problem stringlengths 2 5.64k | solution stringlengths 2 13.5k | answer stringlengths 1 43 | problem_type stringclasses 8
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For each integer $1\le j\le 2017$, let $S_j$ denote the set of integers $0\le i\le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that
\[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1\le j\le 2017} \left(1 - \prod_{i\in S_j} x_i\right)... | 1. **Define the sets \( S_j \):**
For each integer \( 1 \le j \le 2017 \), the set \( S_j \) consists of integers \( 0 \le i \le 2^{2017} - 1 \) such that \( \left\lfloor \frac{i}{2^{j-1}} \right\rfloor \) is an odd integer. This can be written as:
\[
S_j = \{ i \mid 0 \le i \le 2^{2017} - 1, \left\lfloor \fra... | 2 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For any positive integer $n$, let $S_n$ denote the set of positive integers which cannot be written in the form $an+2017b$ for nonnegative integers $a$ and $b$. Let $A_n$ denote the average of the elements of $S_n$ if the cardinality of $S_n$ is positive and finite, and $0$ otherwise. Compute \[\left\lfloor\displaystyl... | 1. **Understanding the Problem:**
We need to compute the sum \(\left\lfloor \sum_{n=1}^{\infty} \frac{A_n}{2^n} \right\rfloor\), where \(A_n\) is the average of the elements of \(S_n\). The set \(S_n\) consists of positive integers that cannot be written in the form \(an + 2017b\) for nonnegative integers \(a\) and ... | 840 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $N$ be the number of functions $f: \mathbb{Z}/16\mathbb{Z} \to \mathbb{Z}/16\mathbb{Z}$ such that for all $a,b \in \mathbb{Z}/16\mathbb{Z}$: \[f(a)^2+f(b)^2+f(a+b)^2 \equiv 1+2f(a)f(b)f(a+b) \pmod{16}.\] Find the remainder when $N$ is divided by 2017.
[i]Proposed by Zack Chroman[/i] | 1. **Understanding the problem**: We need to find the number of functions \( f: \mathbb{Z}/16\mathbb{Z} \to \mathbb{Z}/16\mathbb{Z} \) that satisfy the given functional equation for all \( a, b \in \mathbb{Z}/16\mathbb{Z} \):
\[
f(a)^2 + f(b)^2 + f(a+b)^2 \equiv 1 + 2f(a)f(b)f(a+b) \pmod{16}.
\]
2. **Initial ... | 793 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A [i]simple hyperplane[/i] in $\mathbb{R}^4$ has the form \[k_1x_1+k_2x_2+k_3x_3+k_4x_4=0\] for some integers $k_1,k_2,k_3,k_4\in \{-1,0,1\}$ that are not all zero. Find the number of regions that the set of all simple hyperplanes divide the unit ball $x_1^2+x_2^2+x_3^2+x_4^2\leq 1$ into.
[i]Proposed by Yannick Yao[/i... | 1. **Understanding the Problem:**
We need to find the number of regions that the set of all simple hyperplanes in \(\mathbb{R}^4\) divides the unit ball \(x_1^2 + x_2^2 + x_3^2 + x_4^2 \leq 1\) into. A simple hyperplane in \(\mathbb{R}^4\) is given by the equation \(k_1x_1 + k_2x_2 + k_3x_3 + k_4x_4 = 0\) where \(k_... | 1661981 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
The USAMO is a $6$ question test. For each question, you submit a positive integer number $p$ of pages on which your solution is written. On the $i$th page of this question, you write the fraction $i/p$ to denote that this is the $i$th page out of $p$ for this question. When you turned in your submissions for the $2017... | 1. Let \( a_k \) be the number of pages submitted for the \( k \)-th question, where \( k = 1, 2, \ldots, 6 \).
2. For each question, the sum of the fractions written on the pages is given by:
\[
\sum_{i=1}^{a_k} \frac{i}{a_k}
\]
3. We can simplify this sum as follows:
\[
\sum_{i=1}^{a_k} \frac{i}{a_k} =... | 4028 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
A convex equilateral pentagon with side length $2$ has two right angles. The greatest possible area of the pentagon is $m+\sqrt{n}$, where $m$ and $n$ are positive integers. Find $100m+n$.
[i]Proposed by Yannick Yao[/i] | 1. **Identify the structure of the pentagon:**
- We are given a convex equilateral pentagon with side length \(2\) and two right angles.
- Let the pentagon be labeled as \(ABCDE\).
2. **Case 1: Adjacent right angles:**
- Suppose \(\angle A = \angle B = 90^\circ\).
- Quadrilateral \(ABCE\) has three equal s... | 407 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $a$ and $b$ be positive integers such that $(2a+b)(2b+a)=4752$. Find the value of $ab$.
[i]Proposed by James Lin[/i] | 1. **Expand the given equation:**
\[
(2a + b)(2b + a) = 4752
\]
Expanding the left-hand side, we get:
\[
4ab + 2a^2 + 2b^2 + ab = 4752
\]
Simplifying, we have:
\[
2a^2 + 5ab + 2b^2 = 4752
\]
2. **Divide the equation by 2:**
\[
a^2 + \frac{5}{2}ab + b^2 = 2376
\]
To simplify... | 520 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Senators Sernie Banders and Cedric "Ced" Truz of OMOrica are running for the office of Price Dent. The election works as follows: There are $66$ states, each composed of many adults and $2017$ children, with only the latter eligible to vote. On election day, the children each cast their vote with equal probability to B... | 1. **Understanding the Problem:**
- There are 66 states, each with 2017 children eligible to vote.
- Each child votes for either Banders or Truz with equal probability.
- A candidate wins a state if they receive the majority of votes in that state.
- The candidate who wins the majority of states becomes the... | 96 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a graph $G$ on $n$ vertices, let $P_G(x)$ be the unique polynomial of degree at most $n$ such that for each $i=0,1,2,\dots,n$, $P_G (i)$ equals the number of ways to color the vertices of the graph $G$ with $i$ distinct colors such that no two vertices connected by an edge have the same color. For each integer $3\l... | 1. Recognize that \( P_G(x) \) is the chromatic polynomial \( P(G, x) \). The chromatic polynomial \( P(G, x) \) of a graph \( G \) counts the number of ways to color the vertices of \( G \) with \( x \) colors such that no two adjacent vertices share the same color.
2. Use the well-known fact that for an edge \( uv \i... | 2017 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
We define the bulldozer of triangle $ABC$ as the segment between points $P$ and $Q$, distinct points in the plane of $ABC$ such that $PA\cdot BC=PB\cdot CA=PC\cdot AB$ and $QA\cdot BC=QB\cdot CA=QC\cdot AB$. Let $XY$ be a segment of unit length in a plane $\mathcal{P}$, and let $\mathcal{S}$ be the region of $\mathcal ... | 1. **Part 1: A synthetic observation**
**Lemma 1**: \( O, L \) lie on line \( PQ \).
**Proof**: Recall the \( A \)-Apollonius circle \( \omega_A \), which is the locus of points \( X \) such that \( \frac{BX}{XC} = \frac{BA}{AC} \). Define \( \omega_B, \omega_C \) similarly.
It's easy to prove (by angle cha... | 129 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Steven draws a line segment between every two of the points \[A(2,2), B(-2,2), C(-2,-2), D(2,-2), E(1,0), F(0,1), G(-1,0), H(0,-1).\] How many regions does he divide the square $ABCD$ into?
[i]Proposed by Michael Ren | 1. **Identify the points and the square:**
The points given are \(A(2,2)\), \(B(-2,2)\), \(C(-2,-2)\), \(D(2,-2)\), \(E(1,0)\), \(F(0,1)\), \(G(-1,0)\), and \(H(0,-1)\). The square \(ABCD\) has vertices at \(A\), \(B\), \(C\), and \(D\).
2. **Draw the square and the points:**
Plot the points on a coordinate plan... | 60 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set of $13$ distinct, pairwise relatively prime, positive integers. What is the smallest possible value of $\max_{s \in S} s- \min_{s \in S}s$?
[i]Proposed by James Lin | 1. **Initial Consideration**: We need to find the smallest possible value of $\max_{s \in S} s - \min_{s \in S} s$ for a set $S$ of 13 distinct, pairwise relatively prime, positive integers.
2. **Lower Bound**: Since the integers are pairwise relatively prime, they must be distinct and cannot share any common factors... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?
[i]Proposed by... | 1. **Initial List and Conditions**:
- Henry starts with the list of the first 1000 positive integers: \( \{1, 2, 3, \ldots, 1000\} \).
- At each step, he erases any nonpositive integers or any integers that have a repeated digit.
- After erasing, he decreases every remaining integer in the list by 1.
2. **Und... | 11 | Logic and Puzzles | math-word-problem | Yes | Yes | aops_forum | false |
We define the sets of lattice points $S_0,S_1,\ldots$ as $S_0=\{(0,0)\}$ and $S_k$ consisting of all lattice points that are exactly one unit away from exactly one point in $S_{k-1}$. Determine the number of points in $S_{2017}$.
[i]Proposed by Michael Ren | 1. **Understanding the Problem:**
- We start with the set \( S_0 = \{(0,0)\} \).
- Each subsequent set \( S_k \) consists of all lattice points that are exactly one unit away from exactly one point in \( S_{k-1} \).
2. **Visualizing the Growth of Sets:**
- \( S_0 \) contains the single point \((0,0)\).
- \... | 4096 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{a,b,c,d,e,f,g,h,i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\gcd(c,d)=\gcd(f,g)=1$ and
\[(10a+b)^{c/d}=e^{f/g}.\]
Given that $h>i$, evaluate $10h+i$.
[i]Proposed by James Lin[/i] | 1. Given the permutation $\{a,b,c,d,e,f,g,h,i\}$ of $\{1,2,3,4,5,6,7,8,9\}$, we need to satisfy the conditions $\gcd(c,d)=1$, $\gcd(f,g)=1$, and the equation:
\[
(10a + b)^{c/d} = e^{f/g}.
\]
2. We need to find values for $a, b, c, d, e, f, g, h, i$ that satisfy the given conditions. Let's start by considerin... | 86 | Number Theory | other | Yes | Yes | aops_forum | false |
Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2||\ell_1||\ell_2$, $F_1$ lies on $\mathcal{P}_2$, and $F_2$ lies on $\mathcal{P}_1$. The two parabolas intersect at distinct points $A$ and $B$... | 1. **Assume the equations of the parabolas:**
Without loss of generality, we can assume one of the parabolas, $\mathcal{P}_1$, is given by the equation \( y = \frac{1}{2}x^2 \). The focus of this parabola is at \( F_1 = (0, \frac{1}{2}) \) and the directrix is \( y = -\frac{1}{2} \).
2. **Determine the properties o... | 1504 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Tessa the hyper-ant is at the origin of the four-dimensional Euclidean space $\mathbb R^4$. For each step she moves to another lattice point that is $2$ units away from the point she is currently on. How many ways can she return to the origin for the first time after exactly $6$ steps?
[i]Proposed by Yannick Yao | To solve this problem, we need to determine the number of ways Tessa the hyper-ant can return to the origin in $\mathbb{R}^4$ after exactly 6 steps, where each step is 2 units away from the current point. We will use the principle of inclusion-exclusion (PIE) to count the valid paths.
1. **Understanding the Problem**:... | 725568 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For a positive integer $n$, define $f(n)=\sum_{i=0}^{\infty}\frac{\gcd(i,n)}{2^i}$ and let $g:\mathbb N\rightarrow \mathbb Q$ be a function such that $\sum_{d\mid n}g(d)=f(n)$ for all positive integers $n$. Given that $g(12321)=\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $v_2(p)$.
[i]Proposed by Micha... | 1. **Applying Möbius Inversion**:
Given the function \( f(n) = \sum_{i=0}^{\infty} \frac{\gcd(i,n)}{2^i} \) and the function \( g \) such that \( \sum_{d \mid n} g(d) = f(n) \), we can use Möbius inversion to find \( g(n) \):
\[
g(n) = \sum_{d \mid n} f(d) \mu\left(\frac{n}{d}\right)
\]
where \( \mu \) i... | 12324 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Given a sequence of positive integers $a_1, a_2, a_3, \dots, a_{n}$, define the \emph{power tower function} \[f(a_1, a_2, a_3, \dots, a_{n})=a_1^{a_2^{a_3^{\mathstrut^{ .^{.^{.^{a_{n}}}}}}}}.\] Let $b_1, b_2, b_3, \dots, b_{2017}$ be positive integers such that for any $i$ between 1 and 2017 inclusive, \[f(a_1, a_2, a_... | 1. **Understanding the Problem:**
We need to find the smallest possible value of \( b_1 + b_2 + b_3 + \dots + b_{2017} \) such that for any sequence of positive integers \( a_1, a_2, \dots, a_{2017} \) greater than 2017, the power tower function \( f(a_1, a_2, \dots, a_i, \dots, a_{2017}) \equiv f(a_1, a_2, \dots, a... | 4983 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p=2017$ be a prime. Suppose that the number of ways to place $p$ indistinguishable red marbles, $p$ indistinguishable green marbles, and $p$ indistinguishable blue marbles around a circle such that no red marble is next to a green marble and no blue marble is next to a blue marble is $N$. (Rotations and reflection... | 1. **Fixing a Blue Marble**:
To simplify the problem, we fix one blue marble at the top of the circle. This reduces the problem to arranging the remaining \(2016\) blue marbles, \(2017\) red marbles, and \(2017\) green marbles around the circle. Note that we must multiply by \(3\) at the end to account for the thre... | 3913 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
For an integer $k$ let $T_k$ denote the number of $k$-tuples of integers $(x_1,x_2,...x_k)$ with $0\le x_i < 73$ for each $i$, such that $73|x_1^2+x_2^2+...+x_k^2-1$. Compute the remainder when $T_1+T_2+...+T_{2017}$ is divided by $2017$.
[i]Proposed by Vincent Huang | To solve the problem, we need to compute the number of $k$-tuples $(x_1, x_2, \ldots, x_k)$ such that $0 \le x_i < 73$ for each $i$ and $73 \mid x_1^2 + x_2^2 + \cdots + x_k^2 - 1$. We denote this number by $T_k$. We then need to find the remainder when $T_1 + T_2 + \cdots + T_{2017}$ is divided by 2017.
1. **Counting... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Call a nonempty set $V$ of nonzero integers \emph{victorious} if there exists a polynomial $P(x)$ with integer coefficients such that $P(0)=330$ and that $P(v)=2|v|$ holds for all elements $v\in V$. Find the number of victorious sets.
[i]Proposed by Yannick Yao[/i] | To solve this problem, we need to determine the number of nonempty sets \( V \) of nonzero integers such that there exists a polynomial \( P(x) \) with integer coefficients satisfying \( P(0) = 330 \) and \( P(v) = 2|v| \) for all \( v \in V \).
### Case 1: \( V \) consists entirely of positive integers
By the Ration... | 210 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ABC$ be a triangle with $AB=7, AC=9, BC=10$, circumcenter $O$, circumradius $R$, and circumcircle $\omega$. Let the tangents to $\omega$ at $B,C$ meet at $X$. A variable line $\ell$ passes through $O$. Let $A_1$ be the projection of $X$ onto $\ell$ and $A_2$ be the reflection of $A_1$ over $O$. Suppose that there ... | 1. **Scaling and Complex Numbers:**
We start by scaling the triangle such that the circumcircle \((ABC)\) is the unit circle. This means that the circumradius \(R = 1\). Let the complex numbers corresponding to points \(A\), \(B\), and \(C\) be \(a\), \(b\), and \(c\) respectively. The line \(\ell\) is the real axis... | 567 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Define a sequence of polynomials $P_0,P_1,...$ by the recurrence $P_0(x)=1, P_1(x)=x, P_{n+1}(x) = 2xP_n(x)-P_{n-1}(x)$. Let $S=\left|P_{2017}'\left(\frac{i}{2}\right)\right|$ and $T=\left|P_{17}'\left(\frac{i}{2}\right)\right|$, where $i$ is the imaginary unit. Then $\frac{S}{T}$ is a rational number with fractional p... | 1. **Define the sequence of polynomials**:
The sequence of polynomials \( P_n(x) \) is defined by the recurrence relation:
\[
P_0(x) = 1, \quad P_1(x) = x, \quad P_{n+1}(x) = 2xP_n(x) - P_{n-1}(x)
\]
2. **Transform the polynomials**:
Let \( P_n(ix) = i^n Q_n(x) \). Then we have:
\[
Q_0(x) = 1, \qu... | 4142 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Let $p = 2017$. If $A$ is an $n\times n$ matrix composed of residues $\pmod{p}$ such that $\det A\not\equiv 0\pmod{p}$ then let $\text{ord}(A)$ be the minimum integer $d > 0$ such that $A^d\equiv I\pmod{p}$, where $I$ is the $n\times n$ identity matrix. Let the maximum such order be $a_n$ for every positive integer $n$... | 1. **Understanding the Problem:**
We are given a prime \( p = 2017 \) and an \( n \times n \) matrix \( A \) with entries in \( \mathbb{F}_p \) (the field of integers modulo \( p \)). The matrix \( A \) is invertible, i.e., \( \det A \not\equiv 0 \pmod{p} \). We need to find the maximum order \( a_n \) of such a mat... | 2015 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
For any real numbers $x,y$ that satisfies the equation $$x+y-xy=155$$ and $$x^2+y^2=325$$, Find $|x^3-y^3|$ | 1. Given the equations:
\[
x + y - xy = 155
\]
and
\[
x^2 + y^2 = 325
\]
2. From the first equation, we can express \(x + y\) in terms of \(xy\):
\[
x + y = 155 + xy
\]
3. From the second equation, we use the identity for the sum of squares:
\[
x^2 + y^2 = (x + y)^2 - 2xy
\]
... | 4375 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the last two digits of the number $(11^2 + 15^2 + 19^2 + ... + 2007^2)^2$? | 1. We start by expressing the given sum in a more manageable form. The sequence of numbers is \(11, 15, 19, \ldots, 2007\). This sequence is an arithmetic sequence with the first term \(a = 11\) and common difference \(d = 4\). The general term of the sequence can be written as:
\[
a_n = 11 + (n-1) \cdot 4 = 4n +... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[help me] Let m and n denote the number of digits in $2^{2007}$ and $5^{2007}$ when expressed in base 10. What is the sum m + n? | 1. To determine the number of digits in \(2^{2007}\) and \(5^{2007}\), we need to use the formula for the number of digits of a number \(n\) in base 10, which is given by:
\[
d(n) = \lfloor \log_{10} n \rfloor + 1
\]
2. First, we calculate the number of digits in \(2^{2007}\):
\[
d(2^{2007}) = \lfloor \l... | 2008 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find the maximum value of $M =\frac{x}{2x + y} +\frac{y}{2y + z}+\frac{z}{2z + x}$ , $x,y, z > 0$ | To find the maximum value of \( M = \frac{x}{2x + y} + \frac{y}{2y + z} + \frac{z}{2z + x} \) for \( x, y, z > 0 \), we will use the method of inequalities.
1. **Initial Setup:**
We need to show that:
\[
\frac{x}{2x + y} + \frac{y}{2y + z} + \frac{z}{2z + x} \leq 1
\]
2. **Using the AM-GM Inequality:**
... | 1 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Determine all positive integer $a$ such that the equation $2x^2 - 30x + a = 0$ has two prime roots, i.e. both roots are prime numbers. | 1. Let \( p_1 \) and \( p_2 \) (with \( p_1 \leq p_2 \)) be the prime roots of the quadratic equation \( 2x^2 - 30x + a = 0 \). By Vieta's formulas, we know:
\[
p_1 + p_2 = \frac{30}{2} = 15
\]
and
\[
p_1 p_2 = \frac{a}{2}
\]
2. Since \( p_1 \) and \( p_2 \) are prime numbers and their sum is 15, ... | 52 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of integer $n$ from the set $\{2000,2001,...,2010\}$ such that $2^{2n} + 2^n + 5$ is divisible by $7$
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E) None of the above. | To solve the problem, we need to determine the number of integers \( n \) from the set \(\{2000, 2001, \ldots, 2010\}\) such that \(2^{2n} + 2^n + 5\) is divisible by 7.
1. **Identify the periodicity of \(2^n \mod 7\):**
\[
\begin{aligned}
2^1 &\equiv 2 \pmod{7}, \\
2^2 &\equiv 4 \pmod{7}, \\
2^3 &\equ... | 4 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many real numbers $a \in (1,9)$ such that the corresponding number $a- \frac1a$ is an integer?
(A): $0$, (B): $1$, (C): $8$, (D): $9$, (E) None of the above. | 1. Let \( k = a - \frac{1}{a} \). We need to find the values of \( a \) such that \( k \) is an integer and \( a \in (1, 9) \).
2. Rewrite the equation:
\[
k = a - \frac{1}{a} \implies k = \frac{a^2 - 1}{a}
\]
This implies:
\[
a^2 - ka - 1 = 0
\]
3. Solve the quadratic equation for \( a \):
\[... | 8 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
How many positive integers a less than $100$ such that $4a^2 + 3a + 5$ is divisible by $6$. | 1. Define the function \( f(a) = 4a^2 + 3a + 5 \).
2. We need to find the values of \( a \) such that \( f(a) \) is divisible by \( 6 \). This means \( f(a) \equiv 0 \pmod{6} \).
3. We will check the values of \( f(a) \) for \( a \) modulo \( 6 \).
- If \( a \equiv 0 \pmod{6} \):
\[
f(a) = 4(0)^2 + 3(0) +... | 32 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a, b, c$ be positive integers such that $a + 2b +3c = 100$.
Find the greatest value of $M = abc$ | 1. We start with the given equation:
\[
a + 2b + 3c = 100
\]
where \(a\), \(b\), and \(c\) are positive integers.
2. We aim to maximize the product \(M = abc\). To do this, we can use the Arithmetic Mean-Geometric Mean Inequality (AM-GM Inequality), which states:
\[
\frac{a + 2b + 3c}{3} \geq \sqrt[3... | 6171 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
An integer is called "octal" if it is divisible by $8$ or if at least one of its digits is $8$.
How many integers between $1$ and $100$ are octal?
(A): $22$, (B): $24$, (C): $27$, (D): $30$, (E): $33$ | 1. **Identify numbers with a digit of $8$ between $1$ and $100$:**
- The numbers are: $8, 18, 28, 38, 48, 58, 68, 78, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 98$.
- There are $19$ such numbers.
2. **Identify numbers divisible by $8$ between $1$ and $100$:**
- These numbers are: $8, 16, 24, 32, 40, 48, 56, 64,... | 28 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
A cube with sides of length 3cm is painted red
and then cut into 3 x 3 x 3 = 27 cubes with sides of length 1cm.
If a denotes the number of small cubes (of 1cm x 1cm x 1cm) that
are not painted at all, b the number painted on one sides, c the
number painted on two sides, and d the number painted on three
sides, determin... | 1. **Determine the number of small cubes that are not painted at all ($a$):**
- The small cubes that are not painted at all are those that are completely inside the larger cube, not touching any face.
- For a cube of side length 3 cm, the inner cube that is not painted has side length \(3 - 2 = 1\) cm (since 1 c... | -9 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$ ?
(A) $1$, (B) $2$, (C) $3$, (D) $4$, (E) None of the above. | 1. Given \( x = \frac{\sqrt[3]{2+\sqrt{3}} + \sqrt[3]{2-\sqrt{3}}}{2} \), we start by letting \( y = \sqrt[3]{2+\sqrt{3}} + \sqrt[3]{2-\sqrt{3}} \). Therefore, \( x = \frac{y}{2} \).
2. We need to find the value of \( 4x^3 - 3x \). First, we will find \( y^3 \):
\[
y = \sqrt[3]{2+\sqrt{3}} + \sqrt[3]{2-\sqrt{3}}... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $x=\frac{\sqrt{6+2\sqrt5}+\sqrt{6-2\sqrt5}}{\sqrt{20}}$. The value of $$H=(1+x^5-x^7)^{{2012}^{3^{11}}}$$
is
(A) $1$ (B) $11$ (C) $21$ (D) $101$ (E) None of the above | 1. First, we need to simplify the expression for \( x \):
\[
x = \frac{\sqrt{6 + 2\sqrt{5}} + \sqrt{6 - 2\sqrt{5}}}{\sqrt{20}}
\]
2. We start by simplifying the terms inside the square roots. Notice that:
\[
\sqrt{6 + 2\sqrt{5}} = \sqrt{(\sqrt{5} + 1)^2} = \sqrt{5} + 1
\]
and
\[
\sqrt{6 - 2\... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $f(x)$ be a function such that $f(x)+2f\left(\frac{x+2010}{x-1}\right)=4020 - x$ for all $x \ne 1$.
Then the value of $f(2012)$ is
(A) $2010$, (B) $2011$, (C) $2012$, (D) $2014$, (E) None of the above. | 1. Given the functional equation:
\[
f(x) + 2f\left(\frac{x+2010}{x-1}\right) = 4020 - x
\]
for all \( x \neq 1 \).
2. Let us denote \( y = \frac{x+2010}{x-1} \). Then, we need to express \( x \) in terms of \( y \):
\[
y = \frac{x+2010}{x-1} \implies y(x-1) = x + 2010 \implies yx - y = x + 2010 \imp... | 2010 | Algebra | MCQ | Yes | Yes | aops_forum | false |
[b]Q4.[/b] A man travels from town $A$ to town $E$ through $B,C$ and $D$ with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town $A$ and town $E$ can be classified as horizontal, up slope, down slope and horizontal and total l... | 1. Let the length of each segment \( AB = BC = CD = DE \) be \( x \) km. Then the total distance of the journey from \( A \) to \( E \) is \( 4x \) km.
2. Calculate the time taken for each segment:
- For segment \( AB \) (horizontal road), the speed is \( 3 \) km/h. The time taken is:
\[
t_{AB} = \frac{x}... | 3 | Calculus | MCQ | Yes | Yes | aops_forum | false |
[b]Q11.[/b] Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$. | 1. **Initial Terms and Recurrence Relation:**
Given the sequence:
\[
a_1 = 5, \quad a_2 = 8
\]
and the recurrence relation:
\[
a_{n+1} = a_n + 3a_{n-1} \quad \text{for} \quad n = 1, 2, 3, \ldots
\]
2. **Modulo 3 Analysis:**
We start by examining the sequence modulo 3:
\[
a_1 \equiv 5 \... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]Q12.[/b] Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively. | 1. Let \( P \) be a positive integer. We are given that the sum of all divisors of \( P \) is \( 2P \) and the product of all divisors of \( P \) is \( P^2 \).
2. Let \( \tau(P) \) denote the number of divisors of \( P \). The product of all divisors of \( P \) is known to be \( P^{\tau(P)/2} \). This is because each ... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
[b]Q13.[/b] Determine the greatest value of the sum $M=11xy+3x+2012yz$, where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$ | 1. We start with the given expression \( M = 11xy + 3x + 2012yz \) and the constraint \( x + y + z = 1000 \), where \( x, y, z \) are non-negative integers.
2. To find the maximum value of \( M \), we can express \( x \) in terms of \( y \) and \( z \) using the constraint:
\[
x = 1000 - y - z
\]
3. Substitute... | 503000000 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
[b]Q14.[/b] Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$. | 1. Given a right triangle \( \triangle ABC \) with \( \angle A = 90^\circ \), the incenter \( I \) is the intersection of the angle bisectors of \( \angle B \) and \( \angle C \). The perpendicular from \( I \) to \( BC \) meets \( BC \) at \( H \), with \( HB = 5 \) cm and \( HC = 8 \) cm.
2. The lengths \( HB \) and... | 40 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be an even number but not divisible by $10$. The last two digits of $A^{20}$ are:
(A): $46$, (B): $56$, (C): $66$, (D): $76$, (E): None of the above. | 1. **Understanding the problem**:
We need to find the last two digits of \(A^{20}\) where \(A\) is an even number not divisible by 10. This means \(A\) is not divisible by 2 and 5 simultaneously.
2. **Using Euler's Totient Theorem**:
Euler's Totient Theorem states that \(A^{\phi(n)} \equiv 1 \pmod{n}\) where \(\... | 76 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The number of integer solutions $x$ of the equation below
$(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is
(A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above. | 1. Start with the given equation:
\[
(12x - 1)(6x - 1)(4x - 1)(3x - 1) = 330
\]
2. Notice that the equation can be rearranged in any order due to the commutative property of multiplication:
\[
(12x - 1)(3x - 1)(6x - 1)(4x - 1) = 330
\]
3. To simplify the problem, let's introduce a substitution. Let:... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The positive numbers $a, b, c,d,e$ are such that the following identity hold for all real number $x$:
$(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3$. Find the smallest value of $d$. | 1. Given the identity:
\[
(x + a)(x + b)(x + c) = x^3 + 3dx^2 + 3x + e^3
\]
we can expand the left-hand side and compare coefficients with the right-hand side.
2. Expanding the left-hand side:
\[
(x + a)(x + b)(x + c) = x^3 + (a+b+c)x^2 + (ab+bc+ca)x + abc
\]
3. By comparing coefficients with the... | 1 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
What is the largest integer not exceeding $8x^3 +6x - 1$, where $x =\frac12 \left(\sqrt[3]{2+\sqrt5} + \sqrt[3]{2-\sqrt5}\right)$ ?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above. | 1. Given \( x = \frac{1}{2} \left( \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} \right) \), we need to evaluate \( 8x^3 + 6x - 1 \).
2. First, let's find \( 8x^3 \):
\[
8x^3 = 8 \left( \frac{1}{2} \left( \sqrt[3]{2 + \sqrt{5}} + \sqrt[3]{2 - \sqrt{5}} \right) \right)^3
\]
Simplifying inside the cube:
... | 3 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $x_0 = [a], x_1 = [2a] - [a], x_2 = [3a] - [2a], x_3 = [3a] - [4a],x_4 = [5a] - [4a],x_5 = [6a] - [5a], . . . , $ where $a=\frac{\sqrt{2013}}{\sqrt{2014}}$ .The value of $x_9$ is:
(A): $2$ (B): $3$ (C): $4$ (D): $5$ (E): None of the above. | 1. First, we need to understand the notation $[x]$, which represents the floor function, i.e., the greatest integer less than or equal to $x$.
2. Given $a = \frac{\sqrt{2013}}{\sqrt{2014}} = \sqrt{\frac{2013}{2014}}$.
3. We need to find the value of $x_9 = [10a] - [9a]$.
Let's calculate $10a$ and $9a$ step by step:
4... | 1 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
The number $n$ is called a composite number if it can be written in the form $n = a\times b$, where $a, b$ are positive
integers greater than $1$. Write number $2013$ in a sum of $m$ composite numbers. What is the largest value of $m$?
(A): $500$, (B): $501$, (C): $502$, (D): $503$, (E): None of the above. | To solve the problem, we need to express the number 2013 as a sum of the maximum number of composite numbers.
1. **Understanding Composite Numbers**:
A composite number is a positive integer that has at least one positive divisor other than one or itself. In other words, it can be written as \( n = a \times b \) w... | 502 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let the numbers x and y satisfy the conditions $\begin{cases} x^2 + y^2 - xy = 2 \\
x^4 + y^4 + x^2y^2 = 8 \end{cases}$
The value of $P = x^8 + y^8 + x^{2014}y^{2014}$ is:
(A): $46$, (B): $48$, (C): $50$, (D): $52$, (E) None of the above. | 1. We start with the given system of equations:
\[
\begin{cases}
x^2 + y^2 - xy = 2 \\
x^4 + y^4 + x^2y^2 = 8
\end{cases}
\]
2. From the first equation, we can express \(x^2 + y^2\) in terms of \(xy\):
\[
x^2 + y^2 = 2 + xy
\]
3. Next, we square both sides of the equation \(x^2 + y^2 = 2 + ... | 48 | Algebra | MCQ | Yes | Yes | aops_forum | false |
How many zeros are there in the last digits of the following number $P = 11\times12\times ...\times 88\times 89$ ?
(A): $16$, (B): $17$, (C): $18$, (D): $19$, (E) None of the above. | To determine the number of zeros in the last digits of the product \( P = 11 \times 12 \times \cdots \times 88 \times 89 \), we need to count the number of factors of 10 in \( P \). A factor of 10 is composed of a factor of 2 and a factor of 5. Since there are generally more factors of 2 than factors of 5 in a factoria... | 18 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let be given $a < b < c$ and $f(x) =\frac{c(x - a)(x - b)}{(c - a)(c - b)}+\frac{a(x - b)(x - c)}{(a - b)(a -c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}$.
Determine $f(2014)$. | 1. **Identify the form of \( f(x) \)**:
The given function is:
\[
f(x) = \frac{c(x - a)(x - b)}{(c - a)(c - b)} + \frac{a(x - b)(x - c)}{(a - b)(a - c)} + \frac{b(x - c)(x - a)}{(b - c)(b - a)}
\]
This is a quadratic polynomial in \( x \).
2. **Check the roots of \( f(x) \)**:
Notice that \( f(a) \),... | 2014 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
How many integers are there in $\{0,1, 2,..., 2014\}$ such that $C^x_{2014} \ge C^{999}{2014}$ ?
(A): $15$, (B): $16$, (C): $17$, (D): $18$, (E) None of the above.
Note: $C^{m}_{n}$ stands for $\binom {m}{n}$ | 1. We start by understanding the properties of binomial coefficients. The binomial coefficient $\binom{n}{k}$ is defined as:
\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\]
It is known that $\binom{n}{k}$ is symmetric, meaning:
\[
\binom{n}{k} = \binom{n}{n-k}
\]
Additionally, $\binom{n}{k}$ increases ... | 17 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $a$ and $b$ satisfy the conditions $\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1 \end{cases}$ .
The value of $(a - b)^{2014}$ is:
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above. | 1. We start with the given system of equations:
\[
\begin{cases}
a^3 - 6a^2 + 15a = 9 \\
b^3 - 3b^2 + 6b = -1
\end{cases}
\]
2. Rewrite the first equation:
\[
a^3 - 6a^2 + 15a - 9 = 0
\]
We will try to express this in a form that might reveal a relationship with \(b\). Consider the trans... | 1 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Find the smallest positive integer $n$ such that the number $2^n + 2^8 + 2^{11}$ is a perfect square.
(A): $8$, (B): $9$, (C): $11$, (D): $12$, (E) None of the above. | 1. We start with the expression \(2^n + 2^8 + 2^{11}\) and need to find the smallest positive integer \(n\) such that this expression is a perfect square.
2. Let's rewrite the expression in a more convenient form:
\[
2^n + 2^8 + 2^{11}
\]
Notice that \(2^8 = 256\) and \(2^{11} = 2048\). We can factor out th... | 12 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let $a,b,c$ and $m$ ($0 \le m \le 26$) be integers such that $a + b + c = (a - b)(b- c)(c - a) = m$ (mod $27$) then $m$ is
(A): $0$, (B): $1$, (C): $25$, (D): $26$ (E): None of the above. | 1. **Consider the given conditions:**
\[
a + b + c \equiv (a - b)(b - c)(c - a) \equiv m \pmod{27}
\]
We need to find the possible values of \(m\) under the constraint \(0 \le m \le 26\).
2. **Analyze the equation modulo 3:**
\[
a + b + c \equiv (a - b)(b - c)(c - a) \pmod{3}
\]
Since \(27 \equ... | 0 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let the numbers $a, b,c$ satisfy the relation $a^2+b^2+c^2 \le 8$.
Determine the maximum value of $M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)$ | To determine the maximum value of \( M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4) \) given the constraint \( a^2 + b^2 + c^2 \leq 8 \), we can proceed as follows:
1. **Rewrite the expression for \( M \):**
\[
M = 4(a^3 + b^3 + c^3) - (a^4 + b^4 + c^4)
\]
2. **Introduce the terms \( 4a^2, 4b^2, \) and \( 4c^2 \... | 48 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer.
(A): $30$ (B): $35$ (C): $40$ (D): $45$ (E): None of the above. | To determine the largest number of monkeys that can be made lucky, we need to ensure that each monkey eats three different fruits. We have the following quantities of fruits:
- 20 oranges
- 30 bananas
- 40 peaches
- 50 tangerines
We need to maximize the number of monkeys that can eat three different fruits. Let's brea... | 40 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$. | To determine the greatest value of \( M = |x| + |y| + |z| \) given the inequalities:
\[
\begin{cases}
| x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6
\end{cases}
\]
1. **Understanding the inequalities**:
Each inequality represents a region in 3-dimensional space. The a... | 6 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$
The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above. | 1. **Identify the roots of the polynomial:**
Given the polynomial \( P(x) = x^3 - 6x^2 + 5x + 12 \), we need to find its roots. By the Rational Root Theorem, possible rational roots are the factors of the constant term (12) divided by the factors of the leading coefficient (1). Thus, the possible rational roots are ... | 8 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $a, b, c$ be two-digit, three-digit, and four-digit numbers, respectively. Assume that the sum of all digits of number $a+b$, and the sum of all digits of $b + c$ are all equal to $2$. The largest value of $a + b + c$ is
(A): $1099$ (B): $2099$ (C): $1199$ (D): $2199$ (E): None of the above. | 1. **Understanding the problem:**
- \(a\) is a two-digit number.
- \(b\) is a three-digit number.
- \(c\) is a four-digit number.
- The sum of the digits of \(a + b\) is 2.
- The sum of the digits of \(b + c\) is 2.
- We need to find the largest value of \(a + b + c\).
2. **Analyzing the constraints:... | 10199 | Number Theory | MCQ | Yes | Yes | aops_forum | false |
Let a,b,c be three distinct positive numbers.
Consider the quadratic polynomial $P (x) =\frac{c(x - a)(x - b)}{(c -a)(c -b)}+\frac{a(x - b)(x - c)}{(a - b)(a - c)}+\frac{b(x -c)(x - a)}{(b - c)(b - a)}+ 1$.
The value of $P (2017)$ is
(A): $2015$ (B): $2016$ (C): $2017$ (D): $2018$ (E): None of the above. | 1. Consider the given polynomial:
\[
P(x) = \frac{c(x - a)(x - b)}{(c - a)(c - b)} + \frac{a(x - b)(x - c)}{(a - b)(a - c)} + \frac{b(x - c)(x - a)}{(b - c)(b - a)} + 1
\]
2. We need to show that the expression:
\[
Q(x) = \frac{c(x - a)(x - b)}{(c - a)(c - b)} + \frac{a(x - b)(x - c)}{(a - b)(a - c)} + ... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Let $T=\frac{1}{4}x^{2}-\frac{1}{5}y^{2}+\frac{1}{6}z^{2}$ where $x,y,z$ are real numbers such that $1 \leq x,y,z \leq 4$ and $x-y+z=4$.
Find the smallest value of $10 \times T$. | 1. Given the function \( T = \frac{1}{4}x^2 - \frac{1}{5}y^2 + \frac{1}{6}z^2 \) and the constraints \( 1 \leq x, y, z \leq 4 \) and \( x - y + z = 4 \), we need to find the minimum value of \( 10 \times T \).
2. First, we rewrite \( T \) in a common denominator form:
\[
T = \frac{x^2}{4} - \frac{y^2}{5} + \frac... | 23 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of distinct real roots of the following equation $x^2 +\frac{9x^2}{(x + 3)^2} = 40$.
A. $0$ B. $1$ C. $2$ D. $3$ E. $4$ | 1. Start with the given equation:
\[
x^2 + \frac{9x^2}{(x + 3)^2} = 40
\]
2. Simplify the fraction:
\[
\frac{9x^2}{(x + 3)^2} = \frac{9x^2}{x^2 + 6x + 9}
\]
3. Substitute back into the equation:
\[
x^2 + \frac{9x^2}{x^2 + 6x + 9} = 40
\]
4. Let \( y = x + 3 \), then \( x = y - 3 \). Substi... | 2 | Algebra | MCQ | Yes | Yes | aops_forum | false |
Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^o,\angle B = 105^o,\angle C = 90^o$ and $AB = 2,BC = CD = DE =\sqrt2$. If the length of $AE$ is $\sqrt{a }- b$ where $a, b$ are integers, what is the value of $a + b$?
| 1. **Identify the given information and draw the pentagon:**
- $\angle A = 90^\circ$
- $\angle B = 105^\circ$
- $\angle C = 90^\circ$
- $AB = 2$
- $BC = CD = DE = \sqrt{2}$
2. **Analyze the triangle $\triangle ABD$:**
- Since $\angle A = 90^\circ$ and $\angle B = 105^\circ$, the remaining angle $\ang... | 4 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $k$ be a positive integer such that $1 +\frac12+\frac13+ ... +\frac{1}{13}=\frac{k}{13!}$. Find the remainder when $k$ is divided by $7$. | 1. We start with the given equation:
\[
1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{13} = \frac{k}{13!}
\]
To find \( k \), we multiply both sides by \( 13! \):
\[
13! \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{13}\right) = k
\]
This can be rewritten as:
\[
k = 13! + \fr... | 0 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
| We are given the problem of finding all 3-digit numbers $\overline{abc}$ (where $a, b \neq 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$.
1. **Express the numbers in terms of their digits:**
- Let $\overline{abc} = 100a + 10b + c$
- Let $\overline{bcd} = 100b + 10... | 627 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
In the below figure, there is a regular hexagon and three squares whose sides are equal to $4$ cm. Let $M,N$, and $P$ be the centers of the squares. The perimeter of the triangle $MNP$ can be written in the form $a + b\sqrt3$ (cm), where $a, b$ are integers. Compute the value of $a + b$.
[img]https://cdn.artofproblemso... | 1. **Identify the properties of the shapes involved:**
- The hexagon is regular, meaning all its sides and angles are equal.
- Each square has a side length of 4 cm.
- The centers of the squares, \(M\), \(N\), and \(P\), form the vertices of triangle \(MNP\).
2. **Determine the side length of the equilateral ... | 20 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exact... | To solve this problem, we need to determine the number of ways to distribute 5 dishes over 3 days such that each dish is used exactly once. We will use the principle of inclusion-exclusion (PIE) to count the valid distributions.
1. **Total number of distributions without any restrictions:**
Each of the 5 dishes can... | 150 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
[THE PROBLEM OF PAINTING THE THÁP RÙA (THE CENTRAL TOWER) MODEL]
The following picture illustrates the model of the Tháp Rùa (the Central Tower) in Hanoi, which consists of $3$ levels. For the first and second levels, each has $10$ doorways among which $3$ doorways are located at the front, $3$ at the back, $2$ on the ... | ### Part (a): In how many ways can the first level be painted?
1. **Choosing the color for the three doorways at the front:**
- There are 3 colors available: Blue, Yellow, and Brown.
- Therefore, there are \(3\) ways to choose the color for the three doorways at the front.
2. **Choosing the color for the center... | 27648 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
How many rectangles can be formed by the vertices of a cube? (Note: square is also a special rectangle).
A. $6$ B. $8$ C. $12$ D. $18$ E. $16$ | 1. **Identify the number of square faces in the cube:**
A cube has 6 faces, and each face is a square. Therefore, there are 6 square faces.
2. **Identify the number of rectangles that bisect the cube:**
These rectangles are formed by selecting two opposite edges on one face and two opposite edges on the opposite... | 12 | Combinatorics | MCQ | Yes | Yes | aops_forum | false |
How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$?
A. $4$ B. $6$ C. $8$ D. $10$ E. $12$ | 1. We start with the inequality \( n^4 - n^3 - 3n^2 - 3n - 17 < 0 \).
2. We rewrite the inequality as \( n^4 - n^3 - 3n^2 - 3n - 18 + 1 < 0 \), which simplifies to \( n^4 - n^3 - 3n^2 - 3n - 18 < -1 \).
3. We factorize the polynomial \( n^4 - n^3 - 3n^2 - 3n - 18 \). We find that \( n = -2 \) and \( n = 3 \) are roots ... | 4 | Inequalities | MCQ | Yes | Yes | aops_forum | false |
How many ways of choosing four edges in a cube such that any two among those four choosen edges have no common point. | To solve the problem of choosing four edges in a cube such that any two among those four chosen edges have no common point, we need to consider the geometric properties of the cube and the constraints given.
1. **Case 1: The four edges are pairwise parallel.**
- A cube has 12 edges, and these edges can be grouped i... | 15 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $100$ school students from two clubs $A$ and $B$ standing in circle. Among them $62$ students stand next to at least one student from club $A$, and $54$ students stand next to at least one student from club $B$.
1) How many students stand side-by-side with one friend from club $A$ and one friend from club $B$... | 1. Let \( p \) be the number of students who stand next to exactly one student from club \( A \) and one student from club \( B \).
2. Let \( q \) be the number of students who stand between two students from club \( A \).
3. Let \( r \) be the number of students who stand between two students from club \( B \).
From ... | 16 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $a,b, c$ denote the real numbers such that $1 \le a, b, c\le 2$.
Consider $T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}$.
Determine the largest possible value of $T$. | 1. Given the conditions \(1 \le a, b, c \le 2\), we need to determine the largest possible value of \(T = (a - b)^{2018} + (b - c)^{2018} + (c - a)^{2018}\).
2. First, note that \(|a - b| \le 1\), \(|b - c| \le 1\), and \(|c - a| \le 1\) because \(a, b, c\) are all within the interval \([1, 2]\).
3. Since \(2018\) is... | 2 | Inequalities | math-word-problem | Yes | Yes | aops_forum | false |
Let $f$ be a polynomial such that, for all real number $x$, $f(-x^2-x-1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019$.
Compute $f(2018)$. | 1. We start with the given functional equation for the polynomial \( f \):
\[
f(-x^2 - x - 1) = x^4 + 2x^3 + 2022x^2 + 2021x + 2019
\]
2. To find \( f \), we assume that \( f \) is a polynomial of the form \( f(t) \). We need to express \( f(t) \) in terms of \( t \) such that \( t = -x^2 - x - 1 \).
3. Let ... | -2019 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Given an arbitrary angle $\alpha$, compute
$cos \alpha + cos \big( \alpha +\frac{2\pi }{3 }\big) + cos \big( \alpha +\frac{4\pi }{3 }\big)$ and $sin \alpha + sin \big( \alpha +\frac{2\pi }{3 } \big) + sin \big( \alpha +\frac{4\pi }{3 } \big)$ .
Generalize this result and justify your answer. | 1. **Given Problem:**
We need to compute the following sums:
\[
\cos \alpha + \cos \left( \alpha + \frac{2\pi}{3} \right) + \cos \left( \alpha + \frac{4\pi}{3} \right)
\]
and
\[
\sin \alpha + \sin \left( \alpha + \frac{2\pi}{3} \right) + \sin \left( \alpha + \frac{4\pi}{3} \right).
\]
2. **Gene... | 0 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
$(a)$ Let $x, y$ be integers, not both zero. Find the minimum possible value of $|5x^2 + 11xy - 5y^2|$.
$(b)$ Find all positive real numbers $t$ such that $\frac{9t}{10}=\frac{[t]}{t - [t]}$. | 1. Let \( f(x, y) = 5x^2 + 11xy - 5y^2 \). We need to find the minimum possible value of \( |f(x, y)| \) for integers \( x \) and \( y \) not both zero.
2. Note that \( -f(x, y) = f(y, -x) \). Therefore, it suffices to prove that \( f(x, y) = k \) for \( k \in \{1, 2, 3, 4\} \) has no integral solution.
3. First, consi... | 5 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ u_1$, $ u_2$, $ \ldots$, $ u_{1987}$ be an arithmetic progression with $ u_1 \equal{} \frac {\pi}{1987}$ and the common difference $ \frac {\pi}{3974}$. Evaluate
\[ S \equal{} \sum_{\epsilon_i\in\left\{ \minus{} 1, 1\right\}}\cos\left(\epsilon_1 u_1 \plus{} \epsilon_2 u_2 \plus{} \cdots \plus{} \epsilon_{1987} u... | 1. Given the arithmetic progression \( u_1, u_2, \ldots, u_{1987} \) with \( u_1 = \frac{\pi}{1987} \) and common difference \( \frac{\pi}{3974} \), we can express the general term \( u_n \) as:
\[
u_n = u_1 + (n-1) \cdot \frac{\pi}{3974} = \frac{\pi}{1987} + (n-1) \cdot \frac{\pi}{3974}
\]
2. We need to eval... | 0 | Combinatorics | other | Yes | Yes | aops_forum | false |
$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1,2,3,1,2,3,...$ A student who says $2$ or $3$ must leave the circle. The game is over when there is only one student left. What position was the r... | 1. **Initial Setup**: We start with 1991 students sitting in a circle, each assigned a number from 1 to 1991. The students count off in sequence, saying the numbers 1, 2, 3 repeatedly. Students who say 2 or 3 leave the circle.
2. **First Round**: In the first round, every third student remains. Therefore, the position... | 1093 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Define the sequences $a_{0}, a_{1}, a_{2}, ...$ and $b_{0}, b_{1}, b_{2}, ...$ by $a_{0}= 2, b_{0}= 1, a_{n+1}= 2a_{n}b_{n}/(a_{n}+b_{n}), b_{n+1}= \sqrt{a_{n+1}b_{n}}$. Show that the two sequences converge to the same limit, and find the limit. | 1. Define the sequences \(a_n\) and \(b_n\) as given:
\[
a_0 = 2, \quad b_0 = 1
\]
\[
a_{n+1} = \frac{2a_n b_n}{a_n + b_n}, \quad b_{n+1} = \sqrt{a_{n+1} b_n}
\]
2. Define the ratio sequence \(c_n = \frac{b_n}{a_n}\). Initially, we have:
\[
c_0 = \frac{b_0}{a_0} = \frac{1}{2}
\]
3. Express ... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
Find the number of functions $ f: \mathbb N\rightarrow\mathbb N$ which satisfying:
(i) $ f(1) \equal{} 1$
(ii) $ f(n)f(n \plus{} 2) \equal{} f^2(n \plus{} 1) \plus{} 1997$ for every natural numbers n. | 1. **Initial Setup and Substitution**:
Given the function \( f: \mathbb{N} \rightarrow \mathbb{N} \) satisfying:
\[
f(1) = 1
\]
and
\[
f(n)f(n+2) = f^2(n+1) + 1997 \quad \text{for all natural numbers } n,
\]
we can replace 1997 with a prime number \( p \). Let \( f(2) = m \), then:
\[
f... | 1 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $a\geq 1$ be a real number. Put $x_{1}=a,x_{n+1}=1+\ln{(\frac{x_{n}^{2}}{1+\ln{x_{n}}})}(n=1,2,...)$. Prove that the sequence $\{x_{n}\}$ converges and find its limit. | 1. **Define the function and sequence:**
Let \( f(x) = 1 + \ln\left(\frac{x^2}{1 + \ln x}\right) \). The sequence is defined as \( x_1 = a \) and \( x_{n+1} = f(x_n) \) for \( n \geq 1 \).
2. **Check the derivative of \( f(x) \):**
Compute the derivative \( f'(x) \):
\[
f'(x) = \frac{d}{dx} \left( 1 + \ln\... | 1 | Calculus | math-word-problem | Yes | Yes | aops_forum | false |
The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$ | 1. **Initial Conditions and Recurrence Relation:**
The sequence $\{a_n\}_{n \geq 0}$ is defined by:
\[
a_0 = 20, \quad a_1 = 100, \quad a_{n+2} = 4a_{n+1} + 5a_n + 20 \quad \text{for} \quad n \geq 0
\]
2. **Modulo 3 Analysis:**
We start by analyzing the sequence modulo 3:
\[
a_0 \equiv 20 \equiv 2... | 6 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $ P(x)$ be a nonzero polynomial such that, for all real numbers $ x$, $ P(x^2 \minus{} 1) \equal{} P(x)P(\minus{}x)$. Determine the maximum possible number of real roots of $ P(x)$. | 1. **Define the polynomial and the function:**
Let \( P(x) \) be a nonzero polynomial such that for all real numbers \( x \), \( P(x^2 - 1) = P(x)P(-x) \). Define the function \( f(x) = x^2 - 1 \).
2. **Factorization of \( P(x) \):**
By the Fundamental Theorem of Algebra, \( P(x) \) can be factored as:
\[
... | 4 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ S(n)$ be the sum of decimal digits of a natural number $ n$. Find the least value of $ S(m)$ if $ m$ is an integral multiple of $ 2003$. | 1. **Claim 1:**
- We need to show that \(1001\) is the order of \(10\) modulo \(2003\).
- The order of an integer \(a\) modulo \(n\) is the smallest positive integer \(k\) such that \(a^k \equiv 1 \pmod{n}\).
- To prove this, we need to show that \(10^{1001} \equiv 1 \pmod{2003}\) and that no smaller positive ... | 3 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $A_1A_2A_3A_4A_5A_6A_7A_8$ be convex 8-gon (no three diagonals concruent).
The intersection of arbitrary two diagonals will be called "button".Consider the convex quadrilaterals formed by four vertices of $A_1A_2A_3A_4A_5A_6A_7A_8$ and such convex quadrilaterals will be called "sub quadrilaterals".Find the smalle... | 1. **Understanding the Problem:**
We need to find the smallest number \( n \) such that we can color \( n \) "buttons" (intersections of diagonals) in a convex 8-gon \( A_1A_2A_3A_4A_5A_6A_7A_8 \) so that for any pair of vertices \( A_i \) and \( A_k \), the number of sub-quadrilaterals containing \( A_i \) and \( A... | 14 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let $S$ be a set of 2006 numbers. We call a subset $T$ of $S$ [i]naughty[/i] if for any two arbitrary numbers $u$, $v$ (not neccesary distinct) in $T$, $u+v$ is [i]not[/i] in $T$. Prove that
1) If $S=\{1,2,\ldots,2006\}$ every naughty subset of $S$ has at most 1003 elements;
2) If $S$ is a set of 2006 arbitrary p... | ### Part 1: Prove that if \( S = \{1, 2, \ldots, 2006\} \), every naughty subset of \( S \) has at most 1003 elements.
1. **Definition and Initial Observation**:
- A subset \( T \) of \( S \) is called *naughty* if for any two arbitrary numbers \( u, v \in T \), \( u + v \notin T \).
- Consider the set \( S = \{... | 669 | Combinatorics | proof | Yes | Yes | aops_forum | false |
Given a regular 2007-gon. Find the minimal number $k$ such that: Among every $k$ vertexes of the polygon, there always exists 4 vertexes forming a convex quadrilateral such that 3 sides of the quadrilateral are also sides of the polygon. | 1. **Define the vertices of the polygon**: Let the vertices of the regular 2007-gon be labeled as \( A_1, A_2, \ldots, A_{2007} \).
2. **Objective**: We need to find the minimal number \( k \) such that among any \( k \) vertices of the polygon, there always exist 4 vertices forming a convex quadrilateral with 3 sides... | 1506 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Determine the number of solutions of the simultaneous equations $ x^2 \plus{} y^3 \equal{} 29$ and $ \log_3 x \cdot \log_2 y \equal{} 1.$ | 1. **Rewrite the given equations:**
The given equations are:
\[
x^2 + y^3 = 29
\]
and
\[
\log_3 x \cdot \log_2 y = 1.
\]
2. **Express one variable in terms of the other using the logarithmic equation:**
From the second equation, we have:
\[
\log_3 x \cdot \log_2 y = 1.
\]
Let \( ... | 2 | Algebra | math-word-problem | Yes | Yes | aops_forum | false |
Let $ m \equal{} 2007^{2008}$, how many natural numbers n are there such that $ n < m$ and $ n(2n \plus{} 1)(5n \plus{} 2)$ is divisible by $ m$ (which means that $ m \mid n(2n \plus{} 1)(5n \plus{} 2)$) ? | 1. **Define \( m \) and factorize it:**
\[
m = 2007^{2008}
\]
We can factorize \( 2007 \) as:
\[
2007 = 3^2 \times 223
\]
Therefore,
\[
m = (3^2 \times 223)^{2008} = 3^{4016} \times 223^{2008}
\]
Let \( a = 3^{4016} \) and \( b = 223^{2008} \). Thus, \( m = ab \) and \( \gcd(a, b) = ... | 8 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $\{x\}$ be a sequence of positive reals $x_1, x_2, \ldots, x_n$, defined by: $x_1 = 1, x_2 = 9, x_3=9, x_4=1$. And for $n \geq 1$ we have:
\[x_{n+4} = \sqrt[4]{x_{n} \cdot x_{n+1} \cdot x_{n+2} \cdot x_{n+3}}.\]
Show that this sequence has a finite limit. Determine this limit. | To show that the sequence $\{x_n\}$ has a finite limit and to determine this limit, we will proceed as follows:
1. **Define the sequence and its properties:**
Given the sequence $\{x_n\}$ with initial values:
\[
x_1 = 1, \quad x_2 = 9, \quad x_3 = 9, \quad x_4 = 1
\]
and the recurrence relation for $n \... | 3 | Other | math-word-problem | Yes | Yes | aops_forum | false |
We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without one cell in corner a $P$-rectangle. We call a rectangle of size $2 \times 3$ (or $3 \times 2$) without two cells in opposite (under center of rectangle) corners a $S$-rectangle. Using some squares of size $2 \times 2$, some $P$-rectangles and some $S$-r... | 1. **Understanding the Problem:**
We need to tile a $1993 \times 2000$ rectangle using $2 \times 2$ squares, $P$-rectangles, and $S$-rectangles. We need to find the maximum value of $s$, where $s$ is the sum of the number of $2 \times 2$ squares and $S$-rectangles used.
2. **Area Calculation:**
The total area of... | 996500 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route, then t... | 1. **Define the problem in terms of graph theory:**
- We have 25 towns, which we can represent as vertices in a graph.
- We need to find the smallest \( k \) such that each vertex has exactly \( k \) edges (degree \( k \)).
- Additionally, for any two vertices that are not directly connected, there must be a v... | 6 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
A collection of $2000$ congruent circles is given on the plane such that no
two circles are tangent and each circle meets at least two other circles.
Let $N$ be the number of points that belong to at least two of the circles.
Find the smallest possible value of $N$. | 1. Let \( f(n) \) denote the smallest number of points that belong to at least two of the \( n \) circles. We aim to show that \( f(n) \geq 2(n - 2) + 1 \).
2. **Base Case: \( n = 3 \)**
- For \( n = 3 \), consider three circles arranged such that each circle intersects the other two circles at distinct points. Thi... | 3997 | Geometry | math-word-problem | Yes | Yes | aops_forum | false |
Let $A$ be the set of all permutations $a = (a_1, a_2, \ldots, a_{2003})$ of the 2003 first positive integers such that each permutation satisfies the condition: there is no proper subset $S$ of the set $\{1, 2, \ldots, 2003\}$ such that $\{a_k | k \in S\} = S.$
For each $a = (a_1, a_2, \ldots, a_{2003}) \in A$, let... | To solve the problem, we need to find the least value of \( d(a) \) for permutations \( a \) in the set \( A \), and then identify all permutations that achieve this least value.
### Part I: Finding the Least Value of \( d(a) \)
1. **Understanding the Condition**:
The condition states that there is no proper subse... | 4010006 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
Let us consider a set $S = \{ a_1 < a_2 < \ldots < a_{2004}\}$, satisfying the following properties: $f(a_i) < 2003$ and $f(a_i) = f(a_j) \quad \forall i, j$ from $\{1, 2,\ldots , 2004\}$, where $f(a_i)$ denotes number of elements which are relatively prime with $a_i$. Find the least positive integer $k$ for which in e... | 1. **Define the Set and Function:**
Let \( S = \{ a_1 < a_2 < \ldots < a_{2004} \} \) be a set of 2004 elements. Each element \( a_i \) satisfies \( f(a_i) < 2003 \) and \( f(a_i) = f(a_j) \) for all \( i, j \) in \( \{1, 2, \ldots, 2004\} \), where \( f(a_i) \) denotes the number of elements that are relatively pri... | 1003 | Number Theory | math-word-problem | Yes | Yes | aops_forum | false |
Let $p\in \mathbb P,p>3$. Calcute:
a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$
b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$ | ### Part (a)
Given \( p \in \mathbb{P} \) and \( p > 3 \), we need to calculate:
\[ S = \sum_{k=1}^{\frac{p-1}{2}} \left\lfloor \frac{2k^2}{p} \right\rfloor - 2 \cdot \left\lfloor \frac{k^2}{p} \right\rfloor \]
if \( p \equiv 1 \pmod{4} \).
1. **Identify Quadratic Residues:**
Let \( r_i \) for \( 1 \le i \le \fr... | 0 | Number Theory | other | Yes | Yes | aops_forum | false |
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the giv... | 1. **Define the problem and notation:**
Let \( A(n) \) be the minimum value of a good number if we have \( n \) points in space. We need to find \( A(2006) \).
2. **Initial observation:**
In the minimal case, there is obviously a segment on which the number \( 1 \) is put. Let its endpoints be \( X \) and \( Y \... | 11 | Combinatorics | math-word-problem | Yes | Yes | aops_forum | false |
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