problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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What is the smallest possible natural number $n$ for which the equation $x^2 -nx + 2014 = 0$ has integer roots? | 1. Let $\zeta$ and $\xi$ be the roots of the quadratic equation $x^2 - nx + 2014 = 0$. By Vieta's formulas, we know:
\[
\begin{cases}
\zeta + \xi = n \\
\zeta \xi = 2014
\end{cases}
\]
2. Since $\zeta$ and $\xi$ are roots of the equation and must be integers, they must be divisors of 2014. We need to... | 91 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set of real numbers with mean $M$. If the means of the sets $S\cup \{15\}$ and $S\cup \{15,1\}$ are $M + 2$ and $M + 1$, respectively, then how many elements does $S$ have? | 1. Let \( S \) be a set of \( k \) elements with mean \( M \). Therefore, the sum of the elements in \( S \) is \( Mk \).
2. When we add the number 15 to the set \( S \), the new set \( S \cup \{15\} \) has \( k+1 \) elements. The mean of this new set is given as \( M + 2 \). Therefore, the sum of the elements in \( S... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Natural numbers $k, l,p$ and $q$ are such that if $a$ and $b$ are roots of $x^2 - kx + l = 0$ then $a +\frac1b$ and $b + \frac1a$ are the roots of $x^2 -px + q = 0$. What is the sum of all possible values of $q$? | 1. Assume the roots of the equation \(x^2 - kx + l = 0\) are \(\alpha\) and \(\beta\). By Vieta's formulas, we have:
\[
\alpha + \beta = k \quad \text{and} \quad \alpha \beta = l
\]
2. We need to find an equation whose roots are \(\alpha + \frac{1}{\beta}\) and \(\beta + \frac{1}{\alpha}\). The sum of these r... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$? | 1. Let \( x = ha \) and \( y = hb \) for some natural numbers \( h, a, \) and \( b \). Notice that \(\gcd(x, y) = h\).
2. Substitute \( x = ha \) and \( y = hb \) into the given equation \( xy = x + y + \gcd(x, y) \):
\[
(ha)(hb) = ha + hb + h
\]
3. Simplify the equation:
\[
h^2 ab = h(a + b) + h
\]
4... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle B DC = 90^o$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$ then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$ ? | 1. **Label the tangent points and define variables:**
- Let the tangent of side \( AD \) to the incircle of \(\triangle ABD\) be \( F \).
- Let \( QD = x \).
- Since \( PQ = 200 \), we have \( PD = 200 + x \).
2. **Use the property of tangents from a point to a circle:**
- By the property of tangents from ... | 799 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For how many natural numbers $n$ between $1$ and $2014$ (both inclusive) is $\frac{8n}{9999-n}$ an integer?
| 1. We need to determine for how many natural numbers \( n \) between \( 1 \) and \( 2014 \) (both inclusive) the expression \(\frac{8n}{9999-n}\) is an integer.
2. Let \( k = 9999 - n \). Then, the expression becomes \(\frac{8(9999 - k)}{k} = \frac{8 \cdot 9999 - 8k}{k} = \frac{79992 - 8k}{k}\).
3. For \(\frac{79992 - ... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $XOY$ be a triangle with $\angle XOY = 90^o$. Let $M$ and $N$ be the midpoints of legs $OX$ and $OY$, respectively. Suppose that $XN = 19$ and $YM =22$. What is $XY$? | 1. Given that $\angle XOY = 90^\circ$, triangle $XOY$ is a right triangle with $OX$ and $OY$ as the legs and $XY$ as the hypotenuse.
2. Let $O$ be the origin $(0,0)$, $X$ be $(a,0)$, and $Y$ be $(0,b)$. Therefore, $OX = a$ and $OY = b$.
3. $M$ is the midpoint of $OX$, so $M$ has coordinates $\left(\frac{a}{2}, 0\right)... | 26 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$? | 1. Consider the quadratic equation \(x^2 + ax + b = 0\) with integer roots. Let the roots be \(p\) and \(q\). Then, by Vieta's formulas, we have:
\[
p + q = -a \quad \text{and} \quad pq = b
\]
2. For the quadratic equation to have integer roots, the discriminant must be a perfect square. The discriminant \(D\)... | 240 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A person moves in the $x-y$ plane moving along points with integer co-ordinates $x$ and $y$ only. When she is at a point $(x,y)$, she takes a step based on the following rules:
(a) if $x+y$ is even she moves to either $(x+1,y)$ or $(x+1,y+1)$;
(b) if $x+y$ is odd she moves to either $(x,y+1)$ or $(x+1,y+1)$.
How many d... | 1. **Understanding the Problem:**
- The person starts at \((0,0)\) and needs to reach \((8,8)\).
- The person can move right, up, or diagonally based on the sum \(x + y\):
- If \(x + y\) is even, the person can move to \((x+1, y)\) or \((x+1, y+1)\).
- If \(x + y\) is odd, the person can move to \((x, y... | 462 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any natural number, let $S(n)$ denote sum of digits of $n$. Find the number of $3$ digit numbers for which $S(S(n)) = 2$. | 1. **Understanding the Problem:**
We need to find the number of three-digit numbers \( n \) such that \( S(S(n)) = 2 \), where \( S(n) \) denotes the sum of the digits of \( n \).
2. **Sum of Digits Constraints:**
For a three-digit number \( n \), the sum of its digits \( S(n) \) can be at most \( 27 \) (since t... | 100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given are 100 different positive integers. We call a pair of numbers [i]good[/i] if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
[i]Proposed by Alexander S. Golovanov, Russia[/i] | 1. **Define the problem in terms of graph theory:**
We are given 100 different positive integers and need to find the maximum number of pairs \((a, b)\) such that the ratio \(\frac{a}{b}\) is either 2 or 3. We can model this problem using a directed graph \(G\) where each vertex represents one of the 100 integers, a... | 180 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A basket is called "[i]Stuff Basket[/i]" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice ... | 1. **Define Variables:**
Let \( a_i \) be the amount of rice in basket \( i \) and \( b_i \) be the number of eggs in basket \( i \). We need to ensure that each basket has exactly 10 kg of rice and 30 eggs.
2. **Initial Conditions:**
We are given that the total amount of rice is 1000 kg and the total number of ... | 99 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation
$b$ comes exactly after $a$ | To solve this problem, we need to find the maximum number of permutations of the set $\{1, 2, 3, \ldots, 2014\}$ such that for every pair of different numbers $a$ and $b$ in this set, there exists at least one permutation where $b$ comes exactly after $a$.
1. **Understanding the Problem:**
- We need to ensure that ... | 1007 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
6 points $A,\ B,\ C,\ D,\ E,\ F$ are on a circle in this order and the segments $AD,\ BE,\ CF$ intersect at one point.
If $AB=1,\ BC=2,\ CD=3,\ DE=4,\ EF=5$, then find the length of the segment $FA$. | To solve this problem, we will use **Ceva's Theorem** in its trigonometric form for the cyclic hexagon. Ceva's Theorem states that for a triangle \( \Delta ABC \) with points \( D, E, F \) on sides \( BC, CA, AB \) respectively, the cevians \( AD, BE, CF \) are concurrent if and only if:
\[
\frac{\sin \angle BAD}{\sin... | 6 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Given a grid of $6\times 6$, write down integer which is more than or equal to 1 and less than or equal to 6 in each grid. For integers $1\leq i,\ j\leq 6$, denote
denote by $i \diamondsuit j$ the integer written in the entry in the $i-$ th row and $j-$ th colum of the grid.
How many ways are there to write down integ... | 1. **Define the Problem and Notations:**
We are given a $6 \times 6$ grid where each entry is an integer between 1 and 6. For integers $1 \leq i, j \leq 6$, denote by $i \diamondsuit j$ the integer written in the entry in the $i$-th row and $j$-th column of the grid. The conditions to satisfy are:
- For any integ... | 42 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
How many ways are there to arrange numbers from 1 to 8 in circle in such way the adjacent numbers are coprime?
Note that we consider the case of rotation and turn over as distinct way. | 1. **Identify the constraints**: We need to arrange the numbers 1 to 8 in a circle such that adjacent numbers are coprime. This means:
- Even numbers must alternate with odd numbers.
- The neighbors of 6 cannot be 3, as they are not coprime.
2. **Arrange even and odd numbers alternately**:
- The even numbers... | 72 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a\# b$ be defined as $ab-a-3$. For example, $4\#5=20-4-3=13$ Compute $(2\#0)\#(1\#4)$. | 1. First, compute \(2 \# 0\):
\[
2 \# 0 = 2 \cdot 0 - 2 - 3 = 0 - 2 - 3 = -5
\]
2. Next, compute \(1 \# 4\):
\[
1 \# 4 = 1 \cdot 4 - 1 - 3 = 4 - 1 - 3 = 0
\]
3. Now, use the results from steps 1 and 2 to compute \((-5) \# 0\):
\[
-5 \# 0 = (-5) \cdot 0 - (-5) - 3 = 0 + 5 - 3 = 2
\]
The fin... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write (and submit) before he has written half of the total problems? | 1. Let \( x \) be the number of additional problems Alex needs to write and submit.
2. After writing and submitting \( x \) more problems, Alex will have written \( 61 + x \) problems.
3. The total number of problems submitted will then be \( 187 + x \).
4. We need to find \( x \) such that Alex has written half of the... | 65 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The sum of two integers is $8$. The sum of the squares of those two integers is $34$. What is the product of the two
integers? | 1. Let the two integers be \( x \) and \( y \). We are given two equations:
\[
x + y = 8
\]
\[
x^2 + y^2 = 34
\]
2. To find the product \( xy \), we start by squaring the first equation:
\[
(x + y)^2 = 8^2
\]
\[
x^2 + 2xy + y^2 = 64
\]
3. We already know from the second equation th... | 15 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The side length of a cube is increased by $20\%$. The surface area of the cube then increases by $x\%$ and the volume of the cube increases by $y\%$. Find $5(y -x)$. | 1. **Determine the initial side length of the cube:**
Let the initial side length of the cube be \( s \).
2. **Calculate the initial surface area and volume:**
- The surface area \( A \) of a cube with side length \( s \) is given by:
\[
A = 6s^2
\]
- The volume \( V \) of a cube with side leng... | 144 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
When submitting problems, Steven the troll likes to submit silly names rather than his own. On day $1$, he gives no
name at all. Every day after that, he alternately adds $2$ words and $4$ words to his name. For example, on day $4$ he
submits an $8\text{-word}$ name. On day $n$ he submits the $44\text{-word name}$ “Ste... | 1. On day 1, Steven gives no name at all, so the number of words is 0.
2. From day 2 onwards, he alternately adds 2 words and 4 words to his name. This means:
- On day 2, he adds 2 words.
- On day 3, he adds 4 words.
- On day 4, he adds 2 words.
- On day 5, he adds 4 words.
- And so on.
3. We need to fin... | 16 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Alex is training to make $\text{MOP}$. Currently he will score a $0$ on $\text{the AMC,}\text{ the AIME,}\text{and the USAMO}$. He can expend $3$ units of effort to gain $6$ points on the $\text{AMC}$, $7$ units of effort to gain $10$ points on the $\text{AIME}$, and $10$ units of effort to gain $1$ point on the $\text... | 1. **Identify the requirements and constraints:**
- Alex needs at least 200 points combined on the AMC and AIME.
- Alex needs at least 21 points on the USAMO.
- Effort to points conversion:
- AMC: 3 units of effort for 6 points.
- AIME: 7 units of effort for 10 points.
- USAMO: 10 units of effor... | 320 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
$a_1,a_2,...,a_{2014}$ is a permutation of $1,2,3,...,2014$. What is the greatest number of perfect squares can have a set ${ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?$ | 1. We start by noting that \(a_1, a_2, \ldots, a_{2014}\) is a permutation of \(1, 2, \ldots, 2014\). This means each \(a_i\) is a distinct integer from 1 to 2014.
2. We need to determine the maximum number of perfect squares in the set \(\{a_1^2 + a_2, a_2^2 + a_3, \ldots, a_{2013}^2 + a_{2014}, a_{2014}^2 + a_1\}\).
... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$? | 1. We start with the given equations:
\[
x_i + 2x_{i+1} = 1 \quad \text{for} \quad i = 1, 2, \ldots, 9
\]
2. To solve for \( x_1 + 512x_{10} \), we multiply each equation by \( (-2)^{i-1} \) and then sum them up. This approach leverages the telescoping series property.
3. Multiply each equation by \( (-2)^{i... | 171 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Say that an integer $n \ge 2$ is [i]delicious[/i] if there exist $n$ positive integers adding up to 2014 that have distinct remainders when divided by $n$. What is the smallest delicious integer? | 1. We need to find the smallest integer \( n \ge 2 \) such that there exist \( n \) positive integers adding up to 2014, and these integers have distinct remainders when divided by \( n \).
2. Let \( a_1, a_2, \ldots, a_n \) be the \( n \) positive integers. We know:
\[
a_1 + a_2 + \cdots + a_n = 2014
\]
a... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$. | 1. **Understanding the problem**: We need to find the largest number \( N \) such that there exist \( 2^N - 1 \) distinct positive integers that sum up to 2014. Each possible nonempty group of students is assigned a unique positive integer, and the sum of all these integers is 2014.
2. **Sum of first \( k \) positive ... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
An ant is on one face of a cube. At every step, the ant walks to one of its four neighboring faces with equal probability. What is the expected (average) number of steps for it to reach the face opposite its starting face? | 1. **Define the problem and variables:**
- Let \( A \) be the starting face of the ant.
- Let \( B_1, B_2, B_3, B_4 \) be the faces adjacent to \( A \).
- Let \( C \) be the face opposite \( A \).
- We need to find \( E(A) \), the expected number of steps for the ant to reach \( C \) from \( A \).
2. **Set... | 6 | Other | math-word-problem | Yes | Yes | aops_forum | false |
For how many integers $k$ such that $0 \le k \le 2014$ is it true that the binomial coefficient $\binom{2014}{k}$ is a multiple of 4? | 1. **Understanding the Lemma**: The lemma states that if \( m \) and \( n \) are positive integers and \( p \) is a prime, and \( x \) is the number of carries that occur when \( m \) and \( n \) are added in base \( p \), then \( \nu_p \left( \binom{m+n}{n} \right) = x \). Here, \( \nu_p \) denotes the p-adic valuatio... | 991 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many complex numbers $z$ such that $\left| z \right| < 30$ satisfy the equation
\[
e^z = \frac{z - 1}{z + 1} \, ?
\] | 1. **Substitute \( z = m + ni \) into the equation:**
Given the equation:
\[
e^z = \frac{z - 1}{z + 1}
\]
Substitute \( z = m + ni \):
\[
e^{m+ni} = \frac{(m-1) + ni}{(m+1) + ni}
\]
2. **Separate the exponential term:**
Using Euler's formula \( e^{m+ni} = e^m \cdot e^{ni} \):
\[
e^m ... | 10 | Other | math-word-problem | Yes | Yes | aops_forum | false |
In Happy City there are $2014$ citizens called $A_1, A_2, \dots , A_{2014}$. Each of them is either [i]happy[/i] or [i]unhappy[/i] at any moment in time. The mood of any citizen $A$ changes (from being unhappy to being happy or vice versa) if and only if some other happy citizen smiles at $A$. On Monday morning there w... | 1. **Initial Setup and Definitions**:
- Let $A_1, A_2, \ldots, A_{2014}$ be the citizens of Happy City.
- Each citizen is either happy (denoted by 1) or unhappy (denoted by 0).
- On Monday morning, there are $N$ happy citizens.
- The sequence of smiles is as follows: $A_1$ smiles at $A_2$, $A_2$ smiles at $... | 32 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Define $p(n)$ to be th product of all non-zero digits of $n$. For instance $p(5)=5$, $p(27)=14$, $p(101)=1$ and so on. Find the greatest prime divisor of the following expression:
\[p(1)+p(2)+p(3)+...+p(999).\] | 1. **Define the function \( p(n) \):**
The function \( p(n) \) is defined as the product of all non-zero digits of \( n \). For example:
\[
p(5) = 5, \quad p(27) = 2 \times 7 = 14, \quad p(101) = 1 \times 1 = 1
\]
2. **Observation on concatenation:**
If \( a \) and \( b \) are natural numbers and \( c \... | 103 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many pairs of integers $(m,n)$ are there such that $mn+n+14=\left (m-1 \right)^2$?
$
\textbf{a)}\ 16
\qquad\textbf{b)}\ 12
\qquad\textbf{c)}\ 8
\qquad\textbf{d)}\ 6
\qquad\textbf{e)}\ 2
$ | 1. Start with the given equation:
\[
mn + n + 14 = (m-1)^2
\]
2. Expand and simplify the right-hand side:
\[
(m-1)^2 = m^2 - 2m + 1
\]
3. Substitute this back into the original equation:
\[
mn + n + 14 = m^2 - 2m + 1
\]
4. Rearrange the equation to isolate terms involving \( n \):
\[
... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the $2014^{\text{th}}$ number in that sequence?
$
\textbf{(A)}\ 66480
\qquad\textbf{(B)}\ 64096
\qquad\textbf{(C)}\ 62048
\qquad\textbf{(D)}\ 60288
\qquad... | To find the $2014^{\text{th}}$ number in the sequence of numbers containing only even digits, we need to systematically count the numbers with only even digits in their decimal representations.
1. **Count the single-digit "good" numbers:**
The single-digit "good" numbers are: \(2, 4, 6, 8\).
There are 4 such nu... | 62048 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
If $ (x^2+1)(y^2+1)+9=6(x+y)$ where $x,y$ are real numbers, what is $x^2+y^2$?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 4
\qquad\textbf{(E)}\ 3
$ | 1. Given the equation:
\[
(x^2+1)(y^2+1) + 9 = 6(x+y)
\]
We aim to find \(x^2 + y^2\).
2. By the Cauchy-Schwarz Inequality, we have:
\[
(x^2+1)(y^2+1) \geq (xy + 1)^2
\]
However, a more straightforward approach is to use the AM-GM inequality:
\[
(x^2+1)(y^2+1) \geq (x+y)^2
\]
This i... | 7 | Algebra | MCQ | Yes | Yes | aops_forum | false |
In how many ways can $17$ identical red and $10$ identical white balls be distributed into $4$ distinct boxes such that the number of red balls is greater than the number of white balls in each box?
$
\textbf{(A)}\ 5462
\qquad\textbf{(B)}\ 5586
\qquad\textbf{(C)}\ 5664
\qquad\textbf{(D)}\ 5720
\qquad\textbf{(E)}\ 584... | 1. **Glue each white ball to a red ball**:
- We start by pairing each of the 10 white balls with a red ball. This results in 10 white-and-red pairs.
- We now need to distribute these 10 pairs into 4 distinct boxes.
2. **Distribute the 10 white-and-red pairs**:
- The problem of distributing 10 pairs into 4 ... | 5720 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
What is the product of real numbers $a$ which make $x^2+ax+1$ a negative integer for only one real number $x$?
$
\textbf{(A)}\ -1
\qquad\textbf{(B)}\ -2
\qquad\textbf{(C)}\ -4
\qquad\textbf{(D)}\ -6
\qquad\textbf{(E)}\ -8
$ | 1. We start with the quadratic equation \( f(x) = x^2 + ax + 1 \). We need to find the values of \( a \) such that \( f(x) \) is a negative integer for only one real number \( x \).
2. For a quadratic function \( f(x) = x^2 + ax + 1 \) to be a negative integer at only one point, it must have a minimum value that is a ... | -8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 11
\qquad\textbf{(E)}\ 12
$ | 1. Let \( A \) be the student with the most friends, and let \( B \) be a student who is not a friend of \( A \). Let \( X \) be one of the other 19 students in the class.
2. For the triplet \( A, B, X \), since at least two of any three students are friends, \( X \) must be a friend of either \( A \) or \( B \).
3. ... | 10 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers?
$
\textbf{(A)}\ 59170
\qquad\textbf{(B)}\ 59149
\qquad\textbf{(C)}\ 59130
\qquad\textbf{(D)}\ 59121
\qquad\textbf{(E)}\ 59012
$ | 1. We need to determine which of the given numbers cannot be expressed in the form \( x^2 + y^5 \), where \( x \) and \( y \) are integers.
2. First, we consider the possible values of \( y^5 \mod 11 \). Since \( y^5 \) is a power of \( y \), we can use Fermat's Little Theorem, which states that \( y^{10} \equiv 1 \pmo... | 59121 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many distinct sets are there such that each set contains only non-negative powers of $2$ or $3$ and sum of its elements is $2014$?
$
\textbf{(A)}\ 64
\qquad\textbf{(B)}\ 60
\qquad\textbf{(C)}\ 54
\qquad\textbf{(D)}\ 48
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | To solve the problem, we need to determine how many distinct sets of non-negative powers of $2$ and $3$ sum to $2014$. We will use the properties of binary and ternary representations to achieve this.
1. **Representation of Powers of 3**:
We can represent the sum of powers of $3$ in the form $(A)_{10} = (a_6a_5a_4a... | 64 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
The integers $-1$, $2$, $-3$, $4$, $-5$, $6$ are written on a blackboard. At each move, we erase two numbers $a$ and $b$, then we re-write $2a+b$ and $2b+a$. How many of the sextuples $(0,0,0,3,-9,9)$, $(0,1,1,3,6,-6)$, $(0,0,0,3,-6,9)$, $(0,1,1,-3,6,-9)$, $(0,0,2,5,5,6)$ can be gotten?
$
\textbf{(A)}\ 1
\qquad\textb... | 1. **Initial Setup and Sum Calculation:**
- The initial integers on the blackboard are: $-1, 2, -3, 4, -5, 6$.
- Calculate the sum of these integers:
\[
-1 + 2 - 3 + 4 - 5 + 6 = 3
\]
- Note that the sum is odd.
2. **Sum Parity Analysis:**
- At each move, we erase two numbers $a$ and $b$, and... | 1 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. For how many positive integers $k$ less than $2014$, it is $f(k)=8$?
$
\textbf{(A)}\ 45
\qquad\textbf{(B)}\ 120
\qquad\textbf{(C)}\ 165
\qquad\textbf{(D)}\ 180
\qquad\textbf{(E)}\ ... | 1. We start by analyzing the given function \( f \). The function is defined recursively with the following properties:
- \( f(1) = 4 \)
- \( f(2n) = f(n) \)
- \( f(2n+1) = f(n) + 2 \)
2. To understand the behavior of \( f \), we can explore the values of \( f \) for small \( n \):
- \( f(1) = 4 \)
- \(... | 165 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
If the integers $1,2,\dots,n$ can be divided into two sets such that each of the two sets does not contain the arithmetic mean of its any two elements, what is the largest possible value of $n$?
$
\textbf{(A)}\ 7
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 9
\qquad\textbf{(D)}\ 10
\qquad\textbf{(E)}\ \text{None of the... | To solve this problem, we need to determine the largest integer \( n \) such that the integers \( 1, 2, \ldots, n \) can be divided into two sets where neither set contains the arithmetic mean of any two of its elements.
1. **Understanding the Problem:**
- We need to partition the set \( \{1, 2, \ldots, n\} \) into... | 8 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
Let $f(n)$ be the smallest prime which divides $n^4+1$. What is the remainder when the sum $f(1)+f(2)+\cdots+f(2014)$ is divided by $8$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 5
\qquad\textbf{(D)}\ 7
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | 1. **Understanding the problem**: We need to find the remainder when the sum \( f(1) + f(2) + \cdots + f(2014) \) is divided by 8, where \( f(n) \) is the smallest prime that divides \( n^4 + 1 \).
2. **Analyzing \( n^4 + 1 \) modulo 8**:
- For \( n \equiv 0 \pmod{8} \), \( n^4 \equiv 0 \pmod{8} \), so \( n^4 + 1 \... | 5 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $s(n)$ denote the number of positive divisors of positive integer $n$. What is the largest prime divisor of the sum of numbers $(s(k))^3$ for all positive divisors $k$ of $2014^{2014}$?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 11
\qquad\textbf{(D)}\ 13
\qquad\textbf{(E)}\ \text{None of the pre... | To solve the problem, we need to find the largest prime divisor of the sum of the cubes of the number of positive divisors of all positive divisors of \(2014^{2014}\).
1. **Prime Factorization of 2014:**
\[
2014 = 2 \times 19 \times 53
\]
Therefore, any divisor \(k\) of \(2014^{2014}\) can be written in th... | 31 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
$a_{1}=1$ and for all $n \geq 1$, \[ (a_{n+1}-2a_{n})\cdot \left (a_{n+1} - \dfrac{1}{a_{n}+2} \right )=0.\] If $a_{k}=1$, which of the following can be equal to $k$?
$
\textbf{(A)}\ 6
\qquad\textbf{(B)}\ 8
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$ | Given the recurrence relation:
\[ (a_{n+1} - 2a_{n}) \cdot \left(a_{n+1} - \frac{1}{a_{n} + 2}\right) = 0 \]
This implies that for each \( n \geq 1 \), \( a_{n+1} \) must satisfy one of the following two equations:
1. \( a_{n+1} = 2a_{n} \)
2. \( a_{n+1} = \frac{1}{a_{n} + 2} \)
We start with \( a_1 = 1 \) and explor... | 12 | Logic and Puzzles | MCQ | Yes | Yes | aops_forum | false |
There are $k$ stones on the table. Alper, Betul and Ceyhun take one or two stones from the table one by one. The player who cannot make a move loses the game and then the game finishes. The game is played once for each $k=5,6,7,8,9$. If Alper is always the first player, for how many of the games can Alper guarantee tha... | To determine for how many values of \( k \) Alper can guarantee a win, we need to analyze the game for each \( k \) from 5 to 9. The key is to understand the winning and losing positions. A losing position is one where any move leaves the opponent in a winning position.
1. **For \( k = 5 \):**
- If Alper takes 1 st... | 0 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i] | 1. Start with the given equation:
\[
x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1
\]
2. Simplify the innermost expression step by step:
\[
x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1
\]
Let \( y = 2x - 1 \), then the equa... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine, with proof, the smallest positive integer $c$ such that for any positive integer $n$, the decimal representation of the number $c^n+2014$ has digits all less than $5$.
[i]Proposed by Evan Chen[/i] | 1. **Claim**: The smallest positive integer \( c \) such that for any positive integer \( n \), the decimal representation of the number \( c^n + 2014 \) has digits all less than 5 is \( c = 10 \).
2. **Verification for \( c = 10 \)**:
- For any positive integer \( n \), \( 10^n \) is a power of 10, which means it ... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$.
(b) Show... | ### Part (a)
1. Define the function $\xi : \mathbb{Z}^2 \to \mathbb{Z}$ by:
\[
\xi(n,k) =
\begin{cases}
1 & \text{if } n \le k \\
-1 & \text{if } n > k
\end{cases}
\]
2. Construct the polynomial:
\[
P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right)
... | 0 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define $H_n = 1+\frac{1}{2}+\cdots+\frac{1}{n}$. Let the sum of all $H_n$ that are terminating in base 10 be $S$. If $S = m/n$ where m and n are relatively prime positive integers, find $100m+n$.
[i]Proposed by Lewis Chen[/i] | To solve the problem, we need to determine the sum of all harmonic numbers \( H_n \) that are terminating in base 10. A number is terminating in base 10 if and only if its denominator (in simplest form) has no prime factors other than 2 and 5.
1. **Identify Harmonic Numbers \( H_n \) that are terminating in base 10:**... | 9920 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In the game of Guess the Card, two players each have a $\frac{1}{2}$ chance of winning and there is exactly one winner. Sixteen competitors stand in a circle, numbered $1,2,\dots,16$ clockwise. They participate in an $4$-round single-elimination tournament of Guess the Card. Each round, the referee randomly chooses one... | 1. **Understanding the Problem:**
- We have 16 players in a circle, numbered \(1, 2, \ldots, 16\).
- The game is a 4-round single-elimination tournament.
- Each round, players are paired off randomly and the losers are eliminated.
- We need to find the probability that players 1 and 9 face each other in the... | 164 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Call an integer $k$ [i]debatable[/i] if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)
[i]Proposed by Lewis C... | 1. **Define Debatable Numbers**:
An integer \( k \) is called *debatable* if the number of its odd factors is a power of two. To find the number of odd factors of \( k \), we need to consider the prime factorization of \( k \) excluding any factors of 2.
2. **Prime Factorization**:
Let \( k \) be expressed as \... | 17 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be positive reals for which
\begin{align*}
(a+b)(a+c) &= bc + 2 \\
(b+c)(b+a) &= ca + 5 \\
(c+a)(c+b) &= ab + 9
\end{align*}
If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$.
[i]Proposed by Evan Chen[/i] | 1. **Isolate the constants on the right-hand side (RHS) of the given equations:**
\[
(a+b)(a+c) = bc + 2
\]
\[
(b+c)(b+a) = ca + 5
\]
\[
(c+a)(c+b) = ab + 9
\]
2. **Expand the left-hand side (LHS) of each equation:**
\[
a^2 + ab + ac + bc = bc + 2
\]
\[
b^2 + ba + bc + ca = ca... | 4532 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $\sin A \sin B \sin C = \frac{1}{1000}$ and $AB \cdot BC \cdot CA = 1000$. What is the area of triangle $ABC$?
[i]Proposed by Evan Chen[/i] | 1. **Given Information:**
- \(\sin A \sin B \sin C = \frac{1}{1000}\)
- \(AB \cdot BC \cdot CA = 1000\)
2. **Using the Extended Law of Sines:**
The extended law of sines states that for any triangle \(ABC\) with circumradius \(R\),
\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
\]
... | 5 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose we wish to pick a random integer between $1$ and $N$ inclusive by flipping a fair coin. One way we can do this is through generating a random binary decimal between $0$ and $1$, then multiplying the result by $N$ and taking the ceiling. However, this would take an infinite amount of time. We therefore stopping ... | 1. **Understanding the Problem:**
We need to determine the expected number of coin flips required to pick a random integer between $1$ and $N$ inclusive, using a fair coin. The process involves generating a random binary decimal between $0$ and $1$, multiplying the result by $N$, and taking the ceiling. We stop flip... | 664412 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$, $d$ be complex numbers satisfying
\begin{align*}
5 &= a+b+c+d \\
125 &= (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 \\
1205 &= (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 \\
25 &= a^4+b^4+c^4+d^4
\end{align*}
Compute $abcd$.
[i]Proposed by Evan Chen[/i] | Given the complex numbers \(a, b, c, d\) satisfying the following equations:
\[
\begin{align*}
1. & \quad a + b + c + d = 5, \\
2. & \quad (5-a)^4 + (5-b)^4 + (5-c)^4 + (5-d)^4 = 125, \\
3. & \quad (a+b)^4 + (b+c)^4 + (c+d)^4 + (d+a)^4 + (a+c)^4 + (b+d)^4 = 1205, \\
4. & \quad a^4 + b^4 + c^4 + d^4 = 25.
\end{align*}
\... | 70 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $N=10^6$. For which integer $a$ with $0 \leq a \leq N-1$ is the value of \[\binom{N}{a+1}-\binom{N}{a}\] maximized?
[i]Proposed by Lewis Chen[/i] | 1. We start with the expression \(\binom{N}{a+1} - \binom{N}{a}\). To maximize this expression, we need to analyze its behavior as \(a\) changes.
2. Consider the difference of consecutive terms: \(\left( \binom{N}{a+1} - \binom{N}{a} \right) - \left( \binom{N}{a} - \binom{N}{a-1} \right)\).
3. Simplify the above expres... | 499499 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
You drop a 7 cm long piece of mechanical pencil lead on the floor. A bully takes the lead and breaks it at a random point into two pieces. A piece of lead is unusable if it is 2 cm or shorter. If the expected value of the number of usable pieces afterwards is $\frac{m}n$ for relatively prime positive integers $m$ and $n... | 1. Let's denote the length of the mechanical pencil lead as \( L = 7 \) cm. The lead is broken at a random point \( X \) where \( 0 < X < 7 \).
2. We need to determine the probability that each piece of the broken lead is usable. A piece is considered usable if its length is greater than 2 cm.
3. The left piece has l... | 1007 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In land of Nyemo, the unit of currency is called a [i]quack[/i]. The citizens use coins that are worth $1$, $5$, $25$, and $125$ quacks. How many ways can someone pay off $125$ quacks using these coins?
[i]Proposed by Aaron Lin[/i] | To determine the number of ways to pay off 125 quacks using coins worth 1, 5, 25, and 125 quacks, we need to find the number of non-negative integer solutions to the equation:
\[ a + 5b + 25c + 125d = 125 \]
where \(a\), \(b\), \(c\), and \(d\) are non-negative integers.
1. **Case \(d = 1\)**:
- If \(d = 1\), the... | 82 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of integers which are both a multiple of $70$ and a factor of $630{,}000$. A random element $c$ of $S$ is selected. If the probability that there exists an integer $d$ with $\gcd (c,d) = 70$ and $\operatorname{lcm} (c,d) = 630{,}000$ is $\frac mn$ for some relatively prime integers $m$ and $n$, comp... | 1. **Identify the set \( S \)**:
- \( S \) is the set of integers that are both multiples of \( 70 \) and factors of \( 630,000 \).
- First, factorize \( 70 \) and \( 630,000 \):
\[
70 = 2 \times 5 \times 7
\]
\[
630,000 = 2^3 \times 3^2 \times 5^3 \times 7
\]
- Any element \( c \... | 106 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt3$, find the length of $AB$.
[i]Proposed by Eugene Chen[/i] | 1. Let $ABC$ be an equilateral triangle with side length $s$. Denote $D$ as the midpoint of $\overline{BC}$, and let $\Omega$ be the circle with diameter $\overline{AD}$.
2. Since $D$ is the midpoint of $\overline{BC}$, $BD = DC = \frac{s}{2}$.
3. In an equilateral triangle, the altitude from $A$ to $BC$ (which is also... | 80 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
[i]Proposed by Kevin Sun[/i] | 1. First, we need to compute $\eta(10)$. The divisors of $10$ are $1, 2, 5, 10$. Therefore, the product of all divisors of $10$ is:
\[
\eta(10) = 1 \times 2 \times 5 \times 10 = 100 = 10^2
\]
2. Next, we compute $\eta(10^2)$. The number $10^2 = 100$ has the following divisors: $1, 2, 4, 5, 10, 20, 25, 50, 100... | 450 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of positive integers $n$ with exactly $1974$ factors such that no prime greater than $40$ divides $n$, and $n$ ends in one of the digits $1$, $3$, $7$, $9$. (Note that $1974 = 2 \cdot 3 \cdot 7 \cdot 47$.)
[i]Proposed by Yonah Borns-Weil[/i] | To solve this problem, we need to find the number of positive integers \( n \) with exactly \( 1974 \) factors such that no prime greater than \( 40 \) divides \( n \), and \( n \) ends in one of the digits \( 1, 3, 7, 9 \).
1. **Prime Factorization and Number of Divisors**:
The number of divisors of \( n \) is giv... | 9100 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
A black bishop and a white king are placed randomly on a $2000 \times 2000$ chessboard (in distinct squares). Let $p$ be the probability that the bishop attacks the king (that is, the bishop and king lie on some common diagonal of the board). Then $p$ can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are r... | 1. **Determine the total number of ways to place the bishop and the king:**
Since the bishop and the king must be placed on distinct squares, the total number of ways to place them is:
\[
2000^2 \times (2000^2 - 1)
\]
2. **Calculate the number of ways the bishop can attack the king:**
The bishop attacks... | 1333 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let a positive integer $n$ be $\textit{nice}$ if there exists a positive integer $m$ such that \[ n^3 < 5mn < n^3 +100. \] Find the number of [i]nice[/i] positive integers.
[i]Proposed by Akshaj[/i] | We need to find the number of positive integers \( n \) such that there exists a positive integer \( m \) satisfying the inequality:
\[ n^3 < 5mn < n^3 + 100. \]
1. **Rewriting the inequality:**
\[ n^3 < 5mn < n^3 + 100. \]
Dividing the entire inequality by \( n \) (since \( n \) is positive):
\[ n^2 < 5m < n... | 53 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ denote the number of ordered pairs of sets $(A, B)$ such that $A \cup B$ is a size-$999$ subset of $\{1,2,\dots,1997\}$ and $(A \cap B) \cap \{1,2\} = \{1\}$. If $m$ and $k$ are integers such that $3^m5^k$ divides $N$, compute the the largest possible value of $m+k$.
[i]Proposed by Michael Tang[/i] | 1. **Understanding the Problem:**
We need to find the number of ordered pairs of sets \((A, B)\) such that \(A \cup B\) is a subset of \(\{1, 2, \ldots, 1997\}\) with size 999, and \((A \cap B) \cap \{1, 2\} = \{1\}\). This implies that \(1 \in A \cap B\) and \(2 \notin A \cap B\).
2. **Case Analysis:**
- **Case... | 1004 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$.
[i]Propos... | 1. **Identify the slopes of the sides of triangle \(ABC\)**:
Given the slopes of the sides of triangle \(ABC\) are \(63\), \(73\), and \(97\). Let the vertices of the triangle be \(A\), \(B\), and \(C\).
2. **Equation of the curve \(\mathcal{V}\)**:
The curve \(\mathcal{V}\) is given by the equation \(y = (x+3)(... | 237 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
This is an ARML Super Relay! I'm sure you know how this works! You start from #1 and #15 and meet in the middle.
We are going to require you to solve all $15$ problems, though -- so for the entire task, submit the sum of all the answers, rather than just the answer to #8.
Also, uhh, we can't actually find the slip for... | Let's solve each problem step-by-step.
### Problem 2
We need to find the number of ways to distribute 6 indistinguishable pieces of candy to \( T \) distinguishable schoolchildren, such that each child gets at most one piece of candy.
1. Since each child can get at most one piece of candy, the number of children \( T... | 6286 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Three of the below entries, with labels $a$, $b$, $c$, are blatantly incorrect (in the United States).
What is $a^2+b^2+c^2$?
041. The Gentleman's Alliance Cross
042. Glutamine (an amino acid)
051. Grant Nelson and Norris Windross
052. A compact region at the center of a galaxy
061. The value of \verb+'wat'-1+. (See ... | 1. Identify the incorrect entries among the given list. The problem states that three entries are blatantly incorrect in the context of the United States.
2. The incorrect entries are identified as:
- 052: A compact region at the center of a galaxy (This is not specific to the United States)
- 081: A prank or tri... | 21586 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We know $\mathbb Z_{210} \cong \mathbb Z_2 \times \mathbb Z_3 \times \mathbb Z_5 \times \mathbb Z_7$.
Moreover,\begin{align*}
53 & \equiv 1 \pmod{2} \\
53 & \equiv 2 \pmod{3} \\
53 & \equiv 3 \pmod{5} \\
53 & \equiv 4 \pmod{7}.
\end{align*}
Let
\[ M = \left(
\begin{array}{ccc}
53 & 158 & 53 \\
23 & 93 & 53 \\... | To solve the problem, we need to find the matrix \( \overline{(M \mod{2})(M \mod{3})(M \mod{5})(M \mod{7})} \). We will do this by taking the given matrix \( M \) and reducing it modulo 2, 3, 5, and 7 respectively. Then, we will combine these results to form the final answer.
1. **Matrix \( M \) modulo 2:**
\[
M... | 1234 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be largest number such that \[ \frac{2014^{100!}-2011^{100!}}{3^n} \] is still an integer. Compute the remainder when $3^n$ is divided by $1000$. | 1. We need to find the largest integer \( n \) such that
\[
\frac{2014^{100!} - 2011^{100!}}{3^n}
\]
is still an integer. This means that \( 3^n \) must divide \( 2014^{100!} - 2011^{100!} \).
2. First, observe that \( 2014 \equiv 1 \pmod{3} \) and \( 2011 \equiv 2 \pmod{3} \). Therefore,
\[
2014^{1... | 83 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$, $B$, $C$, $D$ be four points on a line in this order. Suppose that $AC = 25$, $BD = 40$, and $AD = 57$. Compute $AB \cdot CD + AD \cdot BC$.
[i]Proposed by Evan Chen[/i] | 1. Given the points \(A\), \(B\), \(C\), and \(D\) on a line in this order, we know the following distances:
\[
AC = 25, \quad BD = 40, \quad AD = 57
\]
2. First, we calculate the distance \(CD\):
\[
CD = AD - AC = 57 - 25 = 32
\]
3. Next, we calculate the distance \(BC\):
\[
BC = BD - CD = 40... | 1000 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In the Generic Math Tournament, $99$ people participate. One of the participants, Alfred, scores 16th in Algebra, 30th in Combinatorics, and 23rd in Geometry (and does not tie with anyone). The overall ranking is computed by adding the scores from all three tests. Given this information, let $B$ be the best ranking tha... | To determine the best and worst possible rankings for Alfred, we need to consider the sum of his scores in Algebra, Combinatorics, and Geometry, and compare it to the possible sums of the other participants.
1. **Calculate Alfred's total score:**
Alfred's scores are:
- Algebra: 16th
- Combinatorics: 30th
-... | 167 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, we have $AB=AC=20$ and $BC=14$. Consider points $M$ on $\overline{AB}$ and $N$ on $\overline{AC}$. If the minimum value of the sum $BN + MN + MC$ is $x$, compute $100x$.
[i]Proposed by Lewis Chen[/i] | 1. **Reflecting the Triangle:**
Reflect $\triangle ABC$ across $AC$ to get $\triangle AB'C$. Then reflect $\triangle AB'C$ across $AB'$ to get $\triangle AB'C'$. Notice that each path $BNMC$ corresponds to a path from $B$ to $C'$, the shortest being the straight line distance.
2. **Calculating the Distance:**
To... | 693 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define the infinite products \[ A = \prod\limits_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod\limits_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If $\tfrac{A}{B} = \tfrac{m}{n}$ where $m,n$ are relatively prime positive integers, determine $100m+n$.
[i]Proposed by Lewis Chen[/i] | 1. First, let's rewrite the infinite products \( A \) and \( B \) in a more manageable form:
\[
A = \prod_{i=2}^{\infty} \left(1 - \frac{1}{i^3}\right)
\]
\[
B = \prod_{i=1}^{\infty} \left(1 + \frac{1}{i(i+1)}\right)
\]
2. Simplify the expression inside the product for \( B \):
\[
1 + \frac{1}{... | 103 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the largest integer $n$ for which $2^n$ divides \[ \binom 21 \binom 42 \binom 63 \dots \binom {128}{64}. \][i]Proposed by Evan Chen[/i] | 1. **Understanding \( v_2 \):**
The notation \( v_2(n) \) represents the largest power of 2 that divides \( n \). In other words, \( v_2(n) \) is the exponent of the highest power of 2 that divides \( n \) without leaving a remainder. For example, \( v_2(8) = 3 \) because \( 8 = 2^3 \), and \( v_2(12) = 2 \) because... | 193 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Ana and Banana play a game. First, Ana picks a real number $p$ with $0 \le p \le 1$. Then, Banana picks an integer $h$ greater than $1$ and creates a spaceship with $h$ hit points. Now every minute, Ana decreases the spaceship's hit points by $2$ with probability $1-p$, and by $3$ with probability $p$. Ana wins if and ... | 1. **Define the problem and variables:**
- Ana picks a real number \( p \) with \( 0 \le p \le 1 \).
- Banana picks an integer \( h \) greater than 1 and creates a spaceship with \( h \) hit points.
- Every minute, Ana decreases the spaceship's hit points by 2 with probability \( 1-p \), and by 3 with probabil... | 618 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $x$ be a positive real number. Define
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad\text{and}\quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\] Given that $A^3+B^3+C^3 + 8ABC = 2014$, compute $ABC$.
[i]Proposed by Evan Chen[/i] | 1. **Define the series and their sums:**
\[
A = \sum_{k=0}^{\infty} \frac{x^{3k}}{(3k)!}, \quad
B = \sum_{k=0}^{\infty} \frac{x^{3k+1}}{(3k+1)!}, \quad
C = \sum_{k=0}^{\infty} \frac{x^{3k+2}}{(3k+2)!}.
\]
2. **Key Claim:**
We claim that:
\[
A^3 + B^3 + C^3 - 3ABC = 1
\]
for all \( x \).
... | 183 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
We have a five-digit positive integer $N$. We select every pair of digits of $N$ (and keep them in order) to obtain the $\tbinom52 = 10$ numbers $33$, $37$, $37$, $37$, $38$, $73$, $77$, $78$, $83$, $87$. Find $N$.
[i]Proposed by Lewis Chen[/i] | To find the five-digit number \( N \) such that every pair of its digits (in order) forms one of the given pairs \( 33, 37, 37, 37, 38, 73, 77, 78, 83, 87 \), we can follow these steps:
1. **Identify the frequency of each pair:**
- The pair \( 37 \) appears 3 times.
- The pair \( 33 \) appears 1 time.
- The p... | 37837 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Aaron takes a square sheet of paper, with one corner labeled $A$. Point $P$ is chosen at random inside of the square and Aaron folds the paper so that points $A$ and $P$ coincide. He cuts the sheet along the crease and discards the piece containing $A$. Let $p$ be the probability that the remaining piece is a pentagon.... | 1. **Understanding the Problem:**
- We start with a square sheet of paper with one corner labeled \( A \).
- A point \( P \) is chosen randomly inside the square.
- The paper is folded such that \( A \) coincides with \( P \).
- The paper is cut along the crease and the piece containing \( A \) is discarded... | 25 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Among $100$ points in the plane, no three collinear, exactly $4026$ pairs are connected by line segments. Each point is then randomly assigned an integer from $1$ to $100$ inclusive, each equally likely, such that no integer appears more than once. Find the expected value of the number of segments which join two points... | 1. **Understanding the Problem:**
- We have 100 points in the plane with no three collinear.
- There are 4026 pairs of points connected by line segments.
- Each point is assigned a unique integer from 1 to 100.
- We need to find the expected value of the number of segments joining two points whose labels di... | 1037 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Consider real numbers $A$, $B$, \dots, $Z$ such that \[
EVIL = \frac{5}{31}, \;
LOVE = \frac{6}{29}, \text{ and }
IMO = \frac{7}{3}.
\] If $OMO = \tfrac mn$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.
[i]Proposed by Evan Chen[/i] | 1. We start with the given equations:
\[
EVIL = \frac{5}{31}, \quad LOVE = \frac{6}{29}, \quad \text{and} \quad IMO = \frac{7}{3}.
\]
2. To eliminate the variables \(E\), \(L\), and \(V\), we divide \(LOVE\) by \(EVIL\):
\[
\frac{LOVE}{EVIL} = \frac{\frac{6}{29}}{\frac{5}{31}} = \frac{6}{29} \times \fra... | 579 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
[i]Proposed by Lewis Chen[/i] | 1. **Identify the problem and given points:**
We are given four points \( A = (0,0) \), \( B = (-1,-1) \), \( C = (x,y) \), and \( D = (x+1,y) \) that lie on a circle with radius \( r \). We need to find the smallest and second smallest possible values of \( r \) and compute \( r_1^2 + r_2^2 \).
2. **Use the proper... | 1381 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i] | To solve the problem, we need to determine how many $2 \times 2 \times 2$ cubes must be added to an $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube.
1. **Calculate the volume of the $8 \times 8 \times 8$ cube:**
\[
V_{\text{small}} = 8^3 = 512
\]
2. **Calculate the volume of the $12 \times... | 152 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A square and equilateral triangle have the same perimeter. If the triangle has area $16\sqrt3$, what is the area of the square?
[i]Proposed by Evan Chen[/i] | 1. **Determine the side length of the equilateral triangle:**
- Given the area of the equilateral triangle is \(16\sqrt{3}\).
- The formula for the area of an equilateral triangle with side length \(s\) is:
\[
\text{Area} = \frac{s^2 \sqrt{3}}{4}
\]
- Set the given area equal to the formula and ... | 36 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x$ is a random real number between $1$ and $4$, and $y$ is a random real number between $1$ and $9$. If the expected value of \[ \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \] can be expressed as $\frac mn$ where $m$ and $n$ are relatively prime positive integers, compute $100m + n$... | 1. **Define the variables and expressions:**
Let \( x \) be a random real number between 1 and 4, and \( y \) be a random real number between 1 and 9. We need to find the expected value of \( \left\lceil \log_2 x \right\rceil - \left\lfloor \log_3 y \right\rfloor \).
2. **Analyze \( \left\lceil \log_2 x \right\rcei... | 1112 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumcenter $O$ and let $X$, $Y$, $Z$ be the midpoints of arcs $BAC$, $ABC$, $ACB$ on its circumcircle. Let $G$ and $I$ denote the centroid of $\triangle XYZ$ and the incenter of $\triangle ABC$.
Given that $AB = 13$, $BC = 14$, $CA = 15$, and $\frac {GO}{GI} = \frac mn$ for relatively pr... | 1. **Identify Key Points and Definitions:**
- Let \( D, E, F \) be the midpoints of the minor arcs \( BC, CA, AB \) respectively on the circumcircle of \( \triangle ABC \).
- Let \( G' \) be the reflection of \( G \) over \( O \).
- Note that \( X, Y, Z \) are the midpoints of the arcs \( BAC, ABC, ACB \) resp... | 104 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S = \left\{ 1,2, \dots, 2014 \right\}$. Suppose that \[ \sum_{T \subseteq S} i^{\left\lvert T \right\rvert} = p + qi \] where $p$ and $q$ are integers, $i = \sqrt{-1}$, and the summation runs over all $2^{2014}$ subsets of $S$. Find the remainder when $\left\lvert p\right\rvert + \left\lvert q \right\rvert$ is div... | 1. We start by noting that the sum \(\sum_{T \subseteq S} i^{|T|}\) runs over all subsets \(T\) of \(S\), where \(|T|\) denotes the number of elements in \(T\). Since \(S\) has 2014 elements, there are \(2^{2014}\) subsets of \(S\).
2. Each subset \(T\) of \(S\) contributes \(i^{|T|}\) to the sum. We can group these c... | 872 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Points $A$, $B$, $C$, and $D$ lie on a circle such that chords $\overline{AC}$ and $\overline{BD}$ intersect at a point $E$ inside the circle. Suppose that $\angle ADE =\angle CBE = 75^\circ$, $BE=4$, and $DE=8$. The value of $AB^2$ can be written in the form $a+b\sqrt{c}$ for positive integers $a$, $b$, and $c$ such... | 1. **Identify Similar Triangles:**
Given that $\angle ADE = \angle CBE = 75^\circ$, we can infer that $\triangle ADE$ is similar to $\triangle CBE$ by the AA (Angle-Angle) similarity criterion. The ratio of the sides of these triangles is given by:
\[
\frac{BE}{DE} = \frac{4}{8} = \frac{1}{2}
\]
Therefor... | 115 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the prime factors of $67208001$, given that $23$ is one.
[i]Proposed by Justin Stevens[/i] | 1. **Factorization of the Polynomial:**
Consider the polynomial:
\[
P(x) = x^6 + x^5 + x^3 + 1
\]
We can factorize this polynomial as follows:
\[
P(x) = (x+1)(x^2+1)(x^3-x+1)
\]
This factorization can be verified by expanding the right-hand side and confirming it matches the original polynomi... | 781 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For positive integers $a$, $b$, and $c$, define \[ f(a,b,c)=\frac{abc}{\text{gcd}(a,b,c)\cdot\text{lcm}(a,b,c)}. \] We say that a positive integer $n$ is $f@$ if there exist pairwise distinct positive integers $x,y,z\leq60$ that satisfy $f(x,y,z)=n$. How many $f@$ integers are there?
[i]Proposed by Michael Ren[/i] | 1. **Understanding the function \( f(a, b, c) \)**:
\[
f(a, b, c) = \frac{abc}{\text{gcd}(a, b, c) \cdot \text{lcm}(a, b, c)}
\]
By the properties of gcd and lcm, we know:
\[
\text{gcd}(a, b, c) \cdot \text{lcm}(a, b, c) = abc
\]
Therefore:
\[
f(a, b, c) = \frac{abc}{\text{gcd}(a, b, c) \c... | 70 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any interval $\mathcal{A}$ in the real number line not containing zero, define its [i]reciprocal[/i] to be the set of numbers of the form $\frac 1x$ where $x$ is an element in $\mathcal{A}$. Compute the number of ordered pairs of positive integers $(m,n)$ with $m< n$ such that the length of the interval $[m,n]$ is... | 1. **Define the problem mathematically:**
We are given an interval \([m, n]\) on the real number line, where \(m\) and \(n\) are positive integers with \(m < n\). The length of this interval is \(n - m\). The reciprocal of this interval is the set of numbers \(\left\{\frac{1}{x} \mid x \in [m, n]\right\}\). The leng... | 60 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a square with side length $2$. Let $M$ and $N$ be the midpoints of $\overline{BC}$ and $\overline{CD}$ respectively, and let $X$ and $Y$ be the feet of the perpendiculars from $A$ to $\overline{MD}$ and $\overline{NB}$, also respectively. The square of the length of segment $\overline{XY}$ can be writte... | 1. **Define the coordinates of the square:**
- Let the vertices of the square \(ABCD\) be \(A(0, 2)\), \(B(2, 2)\), \(C(2, 0)\), and \(D(0, 0)\).
2. **Find the midpoints \(M\) and \(N\):**
- \(M\) is the midpoint of \(\overline{BC}\), so \(M\left(\frac{2+2}{2}, \frac{2+0}{2}\right) = (2, 1)\).
- \(N\) is the ... | 3225 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be positive real numbers such that $ab=2$ and \[\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac78.\] Find $a^6+b^6$.
[i]Proposed by David Altizio[/i] | 1. Given the equations:
\[
ab = 2
\]
and
\[
\dfrac{a}{a+b^2}+\dfrac{b}{b+a^2}=\dfrac{7}{8},
\]
we start by combining the fractions on the left-hand side.
2. Combine the fractions by creating a common denominator:
\[
\dfrac{a(b+a^2) + b(a+b^2)}{(a+b^2)(b+a^2)} = \dfrac{7}{8}.
\]
3. Sim... | 128 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A positive integer $N$ greater than $1$ is described as special if in its base-$8$ and base-$9$ representations, both the leading and ending digit of $N$ are equal to $1$. What is the smallest special integer in decimal representation?
[i]Proposed by Michael Ren[/i] | 1. **Understanding the Problem:**
We need to find the smallest positive integer \( N \) greater than 1 such that in both its base-8 and base-9 representations, the leading and ending digits are 1.
2. **Base-8 Representation:**
- For \( N \) to end in 1 in base-8, \( N \equiv 1 \pmod{8} \).
- For \( N \) to st... | 793 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Bob is making partitions of $10$, but he hates even numbers, so he splits $10$ up in a special way. He starts with $10$, and at each step he takes every even number in the partition and replaces it with a random pair of two smaller positive integers that sum to that even integer. For example, $6$ could be replaced with... | To solve this problem, we need to determine the expected number of integers in the partition of 10 when all numbers are odd. We will use the concept of expected value and linearity of expectation.
1. **Define the Expected Value Function:**
Let \( E_n \) be the expected number of integers in the partition of \( n \)... | 902 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is [i]monotonically bounded[/i] if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$. We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a [i]mountain[/i] if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$. Evan writes ... | 1. **Understanding the Problem:**
We need to find the remainder when the total number of mountain terms in all possible monotonically bounded sequences of length \(2^{16}\) is divided by \(p = 2^{16} + 1\).
2. **Defining a Mountain Term:**
A term \(a_k\) in the sequence is a mountain if \(a_k > a_{k-1}\) and \(a... | 49153 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In English class, you have discovered a mysterious phenomenon -- if you spend $n$ hours on an essay, your score on the essay will be $100\left( 1-4^{-n} \right)$ points if $2n$ is an integer, and $0$ otherwise. For example, if you spend $30$ minutes on an essay you will get a score of $50$, but if you spend $35$ minute... | 1. Let \( t_1, t_2, t_3, t_4 \) be the times (in hours) that you spend on the essays, respectively. Since there are 4 hours and 5 minutes until class, and you want each essay to actually give you points, the \( t_i \) are multiples of \( \frac{1}{2} \), and they sum to 4 (4 hours).
2. The score for each essay is given... | 75 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Consider two circles of radius one, and let $O$ and $O'$ denote their centers. Point $M$ is selected on either circle. If $OO' = 2014$, what is the largest possible area of triangle $OMO'$?
[i]Proposed by Evan Chen[/i] | 1. **Positioning Point \( M \)**: Without loss of generality (WLOG), let point \( M \) be on the circle centered at \( O \). This simplifies our calculations as we can focus on the geometry involving \( O \), \( O' \), and \( M \).
2. **Area of Triangle \( OMO' \)**: The area of triangle \( OMO' \) can be calculated u... | 1007 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $m$ and $n$ are relatively prime positive integers with $A = \tfrac mn$, where
\[ A = \frac{2+4+6+\dots+2014}{1+3+5+\dots+2013} - \frac{1+3+5+\dots+2013}{2+4+6+\dots+2014}. \] Find $m$. In other words, find the numerator of $A$ when $A$ is written as a fraction in simplest form.
[i]Proposed by Evan Chen[/... | 1. First, we need to find the sum of the even numbers from 2 to 2014. This sequence is an arithmetic series with the first term \(a = 2\), the common difference \(d = 2\), and the last term \(l = 2014\). The number of terms \(n\) in this series can be found using the formula for the \(n\)-th term of an arithmetic seque... | 2015 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
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