problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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In triangle $ABC$ we have $AB=36$, $BC=48$, $CA=60$. The incircle of $ABC$ is centered at $I$ and touches $AB$, $AC$, $BC$ at $M$, $N$, $D$, respectively. Ray $AI$ meets $BC$ at $K$. The radical axis of the circumcircles of triangles $MAN$ and $KID$ intersects lines $AB$ and $AC$ at $L_1$ and $L_2$, respectively. If $... | 1. **Set up the coordinate system:**
Place the triangle \(ABC\) on the coordinate plane with \(B\) at the origin \((0,0)\), \(A\) at \((-36,0)\), and \(C\) at \((0,48)\).
2. **Compute the coordinates of the points where the incircle touches the sides:**
- \(M\) is the point where the incircle touches \(AB\). Sin... | 720 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has $AB = 2$, $BC = 3$, $CA = 4$, and circumcenter $O$. If the sum of the areas of triangles $AOB$, $BOC$, and $COA$ is $\tfrac{a\sqrt{b}}{c}$ for positive integers $a$, $b$, $c$, where $\gcd(a, c) = 1$ and $b$ is not divisible by the square of any prime, find $a+b+c$.
[i]Proposed by Michael Tang[/i] | 1. **Find the circumradius \( R \) of triangle \( ABC \):**
The circumradius \( R \) of a triangle with sides \( a, b, c \) can be found using the formula:
\[
R = \frac{abc}{4K}
\]
where \( K \) is the area of the triangle. First, we need to find the area \( K \) using Heron's formula:
\[
s = \frac... | 152 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For all positive integers $n$, denote by $\sigma(n)$ the sum of the positive divisors of $n$ and $\nu_p(n)$ the largest power of $p$ which divides $n$. Compute the largest positive integer $k$ such that $5^k$ divides \[\sum_{d|N}\nu_3(d!)(-1)^{\sigma(d)},\] where $N=6^{1999}$.
[i]Proposed by David Altizio[/i] | 1. **Understanding the Problem:**
We need to compute the largest positive integer \( k \) such that \( 5^k \) divides the sum
\[
\sum_{d|N} \nu_3(d!)(-1)^{\sigma(d)},
\]
where \( N = 6^{1999} \).
2. **Analyzing \( \sigma(d) \):**
Note that \( \sigma(d) \) is odd precisely when \( d \) is a power of ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $z^{3}=2+2i$, where $i=\sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. **Identify the roots using De Moivre's Theorem:**
Given \( z^3 = 2 + 2i \), we first convert \( 2 + 2i \) to polar form. The magnitude \( r \) and argument \( \theta \) are calculated as follows:
\[
r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}
\]
\[
\theta = \tan^{-1}\left(\frac{2}{2}\right) = \tan^... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Gamma$ be the maximum possible value of $a+3b+9c$ among all triples $(a,b,c)$ of positive real numbers such that
\[ \log_{30}(a+b+c) = \log_{8}(3a) = \log_{27} (3b) = \log_{125} (3c) .\]
If $\Gamma = \frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers, then find $p+q$. | 1. We start by setting all the logarithmic expressions equal to a common variable \( k \):
\[
\log_{30}(a+b+c) = k, \quad \log_{8}(3a) = k, \quad \log_{27}(3b) = k, \quad \log_{125}(3c) = k
\]
2. Converting these logarithmic equations to exponential form, we get:
\[
30^k = a + b + c
\]
\[
8^... | 16 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let the sequence $a_{1}, a_{2}, \cdots$ be defined recursively as follows: $a_{n}=11a_{n-1}-n$. If all terms of the sequence are positive, the smallest possible value of $a_{1}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 1. We start with the recursive sequence defined as \( a_n = 11a_{n-1} - n \). We need to find the smallest possible value of \( a_1 \) such that all terms of the sequence are positive.
2. To ensure all terms \( a_n \) are positive, we need to analyze the behavior of the sequence. Let's consider the first few terms:
... | 121 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Let $f_{0}(x)=x$, and for each $n\geq 0$, let $f_{n+1}(x)=f_{n}(x^{2}(3-2x))$. Find the smallest real number that is at least as large as
\[ \sum_{n=0}^{2017} f_{n}(a) + \sum_{n=0}^{2017} f_{n}(1-a)\]
for all $a \in [0,1]$. | 1. **Base Case:**
We start with the base case where \( k = 0 \). For any \( a \in [0, 1] \), we have:
\[
f_0(a) + f_0(1-a) = a + (1-a) = 1
\]
This satisfies the desired property.
2. **Inductive Step:**
Assume that for some \( k = j \), the property holds:
\[
f_j(a) + f_j(1-a) = 1
\]
We ne... | 2018 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums t... | 1. **Define the roots of unity and the sums:**
Let $\omega = e^{2\pi i / 1000}$ be a primitive 1000th root of unity. The 1000th roots of unity are $\omega^k$ for $k = 0, 1, 2, \ldots, 999$. Define:
\[
X = \sum_{k=0}^{999} \frac{1}{\omega^k - a}
\]
and
\[
Y = \sum_{k=0}^{999} \frac{1}{\omega^k} - 10... | 3 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The sum
\[ \sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}\]
can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 1. **Identify the series and the goal**: We need to find the sum
\[
\sum_{k=0}^{\infty} \frac{2^{k}}{5^{2^{k}}+1}
\]
and express it in the form \(\frac{p}{q}\) where \(p\) and \(q\) are relatively prime positive integers, and then find \(p+q\).
2. **Use the identity for the series**: The given solution us... | 5 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Equilateral triangle $ABC$ has area $1$. $A'$, $B'$, and $C'$ are the midpoints of $BC$, $CA$, and $AB$, respectively. $A''$, $B''$, $C''$ are the midpoints of $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be written as $\frac{m}{n}$ for relative prime positive integers $m$ and $n$. Fin... | 1. **Determine the side length of the equilateral triangle \( \triangle ABC \) given its area:**
The area \( A \) of an equilateral triangle with side length \( s \) is given by:
\[
A = \frac{\sqrt{3}}{4} s^2
\]
Given that the area is 1, we set up the equation:
\[
\frac{\sqrt{3}}{4} s^2 = 1
\]
... | 41 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A kite is inscribed in a circle with center $O$ and radius $60$. The diagonals of the kite meet at a point $P$, and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$. | 1. **Understanding the Problem:**
- A kite is inscribed in a circle with center \( O \) and radius \( 60 \).
- The diagonals of the kite intersect at point \( P \), and \( OP \) is an integer.
- We need to find the minimum possible area of the kite, expressed in the form \( a\sqrt{b} \), where \( a \) and \( b... | 239 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has $AB=BC=10$ and $CA=16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$. | 1. **Identify the given information and draw the necessary elements:**
- Triangle \(ABC\) has sides \(AB = BC = 10\) and \(CA = 16\).
- Circle \(\Omega\) is drawn with diameter \(BC\).
- \(\Omega\) meets \(AC\) at points \(C\) and \(D\).
2. **Label the midpoint of \(BC\) as \(M\):**
- Since \(BC\) is the d... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? | 1. Given that the area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle DAC$ is a right angle, we can infer that $AC$ and $AD$ are the legs of a right triangle $\triangle DAC$.
2. The area of the parallelogram can be expressed as the product of the base and height. Here, $AC$ and $AD$ are perpendicular, so the area... | 90 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the porti... | 1. **Identify the vertices and dimensions of the prism:**
- The bases of the prism are regular hexagons $ABCDEF$ and $A'B'C'D'E'F'$.
- The height of the prism is $5$.
- The side length of the hexagons is $6$.
2. **Determine the coordinates of the vertices:**
- Place the hexagon $ABCDEF$ in the $xy$-plane w... | 405 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Rectangle $HOMF$ has $HO=11$ and $OM=5$. Triangle $ABC$ has orthocenter $H$ and circumcenter $O$. $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$. Find the length of $BC$. | 1. **Define the coordinates:**
- Let $M$ be the midpoint of $BC$, so $M = (0,0)$.
- Let $F$ be the foot of the altitude from $A$ to $BC$, so $F = (-11,0)$.
- Let $H$ be the orthocenter, so $H = (-11,5)$.
- Let $O$ be the circumcenter, so $O = (0,5)$.
- Let $B = (-e,0)$ and $C = (e,0)$.
- Let $A = (11,... | 28 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has $\angle{A}=90^{\circ}$, $AB=2$, and $AC=4$. Circle $\omega_1$ has center $C$ and radius $CA$, while circle $\omega_2$ has center $B$ and radius $BA$. The two circles intersect at $E$, different from point $A$. Point $M$ is on $\omega_2$ and in the interior of $ABC$, such that $BM$ is parallel to $EC$... | 1. **Assigning Coordinates:**
- Let \( A = (0,0) \), \( B = (2,0) \), and \( C = (0,4) \).
- Circle \(\omega_1\) has center \(C\) and radius \(CA = 4\), so its equation is:
\[
x^2 + (y-4)^2 = 16
\]
- Circle \(\omega_2\) has center \(B\) and radius \(BA = 2\), so its equation is:
\[
(x-... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ACDB$ be a cyclic quadrilateral with circumcenter $\omega$. Let $AC=5$, $CD=6$, and $DB=7$. Suppose that there exists a unique point $P$ on $\omega$ such that $\overline{PC}$ intersects $\overline{AB}$ at a point $P_1$ and $\overline{PD}$ intersects $\overline{AB}$ at a point $P_2$, such that $AP_1=3$ and $P_2B=4$... | 1. **Understanding the Problem:**
We are given a cyclic quadrilateral \(ACDB\) with circumcenter \(\omega\). The lengths of the sides are \(AC = 5\), \(CD = 6\), and \(DB = 7\). There are points \(P\) and \(Q\) on \(\omega\) with specific properties related to intersections with line segment \(AB\). We need to find ... | 153 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ with $AB=4$, $BC=5$, $CA=6$ has circumcircle $\Omega$ and incircle $\omega$. Let $\Gamma$ be the circle tangent to $\Omega$ and the sides $AB$, $BC$, and let $X=\Gamma \cap \Omega$. Let $Y$, $Z$ be distinct points on $\Omega$ such that $XY$, $YZ$ are tangent to $\omega$. Find $YZ^2$.
[i]The following fac... | 1. **Define the necessary elements and compute basic properties:**
- Given triangle \( \triangle ABC \) with sides \( AB = 4 \), \( BC = 5 \), and \( CA = 6 \).
- Let \( \Omega \) be the circumcircle and \( \omega \) be the incircle of \( \triangle ABC \).
- Let \( \Gamma \) be the circle tangent to \( \Omega ... | 33 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $2017$ turtles in a room. Every second, two turtles are chosen uniformly at random and combined to form one super-turtle. (Super-turtles are still turtles.) The probability that after $2015$ seconds (meaning when there are only two turtles remaining) there is some turtle that has never been combined with anot... | 1. **Select a random turtle**: Without loss of generality (WLOG), we can consider a specific turtle and calculate the probability that this turtle is never chosen to be combined with another turtle.
2. **Calculate the probability for each step**: The probability that a specific turtle is not chosen in a single step wh... | 1009 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\... | 1. **Define the problem and initial conditions:**
- There are 100 balls in a box, each of which is either orange or black.
- The number of black balls, \( n \), can be any integer from 0 to 100, each with equal probability.
- A random black ball rolls out of the box. We need to find the probability that the ne... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The four faces of a tetrahedral die are labelled $0, 1, 2,$ and $3,$ and the die has the property that, when it is rolled, the die promptly vanishes, and a number of copies of itself appear equal to the number on the face the die landed on. For example, if it lands on the face labelled $0,$ it disappears. If it lands o... | 1. Let \( p \) be the probability that all the dice will eventually disappear. We need to find \( p \).
2. Consider the outcomes when the die is rolled:
- With probability \( \frac{1}{4} \), the die lands on 0 and disappears immediately.
- With probability \( \frac{1}{4} \), the die lands on 1, and we are left w... | 24 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such... | 1. **Define the problem and variables:**
- Jackson starts at position \(1\) on the number line.
- At each step, he remains in place with probability \(0.85\) and moves to the next position with probability \(0.15\).
- Let \(d_n\) be his position after \(n\) steps.
- Let \(E_n\) be the expected value of \(\f... | 13446 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
If $N$ is the number of ways to place $16$ [i]jumping [/i]rooks on an $8 \times 8$ chessboard such that each rook attacks exactly two other rooks, find the remainder when $N$ is divided by $1000$. (A jumping rook is said to [i]attack [/i]a square if the square is in the same row or in the same column as the rook.)
| 1. **Define the problem in terms of a function:**
Let \( f(n) \) be the number of ways to place \( 2n \) jumping rooks on an \( n \times n \) chessboard such that each rook attacks exactly two other rooks. This is equivalent to counting the number of ways to place exactly \( 2n \) rooks such that there are exactly 2... | 530 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Bob chooses a $4$-digit binary string uniformly at random, and examines an infinite sequence of uniformly and independently random binary bits. If $N$ is the least number of bits Bob has to examine in order to find his chosen string, then find the expected value of $N$. For example, if Bob’s string is $0000$ and the st... | To find the expected value of \( N \), the least number of bits Bob has to examine in order to find his chosen 4-digit binary string, we need to delve into the concept of expected value in probability theory.
1. **Define the problem in terms of Markov Chains:**
We can model the problem using a Markov chain with st... | 30 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In how many different orders can the characters $P \ U \ M \ \alpha \ C$ be arranged such that the $M$ is to the left of the $\alpha$ and the $\alpha$ is to the left of the $C?$ | 1. First, we need to determine the total number of ways to arrange the characters \( P, U, M, \alpha, C \). Since there are 5 distinct characters, the total number of permutations is given by:
\[
5! = 120
\]
2. Next, we need to consider the condition that \( M \) is to the left of \( \alpha \) and \( \alpha \... | 20 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Split a face of a regular tetrahedron into four congruent equilateral triangles. How many different ways can the seven triangles of the tetrahedron be colored using only the colors orange and black? (Two tetrahedra are considered to be colored the same way if you can rotate one so it looks like the other.)
| To solve this problem, we need to consider the symmetry of the tetrahedron and the different ways to color the seven triangles using two colors, orange and black. We will use Burnside's Lemma to count the distinct colorings under the rotational symmetries of the tetrahedron.
1. **Identify the symmetries of the tetrahe... | 48 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Shaq sees the numbers $1$ through $2017$ written on a chalkboard. He repeatedly chooses three numbers, erases them, and writes one plus their median. (For instance, if he erased $-2, -1, 0$ he would replace them with $0$.) If $M$ is the maximum possible final value remaining on the board, and if m is the minimum, compu... | 1. **Claim that \( M = 2017 \):**
- Notice that if there was a number greater than \( 2017 \) written on the chalkboard, it would mean that there would have to be another number greater than \( 2017 \) on the chalkboard in order to make that number.
- Since all the numbers start off as \( \leq 2017 \), it is imp... | 2014 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$. | 1. **Understanding the problem**: We need to find the sum of the four smallest possible values of \(a_2\) in a sequence of positive integers \(a_1, a_2, \dots\) such that \(\gcd(a_m, a_n) > 1\) if and only if \(|m - n| = 1\).
2. **Analyzing the condition**: The condition \(\gcd(a_m, a_n) > 1\) if and only if \(|m - n|... | 42 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define the [i]bigness [/i]of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and [i]bigness [/i]$N$ and another one with integer side lengths and [i]bigness [/i]$N + 1$... | 1. Let the sides of the rectangular prism be \(a, b, c\). The volume of the prism is \(abc\), the surface area is \(2(ab + bc + ac)\), and the sum of the lengths of all edges is \(4(a + b + c)\).
2. The *bigness* of the rectangular prism is given by:
\[
\text{bigness} = abc + 2(ab + bc + ac) + 4(a + b + c)
\]... | 55 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any integer $n \ge 2$, let $b_n$ be the least positive integer such that, for any integer $N$, $m$ divides $N$ whenever $m$ divides the digit sum of $N$ written in base $b_n$, for $2 \le m \le n$. Find the integer nearest to $b_{36}/b_{25}$. | 1. **Lemma:** \( b_n > n \)
*Proof:* Assume for the sake of contradiction that \( b_n \leq n \). Then there exists \( m = b_n \). Consider the two-digit number \( N = \overline{1(m-1)}_{b_n} \). Clearly, this is a contradiction since \( m \nmid b_n + (m-1) \). Hence, \( b_n > n \). \(\blacksquare\)
2. **Claim:** \... | 1798 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $N$ such that the only values of $n$ for which $1 + N \cdot 2^n$ is prime are multiples of $12$. | 1. We need to find the least positive integer \( N \) such that \( 1 + N \cdot 2^n \) is prime only when \( n \) is a multiple of 12. This means that for \( n \not\equiv 0 \pmod{12} \), \( 1 + N \cdot 2^n \) must be composite.
2. We start by considering the condition that there exists a prime \( p \) such that \( p \m... | 556 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \{1, 22, 333, \dots , 999999999\}$. For how many pairs of integers $(a, b)$ where $a, b \in S$ and $a < b$ is it the case that $a$ divides $b$? | To solve the problem, we need to determine how many pairs \((a, b)\) exist such that \(a, b \in S\) and \(a < b\) where \(a\) divides \(b\). The set \(S\) is given as \(\{1, 22, 333, \dots, 999999999\}\). For simplicity, we can represent each element in \(S\) as \(n\) repeated \(n\) times, where \(n\) ranges from 1 to ... | 14 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1 \le a < b < c \le 20$. | 1. **Identify the conditions for a feral triple:**
An ordered triple \((a, b, c)\) is feral if \(b - a\), \(c - a\), and \(c - b\) are all prime numbers.
2. **Express \(c - a\) in terms of \(b - a\) and \(c - b\):**
Notice that \(c - a = (c - b) + (b - a)\). This means \(c - a\) is the sum of two primes.
3. **... | 72 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$.
Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
| 1. **Define the functions and the expression:**
Given \( f(x) = (x - 5)(x - 12) \) and \( g(x) = (x - 6)(x - 10) \), we need to find the sum of all integers \( n \) such that \( \frac{f(g(n))}{f(n)^2} \) is defined and an integer.
2. **Simplify the expression:**
First, we compute \( f(g(n)) \):
\[
g(n) = (... | 23 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $
[b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $
[b]b)[/b] Prove that if $ f $ is nondecreasing then $ f... | ### Part (a)
We need to find a continuous function \( f \) such that \( \lim_{x \to \infty} \frac{1}{x^2} \int_0^x f(t) \, dt = 1 \) and \( \frac{f(x)}{x} \) does not have a limit as \( x \to \infty \).
1. Consider the base function \( f(x) = 2x \). For this function:
\[
\int_0^x f(t) \, dt = \int_0^x 2t \, dt =... | 2 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There are $5$ accents in French, each applicable to only specific letters as follows:
[list]
[*] The cédille: ç
[*] The accent aigu: é
[*] The accent circonflexe: â, ê, î, ô, û
[*] The accent grave: à, è, ù
[*] The accent tréma: ë, ö, ü
[/list]
Cédric needs to write down a phrase in French. He knows that there are $3... | 1. **Determine the number of ways to split the letters into 3 words:**
- We have 12 letters: "cesontoiseaux".
- We need to place 2 dividers among these 12 letters to create 3 words.
- The number of ways to place 2 dividers in 12 positions is given by the binomial coefficient:
\[
\binom{12}{2} = \frac... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of queens you could put on $2017 \times 2017$ chess table such that each queen attacks at most $1$ other queen. | To solve the problem of finding the maximum number of queens that can be placed on a $2017 \times 2017$ chessboard such that each queen attacks at most one other queen, we can break down the problem into smaller, manageable parts and use a construction method.
1. **Understanding the $6 \times 6$ Example:**
Consider... | 673359 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares [i]forbidden[/i], meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a for... | 1. **Initial Setup and Strategy**:
- Andile and Zandre play on a \(2017 \times 2017\) board.
- Andile declares some squares as forbidden.
- Zandre places the first coin, and they alternate turns.
- Coins cannot be placed on forbidden squares or in the same row or column as another coin.
- The player who ... | 2017 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A midpoint plotter is an instrument which draws the exact mid point of two point previously drawn. Starting off two points $1$ unit of distance apart and using only the midpoint plotter, you have to get two point which are strictly at a distance between $\frac{1}{2017}$ and $\frac{1}{2016}$ units, drawing the minimum a... | 1. **Initial Setup**:
We start with two points on the real number line, denoted as \(0\) and \(1\), which are \(1\) unit apart.
2. **Using the Midpoint Plotter**:
Each time we use the midpoint plotter, it will place a new point exactly halfway between two existing points. If we denote the coordinates of the poi... | 17 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
At the beginning there are $2017$ marbles in each of $1000$ boxes. On each move Aybike chooses a box, grabs some of the marbles from that box and delivers them one for each to the boxes she wishes. At least how many moves does Aybike have to make to have different number of marbles in each box? | 1. **Define the problem and variables:**
- There are initially \(2017\) marbles in each of \(1000\) boxes.
- Aybike can choose a box, take some marbles from it, and distribute them to other boxes.
- We need to determine the minimum number of moves \(n\) required to ensure each box has a different number of mar... | 176 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a $12\times 12$ square table some stones are placed in the cells with at most one stone per cell. If the number of stones on each line, column, and diagonal is even, what is the maximum number of the stones?
[b]Note[/b]. Each diagonal is parallel to one of two main diagonals of the table and consists of $1,2\ldots,... | 1. **Identify the constraints**: We are given a $12 \times 12$ square table with stones placed such that each row, column, and diagonal has an even number of stones. Each diagonal is parallel to one of the two main diagonals of the table and consists of $1, 2, \ldots, 11$ or $12$ cells.
2. **Analyze the diagonals**: N... | 120 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A circle and a square are drawn on the plane so that they overlap. Together, the two shapes cover an area of $329$ square units. The area common to both shapes is $101$ square units. The area of the circle is $234$ square units. What is the perimeter of the square in units?
$\mathrm{(A) \ } 14 \qquad \mathrm{(B) \ } 4... | 1. Let the area of the square be denoted by \( A_{\text{square}} \) and the area of the circle be denoted by \( A_{\text{circle}} \). We are given:
\[
A_{\text{circle}} = 234 \text{ square units}
\]
and the total area covered by both shapes is:
\[
A_{\text{total}} = 329 \text{ square units}
\]
T... | 56 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Charlie plans to sell bananas for forty cents and apples for fifty cents at his fruit stand, but Dave accidentally reverses the prices. After selling all their fruit they earn a dollar more than they would have with the original prices. How many more bananas than apples did they sell?
$\mathrm{(A) \ } 2 \qquad \mathrm... | 1. Let \( x \) be the number of apples and \( y \) be the number of bananas they sold.
2. If they had sold the fruits at the original prices, they would have earned \( 0.4y + 0.5x \) dollars.
3. With the reversed prices, they earned \( 0.4x + 0.5y \) dollars.
4. According to the problem, the earnings with the reversed ... | 10 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Evaluate the following expression: $$0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000$$ The terms with minus signs are exactly the powers of two.
| 1. **Identify the terms with minus signs:**
The terms with minus signs are exactly the powers of two. The maximum power of two that appears in the range from 0 to 2000 is \(2^{10} = 1024\).
2. **Rewrite the sum:**
We can separate the sum into two parts: the sum of all integers from 0 to 2000 and the sum of the p... | 1996906 | Number Theory | other | Yes | Yes | aops_forum | false |
If $(x + 1) + (x + 2) + ... + (x + 20) = 174 + 176 + 178 + ... + 192$, then what is the value of $x$?
$\mathrm{(A) \ } 80 \qquad \mathrm{(B) \ } 81 \qquad \mathrm {(C) \ } 82 \qquad \mathrm{(D) \ } 83 \qquad \mathrm{(E) \ } 84$
| 1. First, we need to express the left-hand side of the equation, which is the sum of the arithmetic series $(x + 1) + (x + 2) + \ldots + (x + 20)$. The sum of an arithmetic series can be calculated using the formula:
\[
S = \frac{n}{2} \left( a + l \right)
\]
where \( n \) is the number of terms, \( a \) is... | 81 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Lori makes a list of all the numbers between $1$ and $999$ inclusive. She first colors all the multiples of $5$ red. Then she colors blue every number which is adjacent to a red number. How many numbers in her list are left uncolored?
$\mathrm{(A) \ } 400 \qquad \mathrm{(B) \ } 402 \qquad \mathrm {(C) \ } 597 \qquad \... | 1. **Identify the multiples of 5 between 1 and 999:**
- The multiples of 5 are given by the sequence \(5, 10, 15, \ldots, 995\).
- This is an arithmetic sequence where the first term \(a = 5\) and the common difference \(d = 5\).
- To find the number of terms in this sequence, we use the formula for the \(n\)-... | 404 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
For all positive integers $n$ the function $f$ satisfies $f(1) = 1, f(2n + 1) = 2f(n),$ and $f(2n) = 3f(n) + 2$. For how many positive integers $x \leq 100$ is the value of $f(x)$ odd?
$\mathrm{(A) \ } 4 \qquad \mathrm{(B) \ } 5 \qquad \mathrm {(C) \ } 6 \qquad \mathrm{(D) \ } 7 \qquad \mathrm{(E) \ } 10$
| 1. We start by analyzing the given function properties:
- \( f(1) = 1 \)
- \( f(2n + 1) = 2f(n) \)
- \( f(2n) = 3f(n) + 2 \)
2. We need to determine for how many positive integers \( x \leq 100 \) the value of \( f(x) \) is odd.
3. First, let's consider the case when \( x \) is odd:
- If \( x \) is odd, t... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of complex numbers $z$ with the properties that $|z|=1$ and $z^{6!}-z^{5!}$ is a real number. Find the remainder when $N$ is divided by $1000$.
| To solve the problem, we need to find the number of complex numbers \( z \) such that \( |z| = 1 \) and \( z^{720} - z^{120} \) is a real number. Let's break down the solution step by step.
1. **Expressing the condition in terms of complex conjugates:**
Since \( |z| = 1 \), we have \( \overline{z} = \frac{1}{z} \).... | 440 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let \(\triangle ABC\) have side lengths \(AB=30\), \(BC=32\), and \(AC=34\). Point \(X\) lies in the interior of \(\overline{BC}\), and points \(I_1\) and \(I_2\) are the incenters of \(\triangle ABX\) and \(\triangle ACX\), respectively. Find the minimum possible area of \(\triangle AI_1I_2\) as \( X\) varies along \(... | 1. **Determine the inradius of \(\triangle ABC\)**:
- Given side lengths \(AB = 30\), \(BC = 32\), and \(AC = 34\), we can use Heron's formula to find the area of \(\triangle ABC\).
- First, calculate the semi-perimeter \(s\):
\[
s = \frac{AB + BC + AC}{2} = \frac{30 + 32 + 34}{2} = 48
\]
- Usin... | 126 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$. The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2\] are all real can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 1. We start with the given polynomial:
\[
P(x) = x^4 + 2ax^3 + (2a-2)x^2 + (-4a+3)x - 2
\]
We need to determine the conditions under which all the roots of this polynomial are real.
2. We first consider the polynomial in its given form and look for a way to factor it. Notice that the polynomial can be grou... | 37 | Other | math-word-problem | Yes | Yes | aops_forum | false |
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*}
$\textbf{(A) } 1 \qquad
\textbf{(B) } 2 \qquad
\textbf{(C) } 3 \qquad
\textbf{(D) } 4 \qquad
\textbf{(E) } 8 $ | To solve the given system of equations:
\[
\begin{cases}
x + 3y = 3 \\
\big||x| - |y|\big| = 1
\end{cases}
\]
1. **Express \( x \) in terms of \( y \):**
From the first equation, we have:
\[
x = 3 - 3y
\]
2. **Substitute \( x = 3 - 3y \) into the second equation:**
\[
\big||3 - 3y| - |y|\big| = 1
... | 3 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
For a positive integer $n$ and nonzero digits $a$, $b$, and $c$, let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$, and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c$. What is the gre... | 1. Define the integers \(A_n\), \(B_n\), and \(C_n\) as follows:
- \(A_n\) is an \(n\)-digit integer where each digit is \(a\).
- \(B_n\) is an \(n\)-digit integer where each digit is \(b\).
- \(C_n\) is a \(2n\)-digit integer where each digit is \(c\).
2. Express \(A_n\), \(B_n\), and \(C_n\) in terms of \(a... | 18 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many nonnegative integers can be written in the form $$a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2+a_1\cdot3^1+a_0\cdot3^0,$$
where $a_i\in \{-1,0,1\}$ for $0\le i \le 7$?
$\textbf{(A) } 512 \qquad
\textbf{(B) } 729 \qquad
\textbf{(C) } 1094 \qquad
\textbf{(D) } 3281 \qquad
\textbf{... | 1. We are given the expression:
\[
a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0
\]
where \(a_i \in \{-1, 0, 1\}\) for \(0 \leq i \leq 7\).
2. We need to determine how many distinct nonnegative integers can be formed by this ex... | 3281 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$?
$\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \tex... | 1. We start by analyzing the given conditions:
\[
\gcd(a, b) = 24 = 2^3 \cdot 3
\]
\[
\gcd(b, c) = 36 = 2^2 \cdot 3^2
\]
\[
\gcd(c, d) = 54 = 2 \cdot 3^3
\]
\[
70 < \gcd(d, a) < 100
\]
2. We express \(a, b, c,\) and \(d\) in terms of their prime factors:
\[
a = 2^3 \cdot 3 \cd... | 13 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares. A scanning code is called [i]symmetric[/i] if its look does not change when the entire square is rotated by a multi... | 1. **Identify Symmetry Constraints**:
- The grid must be symmetric under rotations by multiples of \(90^\circ\).
- The grid must be symmetric under reflections across lines joining opposite corners and midpoints of opposite sides.
2. **Labeling the Grid**:
- We label the squares in the grid based on their sym... | 1022 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the r... | 1. Given that $\overline{AB}$ is a diameter of a circle with $AB = 24$, we can determine the radius of the circle. Since the diameter is twice the radius, we have:
\[
r = \frac{AB}{2} = \frac{24}{2} = 12
\]
2. Point $C$ lies on the circle, and we need to find the locus of the centroid $G$ of $\triangle ABC$ a... | 50 | Geometry | MCQ | Yes | Yes | aops_forum | false |
A function $f$ is defined recursively by $f(1)=f(2)=1$ and $$f(n)=f(n-1)-f(n-2)+n$$ for all integers $n \geq 3$. What is $f(2018)$?
$\textbf{(A)} \text{ 2016} \qquad \textbf{(B)} \text{ 2017} \qquad \textbf{(C)} \text{ 2018} \qquad \textbf{(D)} \text{ 2019} \qquad \textbf{(E)} \text{ 2020}$ | 1. Define a new function \( g(n) = f(n) - n \). This transformation helps simplify the recursive relation.
2. Substitute \( f(n) = g(n) + n \) into the given recursive formula:
\[
f(n) = f(n-1) - f(n-2) + n
\]
becomes
\[
g(n) + n = (g(n-1) + (n-1)) - (g(n-2) + (n-2)) + n
\]
3. Simplify the equation... | 2017 | Other | MCQ | Yes | Yes | aops_forum | false |
Mary chose an even $4$-digit number $n$. She wrote down all the divisors of $n$ in increasing order from left to right: $1,2,...,\tfrac{n}{2},n$. At some moment Mary wrote $323$ as a divisor of $n$. What is the smallest possible value of the next divisor written to the right of $323$?
$\textbf{(A) } 324 \qquad \textbf... | 1. Let the next divisor be \( k \). We need to find the smallest possible value of \( k \) such that \( k > 323 \) and \( k \) is a divisor of \( n \).
2. Note that \( 323 = 17 \times 19 \). Therefore, \( k \) must be a multiple of either 17 or 19 to ensure that \( n \) remains a 4-digit number.
3. The smallest multipl... | 340 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $ | 1. Let \( P(x) = ax^3 + bx^2 + cx + d \). We are given that \( P(-1) = -9 \).
2. Substituting \( x = -1 \) into \( P(x) \), we get:
\[
P(-1) = a(-1)^3 + b(-1)^2 + c(-1) + d = -a + b - c + d
\]
3. We need \( -a + b - c + d = -9 \). Rearranging, we get:
\[
b + d = a + c + 9
\]
4. Let \( s = a + c + 9 \)... | 220 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $ | To solve the problem, we need to find the smallest positive integer \( q \) such that there exists a positive integer \( p \) satisfying the inequality:
\[
\frac{5}{9} < \frac{p}{q} < \frac{4}{7}
\]
1. **Express the inequalities in terms of \( p \) and \( q \):**
\[
\frac{5}{9} < \frac{p}{q} < \frac{4}{7}
\]
... | 7 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A line with slope $2$ intersects a line with slope $6$ at the point $(40, 30)$. What is the distance between the $x$-intercepts of these two lines?
$\textbf{(A) }5\qquad\textbf{(B) }10\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }50$ | 1. **Determine the equations of the lines:**
- The first line has a slope of \(2\) and passes through the point \((40, 30)\). Using the point-slope form of the line equation \(y - y_1 = m(x - x_1)\), we get:
\[
y - 30 = 2(x - 40)
\]
Simplifying this, we get:
\[
y - 30 = 2x - 80 \implies... | 10 | Geometry | MCQ | Yes | Yes | aops_forum | false |
Given the parallelogram $ABCD$. The circle $S_1$ passes through the vertex $C$ and touches the sides $BA$ and $AD$ at points $P_1$ and $Q_1$, respectively. The circle $S_2$ passes through the vertex $B$ and touches the side $DC$ at points $P_2$ and $Q_2$, respectively. Let $d_1$ and $d_2$ be the distances from $C$ and ... | 1. **Define the heights and distances:**
Let \( h_1 \) and \( h_2 \) denote the perpendicular distances from vertex \( C \) to the sides \( AB \) and \( AD \) of the parallelogram \( ABCD \), respectively. These are the heights of the parallelogram from vertex \( C \).
2. **Express \( d_1 \) in terms of \( h_1 \) a... | 1 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$. | 1. **Understanding the Problem:**
We need to color an infinite table such that any \(2 \times 3\) or \(3 \times 2\) rectangle has all six cells colored differently. We aim to find the smallest number \(n\) of colors required to achieve this.
2. **Initial Consideration:**
Let's consider the constraints. For any \... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2018 Canadian Open Math Challenge Part A Problem 2
-----
Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v\times w\times x\times y\times z.$ What is the sum of the integers? | 1. We start with the given equation:
\[
45 = v \times w \times x \times y \times z
\]
where \(v, w, x, y, z\) are five distinct integers.
2. To find the sum of these integers, we need to factorize 45 into five distinct integers. The prime factorization of 45 is:
\[
45 = 3^2 \times 5 \times 1
\]
3... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2018 Canadian Open Math Challenge Part B Problem 3
-----
The [i]doubling sum[/i] function is defined by
\[D(a,n)=\overbrace{a+2a+4a+8a+...}^{\text{n terms}}.\]
For example, we have
\[D(5,3)=5+10+20=35\]
and
\[D(11,5)=11+22+44+88+176=341.\]
Determine the smallest positive integer $n$ such that for every integer... | 1. The doubling sum function \( D(a, n) \) is defined as the sum of the first \( n \) terms of a geometric series where the first term is \( a \) and the common ratio is \( 2 \). Therefore, we can express \( D(a, n) \) as:
\[
D(a, n) = a + 2a + 4a + \cdots + 2^{n-1}a
\]
2. The sum of a geometric series \( a +... | 9765 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2018 Canadian Open Math Challenge Part B Problem 4
-----
Determine the number of $5$-tuples of integers $(x_1,x_2,x_3,x_4,x_5)$ such that
$\text{(a)}$ $x_i\ge i$ for $1\le i \le 5$;
$\text{(b)}$ $\sum_{i=1}^5 x_i = 25$. | 1. We start with the given conditions:
\[
x_i \ge i \quad \text{for} \quad 1 \le i \le 5
\]
and
\[
\sum_{i=1}^5 x_i = 25.
\]
2. To simplify the problem, we perform a change of variables. Let:
\[
y_i = x_i - i \quad \text{for} \quad 1 \le i \le 5.
\]
This transformation ensures that \( ... | 1001 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Source: 2018 Canadian Open Math Challenge Part C Problem 3
-----
Consider a convex quadrilateral $ABCD$. Let rays $BA$ and $CD$ intersect at $E$, rays $DA$ and $CB$ intersect at $F$, and the diagonals $AC$ and $BD$ intersect at $G$. It is given that the triangles $DBF$ and $DBE$ have the same area.
$\text{(a)}$ Prove... | ### Part (a)
1. **Given**: The triangles \( \triangle DBF \) and \( \triangle DBE \) have the same area.
2. **Observation**: Both triangles share the same base \( BD \).
3. **Conclusion**: Since the areas are equal and the base is the same, the heights from points \( F \) and \( E \) to the line \( BD \) must be equal.... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$
If the sum of the digits of $a$ is 567, what is the value of $n$? | 1. We start with the given expression for \( a \):
\[
a = \frac{10^{2n} - 1}{3(10^n + 1)}
\]
2. We simplify the numerator \( 10^{2n} - 1 \) using the difference of squares:
\[
10^{2n} - 1 = (10^n)^2 - 1^2 = (10^n - 1)(10^n + 1)
\]
3. Substituting this back into the expression for \( a \):
\[
a... | 189 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A [i]string of length $n$[/i] is a sequence of $n$ characters from a specified set. For example, $BCAAB$ is a string of length 5 with characters from the set $\{A,B,C\}$. A [i]substring[/i] of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string $CA... | ### Part (a)
We need to list all strings of length 4 with characters from the set $\{A, B, C\}$ in which both the substrings $AB$ and $BA$ occur.
1. **Identify the positions of $AB$ and $BA$:**
- The string must contain both $AB$ and $BA$ as substrings. This means that $AB$ and $BA$ must overlap in some way within ... | 963 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On a chessboard $8\times 8$, $n>6$ Knights are placed so that for any 6 Knights there are two Knights that attack each other. Find the greatest possible value of $n$. | 1. **Understanding the Problem:**
We need to place \( n \) knights on an \( 8 \times 8 \) chessboard such that for any 6 knights, there are at least two knights that attack each other. We aim to find the maximum value of \( n \).
2. **Initial Construction:**
Consider placing knights on the board in a specific pa... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the cells of an $8\times 8$ board, marbles are placed one by one. Initially there are no marbles on the board. A marble could be placed in a free cell neighboring (by side) with at least three cells which are still free. Find the greatest possible number of marbles that could be placed on the board according to thes... | 1. **Initial Setup and Constraints**:
- We have an $8 \times 8$ board, initially empty.
- A marble can be placed in a cell if it has at least three neighboring cells (by side) that are still free.
2. **Counting Segments**:
- Each cell has up to 4 neighboring cells, but we are only interested in the segments b... | 36 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $P$ be the product of the first $50$ nonzero square numbers. Find the largest integer $k$ such that $7^k$ divides $P$.
[i]2018 CCA Math Bonanza Individual Round #2[/i] | 1. First, we need to express \( P \) as the product of the first 50 nonzero square numbers:
\[
P = 1^2 \cdot 2^2 \cdot 3^2 \cdots 50^2 = (1 \cdot 2 \cdot 3 \cdots 50)^2 = (50!)^2
\]
2. Next, we need to determine the largest integer \( k \) such that \( 7^k \) divides \( (50!)^2 \). This requires finding the h... | 16 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A Louis Vuitton store in Shanghai had a number of pairs of sunglasses which cost an average of \$$900$ per pair. LiAngelo Ball stole a pair which cost \$$2000$. Afterwards, the average cost of sunglasses in the store dropped to \$$890$ per pair. How many pairs of sunglasses were in the store before LiAngelo Ball stole?... | 1. Let \( n \) be the number of pairs of sunglasses in the store before the robbery.
2. The total cost of all the sunglasses before the robbery is \( 900n \) dollars.
3. After LiAngelo Ball stole a pair of sunglasses costing \( 2000 \) dollars, the total cost of the remaining sunglasses is \( 900n - 2000 \) dollars.
4.... | 111 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\log 101$ and $\log 2018$. The last side of his triangle has side length $\log n$, where $n$ is an integer. How many possible values are there for $n$?
[i]2018 CCA Math Bonanza Individual Round #6[/i] | To determine the number of possible values for \( n \) such that the side lengths \(\log 101\), \(\log 2018\), and \(\log n\) form a non-degenerate triangle, we need to apply the triangle inequality. The triangle inequality states that for any triangle with sides \(a\), \(b\), and \(c\):
1. \(a + b > c\)
2. \(a + c > ... | 203797 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
What is the area of the smallest possible square that can be drawn around a regular hexagon of side length $2$ such that the hexagon is contained entirely within the square?
[i]2018 CCA Math Bonanza Individual Round #9[/i] | To find the area of the smallest possible square that can be drawn around a regular hexagon of side length \(2\), we need to determine the side length of the square.
1. **Understanding the Geometry**:
- A regular hexagon can be divided into 6 equilateral triangles.
- The distance from the center of the hexagon ... | 8 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the land of Chaina, people pay each other in the form of links from chains. Fiona, originating from Chaina, has an open chain with $2018$ links. In order to pay for things, she decides to break up the chain by choosing a number of links and cutting them out one by one, each time creating $2$ or $3$ new chains. For e... | 1. To solve this problem, we need to determine the minimum number of links Fiona must cut to be able to pay for any amount from 1 to 2018 links using the resulting chains. The key insight is that any number can be represented as a sum of powers of 2. This is because every integer can be uniquely represented in binary f... | 10 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$P\left(x\right)$ is a polynomial of degree at most $6$ such that such that $P\left(1\right)$, $P\left(2\right)$, $P\left(3\right)$, $P\left(4\right)$, $P\left(5\right)$, $P\left(6\right)$, and $P\left(7\right)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P\left(8\rig... | 1. Given that \( P(x) \) is a polynomial of degree at most 6, we know that the 7th finite difference of \( P \) is zero. This is a property of polynomials of degree \( n \) where the \( (n+1) \)-th finite difference is zero.
2. We are given that \( P(1), P(2), P(3), P(4), P(5), P(6), \) and \( P(7) \) are \( 1, 2, 3, ... | 385 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Brian starts at the point $\left(1,0\right)$ in the plane. Every second, he performs one of two moves: he can move from $\left(a,b\right)$ to $\left(a-b,a+b\right)$ or from $\left(a,b\right)$ to $\left(2a-b,a+2b\right)$. How many different paths can he take to end up at $\left(28,-96\right)$?
[i]2018 CCA Math Bonanza... | 1. **Reinterpret the problem using complex numbers:**
- Brian starts at the point \((1,0)\) in the plane, which can be represented as the complex number \(1\).
- The moves Brian can make are:
- From \((a,b)\) to \((a-b, a+b)\), which corresponds to multiplying by \(1 + i\).
- From \((a,b)\) to \((2a-b, ... | 70 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell, which consists of a $45$-pound bar and any number of plates that he has. Vlad looks at Arnold's bar and is impressed to see him bench-press $600$ pounds. Unfortunately, Vlad mistook each plate on Arnold's bar for the plate one size he... | 1. **Identify the problem and given information:**
- Arnold has plates weighing \(5\), \(15\), \(25\), \(35\), or \(45\) pounds.
- The bar itself weighs \(45\) pounds.
- Vlad mistakenly thought Arnold was lifting \(600\) pounds.
- Arnold was actually lifting \(470\) pounds.
- Vlad mistook each plate for ... | 13 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Call a day a [i]perfect[/i] day if the sum of the digits of the month plus sum of the digits of the day equals the sum of digits of the year. For example, February $28$th, $2028$ is a perfect day because $2+2+8=2+0+2+8$. Find the number of perfect days in $2018$.
[i]2018 CCA Math Bonanza Team Round #5[/i] | 1. First, we need to determine the sum of the digits of the year 2018:
\[
2 + 0 + 1 + 8 = 11
\]
Therefore, for a day to be perfect, the sum of the digits of the month plus the sum of the digits of the day must equal 11.
2. We will analyze each month to find the days that satisfy this condition.
3. **Month... | 36 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Circle $\Gamma$ with radius $1$ is centered at point $A$ on the circumference of circle $\omega$ with radius $7$. Suppose that point $P$ lies on $\omega$ with $AP=4$. Determine the product of the distances from $P$ to the two intersections of $\omega$ and $\Gamma$.
[i]2018 CCA Math Bonanza Team Round #6[/i] | 1. Let $\Gamma$ and $\omega$ intersect at points $X$ and $Y$. We need to determine the product of the distances from point $P$ to these intersection points.
2. Since $A$ is the center of $\Gamma$ and lies on the circumference of $\omega$, the distance $AP$ is given as $4$. The radius of $\omega$ is $7$, and the radius... | 15 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
$21$ Savage has a $12$ car garage, with a row of spaces numbered $1,2,3,\ldots,12$. How many ways can he choose $6$ of them to park his $6$ identical cars in, if no $3$ spaces with consecutive numbers may be all occupied?
[i]2018 CCA Math Bonanza Team Round #9[/i] | 1. Define \( f(n, k) \) as the number of ways to arrange \( k \) cars in \( n \) spaces such that no three consecutive spaces are occupied.
2. Consider the possible endings of a valid arrangement:
- If the arrangement ends in an unoccupied space \( O \), then the rest of the arrangement is \( f(n-1, k) \).
- If t... | 357 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The irrational number $\alpha>1$ satisfies $\alpha^2-3\alpha-1=0$. Given that there is a fraction $\frac{m}{n}$ such that $n<500$ and $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$, find $m$.
[i]2018 CCA Math Bonanza Team Round #10[/i] | 1. First, we need to find the positive root of the quadratic equation \(\alpha^2 - 3\alpha - 1 = 0\). Using the quadratic formula \( \alpha = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \(a = 1\), \(b = -3\), and \(c = -1\), we get:
\[
\alpha = \frac{3 \pm \sqrt{9 + 4}}{2} = \frac{3 \pm \sqrt{13}}{2}
\]
Si... | 199 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The CCA Math Banana$^{\text{TM}}$ costs \$$100$. The cost rises $10$% then drops $10$%. Now what is the cost of the CCA Math Banana$^{\text{TM}}$?
[i]2018 CCA Math Bonanza Lightning Round #1.2[/i] | 1. **Initial Cost Calculation**:
The initial cost of the CCA Math Banana$^{\text{TM}}$ is $100.
2. **First Price Change (Increase by 10%)**:
When the cost rises by 10%, the new cost can be calculated as follows:
\[
\text{New Cost} = 100 + 0.10 \times 100 = 100 \times \left(1 + \frac{10}{100}\right) = 100 \... | 99 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the sum of all distinct values of $x$ that satisfy $x^4-x^3-7x^2+13x-6=0$?
[i]2018 CCA Math Bonanza Lightning Round #1.4[/i] | 1. **Identify the polynomial and the goal**: We are given the polynomial \( P(x) = x^4 - x^3 - 7x^2 + 13x - 6 \) and need to find the sum of all distinct values of \( x \) that satisfy \( P(x) = 0 \).
2. **Test simple roots**: We start by testing \( x = 1 \) and \( x = -1 \) to see if they are roots of the polynomial.... | 0 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of the first $2018$ positive integers, and let $T$ be the set of all distinct numbers of the form $ab$, where $a$ and $b$ are distinct members of $S$. What is the $2018$th smallest member of $T$?
[i]2018 CCA Math Bonanza Lightning Round #2.1[/i] | 1. **Define the sets \( S \) and \( T \):**
- \( S \) is the set of the first 2018 positive integers: \( S = \{1, 2, 3, \ldots, 2018\} \).
- \( T \) is the set of all distinct numbers of the form \( ab \), where \( a \) and \( b \) are distinct members of \( S \).
2. **Understand the structure of \( T \):**
-... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
On January $20$, $2018$, Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$, $3$, $5$, $7$, $9$, $11$, and $13$, with $1+3+5+7+9+11+13=49$. Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January $20$th, and le... | 1. We start by noting that the sum of the ages of Sally's 7 children on January 20, 2018, is given by:
\[
1 + 3 + 5 + 7 + 9 + 11 + 13 = 49
\]
This sum is a perfect square, specifically \(49 = 7^2\).
2. Let \(x\) be the number of years after 2018. The sum of the children's ages \(x\) years later will be:
... | 218 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The number $16^4+16^2+1$ is divisible by four distinct prime numbers. Compute the sum of these four primes.
[i]2018 CCA Math Bonanza Lightning Round #3.1[/i] | 1. We start with the given expression \(16^4 + 16^2 + 1\). Notice that \(16 = 2^4\), so we can rewrite the expression in terms of powers of 2:
\[
16^4 + 16^2 + 1 = (2^4)^4 + (2^4)^2 + 1 = 2^{16} + 2^8 + 1
\]
2. Next, we recognize that \(2^{16} + 2^8 + 1\) can be factored using the sum of cubes formula. Recall... | 264 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many positive integers $n\leq100$ satisfy $\left\lfloor n\pi\right\rfloor=\left\lfloor\left(n-1\right)\pi\right\rfloor+3$? Here $\left\lfloor x\right\rfloor$ is the greatest integer less than or equal to $x$; for example, $\left\lfloor\pi\right\rfloor=3$.
[i]2018 CCA Math Bonanza Lightning Round #3.2[/i] | To solve the problem, we need to find the number of positive integers \( n \leq 100 \) that satisfy the equation:
\[
\left\lfloor n\pi \right\rfloor = \left\lfloor (n-1)\pi \right\rfloor + 3
\]
1. **Understanding the Floor Function**:
The floor function \(\left\lfloor x \right\rfloor\) gives the greatest integer le... | 86 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A subset of $\left\{1,2,3,\ldots,2017,2018\right\}$ has the property that none of its members are $5$ times another. What is the maximum number of elements that such a subset could have?
[i]2018 CCA Math Bonanza Lightning Round #4.2[/i] | To solve this problem, we need to find the maximum number of elements in a subset of $\{1, 2, 3, \ldots, 2018\}$ such that no element in the subset is 5 times another element. We will consider the numbers based on the number of factors of 5 they contain.
1. **Case 1: Numbers with an even number of factors of 5**
-... | 1698 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
What is the maximum number of diagonals of a regular $12$-gon which can be selected such that no two of the chosen diagonals are perpendicular?
Note: sides are not diagonals and diagonals which intersect outside the $12$-gon at right angles are still considered perpendicular.
[i]2018 CCA Math Bonanza Tiebreaker Round... | 1. **Understanding the problem**: We need to find the maximum number of diagonals in a regular 12-gon such that no two diagonals are perpendicular.
2. **Counting the diagonals**: The total number of diagonals in an $n$-gon is given by the formula:
\[
\frac{n(n-3)}{2}
\]
For a 12-gon, this becomes:
\[
... | 24 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Given that $5^{2018}$ has $1411$ digits and starts with $3$ (the leftmost non-zero digit is $3$), for how many integers $1\leq n\leq2017$ does $5^n$ start with $1$?
[i]2018 CCA Math Bonanza Tiebreaker Round #3[/i] | 1. **Understanding the Problem:**
We are given that \(5^{2018}\) has 1411 digits and starts with 3. We need to determine how many integers \(1 \leq n \leq 2017\) make \(5^n\) start with 1.
2. **Number of Digits in \(5^n\):**
The number of digits \(d\) of a number \(x\) can be found using the formula:
\[
d ... | 607 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $32$ students in the class with $10$ interesting group. Each group contains exactly $16$ students. For each couple of students, the square of the number of the groups which are only involved by just one of the two students is defined as their $interests-disparity$. Define $S$ as the sum of the $interests-disp... | 1. **Construct the Incidence Matrix**:
Let's denote the incidence matrix by \( A \), where \( a_{i,j} \) indicates whether student \( j \) is in the \( i \)-th group. Each row of \( A \) corresponds to a group, and each column corresponds to a student. Given that there are 10 groups and 32 students, \( A \) is a \( ... | 14080 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square. | To determine the least positive integer \( n \) such that for every 3-coloring of the numbers \( 1, 2, \ldots, n \), there exist two different numbers \( a \) and \( b \) of the same color such that \( |a - b| \) is a perfect square, we proceed as follows:
1. **Define the sets and the problem:**
Let the set of numb... | 28 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $(a_n)_{n\ge 1}$ be a sequence such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n + 2}$, for any $n\ge 1$. Show that the sequence $(x_n)_{n\ge 1}$ given by $x_n = \log_{a_n} a_{n + 1}$ for $n\ge 1$ is convergent and compute its limit. | 1. Given the sequence $(a_n)_{n \ge 1}$ such that $a_n > 1$ and $a_{n+1}^2 \ge a_n a_{n+2}$ for any $n \ge 1$, we need to show that the sequence $(x_n)_{n \ge 1}$ given by $x_n = \log_{a_n} a_{n+1}$ is convergent and compute its limit.
2. First, take the logarithm of both sides of the inequality $a_{n+1}^2 \ge a_n a_{... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal{F}$ be the set of continuous functions $f : [0, 1]\to\mathbb{R}$ satisfying $\max_{0\le x\le 1} |f(x)| = 1$ and let $I : \mathcal{F} \to \mathbb{R}$,
\[I(f) = \int_0^1 f(x)\, \text{d}x - f(0) + f(1).\]
a) Show that $I(f) < 3$, for any $f \in \mathcal{F}$.
b) Determine $\sup\{I(f) \mid f \in \mathcal{F... | ### Part (a)
1. **Given Conditions:**
- \( f \) is a continuous function on \([0, 1]\).
- \(\max_{0 \le x \le 1} |f(x)| = 1\).
2. **Objective:**
- Show that \( I(f) < 3 \) for any \( f \in \mathcal{F} \).
3. **Expression for \( I(f) \):**
\[
I(f) = \int_0^1 f(x) \, \text{d}x - f(0) + f(1)
\]
4. **B... | 3 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We have $1000$ balls in $40$ different colours, $25$ balls of each colour. Determine the smallest $n$ for which the following holds: if you place the $1000$ balls in a circle, in any arbitrary way, then there are always $n$ adjacent balls which have at least $20$ different colours. | 1. We start by understanding the problem: We have 1000 balls in 40 different colors, with 25 balls of each color. We need to determine the smallest number \( n \) such that in any arrangement of these balls in a circle, there are always \( n \) adjacent balls that include at least 20 different colors.
2. Consider the ... | 352 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We define a sequence of positive integers $a_1,a_2,a_3,\dots$ as follows: Let $a_1=1$ and iteratively, for $k =2,3,\dots$ let $a_k$ be the largest prime factor of $1+a_1a_2\cdots a_{k-1}$. Show that the number $11$ is not an element of this sequence. | 1. **Initial Setup and Sequence Definition:**
We define the sequence \(a_1, a_2, a_3, \ldots\) as follows:
- \(a_1 = 1\)
- For \(k = 2, 3, \ldots\), \(a_k\) is the largest prime factor of \(1 + a_1 a_2 \cdots a_{k-1}\).
2. **Prime Factor Uniqueness:**
If a prime number appears once in the sequence, it cann... | 11 | Number Theory | proof | Yes | Yes | aops_forum | false |
$12$ friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are $B_1, B_2, ..., B_{12}$. Find the largest possible value of the sum $\Sigma_3=B_1^3+B_2^3+ ... + B_{12}^3$ . | 1. **Understanding the Problem:**
We have 12 friends playing a tennis tournament where each player plays exactly one game with each of the other 11 players. The winner of each game gets 1 point, and the loser gets 0 points. We need to find the maximum possible value of the sum \( \Sigma_3 = B_1^3 + B_2^3 + \cdots + ... | 4356 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group. | 1. **Graph Representation**:
Let each person be represented as a vertex in a graph \( G \), and let two vertices share an edge if they are friends. Given that any pair of persons has exactly one common friend, we can set up a double count on every pair of people.
2. **Double Counting**:
We start by counting the ... | 2014 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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