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The integers $1, 2, \dots, n$ are written in order on a long slip of paper. The slip is then cut into five pieces, so that each piece consists of some (nonempty) consecutive set of integers. The averages of the numbers on the five slips are $1234$, $345$, $128$, $19$, and $9.5$ in some order. Compute $n$.
[i]Propose... | 1. **Identify the smallest average**: The smallest average given is \(9.5\). Since the average of a set of consecutive integers is the midpoint of that set, the set must be \(\{1, 2, \ldots, 18\}\). This is because the average of the first 18 integers is:
\[
\frac{1 + 18}{2} = 9.5
\]
Therefore, the first pi... | 2014 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Joe the teacher is bad at rounding. Because of this, he has come up with his own way to round grades, where a [i]grade[/i] is a nonnegative decimal number with finitely many digits after the decimal point.
Given a grade with digits $a_1a_2 \dots a_m.b_1b_2 \dots b_n$, Joe first rounds the number to the nearest $10^{-n... | To solve this problem, we need to understand Joe's rounding process and determine the smallest grade \( x \) that gets rounded to at least 90. We will follow the rounding steps in reverse to find the critical value \( M \).
1. **Final Rounding to Integer:**
- For a grade to be rounded to at least 90, it must be at ... | 814 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
How many integers $n$ with $10 \le n \le 500$ have the property that the hundreds digit of $17n$ and $17n+17$ are different?
[i]Proposed by Evan Chen[/i] | 1. We need to find the number of integers \( n \) such that \( 10 \leq n \leq 500 \) and the hundreds digit of \( 17n \) and \( 17n + 17 \) are different.
2. Let \( 17n = k \). Then \( 17n + 17 = k + 17 \).
3. We need the hundreds digit of \( k \) and \( k + 17 \) to be different. This means that \( k \) must be close ... | 84 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ be real numbers satisfying
\begin{align*}
2a_1+a_2+a_3+a_4+a_5 &= 1 + \tfrac{1}{8}a_4 \\
2a_2+a_3+a_4+a_5 &= 2 + \tfrac{1}{4}a_3 \\
2a_3+a_4+a_5 &= 4 + \tfrac{1}{2}a_2 \\
2a_4+a_5 &= 6 + a_1
\end{align*}
Compute $a_1+a_2+a_3+a_4+a_5$.
[i]Proposed by... | 1. Given the system of equations:
\[
\begin{align*}
2a_1 + a_2 + a_3 + a_4 + a_5 &= 1 + \frac{1}{8}a_4, \\
2a_2 + a_3 + a_4 + a_5 &= 2 + \frac{1}{4}a_3, \\
2a_3 + a_4 + a_5 &= 4 + \frac{1}{2}a_2, \\
2a_4 + a_5 &= 6 + a_1.
\end{align*}
\]
2. Multiply the \(n\)-th equation by \(2^{4-n}\):
\[
... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1A_2 \dots A_{4000}$ be a regular $4000$-gon. Let $X$ be the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$, and let $Y$ be the foot of the altitude from $A_{2014}$ onto $A_{2000}A_{4000}$. If $XY = 1$, what is the area of square $A_{500}A_{1500}A_{2500}A_{3500}$?
[i]Proposed by Evan Chen... | 1. **Understanding the Problem:**
- We have a regular $4000$-gon inscribed in a circle.
- We need to find the area of the square $A_{500}A_{1500}A_{2500}A_{3500}$ given that $XY = 1$.
2. **Analyzing the Given Information:**
- $X$ is the foot of the altitude from $A_{1986}$ onto diagonal $A_{1000}A_{3000}$.
... | 2 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and
\[ g(x) h(x)
= \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \]
holds for all nonzero real numbers $x$. Find $g(2)$.
[i]Proposed by Yang Liu[/i] | 1. We start by analyzing the given polynomial equation:
\[
g(x) h(x) = \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right)
\]
This equation holds for all nonzero real numbers \( x \).
2. We simplify the right-hand side of the equation by combining like... | 2047 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with incenter $I$ and $AB = 1400$, $AC = 1800$, $BC = 2014$. The circle centered at $I$ passing through $A$ intersects line $BC$ at two points $X$ and $Y$. Compute the length $XY$.
[i]Proposed by Evan Chen[/i] | 1. **Define the given parameters and setup the problem:**
- Let $ABC$ be a triangle with incenter $I$.
- Given side lengths: $AB = 1400$, $AC = 1800$, $BC = 2014$.
- The circle centered at $I$ passing through $A$ intersects line $BC$ at points $X$ and $Y$.
- We need to compute the length $XY$.
2. **Calcula... | 1186 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
In Prime Land, there are seven major cities, labelled $C_0$, $C_1$, \dots, $C_6$. For convenience, we let $C_{n+7} = C_n$ for each $n=0,1,\dots,6$; i.e. we take the indices modulo $7$. Al initially starts at city $C_0$.
Each minute for ten minutes, Al flips a fair coin. If the coin land heads, and he is at city $C_... | 1. **Define the problem and initial conditions:**
- We have seven cities labeled \( C_0, C_1, \ldots, C_6 \).
- Al starts at city \( C_0 \).
- Each minute, Al flips a fair coin. If heads, he moves from \( C_k \) to \( C_{2k \mod 7} \); if tails, he moves to \( C_{2k+1 \mod 7} \).
- We need to find the proba... | 147 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Say a positive integer $n$ is [i]radioactive[/i] if one of its prime factors is strictly greater than $\sqrt{n}$. For example, $2012 = 2^2 \cdot 503$, $2013 = 3 \cdot 11 \cdot 61$ and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime f... | To solve the problem, we need to count the number of radioactive numbers that have all prime factors less than 30. A number \( n \) is defined as radioactive if one of its prime factors is strictly greater than \( \sqrt{n} \).
1. **Identify the primes less than 30:**
The prime numbers less than 30 are:
\[
2, ... | 119 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $AXYBZ$ be a convex pentagon inscribed in a circle with diameter $\overline{AB}$. The tangent to the circle at $Y$ intersects lines $BX$ and $BZ$ at $L$ and $K$, respectively. Suppose that $\overline{AY}$ bisects $\angle LAZ$ and $AY=YZ$. If the minimum possible value of \[ \frac{AK}{AX} + \left( \frac{AL}{AB} \... | 1. **Identify Key Angles and Relationships:**
Given that \( \angle LYA = \angle YZA = \angle YAZ = \angle LYA \), we can infer that \( \triangle ALY \) is isosceles, implying \( AL = AY \).
2. **Draw Additional Lines and Use Angle Properties:**
Draw \( YB \). Since:
\[
\angle LAB = \angle LAY + \angle YAZ... | 343 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of pairs $(m,n)$ of integers with $-2014\le m,n\le 2014$ such that $x^3+y^3 = m + 3nxy$ has infinitely many integer solutions $(x,y)$.
[i]Proposed by Victor Wang[/i] | 1. We start with the given equation:
\[
x^3 + y^3 = m + 3nxy
\]
We need to find the pairs \((m, n)\) such that this equation has infinitely many integer solutions \((x, y)\).
2. Let's rearrange the equation by adding \(n^3\) to both sides:
\[
x^3 + y^3 + n^3 = m + n^3 + 3nxy
\]
This can be fact... | 25 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of all positive integers $n$ such that $\tau(n)^2=2n$, where $\tau(n)$ is the number of positive integers dividing $n$.
[i]Proposed by Michael Kural[/i] | To find the sum of all positive integers \( n \) such that \( \tau(n)^2 = 2n \), where \( \tau(n) \) is the number of positive divisors of \( n \), we will proceed as follows:
1. **Understanding the equation**:
\[
\tau(n)^2 = 2n
\]
This implies that \( \tau(n) \) must be an integer, and thus \( 2n \) must ... | 98 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be an acute triangle with circumcenter $O$, and select $E$ on $\overline{AC}$ and $F$ on $\overline{AB}$ so that $\overline{BE} \perp \overline{AC}$, $\overline{CF} \perp \overline{AB}$. Suppose $\angle EOF - \angle A = 90^{\circ}$ and $\angle AOB - \angle B = 30^{\circ}$. If the maximum possible measure of... | 1. Given that $\angle AOB = 2 \angle C$ and $\angle AOB - \angle B = 30^\circ$, we can write:
\[
2 \angle C - \angle B = 30^\circ
\]
Since $\angle A + \angle B + \angle C = 180^\circ$, we can express $\angle B$ as:
\[
\angle B = 180^\circ - \angle A - \angle C
\]
Substituting this into the equat... | 47 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $f(x)$ be a polynomial with integer coefficients such that $f(15) f(21) f(35) - 10$ is divisible by $105$. Given $f(-34) = 2014$ and $f(0) \ge 0$, find the smallest possible value of $f(0)$.
[i]Proposed by Michael Kural and Evan Chen[/i] | 1. **Understanding the problem**: We need to find the smallest possible value of \( f(0) \) given that \( f(x) \) is a polynomial with integer coefficients, \( f(15) f(21) f(35) - 10 \) is divisible by 105, \( f(-34) = 2014 \), and \( f(0) \ge 0 \).
2. **Divisibility conditions**: Since \( f(x) \) has integer coeffici... | 10 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally t... | 1. **Define the centers and radii of the circles:**
- Let $\Gamma_1$ be the circle with center $O_1$ and radius $13$.
- Let $\Gamma_2$ be the circle with center $O_2$ and radius $10$.
- The distance between the centers $O_1$ and $O_2$ is $O_1O_2 = 2$.
2. **Define the fixed circle $\Omega$:**
- $\Omega$ has... | 16909 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal P$ denote the set of planes in three-dimensional space with positive $x$, $y$, and $z$ intercepts summing to one. A point $(x,y,z)$ with $\min \{x,y,z\} > 0$ lies on exactly one plane in $\mathcal P$. What is the maximum possible integer value of $\left(\frac{1}{4} x^2 + 2y^2 + 16z^2\right)^{-1}$?
[i]Pro... | 1. We start by noting that the set \(\mathcal{P}\) consists of planes in three-dimensional space with positive \(x\), \(y\), and \(z\) intercepts summing to one. The equation of such a plane can be written as:
\[
\frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1
\]
where \(a + b + c = 1\).
2. Given a point \((x,... | 21 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
If
\[
\sum_{n=1}^{\infty}\frac{\frac11 + \frac12 + \dots + \frac 1n}{\binom{n+100}{100}} = \frac pq
\]
for relatively prime positive integers $p,q$, find $p+q$.
[i]Proposed by Michael Kural[/i] | To solve the given problem, we need to evaluate the infinite series:
\[
\sum_{n=1}^{\infty}\frac{\frac{1}{1} + \frac{1}{2} + \dots + \frac{1}{n}}{\binom{n+100}{100}}
\]
1. **Rewrite the Binomial Coefficient:**
The binomial coefficient can be rewritten using factorials:
\[
\binom{n+100}{100} = \frac{(n+100)!}... | 9901 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Qing initially writes the ordered pair $(1,0)$ on a blackboard. Each minute, if the pair $(a,b)$ is on the board, she erases it and replaces it with one of the pairs $(2a-b,a)$, $(2a+b+2,a)$ or $(a+2b+2,b)$. Eventually, the board reads $(2014,k)$ for some nonnegative integer $k$. How many possible values of $k$ are t... | 1. **Initial Setup and Transformation:**
- Qing starts with the pair \((1,0)\).
- The allowed transformations are:
\[
(a, b) \rightarrow (2a - b, a), \quad (a, b) \rightarrow (2a + b + 2, a), \quad (a, b) \rightarrow (a + 2b + 2, b)
\]
- To simplify, we add 1 to each element of the pair, transfo... | 720 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
In the game of Nim, players are given several piles of stones. On each turn, a player picks a nonempty pile and removes any positive integer number of stones from that pile. The player who removes the last stone wins, while the first player who cannot move loses.
Alice, Bob, and Chebyshev play a 3-player version of Ni... | To solve this problem, we need to understand the concept of Nim-Sum in the game of Nim. The Nim-Sum is the bitwise XOR of the sizes of all piles. For the first player to lose when all players play optimally, the Nim-Sum of the initial pile sizes must be zero.
1. **Convert the known pile sizes to binary:**
\[
43_... | 7704 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, an [i]$n$-branch[/i] $B$ is an ordered tuple $(S_1, S_2, \dots, S_m)$ of nonempty sets (where $m$ is any positive integer) satisfying $S_1 \subset S_2 \subset \dots \subset S_m \subseteq \{1,2,\dots,n\}$. An integer $x$ is said to [i]appear[/i] in $B$ if it is an element of the last set $S_m... | 1. **Define the necessary terms and functions:**
- Let \( B_n \) be the number of \( n \)-branches.
- Let \( C_n \) be the number of \( n \)-branches which contain \(\{1, 2, \ldots, n\}\). It can be seen that \( C_n = \frac{B_n + 1}{2} \).
- Let \( T_n \) be the number of distinct perfect \( n \)-plants.
2. *... | 76 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Carl has a rectangle whose side lengths are positive integers. This rectangle has the property that when he increases the width by 1 unit and decreases the length by 1 unit, the area increases by $x$ square units. What is the smallest possible positive value of $x$?
[i]Proposed by Ray Li[/i] | 1. Let the original dimensions of the rectangle be \( l \) (length) and \( w \) (width), where both \( l \) and \( w \) are positive integers.
2. The original area of the rectangle is \( A = l \times w \).
3. When the width is increased by 1 unit and the length is decreased by 1 unit, the new dimensions become \( l-1 \... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $(a_n)$, $(b_n)$, $(c_n)$ are arithmetic progressions. Given that $a_1+b_1+c_1 = 0$ and $a_2+b_2+c_2 = 1$, compute $a_{2014}+b_{2014}+c_{2014}$.
[i]Proposed by Evan Chen[/i] | 1. Given that \( (a_n) \), \( (b_n) \), and \( (c_n) \) are arithmetic progressions, we can express them as follows:
\[
a_n = a_1 + (n-1)d_a, \quad b_n = b_1 + (n-1)d_b, \quad c_n = c_1 + (n-1)d_c
\]
where \( d_a \), \( d_b \), and \( d_c \) are the common differences of the sequences \( (a_n) \), \( (b_n) ... | 2013 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $B = (20, 14)$ and $C = (18, 0)$ be two points in the plane. For every line $\ell$ passing through $B$, we color red the foot of the perpendicular from $C$ to $\ell$. The set of red points enclose a bounded region of area $\mathcal{A}$. Find $\lfloor \mathcal{A} \rfloor$ (that is, find the greatest integer not exc... | 1. **Determine the distance \( BC \):**
- The coordinates of points \( B \) and \( C \) are \( B = (20, 14) \) and \( C = (18, 0) \).
- The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]
- Applying this formula to... | 157 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A crazy physicist has discovered a new particle called an omon. He has a machine, which takes two omons of mass $a$ and $b$ and entangles them; this process destroys the omon with mass $a$, preserves the one with mass $b$, and creates a new omon whose mass is $\frac 12 (a+b)$. The physicist can then repeat the process ... | 1. **Initial Setup**: We start with two omons of masses \(a\) and \(b\), where \(a\) and \(b\) are distinct positive integers less than 1000. Without loss of generality, assume \(a < b\).
2. **Entanglement Process**: The machine destroys the omon with mass \(a\), preserves the omon with mass \(b\), and creates a new o... | 9 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^{\circ}$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^{\circ}$ or $60^{\circ}$, and the triangle should definitely not be isosceles.
How many differ... | 1. **Identify the possible angle measures:**
Since each angle must be a multiple of \(10^\circ\) and the triangle must be acute, the possible angles are:
\[
10^\circ, 20^\circ, 30^\circ, 40^\circ, 50^\circ, 60^\circ, 70^\circ, 80^\circ
\]
However, the problem states that none of the angles can measure \(... | 0 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Find the sum of the decimal digits of \[ \left\lfloor \frac{51525354555657\dots979899}{50} \right\rfloor. \] Here $\left\lfloor x \right\rfloor$ is the greatest integer not exceeding $x$.
[i]Proposed by Evan Chen[/i] | 1. Let \( S(n) \) denote the sum of the digits of \( n \). We need to find the sum of the decimal digits of
\[
\left\lfloor \frac{51525354555657 \dots 979899}{50} \right\rfloor.
\]
Here, \( \left\lfloor x \right\rfloor \) is the greatest integer not exceeding \( x \).
2. We can express the large number \(... | 457 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a triangle $ABC$, consider the semicircle with diameter $\overline{EF}$ on $\overline{BC}$ tangent to $\overline{AB}$ and $\overline{AC}$. If $BE=1$, $EF=24$, and $FC=3$, find the perimeter of $\triangle{ABC}$.
[i]Proposed by Ray Li[/i] | 1. **Identify the given values and setup the problem:**
- We are given a triangle \( \triangle ABC \) with a semicircle on diameter \( \overline{EF} \) tangent to \( \overline{AB} \) and \( \overline{AC} \).
- Given values: \( BE = 1 \), \( EF = 24 \), and \( FC = 3 \).
2. **Calculate the length of \( \overline{... | 175 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be positive real numbers for which \[
\frac{5}{a} = b+c, \quad
\frac{10}{b} = c+a, \quad \text{and} \quad
\frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Evan Chen[/i] | 1. We start with the given equations:
\[
\frac{5}{a} = b + c, \quad \frac{10}{b} = c + a, \quad \frac{13}{c} = a + b.
\]
Let's rewrite these equations in a more convenient form:
\[
b + c = \frac{5}{a}, \quad c + a = \frac{10}{b}, \quad a + b = \frac{13}{c}.
\]
2. Next, we multiply all three equati... | 55 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.)
[i]Proposed by Robin Park[/i] | 1. **Define Variables and Given Information:**
Let \( a = AO \), \( b = BO \), \( c = CO \), \( x = BC \), \( y = CA \), and \( z = AB \). We are given:
\[
[OAB] = 20 \quad \text{and} \quad [OBC] = 14
\]
Since \(\angle AOB = \angle BOC = \angle COA = 90^\circ\), the areas of the triangles can be expresse... | 22200 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with area $5$ and $BC = 10.$ Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G.$ Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2+AC^2.$
[i]Proposed by Ray Li[/i] | 1. Given that \( \triangle ABC \) has an area of 5 and \( BC = 10 \). Let \( E \) and \( F \) be the midpoints of sides \( AC \) and \( AB \) respectively, and let \( BE \) and \( CF \) intersect at \( G \). Since \( G \) is the centroid of \( \triangle ABC \), it divides each median in the ratio 2:1.
2. Since \( E \)... | 200 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define \[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \] Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.
[i]Proposed by Evan Chen[/i] | 1. **Switch the order of summation:**
\[
S = \frac{1}{\alpha} \sum_{n=1}^{1000} \sum_{m=1}^{n} \left\lfloor \frac{m+\alpha}{n} \right\rfloor
\]
This can be rewritten as:
\[
S = \frac{1}{\alpha} \sum_{n=1}^{1000} \left[ \sum_{m=1}^{n} \frac{m+\alpha}{n} - \sum_{m=1}^{n} \left\{ \frac{m+\alpha}{n} \righ... | 5 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
In triangle $ABC$, $AB=3$, $AC=5$, and $BC=7$. Let $E$ be the reflection of $A$ over $\overline{BC}$, and let line $BE$ meet the circumcircle of $ABC$ again at $D$. Let $I$ be the incenter of $\triangle ABD$. Given that $\cos ^2 \angle AEI = \frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers, deter... | 1. **Reflecting Point \( A \) Over Line \( BC \)**:
- Given \( AB = 3 \), \( AC = 5 \), and \( BC = 7 \), we start by reflecting point \( A \) over line \( BC \) to get point \( E \).
- The reflection of \( A \) over \( BC \) means that \( E \) is such that \( BE = BA \) and \( CE = CA \).
2. **Finding Coordinat... | 55 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, ..., A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$ . For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \[ \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1... | 1. **Convert the problem to the complex plane:**
- Let $\omega = \operatorname{cis}\left(\frac{2\pi}{n}\right)$, where $\operatorname{cis}(\theta) = \cos(\theta) + i\sin(\theta)$.
- The vertices of the regular $n$-gon can be represented as $A_j^{(0)} = \omega^j$ for $j = 0, 1, \ldots, n-1$.
2. **Determine the ce... | 2188 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a>... | 1. **Rearrange the given condition:**
Given the recurrence relation:
\[
x_{n+2} = \frac{(x_{n+1} + 1)(x_{n+1} - 1)}{x_n}
\]
We can rewrite it as:
\[
x_{n+2} = \frac{x_{n+1}^2 - 1}{x_n}
\]
Multiplying both sides by \(x_n\) and rearranging terms, we get:
\[
x_n x_{n+2} + 1 = x_{n+1}^2
... | 622 | Other | math-word-problem | Yes | Yes | aops_forum | false |
Find the smallest positive integer $c$ for which the following statement holds: Let $k$ and $n$ be positive integers. Suppose there exist pairwise distinct subsets $S_1$, $S_2$, $\dots$, $S_{2k}$ of $\{1, 2, \dots, n\}$, such that $S_i \cap S_j \neq \varnothing$ and $S_i \cap S_{j+k} \neq \varnothing$ for all $1 \le i,... | 1. We are given that there exist \(2k\) pairwise distinct subsets \(S_1, S_2, \dots, S_{2k}\) of \(\{1, 2, \dots, n\}\) such that \(S_i \cap S_j \neq \varnothing\) and \(S_i \cap S_{j+k} \neq \varnothing\) for all \(1 \le i,j \le k\).
2. We need to find the smallest positive integer \(c\) such that \(1000k \le c \cdot ... | 334 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a prime $q$, let $\Phi_q(x)=x^{q-1}+x^{q-2}+\cdots+x+1$.
Find the sum of all primes $p$ such that $3 \le p \le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying
\[\dbinom{N}{\Phi_q(p)}\equiv \dbinom{2\Phi_q(p)}{N} \not \equiv 0 \pmod p. \][i]Proposed by Sammy Luo[/i] | 1. **Understanding the Problem:**
We need to find the sum of all primes \( p \) such that \( 3 \le p \le 100 \) and there exists an odd prime \( q \) and a positive integer \( N \) satisfying:
\[
\binom{N}{\Phi_q(p)} \equiv \binom{2\Phi_q(p)}{N} \not\equiv 0 \pmod{p}
\]
where \(\Phi_q(x) = x^{q-1} + x^{q... | 420 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathcal A = A_0A_1A_2A_3 \cdots A_{2013}A_{2014}$ be a [i]regular 2014-simplex[/i], meaning the $2015$ vertices of $\mathcal A$ lie in $2014$-dimensional Euclidean space and there exists a constant $c > 0$ such that $A_iA_j = c$ for any $0 \le i < j \le 2014$. Let $O = (0,0,0,\dots,0)$, $A_0 = (1,0,0,\dots,0)$, a... | 1. **Understanding the Problem:**
We are given a regular 2014-simplex $\mathcal{A}$ with vertices $A_0, A_1, \ldots, A_{2014}$ in a 2014-dimensional Euclidean space. The distance between any two vertices is constant, $c$. The origin $O$ is at $(0,0,\ldots,0)$, and $A_0$ is at $(1,0,0,\ldots,0)$. Each vertex $A_i$ is... | 600572 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 2^{16}+1$ be a prime, and let $S$ be the set of positive integers not divisible by $p$.
Let $f: S \to \{0, 1, 2, ..., p-1\}$ be a function satisfying
\[ f(x)f(y) \equiv f(xy)+f(xy^{p-2}) \pmod{p} \quad\text{and}\quad f(x+p) = f(x) \]
for all $x,y \in S$.
Let $N$ be the product of all possible nonzero values of... | 1. **Initial Setup and Function Properties:**
Given \( p = 2^{16} + 1 \) is a prime, and \( S \) is the set of positive integers not divisible by \( p \). The function \( f: S \to \{0, 1, 2, \ldots, p-1\} \) satisfies:
\[
f(x)f(y) \equiv f(xy) + f(xy^{p-2}) \pmod{p}
\]
and
\[
f(x+p) = f(x)
\]
... | 16384 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where \[
A = \prod_{(a,b) \in S} a
, \quad
B = \prod_{(a,b) \in S} b
, \quad \text{and} \quad
C = \sum_{(a,b) \in S} ab.
\][i]Proposed by Evan Chen[/i] | Given the equations:
\[ 1 + a + a^2 + a^3 = b^2(1 + 3a) \]
\[ 1 + 2a + 3a^2 = b^2 - \frac{5}{b} \]
We need to find \( A + B + C \), where:
\[ A = \prod_{(a,b) \in S} a \]
\[ B = \prod_{(a,b) \in S} b \]
\[ C = \sum_{(a,b) \in S} ab \]
1. **Rewrite the first equation:**
\[ 1 + a + a^2 + a^3 = b^2(1 + 3a) \]
\[ a... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with circumcenter $O$, incenter $I$, and circumcircle $\Gamma$. It is known that $AB = 7$, $BC = 8$, $CA = 9$. Let $M$ denote the midpoint of major arc $\widehat{BAC}$ of $\Gamma$, and let $D$ denote the intersection of $\Gamma$ with the circumcircle of $\triangle IMO$ (other than $M$). Let $E$ ... | 1. **Identify the given elements and their relationships:**
- Triangle \(ABC\) with sides \(AB = 7\), \(BC = 8\), and \(CA = 9\).
- \(O\) is the circumcenter, \(I\) is the incenter, and \(\Gamma\) is the circumcircle of \(\triangle ABC\).
- \(M\) is the midpoint of the major arc \(\widehat{BAC}\) of \(\Gamma\)... | 467 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
There are $n$ cards numbered and stacked in increasing order from up to down (i.e. the card in the top is the number 1, the second is the 2, and so on...). With this deck, the next steps are followed:
-the first card (from the top) is put in the bottom of the deck.
-the second card (from the top) is taken away of the ... | To determine the last remaining card after following the described process, we can use a known result related to the Josephus problem for step size 2. The Josephus problem is a theoretical problem related to a certain elimination game.
1. **Understanding the Josephus Problem**:
The Josephus problem for step size 2 ... | 41 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Consider a square of side length $12$ centimeters. Irina draws another square that has $8$ centimeters more of perimeter than the original square. What is the area of the square drawn by Irina? | 1. Calculate the perimeter of the original square:
\[
\text{Perimeter of original square} = 4 \times 12 = 48 \text{ cm}
\]
2. Determine the perimeter of Irina's square:
\[
\text{Perimeter of Irina's square} = 48 \text{ cm} + 8 \text{ cm} = 56 \text{ cm}
\]
3. Calculate the side length of Irina's squ... | 196 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Juan chooses a five-digit positive integer. Maria erases the ones digit and gets a four-digit number. The sum of this four-digit number and the original five-digit number is $52,713$. What can the sum of the five digits of the original number be? | 1. Let the original five-digit number be represented as \(10x + y\), where \(x\) is a four-digit number and \(y\) is the units digit (0 through 9).
2. When Maria erases the ones digit, the resulting four-digit number is \(x\).
3. According to the problem, the sum of the original five-digit number and the four-digit num... | 23 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$. | 1. **Understanding the Problem:**
We are given a triangle \(ABC\) with an area of \(92\) square centimeters. We need to find the area of another triangle whose sides are the medians of triangle \(ABC\).
2. **Key Concept:**
The area of a triangle formed by the medians of another triangle is \(\frac{3}{4}\) of the... | 69 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$. | To solve this problem, we need to determine the smallest integer \( k \) such that the set \( A = \{1, 2, 3, \ldots, 40\} \) can be partitioned into \( k \) disjoint subsets with the property that if \( a, b, c \) (not necessarily distinct) are in the same subset, then \( a \neq b + c \).
1. **Understanding Schur's Pr... | 4 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABCD$ be a cyclic quadrilateral with $AB = 1$, $BC = 2$, $CD = 3$, $DA = 4$. Find the square of the area of quadrilateral $ABCD$. | 1. **Identify the sides and semiperimeter:**
Given the sides of the cyclic quadrilateral \(ABCD\) are \(AB = 1\), \(BC = 2\), \(CD = 3\), and \(DA = 4\).
The semiperimeter \(s\) is calculated as:
\[
s = \frac{AB + BC + CD + DA}{2} = \frac{1 + 2 + 3 + 4}{2} = 5
\]
2. **Apply Brahmagupta's formula:**
... | 24 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $S=2^3+3^4+5^4+7^4+\cdots+17497^4$ be the sum of the fourth powers of the first $2014$ prime numbers. Find the remainder when $S$ is divided by $240$. | 1. **Using Fermat's Little Theorem**:
Fermat's Little Theorem states that for any integer \(a\) and a prime \(p\), \(a^{p-1} \equiv 1 \pmod{p}\) if \(a\) is not divisible by \(p\). For the fourth power, we have:
\[
a^4 \equiv 1 \pmod{p} \quad \text{for primes } p > 3
\]
This means that for all primes \(p... | 168 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be positive integers such that $29a + 30b + 31c = 366$. Find $19a + 20b + 21c$. | 1. We start with the given equation:
\[
29a + 30b + 31c = 366
\]
This equation represents the total number of days in a leap year, where \(a\), \(b\), and \(c\) are the number of months with 29, 30, and 31 days respectively.
2. In a leap year, there are 12 months. We know that:
- February has 29 days (s... | 246 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, fi nd $A$. | 1. Let \( A = 100a + 10b + c \), where \( a, b, c \) are digits and \( a \neq 0 \) since \( A \) is a three-digit number.
2. By interchanging the two leftmost digits of \( A \), we obtain \( B = 100b + 10a + c \).
3. Doubling \( B \) gives \( C = 2B = 2(100b + 10a + c) = 200b + 20a + 2c \).
4. Subtracting 500 from \( C... | 344 | Algebra | other | Yes | Yes | aops_forum | false |
Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$. | 1. We start with the given equation:
\[
abc + bc + c = 2014
\]
Factoring out \(c\) from the left-hand side, we get:
\[
c(ab + b + 1) = 2014
\]
This implies:
\[
ab + b + 1 = \frac{2014}{c}
\]
Since \(a\), \(b\), and \(c\) are positive integers, \(\frac{2014}{c}\) must also be an integ... | 40 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One morning a baker notices that she has $188$ cups of flour and $113$ cups of sugar available. Each loaf of bread that the baker makes takes three cups of flour and a half cup of sugar. Each cake that the baker makes takes two cups of flour and two cups of sugar. The baker decides to make some loaves of bread and some... | 1. Let \( c \) be the number of cakes and \( b \) be the number of loaves of bread.
2. Each loaf of bread requires 3 cups of flour and 0.5 cups of sugar.
3. Each cake requires 2 cups of flour and 2 cups of sugar.
4. We are given that the baker has 188 cups of flour and 113 cups of sugar.
We can set up the following sy... | 49 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). | To solve this problem, we need to find the smallest positive integer \( n \) such that \( n \), \( n+1 \), and \( n+2 \) each have exactly two prime factors. This means each of these numbers must be the product of exactly two prime numbers (i.e., they must be semiprimes).
1. **Understanding the Problem:**
- A numbe... | 33 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Andrea is three times as old as Jim was when Jim was twice as old as he was when the sum of their ages was $47$. If Andrea is $29$ years older than Jim, what is the sum of their ages now? | 1. Let \( A \) be Andrea's current age and \( J \) be Jim's current age. We are given that \( A = J + 29 \).
2. We need to find the ages when the sum of their ages was 47. Let \( x \) be the number of years ago when the sum of their ages was 47. Then:
\[
(A - x) + (J - x) = 47
\]
Substituting \( A = J + 29... | 79 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer such that $\lfloor\sqrt n\rfloor-2$ divides $n-4$ and $\lfloor\sqrt n\rfloor+2$ divides $n+4$. Find the greatest such $n$ less than $1000$. (Note: $\lfloor x\rfloor$ refers to the greatest integer less than or equal to $x$.) | 1. Let \( n \) be a positive integer such that \( \lfloor \sqrt{n} \rfloor = k \). This means \( k \leq \sqrt{n} < k+1 \), or equivalently, \( k^2 \leq n < (k+1)^2 \).
2. Given that \( k-2 \) divides \( n-4 \), we can write:
\[
n - 4 = m(k-2) \quad \text{for some integer } m.
\]
3. Similarly, given that \( k... | 956 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions. The Bell Zoo has three more elephants than the Carlton Zoo has lions. The Bell Zoo has the same number of elephants as the Carlton Zoo has rhinoceroses. The Carlton Zoo has two more elephants than rhinoceroses. The Carlton Zoo has twice as m... | 1. Let \( L \) be the number of lions in the Carlton Zoo. According to the problem:
- The Bell Zoo has the same number of rhinoceroses as the Carlton Zoo has lions, so let the number of rhinoceroses in the Bell Zoo be \( L \).
- The Bell Zoo has three more elephants than the Carlton Zoo has lions, so the number o... | 57 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
The first number in the following sequence is $1$. It is followed by two $1$'s and two $2$'s. This is followed by three $1$'s, three $2$'s, and three $3$'s. The sequence continues in this fashion.
\[1,1,1,2,2,1,1,1,2,2,2,3,3,3,1,1,1,1,2,2,2,2,3,3,3,3,4,4,4,4,\dots.\]
Find the $2014$th number in this sequence. | 1. **Identify the pattern in the sequence:**
The sequence is constructed by repeating numbers in increasing order, where each number \( k \) is repeated \( k \) times. For example:
- The first group is \( 1 \) repeated \( 1 \) time.
- The second group is \( 1 \) repeated \( 2 \) times, followed by \( 2 \) repe... | 13 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address, but he forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\tfrac m n$ where $m$ and $n$ are relatively... | 1. **Determine the total number of permutations of the digits:**
The address is 2743, which has 4 digits. The total number of permutations of these 4 digits is given by:
\[
4! = 24
\]
2. **Calculate the number of ways to choose 2 correct positions out of 4:**
We need to choose 2 positions out of 4 where... | 5 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have. | 1. **Formulate the problem as a linear Diophantine equation:**
Given that the side lengths of the tiles are \(5 \text{ cm}\) and \(3 \text{ cm}\), the areas of these tiles are \(25 \text{ cm}^2\) and \(9 \text{ cm}^2\) respectively. Let \(x\) be the number of \(25 \text{ cm}^2\) tiles and \(y\) be the number of \(9 ... | 94 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
One-third of the students who attend Grant School are taking Algebra. One-quarter of the students taking Algebra are also on the track team. There are $15$ students on the track team who take Algebra. Find the number of students who attend Grant School. | 1. Let \( S \) be the total number of students who attend Grant School.
2. According to the problem, one-third of the students are taking Algebra. Therefore, the number of students taking Algebra is:
\[
\frac{S}{3}
\]
3. One-quarter of the students taking Algebra are also on the track team. Therefore, the numb... | 180 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Determine the smallest real constant $c$ such that
\[\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2\]
for all positive integers $n$ and all positive real numbers $x_1,\cdots ,x_n$. | To determine the smallest real constant \( c \) such that
\[
\sum_{k=1}^{n}\left ( \frac{1}{k}\sum_{j=1}^{k}x_j \right )^2\leq c\sum_{k=1}^{n}x_k^2
\]
for all positive integers \( n \) and all positive real numbers \( x_1, x_2, \ldots, x_n \), we will use the Cauchy-Schwarz inequality and some integral approximations.
... | 4 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
There are cities in country, and some cities are connected by roads. Not more than $100$ roads go from every city. Set of roads is called as ideal if all roads in set have not common ends, and we can not add one more road in set without breaking this rule. Every day minister destroy one ideal set of roads.
Prove, that... | 1. **Assume the contrary**: Suppose that after 199 days, there is still a road between two cities, say $A$ and $B$.
2. **Count the roads**: Since each city has at most 100 roads, the total number of roads connected to either $A$ or $B$ (excluding the road $AB$ itself) is at most $198$.
3. **Ideal set definition**: An i... | 199 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$. | To solve the problem, we need to find all possible values of \( n \) such that \( \frac{1}{n} = 0.a_1a_2a_3\ldots \) and \( n = a_1 + a_2 + a_3 + \cdots \).
1. **Express \( n \) in terms of its prime factors:**
Since \( \frac{1}{n} \) is a terminating decimal, \( n \) must be of the form \( n = 2^a \cdot 5^b \) for... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\mathbb{N}$ denote the set of positive integers, and let $S$ be a set. There exists a function $f :\mathbb{N} \rightarrow S$ such that if $x$ and $y$ are a pair of positive integers with their difference being a prime number, then $f(x) \neq f(y)$. Determine the minimum number of elements in $S$. | To determine the minimum number of elements in \( S \) such that there exists a function \( f: \mathbb{N} \rightarrow S \) with the property that if \( x \) and \( y \) are positive integers whose difference is a prime number, then \( f(x) \neq f(y) \), we will proceed as follows:
1. **Construct a function \( f \) wit... | 4 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
In a tournament each player played exactly one game against each of the other players. In each game the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\tfrac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the poi... | 1. **Total Points Calculation:**
In a tournament with \( n \) players, each player plays exactly one game against each of the other players. Therefore, the total number of games played is given by the combination formula:
\[
\binom{n}{2} = \frac{n(n-1)}{2}
\]
Each game results in 1 point being distribute... | 19 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$.
$ \textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }3\qquad\textbf{(E) }4 $ | To solve the equation \( x|x-1| - 4|x| + 3 = 0 \), we need to consider different cases based on the value of \( x \) because of the absolute value functions. We will split the problem into three cases: \( x \geq 1 \), \( 0 \leq x < 1 \), and \( x < 0 \).
1. **Case 1: \( x \geq 1 \)**
- Here, \( |x-1| = x-1 \) and \... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$ | 1. Start with the given equation:
\[
\frac{27 \times 9^x}{4^x} = \frac{3^x}{8^x}
\]
2. Express all terms with the same base:
\[
27 = 3^3, \quad 9^x = (3^2)^x = 3^{2x}, \quad 4^x = (2^2)^x = 2^{2x}, \quad 8^x = (2^3)^x = 2^{3x}
\]
3. Substitute these expressions into the equation:
\[
\frac{3^3 ... | 216 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer such that $12n^2+12n+11$ is a $4$-digit number with all $4$ digits equal. Determine the value of $n$. | 1. We start with the given expression \(12n^2 + 12n + 11\) and note that it must be a 4-digit number with all digits equal. The possible 4-digit numbers with all digits equal are \(1111, 2222, 3333, 4444, 5555, 6666, 7777, 8888, 9999\).
2. We rewrite the expression \(12n^2 + 12n + 11\) in a more convenient form:
\[... | 21 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be an integer, and let $\triangle ABC$ be a right-angles triangle with right angle at $C$. It is given that $\sin A$ and $\sin B$ are the roots of the quadratic equation \[(5n+8)x^2-(7n-20)x+120=0.\] Find the value of $n$ | 1. Given that $\triangle ABC$ is a right-angled triangle with the right angle at $C$, we know that $\angle A + \angle B = \frac{\pi}{2}$. Therefore, $\sin B = \sin\left(\frac{\pi}{2} - A\right) = \cos A$.
2. Using the Pythagorean identity, we have $\sin^2 A + \cos^2 A = 1$. Since $\sin B = \cos A$, it follows that $\si... | 66 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $S_1$ and $S_2$ be sets of points on the coordinate plane $\mathbb{R}^2$ defined as follows
\[S_1={(x,y)\in \mathbb{R}^2:|x+|x||+|y+|y||\le 2}\]
\[S_2={(x,y)\in \mathbb{R}^2:|x-|x||+|y-|y||\le 2}\]
Find the area of the intersection of $S_1$ and $S_2$ | To find the area of the intersection of \( S_1 \) and \( S_2 \), we need to analyze the regions defined by these sets. We will consider the four cases based on the signs of \( x \) and \( y \).
1. **Case 1: \( x \geq 0 \) and \( y \geq 0 \)**
For \( S_1 \):
\[
|x + |x|| + |y + |y|| \leq 2 \implies 2x + 2y \l... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n$ be a positive integer, and let $x=\frac{\sqrt{n+2}-\sqrt{n}}{\sqrt{n+2}+\sqrt{n}}$ and $y=\frac{\sqrt{n+2}+\sqrt{n}}{\sqrt{n+2}-\sqrt{n}}$.
It is given that $14x^2+26xy+14y^2=2014$. Find the value of $n$. | 1. Given the expressions for \( x \) and \( y \):
\[
x = \frac{\sqrt{n+2} - \sqrt{n}}{\sqrt{n+2} + \sqrt{n}}, \quad y = \frac{\sqrt{n+2} + \sqrt{n}}{\sqrt{n+2} - \sqrt{n}}
\]
We need to find the value of \( n \) given that:
\[
14x^2 + 26xy + 14y^2 = 2014
\]
2. First, simplify the given equation by... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$ | 1. We need to find the maximum value of the expression
\[
\frac{\sin 2x + \sin 4x + \sin 6x}{\cos 2x + \cos 4x + \cos 6x}
\]
for \(0 \le x \le \frac{\pi}{16}\).
2. First, observe that the numerator and the denominator are sums of sine and cosine functions, respectively. We can use trigonometric identities... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of ways of colouring a $10\times 10$ square board using two colours black and white such that each $2\times 2$ subsquare contains 2 black squares and 2 white squares. | To determine the number of ways to color a $10 \times 10$ square board using two colors (black and white) such that each $2 \times 2$ subsquare contains 2 black squares and 2 white squares, we can generalize the problem to an $n \times n$ board and then apply the result to $n = 10$.
1. **Establish a Coordinate System:... | 2046 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $X={1,2,3,4,5,6,7,8,9,10}$ and $A={1,2,3,4}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10\in Y$ and the intersection of $Y$ and $A$ is not empty. | 1. We start with the set \( X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \) and the subset \( A = \{1, 2, 3, 4\} \). We need to find the number of 4-element subsets \( Y \subseteq X \) such that \( 10 \in Y \) and \( Y \cap A \neq \emptyset \).
2. Since \( 10 \in Y \), we can write \( Y \) as \( Y = \{10, a, b, c\} \) where ... | 74 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$. | 1. **Given Information and Setup:**
- Two circles \(\Gamma_1\) and \(\Gamma_2\) intersect at points \(C\) and \(D\).
- A line intersects \(\Gamma_1\) at points \(A\) and \(Y\), intersects segment \(CD\) at point \(Z\), and intersects \(\Gamma_2\) at points \(X\) and \(B\) in the order \(A, X, Z, Y, B\).
- The ... | 10 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $n\geq 4$ be a positive integer.Out of $n$ people,each of two individuals play table tennis game(every game has a winner).Find the minimum value of $n$,such that for any possible outcome of the game,there always exist an ordered four people group $(a_{1},a_{2},a_{3},a_{4})$,such that the person $a_{i}$ wins against... | 1. **Graph Interpretation**: We interpret the problem using a directed graph where each vertex represents a person, and a directed edge from vertex \( u \) to vertex \( v \) indicates that person \( u \) wins against person \( v \).
2. **Total Out-Degree Calculation**: For \( n = 8 \), the total number of directed edg... | 8 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $(x_n)$ be a sequence of positive integers defined by $x_1=2$ and $x_{n+1}=2x_n^3+x_n$ for all integers $n\ge1$. Determine the largest power of $5$ that divides $x_{2014}^2+1$. | 1. **Base Case:**
We start by verifying the base case for \( n = 1 \):
\[
x_1 = 2 \implies x_1^2 + 1 = 2^2 + 1 = 4 + 1 = 5
\]
Clearly, \( 5^1 \mid 5 \) and \( 5^2 \nmid 5 \). Thus, \( 5^1 \mid \mid x_1^2 + 1 \).
2. **Inductive Hypothesis:**
Assume that for some \( n \geq 1 \), \( 5^n \mid \mid x_n^2 ... | 2014 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a_n\}_{n\geq 1}$ be a sequence defined by $a_n=\int_0^1 x^2(1-x)^ndx$.
Find the real value of $c$ such that $\sum_{n=1}^{\infty} (n+c)(a_n-a_{n+1})=2.$ | 1. First, we start with the given sequence \( a_n = \int_0^1 x^2 (1-x)^n \, dx \).
2. We need to find the value of \( c \) such that \( \sum_{n=1}^{\infty} (n+c)(a_n - a_{n+1}) = 2 \).
3. To proceed, we need to understand the behavior of \( a_n \) and \( a_{n+1} \). Let's compute \( a_n - a_{n+1} \):
\[
a_n = \i... | 22 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
There are $2014$ balls with $106$ different colors, $19$ of each color. Determine the least possible value of $n$ so that no matter how these balls are arranged around a circle, one can choose $n$ consecutive balls so that amongst them, there are $53$ balls with different colors. | 1. **Understanding the Problem:**
We have 2014 balls with 106 different colors, and there are 19 balls of each color. We need to determine the smallest number \( n \) such that in any arrangement of these balls around a circle, there are \( n \) consecutive balls that include at least 53 different colors.
2. **Init... | 971 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?
[i](A. Golovanov)[/i] | 1. Let \( P(x) = ax^2 + bx + c \) and \( Q(x) = dx^2 + ex + f \) be two quadratic trinomials, where \( a \neq 0 \) and \( d \neq 0 \).
2. Suppose there exists a linear function \( \ell(x) = mx + n \) such that \( P(x) = Q(\ell(x)) \) for all real \( x \).
3. Substituting \( \ell(x) \) into \( Q \), we get:
\[
P(x... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?
[i](A. Golovanov)[/i] | 1. Let the four three-digit numbers be \( x, x+1, x+2, x+3 \) and the moduli be \( a, a+1, a+2, a+3 \). We are looking for the minimum number of different remainders when these numbers are divided by the moduli.
2. Assume that the common remainder is \( r \). Then we have:
\[
x \equiv r \pmod{a}, \quad x+1 \equi... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
There are $100$ cards with numbers from $1$ to $100$ on the table.Andriy and Nick took the same number of cards in a way such that the following condition holds:if Andriy has a card with a number $n$ then Nick has a card with a number $2n+2$.What is the maximal number of cards that could be taken by the two guys? | 1. **Identify the relationship between the cards:**
If Andriy has a card with number \( n \), then Nick has a card with number \( 2n + 2 \). This relationship forms a sequence where each term is derived from the previous term using the formula \( a_{k+1} = 2a_k + 2 \).
2. **Form sequences (chains) based on the give... | 50 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
The expressions $A=1\times2+3\times4+5\times6+\cdots+37\times38+39$ and $B=1+2\times3+4\times5+\cdots+36\times37+38\times39$ are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between integers $A$ and $B$. | 1. **Expression for \( A \):**
The expression \( A \) is given by:
\[
A = 1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39
\]
We can rewrite this as:
\[
A = \sum_{k=1}^{19} (2k-1) \times (2k) + 39
\]
Each term in the sum is of the form \((2k-1) \times (2k)\).
2. **Simplify e... | 722 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
The nine delegates to the Economic Cooperation Conference include 2 officials from Mexico, 3 officials from Canada, and 4 officials from the United States. During the opening session, three of the delegates fall asleep. Assuming that the three sleepers were determined randomly, the probability that exactly two of the s... | To solve this problem, we need to calculate the probability that exactly two of the three delegates who fall asleep are from the same country. We will break this into cases based on the different combinations of countries the sleepers can be from.
1. **Total number of ways to choose 3 sleepers out of 9 delegates:**
... | 139 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Find the smallest positive integer $n$ satisfying $s(n)=s(n+864)=20$. | 1. **Identify the problem constraints:**
We need to find the smallest positive integer \( n \) such that the sum of its digits \( s(n) \) and the sum of the digits of \( n + 864 \) both equal 20.
2. **Analyze the digit sum properties:**
Let \( n = \overline{abc} \) where \( a, b, c \) are the digits of \( n \). ... | 695 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be the set of all ordered triples of integers $(a_1,a_2,a_3)$ with $1 \le a_1,a_2,a_3 \le 10$. Each ordered triple in $S$ generates a sequence according to the rule $a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} |$ for all $n\ge 4$. Find the number of such sequences for which $a_n=0$ for some $n$. | 1. **Identify the condition for \(a_n = 0\):**
Given the sequence rule \(a_n = a_{n-1} \cdot |a_{n-2} - a_{n-3}|\), for \(a_n\) to be zero, we need \(a_{n-1} = 0\) or \(|a_{n-2} - a_{n-3}| = 0\). Since \(a_n\) is the first term to be zero, \(a_{n-1} \neq 0\), so \(|a_{n-2} - a_{n-3}| = 0\), implying \(a_{n-2} = a_{n... | 594 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Triangle $ABC$ has side lengths $AB=12$, $BC=25$, and $CA=17$. Rectangle $PQRS$ has vertex $P$ on $\overline{AB}$, vertex $Q$ on $\overline{AC}$, and vertices $R$ and $S$ on $\overline{BC}$. In terms of the side length $PQ=w$, the area of $PQRS$ can be expressed as the quadratic polynomial
\[\text{Area}(PQRS)=\alpha w-... | 1. **Calculate the area of triangle \( \triangle ABC \) using Heron's formula:**
Given side lengths are \( AB = 12 \), \( BC = 25 \), and \( CA = 17 \).
First, calculate the semi-perimeter \( s \):
\[
s = \frac{AB + BC + CA}{2} = \frac{12 + 25 + 17}{2} = 27
\]
Using Heron's formula for the area \( ... | 161 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequence $a_1,a_2,a_3,\ldots$ by $a_n=\sum_{k=1}^n\sin(k)$, where $k$ represents radian measure. Find the index of the $100$th term for which $a_n<0$. | 1. Define the sequence \( a_n \) by \( a_n = \sum_{k=1}^n \sin(k) \), where \( k \) is in radians.
2. Multiply the entire sum by \( 2\cos(1) \):
\[
2a_n \cos(1) = 2\cos(1) (\sin(1) + \sin(2) + \cdots + \sin(n))
\]
3. Use the product-to-sum identities to simplify:
\[
2\sin(k)\cos(1) = \sin(k+1) + \sin(k-1... | 628 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,x,y$ be positive integer such that $a>100,x>100,y>100$ and $y^2-1=a^2(x^2-1)$ . Find the minimum value of $\frac{a}{x}$. | Given the equation \( y^2 - 1 = a^2(x^2 - 1) \), we need to find the minimum value of \( \frac{a}{x} \) under the conditions \( a > 100 \), \( x > 100 \), and \( y > 100 \).
1. **Rewrite the given equation:**
\[
y^2 - 1 = a^2(x^2 - 1)
\]
This can be rearranged as:
\[
y^2 = a^2x^2 - a^2 + 1
\]
2. ... | 2 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
$100$ integers are arranged in a circle. Each number is greater than the sum of the two subsequent numbers (in a clockwise order). Determine the maximal possible number of positive numbers in such circle. [i](S.Berlov)[/i] | 1. **Label the numbers**: Let the numbers be labeled as \(a_1, a_2, \dotsc, a_{100}\). According to the problem, each number is greater than the sum of the two subsequent numbers, i.e.,
\[
a_i > a_{i+1} + a_{i+2} \quad \text{for all } i \pmod{100}.
\]
2. **Assume at least 50 positive numbers**: Suppose there... | 49 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
A field has a shape of checkboard $\text{41x41}$ square. A tank concealed in one of the cells of the field. By one shot, a fighter airplane fires one of the cells. If a shot hits the tank, then the tank moves to a neighboring cell of the field, otherwise it stays in its cell (the cells are neighbours if they share a si... | 1. **Understanding the Problem:**
- We have a $41 \times 41$ checkerboard grid.
- A tank is hidden in one of the cells.
- If a shot hits the tank, it moves to a neighboring cell (sharing a side).
- We need to hit the tank twice to destroy it.
- We need to find the minimum number of shots required to guar... | 2521 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$?
$\textbf{(A) ... | 1. Let the square be \(ABCD\) with side length 1. Suppose the two points chosen are \(P\) and \(Q\). Without loss of generality, let \(P\) lie on \(\overline{AB}\) with \(P\) closer to \(A\) than \(B\). Denote the length \(AP = x\).
2. To find the probability that the straight-line distance between \(P\) and \(Q\) is ... | 59 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. How many different values of $p<2015$ are possible?
$\textbf{(A) }30\qquad\textbf{(B) }31\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$ | 1. **Define Variables and Setup the Problem:**
- Let \(AB = 2\), \(BC = h\), \(CD = AD = x\).
- Let \(H\) be the foot of the perpendicular from \(A\) to \(\overline{CD}\).
- We need \(h\) and \(x\) to be positive integers, and the perimeter \(p = 2 + h + 2x < 2015\).
2. **Use the Pythagorean Theorem:**
- S... | 31 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$ | 1. Let \( p \) and \( q \) denote the roots of the equation \( x^2 - ax + 2a = 0 \). By Vieta's formulas, we have:
\[
p + q = a
\]
\[
pq = 2a
\]
2. Substitute \( a \) from the first equation into the second equation:
\[
pq = 2(p + q)
\]
3. Rearrange the equation:
\[
pq - 2p - 2q = 0
... | 16 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Integers $x$ and $y$ with $x>y>0$ satisfy $x+y+xy=80$. What is $x$?
$\textbf{(A) }8\qquad\textbf{(B) }10\qquad\textbf{(C) }15\qquad\textbf{(D) }18\qquad\textbf{(E) }26$ | 1. We start with the given equation:
\[
x + y + xy = 80
\]
2. To simplify this equation, we use Simon's Favorite Factoring Trick (SFFT). We add 1 to both sides of the equation:
\[
x + y + xy + 1 = 81
\]
3. Notice that the left-hand side can be factored:
\[
(x + 1)(y + 1) = 81
\]
4... | 26 | Algebra | MCQ | Yes | Yes | aops_forum | false |
A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$,... | 1. Define \(\omega = e^{\frac{i\pi}{6}}\). Note that \(P_{2015}\) is the sum of the points \(1\omega^0, 2\omega^1, \ldots, 2015\omega^{2014}\) by translating each move to the origin. Hence, we want to evaluate \(1\omega^0 + 2\omega^1 + \ldots + 2015\omega^{2014}\).
2. Consider the polynomial \(P(x) = \frac{x^{2016} - ... | 2024 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Consider the function $f(x)=5x^4-12x^3+30x^2-12x+5$. Let $f(x_1)=p$, wher $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$.
[i]Proposed by tkhalid[/i] | 1. **Rewrite the polynomial using Sophie Germain Identity:**
The given polynomial is \( f(x) = 5x^4 - 12x^3 + 30x^2 - 12x + 5 \). We can rewrite this polynomial using the Sophie Germain Identity:
\[
f(x) = (x+1)^4 + 4(x-1)^4
\]
This identity states that \( a^4 + 4b^4 = (a^2 - 2ab + 2b^2)(a^2 + 2ab + 2b^2... | 5 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$. | 1. **Identify the possible forms of \( M \)**:
- Since \( M \) has exactly 6 divisors, it must be of the form \( p^5 \) or \( p^2q \), where \( p \) and \( q \) are prime numbers. This is because:
- If \( M = p^5 \), the divisors are \( 1, p, p^2, p^3, p^4, p^5 \).
- If \( M = p^2q \), the divisors are \( ... | 1996 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For $n\geq 2$ , an equilateral triangle is divided into $n^2$ congruent smaller equilateral triangles. Detemine all ways in which real numbers can be assigned to the $\frac{(n+1)(n+2)}{2}$ vertices so that three such numbers sum to zero whenever the three vertices form a triangle with edges parallel to the sides of the... | To solve the problem, we need to determine how to assign real numbers to the vertices of an equilateral triangle divided into \(n^2\) smaller equilateral triangles such that the sum of the numbers at the vertices of any smaller triangle is zero. We will start by examining the case for \(n=2\) and then generalize our fi... | 0 | Combinatorics | other | Yes | Yes | aops_forum | false |
With inspiration drawn from the rectilinear network of streets in [i]New York[/i] , the [i]Manhattan distance[/i] between two points $(a,b)$ and $(c,d)$ in the plane is defined to be \[|a-c|+|b-d|\] Suppose only two distinct [i]Manhattan distance[/i] occur between all pairs of distinct points of some point set. What is... | 1. **Define the Problem and Setup:**
We are given a set of points in the plane such that the Manhattan distance between any two distinct points is either 'long' or 'short'. We need to determine the maximum number of points in such a set.
2. **Understanding Manhattan Distance:**
The Manhattan distance between two... | 3 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
A family wears clothes of three colors: red,blue and green,with a separate,identical laundry bin for each color. At the beginning of the first week,all bins are empty.Each week,the family generates a total of $10 kg $ of laundry(the proportion of each color is subject to variation).The laundry is sorted by color and pl... | 1. **Initial Setup and Assumptions:**
- The family generates 10 kg of laundry each week.
- There are three bins for red, blue, and green clothes.
- At the beginning of the first week, all bins are empty.
- Each week, the heaviest bin is emptied.
2. **Objective:**
- Determine the minimal possible storing... | 25 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
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