has the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$. Find $D$.
[i]Proposed by Levi Iszler[/i] | Given a nine-digit number of the form $\overline{6ABCDEFG3}$, where every three consecutive digits sum to $13$, we need to find the value of $D$.
1. **Set up the equations based on the given condition:**
- The first three digits sum to $13$: $6 + A + B = 13$
- The next three digits sum to $13$: $A + B + C = 13$
... | 4 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $b$ and $c$ be real numbers not both equal to $1$ such that $1,b,c$ is an arithmetic progression and $1,c,b$ is a geometric progression. What is $100(b-c)$?
[i]Proposed by Noah Kravitz[/i] | 1. Let \( n \) be the common difference of the arithmetic progression. Since \( 1, b, c \) is an arithmetic progression, we have:
\[
b = 1 + n
\]
\[
c = 1 + 2n
\]
2. Since \( 1, c, b \) forms a geometric progression, the ratio between consecutive terms must be the same. Therefore:
\[
\frac{c}{1... | 75 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that three prime numbers $p,q,$ and $r$ satisfy the equations $pq + qr + rp = 191$ and $p + q = r - 1$. Find $p + q + r$.
[i]Proposed by Andrew Wu[/i] | 1. We start with the given equations:
\[
pq + qr + rp = 191
\]
\[
p + q = r - 1
\]
2. We will assume without loss of generality (WLOG) that \( p \leq q \leq r \). We will test small prime values for \( p \) to find a solution.
3. **Case \( p = 2 \)**:
\[
p = 2 \implies q + 2 = r - 1 \implies r... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square of side length $4$. Points $E$ and $F$ are chosen on sides $BC$ and $DA$, respectively, such that $EF = 5$. Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$.
[i]Proposed by Andrew Wu[/i] | 1. **Define the problem and set up coordinates:**
Let the square \(ABCD\) have vertices at \(A(0, 4)\), \(B(0, 0)\), \(C(4, 0)\), and \(D(4, 4)\). Points \(E\) and \(F\) are on sides \(BC\) and \(DA\) respectively, such that \(EF = 5\).
2. **Determine the coordinates of \(E\) and \(F\):**
Let \(E\) be at \((4, y... | 16 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integers $m>n$, the quantities $a=\text{lcm}(m,n)$ and $b=\gcd(m,n)$ satisfy $a=30b$. If $m-n$ divides $a$, then what is the value of $\frac{m+n}{b}$?
[i]Proposed by Andrew Wu[/i] | 1. Given the quantities \( a = \text{lcm}(m, n) \) and \( b = \gcd(m, n) \) satisfy \( a = 30b \). We need to find the value of \( \frac{m+n}{b} \) under the condition that \( m-n \) divides \( a \).
2. Recall the relationship between the least common multiple and greatest common divisor:
\[
\text{lcm}(m, n) \cd... | 11 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Prair has a box with some combination of red and green balls. If she randomly draws two balls out of the box (without replacement), the probability of drawing two balls of the same color is equal to the probability of drawing two balls of different colors! How many possible values between $200$ and $1000$ are there f... | 1. Let the number of red balls be \( x \) and the number of green balls be \( y \). The total number of balls is \( x + y \).
2. The probability of drawing two balls of the same color is:
\[
\frac{\binom{x}{2} + \binom{y}{2}}{\binom{x+y}{2}} = \frac{\frac{x(x-1)}{2} + \frac{y(y-1)}{2}}{\frac{(x+y)(x+y-1)}{2}} = ... | 17 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $ABC$ is a triangle with $AB = 6, BC = 12$, and $\angle B = 90^{\circ}$. Point $D$ lies on side $BC$, and point $E$ is constructed on $AC$ such that $\angle ADE = 90^{\circ}$. Given that $DE = EC = \frac{a\sqrt{b}}{c}$ for positive integers $a, b,$ and $c$ with $b$ squarefree and $\gcd(a,c) = 1$, find $a+ ... | 1. Given the triangle \(ABC\) with \(AB = 6\), \(BC = 12\), and \(\angle B = 90^\circ\), we can use the Pythagorean theorem to find \(AC\):
\[
AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 12^2} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5}
\]
2. Point \(D\) lies on side \(BC\), and point \(E\) is constructed on \(AC\) ... | 76 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_ib_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?
[i]Proposed by Andrew Wu[/i] | 1. Given the sequences \(a_1, a_2, \ldots\) and \(b_1, b_2, \ldots\) such that \(a_i b_i - a_i - b_i = 0\) and \(a_{i+1} = \frac{2 - a_i b_i}{1 - b_i}\) for all \(i \ge 1\), we start by rearranging the first condition:
\[
a_i b_i - a_i - b_i = 0 \implies a_i b_i = a_i + b_i
\]
This can be rewritten as:
\... | 257 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $p(x)$ be the monic cubic polynomial with roots $\sin^2(1^{\circ})$, $\sin^2(3^{\circ})$, and $\sin^2(9^{\circ})$. Suppose that $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$, where $0 <a,b \le 90$ and $a,b,n$ are positive integers. What is $a+b+n$?
[i]Proposed by Andrew Yuan[/i] | 1. Let \( z = e^{\frac{i\pi}{180}} \). Using Euler's formula, we have \( \sin(1^\circ) = \frac{z - \frac{1}{z}}{2i} \). Similarly, \( \sin(3^\circ) = \frac{z^3 - \frac{1}{z^3}}{2i} \), and \( \sin(9^\circ) = \frac{z^9 - \frac{1}{z^9}}{2i} \).
2. We can compute their squares as follows:
\[
\sin^2(1^\circ) = \left... | 216 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that points $A$ and $B$ lie on circle $\Omega$, and suppose that points $C$ and $D$ are the trisection points of major arc $AB$, with $C$ closer to $B$ than $A$. Let $E$ be the intersection of line $AB$ with the line tangent to $\Omega$ at $C$. Suppose that $DC = 8$ and $DB = 11$. If $DE = a\sqrt{b}$ for intege... | 1. **Identify the given points and their relationships:**
- Points \( A \) and \( B \) lie on circle \( \Omega \).
- Points \( C \) and \( D \) are the trisection points of the major arc \( AB \), with \( C \) closer to \( B \) than \( A \).
- \( E \) is the intersection of line \( AB \) with the line tangent ... | 37 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$. Find $\frac{100bdf}{ace}$.
[i]Proposed by Andrew Wu[/i] | 1. We start with the given equation \( 20^{21} = 2^a 5^b \). We know that \( 20 = 2^2 \cdot 5 \), so:
\[
20^{21} = (2^2 \cdot 5)^{21} = 2^{42} \cdot 5^{21}
\]
Therefore, we can identify:
\[
a = 42 \quad \text{and} \quad b = 21
\]
2. Next, we consider the equation \( 20^{21} = 4^c 5^d \). We know t... | 75 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define the [i]digital reduction[/i] of a two-digit positive integer $\underline{AB}$ to be the quantity $\underline{AB} - A - B$. Find the greatest common divisor of the digital reductions of all the two-digit positive integers. (For example, the digital reduction of $62$ is $62 - 6 - 2 = 54.$)
[i]Proposed by Andrew W... | 1. Let's denote a two-digit number as $\underline{AB}$, where $A$ is the tens digit and $B$ is the units digit. Therefore, the number can be expressed as $10A + B$.
2. The digital reduction of $\underline{AB}$ is defined as $\underline{AB} - A - B$. Substituting the expression for $\underline{AB}$, we get:
\[
\un... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCDEF$ be a regular hexagon with sidelength $6$, and construct squares $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ outside the hexagon. Find the perimeter of dodecagon $HGJILKNMPORQ$.
[i]Proposed by Andrew Wu[/i] | 1. **Identify the side lengths of the squares:**
Since $ABGH$, $BCIJ$, $CDKL$, $DEMN$, $EFOP$, and $FAQR$ are squares constructed outside the regular hexagon $ABCDEF$ with side length $6$, each side of these squares is also $6$ units long. Therefore, we have:
\[
HG = JI = LK = NM = PO = RQ = 6
\]
2. **Dete... | 72 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Cat and Claire are having a conversation about Cat's favorite number. Cat says, "My favorite number is a two-digit positive prime integer whose first digit is less than its second, and when you reverse its digits, it's still a prime number!"
Claire asks, "If you picked a digit of your favorite number at random and re... | 1. **Identify the two-digit prime numbers where the first digit is less than the second digit and the number remains prime when reversed:**
- The two-digit prime numbers are: 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
- From these, we need to filter out the numbers where t... | 13 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $a_1 = 1$, and that for all $n \ge 2$, $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$. If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$, find $k$.
[i]Proposed by Andrew Wu[/i] | 1. We start with the given recurrence relation for \(a_n\):
\[
a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1
\]
for \(n \geq 2\), and \(a_1 = 1\).
2. We need to express \(a_n\) in terms of the sum of previous terms. Notice that:
\[
a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1
... | 2022 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Ben flips a coin 10 times and then records the absolute difference between the total number of heads and tails he's flipped. He then flips the coin one more time and records the absolute difference between the total number of heads and tails he's flipped. If the probability that the second number he records is greater ... | 1. **Calculate the probability of equal heads and tails in the first 10 flips:**
The number of ways to get exactly 5 heads and 5 tails in 10 flips is given by the binomial coefficient:
\[
\binom{10}{5} = 252
\]
The total number of possible outcomes when flipping a coin 10 times is:
\[
2^{10} = 102... | 831 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider a hexagon with vertices labeled $M$, $M$, $A$, $T$, $H$, $S$ in that order. Clayton starts at the $M$ adjacent to $M$ and $A$, and writes the letter down. Each second, Clayton moves to an adjacent vertex, each with probability $\frac{1}{2}$, and writes down the corresponding letter. Clayton stops moving when t... | 1. **Define the states and transitions:**
- Let \( S_0 \) be the state where Clayton has not yet written any of the letters \( M, A, T, H \).
- Let \( S_1 \) be the state where Clayton has written \( M \).
- Let \( S_2 \) be the state where Clayton has written \( M \) and \( A \).
- Let \( S_3 \) be the sta... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $P(x)$ is a monic cubic polynomial with integer roots, and suppose that $\frac{P(a)}{a}$ is an integer for exactly $6$ integer values of $a$. Suppose furthermore that exactly one of the distinct numbers $\frac{P(1) + P(-1)}{2}$ and $\frac{P(1) - P(-1)}{2}$ is a perfect square. Given that $P(0) > 0$, find t... | 1. Let \( P(x) = x^3 + dx^2 + ex + f \), where \( d \), \( e \), and \( f \) are integers. Since \( P(x) \) is a monic cubic polynomial with integer roots, it can be written as \( P(x) = (x - r_1)(x - r_2)(x - r_3) \) for some integers \( r_1, r_2, r_3 \).
2. We are given that \( \frac{P(a)}{a} \) is an integer for ex... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumcenter $O$ and incenter $I$, and suppose that $OI$ meets $AB$ and $AC$ at $P$ and $Q$, respectively. There exists a point $R$ on arc $\widehat{BAC}$ such that the circumcircles of triangles $PQR$ and $ABC$ are tangent. Given that $AB = 14$, $BC = 20$, and $CA = 26$, find $\frac{RC}{RB... | 1. **Given Information and Setup:**
- We have a triangle \(ABC\) with circumcenter \(O\) and incenter \(I\).
- The line \(OI\) intersects \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively.
- There exists a point \(R\) on arc \(\widehat{BAC}\) such that the circumcircles of triangles \(PQR\) and \(ABC\)... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $\displaystyle\prod_{i=6}^{2021} (1-\tan^2((2^i)^\circ))$ can be written in the form $a^b$ for positive integers $a,b$ with $a$ squarefree, find $a+b$.
[i]Proposed by Deyuan Li and Andrew Milas[/i] | 1. **Simplify the expression \(1 - \tan^2 \theta\):**
For any angle \(\theta\), we have:
\[
1 - \tan^2 \theta = \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta} = \frac{\cos 2\theta}{\cos^2 \theta}
\]
Therefore, the given product can be rewritten as:
\[
\prod_{i=6}^{2021} (1 - \tan^2((2^i)^\cir... | 2018 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b,c$ be the roots of the polynomial $x^3 - 20x^2 + 22.$ Find \[\frac{bc}{a^2}+\frac{ac}{b^2}+\frac{ab}{c^2}.\]
[i]Proposed by Deyuan Li and Andrew Milas[/i] | 1. Given the polynomial \(x^3 - 20x^2 + 22 = 0\), let \(a, b, c\) be its roots. We need to find the value of \(\frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}\).
2. By Vieta's formulas, for the polynomial \(x^3 - 20x^2 + 22 = 0\), we have:
\[
a + b + c = 20,
\]
\[
ab + bc + ca = 0,
\]
\[
abc =... | -40 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it; this grid's [i]score[/i] is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n\times n$ grid with probability $k$; he notices ... | 1. **Understanding the Problem:**
We need to find the minimum value of \( n \) such that the expected score of an \( n \times n \) grid, where each square is shaded with probability \( k \) and \( k > 0.9999 \), is equal to \( k \).
2. **Score Calculation:**
The score of the grid is defined as the sum of the num... | 51 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod_{n=2}^x \log_{n^n}(n+1)^{n+2}$$ is an integer.
[i]Proposed by Deyuan Li and Andrew Milas[/i] | 1. We start with the given product:
\[
\prod_{n=2}^x \log_{n^n}(n+1)^{n+2}
\]
2. We simplify the logarithmic term:
\[
\log_{n^n} (n+1)^{n+2} = \frac{\log (n+1)^{n+2}}{\log (n^n)} = \frac{(n+2) \log (n+1)}{n \log n}
\]
This uses the change of base formula for logarithms: \(\log_a b = \frac{\log b}{\... | 739 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}, BC = 14,$ and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed ... | 1. **Identify the given elements and construct the points \( P \) and \( Q \):**
- Given triangle \( ABC \) with \( AB = 6\sqrt{3} \), \( BC = 14 \), and \( CA = 22 \).
- Incenter \( I \) and circumcircle \( \Gamma \).
- Points \( P \) and \( Q \) are constructed on rays \( BA \) and \( CA \) such that \( BP =... | 31 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[u]Round 6[/u]
[b]p16.[/b] Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. There exist two possible points $X$ on $CA$ such that if $Y$ and $Z$ are the feet of the perpendiculars from $X$ to $AB$ and $BC,$ respectively, then the area of triangle $XY Z$ is $1$. If the distance between those two possible... | 1. Let us construct a desired order of \(1, 2, 3, \ldots, n\) from a desired ordering of \(1, 2, 3, \ldots, n-1\). Note that we can insert an \(n\) behind any of the numbers
\[
\left\lceil \frac{n}{2} \right\rceil, \left\lceil \frac{n}{2} \right\rceil + 1, \ldots, n-1,
\]
or at the front of the line. There... | 29 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in he... | To solve the problem, we need to find the value of the expression:
\[
\frac{bc}{a^2} + \frac{ac}{b^2} + \frac{ab}{c^2}
\]
where \(a\), \(b\), and \(c\) are the roots of the polynomial \(x^3 - 20x^2 + 22\).
1. **Apply Vieta's Formulas:**
By Vieta's formulas for the polynomial \(x^3 - 20x^2 + 22\), we have:
\[
... | 3 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Rectangle $ABCD$ has $AB = 8$ and $BC = 13$. Points $P_1$ and $P_2$ lie on $AB$ and $CD$ with $P_1P_2 \parallel BC$. Points $Q_1$ and $Q_2$ lie on $BC$ and $DA$ with $Q_1Q_2 \parallel AB$. Find the area of quadrilateral $P_1Q_1P_2Q_2$. | 1. Given a rectangle \(ABCD\) with \(AB = 8\) and \(BC = 13\), we need to find the area of the quadrilateral \(P_1Q_1P_2Q_2\).
2. Points \(P_1\) and \(P_2\) lie on \(AB\) and \(CD\) respectively, with \(P_1P_2 \parallel BC\). Points \(Q_1\) and \(Q_2\) lie on \(BC\) and \(DA\) respectively, with \(Q_1Q_2 \parallel AB\)... | 52 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $522$ people at a beach, each of whom owns a cat, a dog, both, or neither. If $20$ percent of cat-owners also own a dog, $70$ percent of dog-owners do not own a cat, and $50$ percent of people who don’t own a cat also don’t own a dog, how many people own neither type of pet? | 1. Let \( x \) be the number of people who only own a cat.
2. Let \( y \) be the number of people who own both a cat and a dog.
3. Let \( z \) be the number of people who only own a dog.
4. Let \( w \) be the number of people who own neither a cat nor a dog.
From the given information, we can set up the following equa... | 126 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to choose three digits $A,B,C$ with $1 \le A \le 9$ and $0 \le B,C \le 9$ such that $\overline{ABC}_b$ is even for all choices of base $b$ with $b \ge 10$? | To solve the problem, we need to count the number of ways to choose three digits \( A, B, C \) such that \( \overline{ABC}_b \) is even for all choices of base \( b \ge 10 \). This means that the last digit \( C \) must be even, as the parity of the number in any base is determined by the last digit.
1. **Determine th... | 450 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$\vartriangle ABC$ satisfies $AB = 16$, $BC = 30$, and $\angle ABC = 90^o$. On the circumcircle of $\vartriangle ABC$, let $P$ be the midpoint of arc $AC$ not containing $B$, and let $X$ and $Y$ lie on lines $AB$ and $BC$, respectively, with $PX \perp AB$ and $PY \perp BC$. Find $XY^2$. | 1. **Identify the coordinates of points \(A\), \(B\), and \(C\):**
Given that \(\triangle ABC\) is a right triangle with \(\angle ABC = 90^\circ\), we can place the points in a coordinate system:
- \(A(0, 16)\)
- \(B(0, 0)\)
- \(C(30, 0)\)
2. **Find the circumcenter and circumradius of \(\triangle ABC\):**... | 850 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a monic cubic polynomial such that the sum of the coefficients of $f$ is $5$ and such that the sum of the roots of $f$ is $1$. Find the absolute value of the sum of the cubes of the roots of $f$. | 1. Let \( f(x) \) be a monic cubic polynomial. Since it is monic, it can be written as:
\[
f(x) = x^3 + ax^2 + bx + c
\]
Given that the sum of the coefficients of \( f \) is 5, we have:
\[
1 + a + b + c = 5 \implies a + b + c = 4
\]
2. The sum of the roots of \( f \) is given as 1. By Vieta's form... | 14 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $ a + b = 20$, $b + c = 22$, and $c + a = 2022$. Compute $\frac{a-b}{c-a}$ . | 1. We are given the equations:
\[
a + b = 20
\]
\[
b + c = 22
\]
\[
c + a = 2022
\]
2. To find \(a - b\), we subtract the second equation from the third equation:
\[
(c + a) - (b + c) = 2022 - 22
\]
Simplifying, we get:
\[
a - b = 2000
\]
3. Next, to find \(c - a\), we ... | 1000 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Luke and Carissa are finding the sum of the first $20$ positive integers by adding them one at a time. Luke forgets to add a number and gets an answer of $207$. Carissa adds a number twice by mistake and gets an answer of $225$. What is the sum of the number that Luke forgot and the number that Carissa added twice? | 1. First, we need to find the sum of the first 20 positive integers. This can be calculated using the formula for the sum of an arithmetic series:
\[
S_n = \frac{n(n+1)}{2}
\]
where \( n \) is the number of terms. For \( n = 20 \):
\[
S_{20} = \frac{20 \cdot 21}{2} = \frac{420}{2} = 210
\]
2. Luke... | 18 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
... | 1. **List the two-digit perfect squares:**
The two-digit perfect squares are:
\[
16, 25, 36, 49, 64, 81
\]
2. **Analyze the condition about revealing a digit:**
If Cat reveals a digit and Claire can be certain of the number, it means that the digit must be unique among the digits of all the two-digit pe... | 25 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Holding a rectangular sheet of paper $ABCD$, Prair folds triangle $ABD$ over diagonal $BD$, so that the new location of point $A$ is $A'$. She notices that $A'C =\frac13 BD$. If the area of $ABCD$ is $27\sqrt2$, find $BD$. | 1. Let \( E \) be the intersection of \( A'D \) and \( BC \), and let \( AB = x \) and \( AD = y \). Since the area of \( ABCD \) is \( 27\sqrt{2} \), we have:
\[
xy = 27\sqrt{2}
\]
2. Since \( \triangle A'EC \sim \triangle DEB \), we have:
\[
\frac{ED}{EA'} = \frac{A'C}{BD} = \frac{1}{3}
\]
Given... | 9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Siva has the following expression, which is missing operations:
$$\frac12 \,\, \_ \,\,\frac14 \,\, \_ \,\, \frac18 \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}.$$ For each blank, he flips a fair coin: if it comes up heads, he fills it with a plus, and if it comes up tails, he fills it with a minus. Afterwards, ... | 1. **Understanding the Problem:**
We need to determine the probability that the positive difference between two computed values of the expression
\[
\frac{1}{2} \,\, \_ \,\,\frac{1}{4} \,\, \_ \,\, \frac{1}{8} \,\, \_ \,\,\frac{1}{16} \,\, \_ \,\,\frac{1}{32}
\]
is greater than $\frac{1}{2}$. Each bl... | 39 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Katherine makes Benj play a game called $50$ Cent. Benj starts with $\$0.50$, and every century thereafter has a $50\%$ chance of doubling his money and a $50\%$ chance of having his money reset to $\$0.50$. What is the expected value of the amount of money Benj will have, in dollars, after $50$ centuries? | 1. Let \( e_n \) be the expected amount of money that Benj will have after \( n \) centuries where \( n \in \mathbb{Z}_{\geq0} \). We wish to find \( e_{50} \).
2. From the given information, we have the following recurrence relation for the expected value:
\[
e_n = \frac{1}{2}(2e_{n-1}) + \frac{1}{2} \cdot 0.50... | 13 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let triangle $ABC$ with incenter $I$ satisfy $AB = 10$, $BC = 21$, and $CA = 17$. Points $D$ and E lie on side $BC$ such that $BD = 4$, $DE = 6$, and $EC = 11$. The circumcircles of triangles $BIE$ and $CID$ meet again at point $P$, and line $IP$ meets the altitude from $A$ to $BC$ at $X$. Find $(DX \cdot EX)^2$. | 1. **Determine the coordinates of points \(A\), \(B\), and \(C\)**:
- Given \(AB = 10\), \(BC = 21\), and \(CA = 17\), we can place \(B\) at \((0,0)\) and \(C\) at \((21,0)\) on the coordinate plane.
- To find the coordinates of \(A\), we use the distance formula:
\[
AB = 10 \implies \sqrt{(x-0)^2 + (y-... | 85 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
If $a, b, c$ are real numbers such that $a+b+c=6$ and $ab+bc+ca = 9$, find the sum of all possible values of the expression $\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor$. | 1. Given the conditions \(a + b + c = 6\) and \(ab + bc + ca = 9\), we need to find the sum of all possible values of the expression \(\lfloor a \rfloor + \lfloor b \rfloor + \lfloor c \rfloor\).
2. First, we note that the floor function \(\lfloor x \rfloor\) gives the greatest integer less than or equal to \(x\). The... | 15 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer? | 1. **Rewrite the function in vertex form:**
The given quadratic function is \( f(x) = -x^2 + 4px - p + 1 \). To rewrite it in vertex form, we complete the square:
\[
f(x) = -x^2 + 4px - p + 1 = -\left(x^2 - 4px\right) - p + 1
\]
\[
= -\left(x^2 - 4px + 4p^2 - 4p^2\right) - p + 1
\]
\[
= -\lef... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Hugo, Evo, and Fidel are playing Dungeons and Dragons, which requires many twenty-sided dice. Attempting to slay Evo's [i]vicious hobgoblin +1 of viciousness,[/i] Hugo rolls $25$ $20$-sided dice, obtaining a sum of (alas!) only $70$. Trying to console him, Fidel notes that, given that sum, the product of the numbers wa... | 1. **Understanding the Problem:**
Hugo rolls 25 twenty-sided dice, and the sum of the numbers rolled is 70. We need to determine how many 2's Hugo rolled if the product of the numbers is maximized.
2. **Formulating the Problem:**
Let \( x_1, x_2, \ldots, x_{25} \) be the numbers rolled on the 25 dice. We know:
... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$? | 1. **Identify the given values and calculate the semi-perimeter:**
- Given side lengths: \(AB = 13\), \(BC = 13\), \(CD = 37\), \(AD = 47\).
- The semi-perimeter \(s\) of the quadrilateral is calculated as:
\[
s = \frac{AB + BC + CD + AD}{2} = \frac{13 + 13 + 37 + 47}{2} = \frac{110}{2} = 55
\]
2.... | 504 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. $D$ is chosen so that $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$. What is $DE$? | 1. Given that $A$ and $B$ are on a circle of radius $20$ centered at $C$, and $\angle ACB = 60^\circ$. We need to find the length of $DE$ where $D$ is also on the circle, $\angle ACD = 160^\circ$, and $\angle DCB = 100^\circ$. Let $E$ be the intersection of lines $AC$ and $BD$.
2. First, we note that $\angle ACB = 60^... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A sphere of radius $\sqrt{85}$ is centered at the origin in three dimensions. A tetrahedron with vertices at integer lattice points is inscribed inside the sphere. What is the maximum possible volume of this tetrahedron? | To find the maximum possible volume of a tetrahedron inscribed in a sphere of radius $\sqrt{85}$ centered at the origin, we need to follow these steps:
1. **Determine the maximum distance from the origin to any vertex of the tetrahedron:**
Since the radius of the sphere is $\sqrt{85}$, the distance from the origin ... | 243 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$. Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$. | 1. **Identify the center of the rectangle:**
The center of the rectangle \(ABCD\) with sides of length 12 and 18 can be found by taking the midpoint of the diagonals. The coordinates of the center \(O\) are:
\[
O = \left(\frac{12}{2}, \frac{18}{2}\right) = (6, 9)
\]
2. **Determine the distance from the cen... | 54 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits. | 1. We start by considering the problem statement: we need to find the largest integer \( n \) such that \( n \) equals the product of its leading digit and the sum of its digits.
2. Let's first consider single-digit numbers. Clearly, \( n = 1 \) works because:
\[
1 = 1 \times 1
\]
However, we need to find ... | 48 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can $1 + 2 + \cdots + 2007$ be expressed as a sum of consecutive positive integers? | To solve the problem of expressing the sum \(1 + 2 + \cdots + 2007\) as a sum of consecutive positive integers, we need to find the number of ways to express the sum \( \frac{2007 \cdot 2008}{2} \) as a sum of consecutive integers.
1. **Calculate the sum \(1 + 2 + \cdots + 2007\):**
\[
S = \frac{2007 \cdot 2008}... | 24 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$? | 1. **Understanding the problem**: We need to find the number of pairs of integers \( (a, b) \) such that \( 1 \leq a, b \leq 42 \) and \( a^9 \equiv b^7 \pmod{43} \).
2. **Using Fermat's Little Theorem**: Since \( 43 \) is a prime number, Fermat's Little Theorem tells us that for any integer \( x \) not divisible by \... | 42 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs of integers $x, y$ with different parities such that $\frac{1}{x}+\frac{1}{y} = \frac{1}{2520}$. | 1. Start with the given equation:
\[
\frac{1}{x} + \frac{1}{y} = \frac{1}{2520}
\]
Combine the fractions on the left-hand side:
\[
\frac{x + y}{xy} = \frac{1}{2520}
\]
Cross-multiply to clear the fractions:
\[
2520(x + y) = xy
\]
Rearrange the equation:
\[
xy - 2520x - 2520y = ... | 90 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum number $n$ such that for any coloring of the integers from $1$ to $n$ into two colors, one can find monochromatic $a$, $b$, $c$, and $d$ (not necessarily distinct) such that $a+b+c=d$. | To find the minimum number \( n \) such that for any coloring of the integers from \( 1 \) to \( n \) into two colors, one can find monochromatic \( a \), \( b \), \( c \), and \( d \) (not necessarily distinct) such that \( a + b + c = d \), we can use the following steps:
1. **Construct a coloring and analyze when i... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a $7 \times 7$ square table, some of the squares are colored black and the others white, such that each white square is adjacent (along an edge) to an edge of the table or to a black square. Find the minimum number of black squares on the table. | To solve this problem, we need to ensure that each white square is adjacent to either the edge of the table or a black square. We aim to find the minimum number of black squares required to satisfy this condition.
1. **Outer Layer Analysis**:
- The outermost layer of the $7 \times 7$ table consists of 24 squares (s... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In how many distinguishable ways can $10$ distinct pool balls be formed into a pyramid ($6$ on the bottom, $3$ in the middle, one on top), assuming that all rotations of the pyramid are indistinguishable? | 1. **Identify the total number of ways to arrange 10 distinct pool balls:**
The total number of ways to arrange 10 distinct pool balls is given by the factorial of 10:
\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 3,628,800
\]
2. **Determine the number of in... | 403200 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the minimum number of colors necessary to color the integers from $1$ to $2007$ such that if distinct integers $a$, $b$, and $c$ are the same color, then $a \nmid b$ or $b \nmid c$. | 1. **Understanding the Problem:**
We need to color the integers from \(1\) to \(2007\) such that if three distinct integers \(a\), \(b\), and \(c\) are the same color, then \(a \nmid b\) or \(b \nmid c\). This means that no three integers in the same color class can form a chain of divisibility.
2. **Using the Pige... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Integers $x_1,x_2,\cdots,x_{100}$ satisfy \[ \frac {1}{\sqrt{x_1}} + \frac {1}{\sqrt{x_2}} + \cdots + \frac {1}{\sqrt{x_{100}}} = 20. \]Find $ \displaystyle\prod_{i \ne j} \left( x_i - x_j \right) $. | 1. **Assume the integers \( x_1, x_2, \ldots, x_{100} \) are distinct.**
- If \( x_i \) are distinct, then \( x_i \) must be at least \( 1, 2, \ldots, 100 \) in some order.
- The function \( \frac{1}{\sqrt{x}} \) is decreasing, so the sum \( \sum_{i=1}^{100} \frac{1}{\sqrt{x_i}} \) is maximized when \( x_i = i \... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
If integers $a$, $b$, $c$, and $d$ satisfy $ bc + ad = ac + 2bd = 1 $, find all possible values of $ \frac {a^2 + c^2}{b^2 + d^2} $. | 1. We start with the given equations:
\[
bc + ad = 1
\]
\[
ac + 2bd = 1
\]
2. Subtract the first equation from the second:
\[
ac + 2bd - (bc + ad) = 1 - 1
\]
Simplifying, we get:
\[
ac + 2bd - bc - ad = 0
\]
\[
ac - bc + 2bd - ad = 0
\]
Factor out common terms:
\[
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...,$ find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$. | 1. **Group the terms**: We observe that the sequence can be grouped as follows: $(1), (2,1), (2,2,1), (2,2,2,1), \ldots$. Each group ends with a $1$ and contains an increasing number of $2$'s before the $1$.
2. **Sum of terms in each group**: The sum of the terms in the $n$-th group is $2(n-1) + 1 = 2n - 1$. For examp... | 1026 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ be the largest root of $x^4 - 2009x + 1$. Find the nearest integer to $\frac{1}{x^3-2009}$ . | 1. Let \( f(x) = x^4 - 2009x + 1 \). We need to find the largest root of this polynomial and then determine the nearest integer to \( \frac{1}{x^3 - 2009} \).
2. If \( x \) is a root of \( f(x) \), then:
\[
x^4 - 2009x + 1 = 0 \implies x^4 - 2009x = -1 \implies x(x^3 - 2009) = -1 \implies \frac{1}{x^3 - 2009} = ... | -13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In how many ways can Alice, Bob, Charlie, David, and Eve split $18$ marbles among themselves so that no two of them have the same number of marbles? | 1. **Understanding the problem**: We need to distribute 18 marbles among 5 people (Alice, Bob, Charlie, David, and Eve) such that no two people have the same number of marbles. This means each person must have a unique number of marbles.
2. **Assigning unique numbers**: Since no two people can have the same number of ... | 120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the remainder, in base $10$, when $24_7 + 364_7 + 43_7 + 12_7 + 3_7 + 1_7$ is divided by $6$?
| 1. Convert each base \(7\) number to base \(10\):
- \(24_7\):
\[
24_7 = 2 \cdot 7^1 + 4 \cdot 7^0 = 2 \cdot 7 + 4 = 14 + 4 = 18_{10}
\]
- \(364_7\):
\[
364_7 = 3 \cdot 7^2 + 6 \cdot 7^1 + 4 \cdot 7^0 = 3 \cdot 49 + 6 \cdot 7 + 4 = 147 + 42 + 4 = 193_{10}
\]
- \(43_7\):
\[
... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $n$ are there such that $0 \le n \le 720$ and $n^2 \equiv 1$ (mod $720$)? | To solve the problem, we need to find the number of integers \( n \) such that \( 0 \le n \le 720 \) and \( n^2 \equiv 1 \pmod{720} \).
1. **Factorize 720**:
\[
720 = 2^4 \cdot 3^2 \cdot 5
\]
We will solve the congruence \( n^2 \equiv 1 \pmod{720} \) by solving it modulo \( 16 \), \( 9 \), and \( 5 \) sepa... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of real numbers $(x, y)$ are there such that $x^2+y^2 = 200$ and
\[\sqrt{(x-5)^2+(y-5)^2}+\sqrt{(x+5)^2+(y+5)^2}\]
is an integer? | 1. Let \( P = \sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \). We need to find the number of ordered pairs \((x, y)\) such that \( x^2 + y^2 = 200 \) and \( P \) is an integer.
2. First, we apply the Root Mean Square (RMS) inequality:
\[
\sqrt{(x-5)^2 + (y-5)^2} + \sqrt{(x+5)^2 + (y+5)^2} \geq \sqrt{4(x^2... | 12 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
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