question_answer_string stringlengths 2.71k 77.1k |
|---|
Question: Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$ .)
Answer: Okay, I need to find the arithmetic mean of all three-digit palindromes. Let me start by understanding what a three-digit palindrome is... |
Question: The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
Answer: Okay, so I need to solve this logarithmic equation: log base 2^x of 3^20 equals log base 2^(x+3) of 3^2020. Hmm, that l... |
Question: Find the number of positive integers $m$ for which there exist nonnegative integers $x_0$ , $x_1$ , $\dots$ , $x_{2011}$ such that \[m^{x_0} = \sum_{k = 1}^{2011} m^{x_k}.\]
Answer: Okay, so I need to figure out the number of positive integers m for which there exist nonnegative integers x₀, x₁, ..., x₂₀₁₁ su... |
Question: Let $\mathcal{S}$ be the set $\lbrace1,2,3,\ldots,10\rbrace$ Let $n$ be the number of sets of two non-empty disjoint subsets of $\mathcal{S}$ . (Disjoint sets are defined as sets that have no common elements.) Find the remainder obtained when $n$ is divided by $1000$ .
Answer: Okay, let me try to figure out t... |
Question: Let $u$ and $v$ be integers satisfying $0 < v < u$ . Let $A = (u,v)$ , let $B$ be the reflection of $A$ across the line $y = x$ , let $C$ be the reflection of $B$ across the y-axis, let $D$ be the reflection of $C$ across the x-axis, and let $E$ be the reflection of $D$ across the y-axis. The area of pentagon... |
Question: Let $S$ be the set of integers between 1 and $2^{40}$ whose binary expansions have exactly two 1's. If a number is chosen at random from $S,$ the probability that it is divisible by 9 is $p/q,$ where $p$ and $q$ are relatively prime positive integers. Find $p+q.$
Answer: Alright, so I need to solve this probl... |
Question: Alice knows that $3$ red cards and $3$ black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive ... |
Question: Circles $C_1$ and $C_2$ are externally tangent, and they are both internally tangent to circle $C_3.$ The radii of $C_1$ and $C_2$ are 4 and 10, respectively, and the centers of the three circles are all collinear. A chord of $C_3$ is also a common external tangent of $C_1$ and $C_2.$ Given that the length of... |
Question: Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is $1/29$ of the original integer.
Answer: Alright, I need to find the smallest positive integer where if I delete the leftmost digit, the remaining number is 1/29th of the original number. Let's try to think t... |
Question: In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$ . Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB$ . Find the greatest integer that does not exceed $1000r$ .
Answer: Okay, so I need to solv... |
Question: A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},... |
Question: Compute $\sqrt{(31)(30)(29)(28)+1}$ .
Answer: Okay, so I need to compute the square root of (31×30×29×28) plus 1. That looks like a big number, maybe there's a pattern or a shortcut to simplify it instead of just calculating each step normally. Hmm. Let me think...
First, let me write down the expression ins... |
Question: The numbers in the sequence $101$ , $104$ , $109$ , $116$ , $\ldots$ are of the form $a_n=100+n^2$ , where $n=1,2,3,\ldots$ . For each $n$ , let $d_n$ be the greatest common divisor of $a_n$ and $a_{n+1}$ . Find the maximum value of $d_n$ as $n$ ranges through the positive integers .
Answer: Okay, so I have t... |
Question: Let $P$ be a point chosen uniformly at random in the interior of the unit square with vertices at $(0,0), (1,0), (1,1)$ , and $(0,1)$ . The probability that the slope of the line determined by $P$ and the point $\left(\frac58, \frac38 \right)$ is greater than or equal to $\frac12$ can be written as $\frac{m}{... |
Question: For $n\ge1$ call a finite sequence $(a_1,a_2,\ldots,a_n)$ of positive integers progressive if $a_i<a_{i+1}$ and $a_i$ divides $a_{i+1}$ for $1\le i\le n-1$ . Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360.$
Answer: Okay, so I need to find the number of... |
Question: Define a sequence recursively by $t_1 = 20$ , $t_2 = 21$ , and \[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\] for all $n \ge 3$ . Then $t_{2020}$ can be written as $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .
Answer: Okay, so I need to find t_{2020} in this recursively define... |
Question: Find the least positive integer $n$ for which $2^n + 5^n - n$ is a multiple of $1000$ .
Answer: Okay, so I need to find the smallest positive integer n such that 2^n + 5^n - n is divisible by 1000. That means 2^n + 5^n - n ≡ 0 mod 1000. Hmm, 1000 is 8×125, which are 2^3 and 5^3. Maybe I can use the Chinese Re... |
Question: Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP > CP$ . Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP$ , respectively. Given that $AB = 12$ and $\angle O_1PO_2 = 120 ^{\circ}$ , then $AP = \sqrt{a} + \sqrt{b}$ , where $a$ and $b$ are positive integers. Find $a + b$ .
Answe... |
Question: For any positive integer $x_{}$ , let $S(x)$ be the sum of the digits of $x_{}$ , and let $T(x)$ be $|S(x+2)-S(x)|.$ For example, $T(199)=|S(201)-S(199)|=|3-19|=16.$ How many values of $T(x)$ do not exceed 1999?
Answer: Okay, let's try to figure out this problem. So, the question is about functions S(x) and T... |
Question: Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$ , $BC = 14$ , and $AD = 2\sqrt{65}$ . Assume that the diagonals of $ABCD$ intersect at point $P$ , and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$ . Find the area of quadrilateral $ABCD... |
Question: In a certain circle, the chord of a $d$ -degree arc is 22 centimeters long, and the chord of a $2d$ -degree arc is 20 centimeters longer than the chord of a $3d$ -degree arc, where $d < 120.$ The length of the chord of a $3d$ -degree arc is $- m + \sqrt {n}$ centimeters, where $m$ and $n$ are positive integer... |
Question: Point $B$ lies on line segment $\overline{AC}$ with $AB=16$ and $BC=4$ . Points $D$ and $E$ lie on the same side of line $AC$ forming equilateral triangles $\triangle ABD$ and $\triangle BCE$ . Let $M$ be the midpoint of $\overline{AE}$ , and $N$ be the midpoint of $\overline{CD}$ . The area of $\triangle BMN... |
Question: Define a $\text{good word}$ as a sequence of letters that consists only of the letters $A$ , $B$ , and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$ , $B$ is never immediately followed by $C$ , and $C$ is never immediately followed by $A$ . ... |
Question: A circle with diameter $\overline{PQ}\,$ of length 10 is internally tangent at $P^{}_{}$ to a circle of radius 20. Square $ABCD\,$ is constructed with $A\,$ and $B\,$ on the larger circle, $\overline{CD}\,$ tangent at $Q\,$ to the smaller circle, and the smaller circle outside $ABCD\,$ . The length of $\overl... |
Question: The perimeter of triangle $APM$ is $152$ , and the angle $PAM$ is a right angle. A circle of radius $19$ with center $O$ on $\overline{AP}$ is drawn so that it is tangent to $\overline{AM}$ and $\overline{PM}$ . Given that $OP=m/n$ where $m$ and $n$ are relatively prime positive integers, find $m+n$ .
Answer:... |
Question: Point $P_{}$ is located inside triangle $ABC$ so that angles $PAB, PBC,$ and $PCA$ are all congruent. The sides of the triangle have lengths $AB=13, BC=14,$ and $CA=15,$ and the tangent of angle $PAB$ is $m/n,$ where $m_{}$ and $n_{}$ are relatively prime positive integers. Find $m+n.$
Answer: Alright, so I... |
Question: Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$ , $b$ , and $c$ , and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$ , $b+c$ , and $c+a$ . Find $t$ .
Answer: Okay, so I need to find the value of \( t \) in the cubic equation \( x^3 + rx^2 + sx + t = 0 \), whose roots are \( a+b \), \( b+c \), and \( c+a \... |
Question: Let $a_{10} = 10$ , and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$ . Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$ .
Answer: Okay, so I need to find the least positive integer n greater than 10 such that a_n is divisible by 99. The sequence is defined as a_{10} = 1... |
Question: Let $S$ be the set of all polynomials of the form $z^3 + az^2 + bz + c$ , where $a$ , $b$ , and $c$ are integers. Find the number of polynomials in $S$ such that each of its roots $z$ satisfies either $|z| = 20$ or $|z| = 13$ .
Answer: Okay, so I need to find the number of polynomials in the set S, where each... |
Question: Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is $\dfrac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .
Answer: Okay, so I need to find the probability ... |
Question: When each of $702$ , $787$ , and $855$ is divided by the positive integer $m$ , the remainder is always the positive integer $r$ . When each of $412$ , $722$ , and $815$ is divided by the positive integer $n$ , the remainder is always the positive integer $s \neq r$ . Find $m+n+r+s$ .
Answer: Okay, let's try ... |
Question: For distinct complex numbers $z_1,z_2,\dots,z_{673}$ , the polynomial \[(x-z_1)^3(x-z_2)^3 \cdots (x-z_{673})^3\] can be expressed as $x^{2019} + 20x^{2018} + 19x^{2017}+g(x)$ , where $g(x)$ is a polynomial with complex coefficients and with degree at most $2016$ . The value of \[\left| \sum_{1 \le j <k \le 6... |
Question: The numbers $1447$ , $1005$ and $1231$ have something in common: each is a $4$ -digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?
Answer: Okay, let's see. The problem is asking about how many 4-digit numbers start with 1 and have exactly two identical digi... |
Question: Find the eighth term of the sequence $1440,$ $1716,$ $1848,\ldots,$ whose terms are formed by multiplying the corresponding terms of two arithmetic sequences.
Answer: Okay, so I need to find the eighth term of this sequence: 1440, 1716, 1848,… Each term in this sequence is formed by multiplying the correspond... |
Question: Find $3x^2 y^2$ if $x$ and $y$ are integers such that $y^2 + 3x^2 y^2 = 30x^2 + 517$ .
Answer: Okay, so I have this problem here: Find \(3x^2 y^2\) if \(x\) and \(y\) are integers satisfying the equation \(y^2 + 3x^2 y^2 = 30x^2 + 517\). Hmm, let me see how to approach this.
First, I need to understand the ... |
Question: Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N$ . For example, $51$ is a 7-10 double because its base-7 representation is $102$ . What is the largest 7-10 double?
Answer: Okay, so I need to find the largest positive... |
Question: Find the sum of the roots, real and non-real, of the equation $x^{2001}+\left(\frac 12-x\right)^{2001}=0$ , given that there are no multiple roots.
Answer: Okay, so I need to find the sum of all the roots, real and non-real, of the equation \( x^{2001} + \left( \frac{1}{2} - x \right)^{2001} = 0 \). The probl... |
Question: A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_n$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_n\le.4$ for all $n$ such that $1\le n\le9$ is given to be $p^aq^br/\left(s^c\right)$ ... |
Question: Triangles $\triangle ABC$ and $\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$ , $B(0,12)$ , $C(16,0)$ , $A'(24,18)$ , $B'(36,18)$ , $C'(24,2)$ . A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$ , will transform $\triangle ABC$ to $\triangle A'B'C'$ . Find $m+x... |
Question: There is a set of 1000 switches, each of which has four positions, called $A, B, C$ , and $D$ . When the position of any switch changes, it is only from $A$ to $B$ , from $B$ to $C$ , from $C$ to $D$ , or from $D$ to $A$ . Initially each switch is in position $A$ . The switches are labeled with the 1000 di... |
Question: A real number $a$ is chosen randomly and uniformly from the interval $[-20, 18]$ . The probability that the roots of the polynomial \[x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2\] are all real can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$ .
A... |
Question: How many of the first $1000$ positive integers can be expressed in the form $\lfloor 2x \rfloor + \lfloor 4x \rfloor + \lfloor 6x \rfloor + \lfloor 8x \rfloor$ , where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$ ?
Answer: Okay, so I have this problem wh... |
Question: The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\frac{1}{2}\left(\sqrt... |
Question: There exists a unique positive integer $a$ for which the sum \[U=\sum_{n=1}^{2023}\left\lfloor\dfrac{n^{2}-na}{5}\right\rfloor\] is an integer strictly between $-1000$ and $1000$ . For that unique $a$ , find $a+U$ . (Note that $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$ .... |
Question: If $a<b<c<d<e^{}_{}$ are consecutive positive integers such that $b+c+d$ is a perfect square and $a+b+c+d+e^{}_{}$ is a perfect cube, what is the smallest possible value of $c^{}_{}$ ?
Answer: Okay, so I need to find five consecutive positive integers a, b, c, d, e where a < b < c < d < e. The conditions are... |
Question: The graph of $y^2 + 2xy + 40|x|= 400$ partitions the plane into several regions. What is the area of the bounded region?
Answer: Okay, so the problem is asking for the area of the bounded region formed by the graph of the equation y² + 2xy + 40|x| = 400. Hmm. Let me start by trying to understand this equatio... |
Question: One of Euler's conjectures was disproved in the 1960s by three American mathematicians when they showed there was a positive integer such that \[133^5+110^5+84^5+27^5=n^{5}.\] Find the value of $n$ .
Answer: Okay, so I need to find the value of n such that 133^5 + 110^5 + 84^5 + 27^5 equals n^5. This was a co... |
Question: For any finite set $X,$ let $|X|$ denote the number of elements in $X.$ Define \[S_n = \sum |A \cap B|,\] where the sum is taken over all ordered pairs $(A,B)$ such that $A$ and $B$ are subsets of $\{1,2,3,\ldots,n\}$ with $|A|=|B|.$ For example, $S_2 = 4$ because the sum is taken over the pairs of subsets \[... |
Question: The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C^{}_{}$ ?
Answer: Okay, so I have this problem where I need to find the n... |
Question: A token starts at the point $(0,0)$ of an $xy$ -coordinate grid and then makes a sequence of six moves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each move is selected randomly from the four possible directions and independently of the other moves. The probability the token en... |
Question: In rectangle $ABCD$ , $AB=100$ . Let $E$ be the midpoint of $\overline{AD}$ . Given that line $AC$ and line $BE$ are perpendicular, find the greatest integer less than $AD$ .
Answer: Alright, so I have this geometry problem here. Let me try to parse it properly. First, we're dealing with a rectangle ABCD, whe... |
Question: A regular icosahedron is a $20$ -faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizo... |
Question: Given eight distinguishable rings, let $n$ be the number of possible five-ring arrangements on the four fingers (not the thumb) of one hand. The order of rings on each finger is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of $n$ .
Answer: Alright, ... |
Question: A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of t... |
Question: Let $p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $p_{}$ can be written in the form $m/n$ where $m_{}$ and $n_{}$ are relatively prime positive integers, find $m+n$ .
Answer: Okay, so I... |
Question: Circles $\omega_1$ and $\omega_2$ with radii $961$ and $625$ , respectively, intersect at distinct points $A$ and $B$ . A third circle $\omega$ is externally tangent to both $\omega_1$ and $\omega_2$ . Suppose line $AB$ intersects $\omega$ at two points $P$ and $Q$ such that the measure of minor arc $\widehat... |
Question: Let $P_0(x) = x^3 + 313x^2 - 77x - 8\,$ . For integers $n \ge 1\,$ , define $P_n(x) = P_{n - 1}(x - n)\,$ . What is the coefficient of $x\,$ in $P_{20}(x)\,$ ?
Answer: Okay, let's see. The problem is about polynomials defined recursively, and we need to find the coefficient of x in P_{20}(x). Hmm. Let's sta... |
Question: A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let $R$ be the region outside the hexagon, and let $S = \left\lbrace\frac{1}{z}|z \in R\right\rbrace$ . Then the area of $S$ has the form $a\pi + \s... |
Question: Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$ .
Answer: Okay, so I need to find the number of ordered pairs of positive integers (m, n) such that m²n = 20²⁰. Hmm, let me think about how to approach this.
First, I remember that problems like this often involve pr... |
Question: Let $A$ be an acute angle such that $\tan A = 2 \cos A.$ Find the number of positive integers $n$ less than or equal to $1000$ such that $\sec^n A + \tan^n A$ is a positive integer whose units digit is $9.$
Answer: Alright, so I have this problem here: Let A be an acute angle such that tan A = 2 cos A. I need... |
Question: A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of a shadow, which does not include the area beneath the cube is 48 squa... |
Question: Let $[r,s]$ denote the least common multiple of positive integers $r$ and $s$ . Find the number of ordered triples $(a,b,c)$ of positive integers for which $[a,b] = 1000$ , $[b,c] = 2000$ , and $[c,a] = 2000$ .
Answer: Alright, so I need to solve this problem where I have to find the number of ordered triple... |
Question: In a rectangular array of points, with 5 rows and $N$ columns, the points are numbered consecutively from left to right beginning with the top row. Thus the top row is numbered 1 through $N,$ the second row is numbered $N + 1$ through $2N,$ and so forth. Five points, $P_1, P_2, P_3, P_4,$ and $P_5,$ are sel... |
Question: Triangle $ABC$ has right angle at $B$ , and contains a point $P$ for which $PA = 10$ , $PB = 6$ , and $\angle APB = \angle BPC = \angle CPA$ . Find $PC$ . AIME 1987 Problem 9.png
Answer: Okay, so I need to solve this geometry problem from the AIME. Let me read it again carefully to make sure I understand all... |
Question: The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes doe... |
Question: Consider the region $A^{}_{}$ in the complex plane that consists of all points $z^{}_{}$ such that both $\frac{z^{}_{}}{40}$ and $\frac{40^{}_{}}{\overline{z}}$ have real and imaginary parts between $0^{}_{}$ and $1^{}_{}$ , inclusive. What is the integer that is nearest the area of $A^{}_{}$ ?
Answer: Okay, ... |
Question: Consider the integer \[N = 9 + 99 + 999 + 9999 + \cdots + \underbrace{99\ldots 99}_\text{321 digits}.\] Find the sum of the digits of $N$ .
Answer: Okay, let's try to figure out this problem step by step. So, we need to find the sum of the digits of the number N, which is the sum of a series of numbers: 9 + 9... |
Question: A sequence of positive integers with $a_1=1$ and $a_9+a_{10}=646$ is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all $n\ge1,$ the terms $a_{2n-1}, a_{2n}, a_{2n+1}$ are in geometric progression, and ... |
Question: How many different $4\times 4$ arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0?
Answer: Okay, so I have this problem: I need to find the number of different 4x4 arrays filled with 1's and -1's such that the... |
Question: Let $S$ be the number of ordered pairs of integers $(a,b)$ with $1 \leq a \leq 100$ and $b \geq 0$ such that the polynomial $x^2+ax+b$ can be factored into the product of two (not necessarily distinct) linear factors with integer coefficients. Find the remainder when $S$ is divided by $1000$ .
Answer: Alright... |
Question: Let $S$ be the increasing sequence of positive integers whose binary representation has exactly $8$ ones. Let $N$ be the 1000th number in $S$ . Find the remainder when $N$ is divided by $1000$ .
Answer: Okay, I need to solve this problem where we're looking for the 1000th number in the sequence S, which consi... |
Question: For $\pi \le \theta < 2\pi$ , let \[P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots\] and \[Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\... |
Question: How many positive perfect squares less than $10^6$ are multiples of 24?
Answer: Okay, so the problem is: How many positive perfect squares less than 10^6 are multiples of 24? Hmm, let me think. Alright, first, I need to understand what exactly is being asked here. We need to find all perfect squares that are ... |
Question: Find the number of positive integers that are divisors of at least one of $10^{10},15^7,18^{11}.$
Answer: Okay, so I need to find the number of positive integers that are divisors of at least one of these three numbers: \(10^{10}\), \(15^7\), and \(18^{11}\). Hmm, let's see. This sounds like a problem involvi... |
Question: A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees t... |
Question: A game uses a deck of $n$ different cards, where $n$ is an integer and $n \geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$
Answer: Okay, so I need to find the number of cards, n, in a deck where the n... |
Question: Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\times$ $27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH}$ , and partition ... |
Question: A particle is located on the coordinate plane at $(5,0)$ . Define a move for the particle as a counterclockwise rotation of $\pi/4$ radians about the origin followed by a translation of $10$ units in the positive $x$ -direction. Given that the particle's position after $150$ moves is $(p,q)$ , find the greate... |
Question: For how many positive integers $n$ less than or equal to 1000 is $(\sin t + i \cos t)^n = \sin nt + i \cos nt$ true for all real $t$ ?
Answer: Okay, so I need to figure out for how many positive integers n ≤ 1000 the equation (sin t + i cos t)^n = sin nt + i cos nt holds true for all real t. Hmm, complex numb... |
Question: Let $P(x) = x^2 - 3x - 7$ , and let $Q(x)$ and $R(x)$ be two quadratic polynomials also with the coefficient of $x^2$ equal to $1$ . David computes each of the three sums $P + Q$ , $P + R$ , and $Q + R$ and is surprised to find that each pair of these sums has a common root, and these three common roots are d... |
Question: Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms.
Answer: Okay, so I need to find the sum of all positive rational numbers less than 10 that have a denominator of 30 when written in lowest terms. Hmm, let's break this down step by st... |
Question: The number $n$ can be written in base $14$ as $\underline{a}\text{ }\underline{b}\text{ }\underline{c}$ , can be written in base $15$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{b}$ , and can be written in base $6$ as $\underline{a}\text{ }\underline{c}\text{ }\underline{a}\text{ }\underline{c}\t... |
Question: Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8$ , and the second term of both series can be written in the form $\frac{\sqrt{m}-n}p$ , where $m$ , $n$ , and $p$ are positive integers and $m$ is not divisible by th... |
Question: What is that largest positive integer $n$ for which $n^3+100$ is divisible by $n+10$ ?
Answer: Okay, so I have this problem here: What is the largest positive integer \( n \) for which \( n^3 + 100 \) is divisible by \( n + 10 \)? Hmm, divisibility... Let me think. If \( n + 10 \) divides \( n^3 + 100 \), tha... |
Question: Dave arrives at an airport which has twelve gates arranged in a straight line with exactly $100$ feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Da... |
Question: Let $\overline{MN}$ be a diameter of a circle with diameter $1$ . Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB=\dfrac 35$ . Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment... |
Question: Let $\triangle ABC$ be a right triangle with right angle at $C.$ Let $D$ and $E$ be points on $\overline{AB}$ with $D$ between $A$ and $E$ such that $\overline{CD}$ and $\overline{CE}$ trisect $\angle C.$ If $\frac{DE}{BE} = \frac{8}{15},$ then $\tan B$ can be written as $\frac{m \sqrt{p}}{n},$ where $m$ and ... |
Question: Let $R$ be the set of all possible remainders when a number of the form $2^n$ , $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$ . Find the remainder when $S$ is divided by 1000.
Answer: Okay, so I need to find the set R which consists of all possible remainders when a ... |
Question: Euler's formula states that for a convex polyhedron with $V\,$ vertices, $E\,$ edges, and $F\,$ faces, $V-E+F=2\,$ . A particular convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. At each of its $V\,$ vertices, $T\,$ triangular faces and $P^{}_{}$ pentagonal faces meet. What is... |
Question: A small square is constructed inside a square of area $1$ by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices, as shown in the figure. Find the value of $n$ if the the area of the small square is exactly $\fr... |
Question: Let $a = \pi/2008$ . Find the smallest positive integer $n$ such that \[2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]\] is an integer.
Answer: Alright, let's try to figure out this problem. So, we need to find the smallest positive integer n such that the expression:
2... |
Question: The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\mbox{th}}$ term of this sequence.
Answer: Okay, so I need to find the 100th term in this sequence: 1, 3, 4, 9, 10, 12, 13,… which includes all positiv... |
Question: For certain real values of $a, b, c,$ and $d_{},$ the equation $x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $13+i$ and the sum of the other two roots is $3+4i,$ where $i=\sqrt{-1}.$ Find $b.$
Answer: Okay, so I need to find the value of \( b \) in the quartic equation ... |
Question: The positive integers $N$ and $N^2$ both end in the same sequence of four digits $abcd$ when written in base 10, where digit $a$ is not zero. Find the three-digit number $abc$ .
Answer: Okay, so I need to find a positive integer N such that both N and N² end with the same four digits abcd in base 10, and the ... |
Question: Find the number of ordered pairs $(x,y)$ of positive integers that satisfy $x \le 2y \le 60$ and $y \le 2x \le 60$ .
Answer: Okay, so I need to find the number of ordered pairs (x, y) of positive integers that satisfy two conditions: x ≤ 2y ≤ 60 and y ≤ 2x ≤ 60. Hmm, both conditions involve inequalities with ... |
Question: Let $x$ , $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\log_xw=24$ , $\log_y w = 40$ and $\log_{xyz}w=12$ . Find $\log_zw$ .
Answer: Okay, so I have this problem here about logarithms. Let me try to parse it out step by step.
First, the problem says that x, y, and z all exceed 1, a... |
Question: Let $m$ be a positive integer, and let $a_0, a_1,\ldots,a_m$ be a sequence of reals such that $a_0 = 37, a_1 = 72, a_m = 0,$ and $a_{k+1} = a_{k-1} - \frac 3{a_k}$ for $k = 1,2,\ldots, m-1.$ Find $m.$
Answer: Okay, so I need to find the positive integer m such that we have this sequence starting with a₀ = 37,... |
Question: The equation $2000x^6+100x^5+10x^3+x-2=0$ has exactly two real roots, one of which is $\frac{m+\sqrt{n}}r$ , where $m$ , $n$ and $r$ are integers, $m$ and $r$ are relatively prime, and $r>0$ . Find $m+n+r$ .
Answer: Okay, so I have this equation: 2000x⁶ + 100x⁵ + 10x³ + x - 2 = 0. It says it has exactly two r... |
Question: Let $P(x)$ be a polynomial with integer coefficients that satisfies $P(17)=10$ and $P(24)=17.$ Given that $P(n)=n+3$ has two distinct integer solutions $n_1$ and $n_2,$ find the product $n_1\cdot n_2.$
Answer: Okay, let's see. I need to find the product of two distinct integer solutions, n₁ and n₂, to the equ... |
Question: A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.