prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A... | 204 |
Find the two $10$-digit numbers which become nine times as large if the order of the digits is reversed. Please provide the sum of these two numbers. | 1287128711 |
In a language college, $72\%$ students can speak Chinese, $65\%$ students can speak English, and $10\%$ students can speak neither Chinese nor English. Find the percentage of students who can speak both Chinese and English. Please provide the sum of the original percentage and 100. | 147 |
Replace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetition: $a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}$ Find the difference between the maximum value and the minimum value of the expression. The original answer is in \frac{m}{n} format, please give th... | 67 |
Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a $2$-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$. The original answer is ... | 3 |
Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Please provide the value of $p + q + pq$. | 11 |
Find all triples of positive integers $(x,y,z)$ that satisfy the equation $$\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*}$$ Please provide the sum of all elements in one such triple. | 6 |
Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes, where each domino covers exactly two neighboring squares and the dominoes don’t overlap: $C$ th... | 11 |
A random number selector can only select one of the nine integers $1, 2, ..., 9$, and it makes these selections with equal probability. Determine the probability that after $n$ selections ($n>1$), the product of the $n$ numbers selected will be divisible by $10$. Please calculate the probability when n = 2 and give the... | 89 |
Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? The original answer is in t... | 4 |
On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. The original answer is in \frac{m}{n} format, please give the value of ... | 3 |
Determine the smallest number $M$ such that the inequality $$\begin{vmatrix}ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\end{vmatrix}\le M\bigl(a^2+b^2+c^2\bigr)^2$$ holds for all real numbers $a,b,c.$ The original answer is in the format \frac{m}{n}\sqrt{2}, please give the value of m + n. | 11 |
Determine all pairs $(x,y)$ of integers satisfying the equation $1+2^x+2^{2x+1}=y^2$. Please provide the sum of all individual elements from each solution pair. | 0 |
Let $n>1$ be an integer. In the space, consider the set $S=\{(x,y,z)\mid x,y,z\in\{0,1,\cdots,n\}, x+y+z>0\}.$ Find the smallest number of planes that jointly contain all $(n+1)^3-1$ points of $S$ but none of them passes through the origin. Provide the value of this expression when n = 2. | 6 |
Find all pairs $(k,n)$ of positive integers for which $7^k-3^n$ divides $k^4+n^2$. Please provide the sum of all values of $k$ and $n$ in the pairs. | 4 |
Find all functions $f\colon(0,\infty)\to(0,\infty)$ such that $\frac{f(p)^2+f(q)^2}{f(r^2)+f(s^2)}=\frac{p^2+q^2}{r^2+s^2}$ for all $p,q, r, s > 0$ with $pq = rs$. There are multiple solutions for $f(x)$, represented by their respective function numbers. Please provide the sum of these numbers. | 0 |
Let $n$ and $k$ be fixed positive integers of the same parity, $k \ge n$. We are given $2n$ lamps numbered $1$ through $2n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off). Let $N$ be the... | 1 |
Find all positive integers n such that there exists a sequence of positive integers $a_{1},a_{2},\\ldots,a_{n}$ satisfying $a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$ for every k with $2\leq k\leq n-1.$ When n = 4, a sequence exists with specific values for $a_1$, $a_2$, $a_3$, and $a_4$. Please provide the sum of these valu... | 3 |
Determine all functions f : $\mathbb{R}\to\mathbb{R}$ such that the equality $f([x]y)=f(x)[f(y)].\quad(1)$ holds for all $x,y\in\mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. Please provide the sum of all possible integer values of the constant C. | 1 |
Denote by $\mathbb{Q}^{+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^{+}\to\mathbb{Q}^{+}$ which satisfy the following equation for all $x,y\in\mathbb{Q}^{+}:$ $f\big(f(x)^2y\big)=x^3f(xy).\quad(1)$ The original answer is in \frac{m}{n} format for a specific x, please provide the v... | 2 |
On some planet, there are $2^N$ countries ($N \ge 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1$, each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \t... | 17 |
Find all pairs $( m, n)$ of nonnegative integers for which $$m^2+2\cdot3^n=m\big(2^{n+1}-1\big).\quad(1)$$ Please provide the sum of all values of $m + n$ for each pair $(m, n)$. | 21 |
For any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers. | 4 |
Determine all sequences $(x_1,x_2,\cdots,x_{2011})$ of positive integers such that for every positive integer $n$ there is an integer $a$ with $$x_1^n+2x_2^n+\cdots+2011x_{2011}^n=a^{n+1}+1.$$ Please provide the value of $x_1 + x_2 + \cdots + x_{2011}$ for the sequence that satisfies the condition. | 2011 |
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1$ for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\cdots f}_k(n)\cdots)).$ Provide the value of $f(1) + g(1)$. | 1 |
Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself such that for all integers $x$ and $y$, the difference $f(x) − f(y)$ divides $x^n − y^n$. The original answer involves parameters \varepsilon, d, and c. Provide the sum of all distinct possible values of \varepsilon + d + c... | 0 |
For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences $\begin{aligned}t(n+a)-t(n), t(n+a+1)-t(n+1),\cdots,t(n+2a-1)-t(n+a-1)\end{aligned}$ are divisible by $4$. Please provide the sum o... | 9 |
Find all the functions $f: \mathbb{Z} \to \mathbb{Z}$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for all integers $a,b,c$ satisfying $a+b+c=0.$ Please provide the sum of all elements within each tuple of possible values for f(a), f(b), and f(c). | 0 |
In a $999 × 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1, C_2, C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1$ and $C_3$ white and $C_2$ red. Find the maximum value T can attain. The original answer is in \fra... | 3980000000000 |
Call admissible a set $A$ of integers that has the following property: If $x,y\in A$ (possibly $x=y)$ then $x^2+kxy+y^2\in A$ for every integer $k.$ Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. Please provide the sum of the pair ... | 1 |
Find all triples $(x, y, z)$ of positive integers such that $x ≤ y ≤ z$ and $x^3(y^3 + z^3) = 2012(xyz + 2)$. Please provide the sum of the values of x, y, and z for all such triples. | 505 |
Find all $n\in\mathbb{N}$ for which there exist nonnegative integers $a_1,a_2,\cdots,a_n$ such that $\frac{1}{2^{a_{1}}}+\frac{1}{2^{a_{2}}}+\cdots+\frac{1}{2^{a_{n}}}=\frac{1}{3^{a_{1}}}+\frac{2}{3^{a_{2}}}+\cdots+\frac{n}{3^{a_{n}}}=1$. Please provide the sum of all such possible values of $n$. | 37 |
Find the least positive integer $n$ for which there exists a set $\{s_1,s_2,\ldots,s_n\}$ consisting of $n$ distinct positive integers such that $\left(1-\dfrac{1}{s_1}\right)\left(1-\dfrac{1}{s_2}\right)\dots\left(1-\dfrac{1}{s_n}\right)=\dfrac{51}{2010}.$ Please provide the value of $n$. | 39 |
In the plane, $2013$ red points and $2014$ blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the... | 2013 |
$A \pm1-sequence$ is a sequence of 2022 numbers $a_{1},\ldots,a_{2022}$, each equal to either +1 or -1. Determine the largest $C$ so that, for any $\pm1$ sequence, there exists an integer $k$ and indices $1\leqslant t_{1}<\ldots<t_{k}\leqslant2022$ so that $t_{i+1}-t_{i}\leqslant2$ for all $i$, and $$\left|\sum_{i=1}^k... | 506 |
Solve in integers the equation $xy+3x-5y=-3.$ Please provide the value of the number of solutions multiplied by 2. | 24 |
Find integers $x, y, z, u$ such that $$x^2 + y^2 + z^2 + u^2 = 2xyzu.$$ Provide the sum of all integers $x, y, z, u$ that satisfy the equation. | 0 |
Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $f\left(f(m)+n\right)+f(m)=f(n)+f(3m)+2014\quad(1)$ for all integers $m$ and $n$. The original answer is in the form of a function $n\longmapsto2n+1007$. Please provide the value of the function when $n = 1000$. Directly give the resulting integer value. | 3007 |
Consider all polynomials $P(x)$ with real coefficients that have the following property: for any two real numbers $x$ and $y$ one has $|y^2-P(x)|\leqslant2|x|$ if and only if $|x^2-P(y)|\leqslant2|y|$. Determines the positive integer value in the set of possible values for P(0). | 1 |
Find all functions $f: \mathbb{Z} \to \mathbb{Z}$ such that $$n^2+4f(n)=f(f(n))^2\quad(1)$$ for all $n \in \mathbb{Z}.$ Provide the number of distinct functional forms for $f(n)$. | 3 |
Let $n\geqslant2$ be an integer. Consider an $n \times n$ chessboard divided into $n^ 2$ unit squares. We call a configuration of n rooks on this board happy if every row and every column contains exactly one rook. Find the greatest positive integer $k$ such that for every happy configuration of rooks, we can find a $k... | 3 |
$A$ card deck consists of $1024$ cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one o... | 1023 |
Let $n\geqslant 2$ be an integer, and let $A_n$ be the set $$A_n=\{2^n-2^k\mid k\in\mathbb{Z},0\leqslant k<n\}.$$ Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_n.$ Please provide the value when n = 9. | 3585 |
Find all triples $( p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^p-1+y$ and $x+y^p-1$ are both powers of $p.$ Provide the sum of the elements in all triples when k=1. | 24 |
Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ with the property that $$f\big(x-f(y)\big)=f\big(f(x)\big)-f(y)-1\quad(1)$$ holds for all $x,y\in\mathbb{Z}.$ Please provide several such functions. | 2 |
Let $n$ be a fixed positive integer. Find the maximum possible value of $$\sum\limits_{1\leqslant r<s\leqslant2n}(s-r-n)x_rx_s,$$ where $-1\leqslant x_{i}\leqslant1$ for all $i=1,2,\cdots,2n.$ Please provide the value of the expression when $n$ is a specific integer, such as $n=5$. The answer should be given in integer... | 20 |
Evaluate the expression: $\tan \left(\arctan 2+\arctan \frac{2}{2^{2}}+\cdots+\arctan \frac{2}{12^{2}}\right)$. The original answer is in the form $-\frac{m}{n}$. Please provide the value of $m + n$. | 31 |
Given the functions $$ f(x)=\frac{x}{\ln x}, $$ and $g(x)=k(x-1)$ where $k \in \mathbf{R}$. (1) Prove that for any $k \in \mathbf{R}$, the line $y=g(x)$ is not a tangent to the curve $y=f(x)$; (2) If there exists $x \in[\mathrm{e}, \mathrm{e}^{2}]$, such that $$ f(x) \leqslant g(x)+\frac{1}{2}, $$ find the range of val... | 1 |
Let $a_{1}, a_{2} \ldots, a_{51}$ be non-zero elements of a field. We simultaneously replace each element with the sum of the 50 remaining ones. In this way we get a sequence $b_{1} \ldots, b_{51}$. If this new sequence is a permutation of the original one, what can be the characteristic of the field? Please provide th... | 9 |
Let $A$ be a real $4 \times 2$ matrix and $B$ be a real $2 \times 4$ matrix such that $$ A B=\left(\begin{array}{cccc} 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1 \\ -1 & 0 & 1 & 0 \\ 0 & -1 & 0 & 1 \end{array}\right) $$ Find $B A$. Provide the sum of all elements in the resulting matrix. | 4 |
Find all polynomials $P(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots+a_{1} x+a_{0}\left(a_{n} \neq 0\right)$ satisfying the following two conditions: (i) $(a_{0}, a_{1}, \ldots, a_{n})$ is a permutation of the numbers $(0,1, \ldots, n)$ and (ii) all roots of $P(x)$ are rational numbers. Please provide the sum of all coefficie... | 17 |
Find all $r>0$ such that whenever $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a differentiable function such that $|\operatorname{grad} f(0,0)|=1$ and $|\operatorname{grad} f(u)-\operatorname{grad} f(v)| \leq|u-v|$ for all $u, v \in \mathbb{R}^{2}$, then the maximum of $f$ on the disk $\left\{u \in \mathbb{R}^{2}:|u|... | 3 |
Find the number of positive integers $x$ satisfying the following two conditions: 1. $x<10^{2006}$; 2. $x^{2}-x$ is divisible by $10^{2006}$. Please provide the sum of all such positive integers. | 12 |
Call a polynomial $P\left(x_{1}, \ldots, x_{k}\right)$ good if there exist $2 \times 2$ real matrices $A_{1}, \ldots, A_{k}$ such that $$ P\left(x_{1}, \ldots, x_{k}\right)=\operatorname{det}\left(\sum_{i=1}^{k} x_{i} A_{i}\right) $$ Find all values of $k$ for which all homogeneous polynomials with $k$ variables of deg... | 3 |
How many nonzero coefficients can a polynomial $P(z)$ have if its coefficients are integers and $|P(z)| \leq 2$ for any complex number $z$ of unit length? Please provide the sum of all possible values. | 3 |
Determine the value of $$ \sum_{n=1}^{\infty} \ln \left(1+\frac{1}{n}\right) \cdot \ln \left(1+\frac{1}{2 n}\right) \cdot \ln \left(1+\frac{1}{2 n+1}\right) $$ The original answer is in the form of \frac{m}{n} \ln^3(2). Please provide the value of m + n. | 3 |
Consider the following sequence \(\left(a_{n}\right)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1, \ldots)\). Find all pairs \((\alpha, \beta)\) of positive real numbers such that \(\lim _{n \rightarrow \infty} \frac{\sum_{k=1}^{n} a_{k}}{n^{\alpha}}=\beta\). Please provide the integer value of \(\alpha + \beta\). | 6 |
Determine all pairs $(a, b)$ of real numbers for which there exists a unique symmetric $2 \times 2$ matrix $M$ with real entries satisfying $\operatorname{trace}(M)=a$ and $\operatorname{det}(M)=b$. Please provide the value of $a + b$ for the unique pair $(a, b)$. | 4 |
Let $\left\{b_{n}\right\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2 \sqrt{1+\sqrt{b_{n-1}}}$. Calculate the sum of $b_{n} 2^{n}$ from $n=1$ to $\infty$. The original answer includes a constant term and a logarithmic term. Please provide the integer part of the... | 2 |
An $n \times n$ complex matrix $A$ is called $t$-normal if $A A^{t}=A^{t} A$ where $A^{t}$ is the transpose of $A$. For each $n$, determine the maximum dimension of a linear space of complex $n \times n$ matrices consisting of t-normal matrices. Please provide the value of the dimension when $n = 10$. The answer should... | 56 |
Determine whether there exist an odd positive integer $n$ and $n \times n$ matrices $A$ and $B$ with integer entries, that satisfy the following conditions: (1) $\operatorname{det}(B)=1$; (2) $A B=B A$; (3) $A^{4}+4 A^{2} B^{2}+16 B^{4}=2019 I$. Give the sum of all possible values of n. (Here $I$ denotes the $n \times ... | 0 |
Let $n, k \geq 3$ be integers, and let $S$ be a circle. Let $n$ blue points and $k$ red points be chosen uniformly and independently at random on the circle $S$. Denote by $F$ the intersection of the convex hull of the red points and the convex hull of the blue points. Let $m$ be the number of vertices of the convex po... | 2 |
Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\ldots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_{2}$ upon division by $p$? Please provide the sum of the possible remainders. | 5 |
We say that a real number $V$ is good if there exist two closed convex subsets $X, Y$ of the unit cube in $\mathbb{R}^{3}$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are di... | 5 |
For $n=1,2, \ldots$ let $$ S_{n}=-\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)+\log (\sqrt{n}) $$ where $\log$ denotes the natural logarithm. Find $\lim _{n \rightarrow \infty} S_{n}$. If the value can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q... | 5 |
Let for $j=0, \ldots, n, a_{j}=a_{0}+j d$, where $a_{0}, d$ are fixed real numbers. Put $$ A=\left(\begin{array}{ccccc} a_{0} & a_{1} & a_{2} & \ldots & a_{n} \\ a_{1} & a_{0} & a_{1} & \ldots & a_{n-1} \\ a_{2} & a_{1} & a_{0} & \ldots & a_{n-2} \\ \ldots & \ldots & \ldots \ldots \ldots & \ldots & \ldots \\ a_{n} & a_... | 8 |
Let $V$ be a 10 -dimensional real vector space and $U_{1}$ and $U_{2}$ two linear subspaces such that $U_{1} \subseteq U_{2}, \operatorname{dim}_{\mathbb{R}} U_{1}=3$ and $\operatorname{dim}_{\mathbb{R}} U_{2}=6$. Let $\mathcal{E}$ be the set of all linear maps $T: V \longrightarrow V$ which have $U_{1}$ and $U_{2}$ as... | 67 |
Let $A_{n}=\{1,2, \ldots, n\}$, where $n \geq 3$. Let $\mathcal{F}$ be the family of all non-constant functions $f: A_{n} \rightarrow A_{n}$ satisfying the following conditions:\n(1) $f(k) \leq f(k+1)$ for $k=1,2, \ldots, n-1$,\n(2) $f(k)=f(f(k+1))$ for $k=1,2, \ldots, n-1$.\nFind the number of functions in $\mathcal{F... | 1 |
Evaluate the expression $\tan \left(\arctan 2+\arctan \frac{2}{2^{2}}+\cdots+\arctan \frac{2}{12^{2}}\right)$. The original answer is in the form of $-\frac{m}{n}$. Please provide the value of m + n. | 31 |
Given that the line $y=kx+b$ intersects the curve $y=x^{3}-x$ at three distinct points $A, B, C$, and that $|AB|=|BC|=2$, find the integer value of $k$. | 1 |
In how many ways can the integers from 1 to $5$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it? | 16 |
A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ( $a$ and $b$ are positive integers with $a$ greater than $b$ ), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that on... | 19 |
Evaluate$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]-\ln4\right)$$ and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers. Here $[x]$ is defined to be the integer such that $[x]\leqq x<[x]+1$ and $\log x$ is the logarithm of $x... | -1 |
Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_1,\ldots,n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^{4}=B^{7}=ABA^{-1}B=1,A^{2}\neq1$, and $B\neq1$. How many element... | 8 |
A particle moving on a straight line starts from rest and attains a velocity $2$ after traversing a distance $2$. If the motion is such that the acceleration was never increasing, find the maximum time for the traverse. | 2 |
Evaluate $$ \sum_{k=1}^{n}(-1)^{k}\binom{n}{k}(x-k)^{n} $$，when $n=2$ | 2 |
Let $p$ be a prime number. Let $J$ be the set of all $2 \times 2$ matrices $\binom{c b}{c d}$ whose entries are chosen from $\{0,1,2, \cdots, p-1\}$ and satisfy the conditions $a+d \equiv 1(\bmod p), a d-b c \equiv 0(\bmod p)$. Determine how many members $J$ has. Please provide the number of members when $p=2$. | 6 |
For a set with $n$ elements, how many subsets are there whose cardinality (the number of elements in the subset) is respectively $\equiv 0(\bmod 3), \equiv 1(\bmod 3), \equiv 2(\bmod 3)$? In other words, calculate $$ s_{(, n}=\sum_{k-1(\text { mod })}\binom{n}{k} \quad \text { for } i=0.1,2 $$ Your result should be str... | 667 |
The lengths of the hour hand and the minute hand of an accurate clock are 3 and 4 respectively. Find the square of the distance between the tips of the hour hand and the minute hand when the rate of increase of this distance is the fastest. | 7 |
Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. Give the answer in the form $r+s\sqrt{t}$ with $r$, $s$, and $t$ positive integers，writing r. | 13 |
Let $v$ be a vertex (corner) of a cube $C$ with edges of length 4. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R+\frac{4\pi}{3}$. | 8 |
Find the value of \(3(A + B)\) where \(A\) and \(B\) are constants such that \(f(x)\) is a solution to the differential equation. | 4 |
Let $P$ be a polynomial, with real coefficients, in three variables and $F$ be a function of two variables such that $P(u x, u y, u z)=u^{2} F(y-x, z-x)$ for all real $x, y, z, u$, and such that $P(1,0,0)=4, P(0,1,0)=5$, and $P(0,0,1)=6$. Also let $A, B, C$ be complex numbers with $P(A, B, C)=0$ and $|B-A|=10$. Find $\... | 3 |
For each positive integer $n$, let $a(n)$ be the number of zeros in the base $3$ representation of $n$. For which positive real numbers $x$ does the series $$ \sum_{n=1}^{\infty} \frac{x^{a(n)}}{n^{3}} $$ converge? Please provide the sum of the bounds of the interval where x satisfies the condition. | 25 |
Develop necessary and sufficient conditions which ensure that $r_{1},r_{2},r_{3}$ and $r_{1}{}^{2},r_{2}{}^{2}, r_{3}{}^{2}$ are simultaneously roots of the equation $x^{3}+ax^{2}+bx+c=0$.How many possible situations are there in total? | 8 |
In a solid sphere of radius $R$ the density $\rho$ is a function of $r$, the distance from the center of the sphere. If the magnitude of the gravitational force of attraction due to the sphere at any point inside the sphere is $k r^{2}$, where $k$ is a constant, find $\rho$ as a function of $r$. Find also the magnitude... | 6 |
Let $\min (x, y)$ denote the smaller of the numbers $x$ and $y$. For what $\lambda$'s does the equation $$\int_{0}^{1} \min (x, y) f(y) d y=\lambda f(x)$$ have continuous solutions which do not vanish identically in ( 0,1$)$? The largest $\lambda$ can be expressed in the form of $\frac{m}{n\pi^{2}}$. Find the value of ... | 5 |
Find the equations of the two straight lines each of which cuts all the four straight lines $x=1, y=0 ; \quad y=1, z=0 ; \quad z=1, x=0 ; \quad x=y=-6 z$. If \(L_{1}\) is given by $$y = a_{1}x + b_{1} = c_{1}z + d_{1}$$ and \(L_{2}\) is given by $$y = a_{2}x + b_{2} = c_{2}z + d_{2}$$ Find the expression of \(a_{1} + b... | 1 |
In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$AA, AB, AC, BB, BC, AD, CC, BD, CD, DD$$ in which $AA$ means two seniors, $AB$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A+A, A+B, A+C, B+B$, etc., corr... | 23 |
Given the parabola $y^{2}=2 m x$, what is the length of the shortest chord that is normal to the curve at one end?Write down the value when \(m = \sqrt{3}\). | 9 |
Find the equation of the parabola which touches the $x$-axis at the point $(1,0)$ and the $y$-axis at the point $(0,2)$. Find the coordinates of its vertex.If the coordinates of the vertex are $(\frac{a}{b},\frac{c}{d})$, find the value of $a + b + c + d$. | 68 |
A particle moves on a straight line, the only force acting on it being a resistance proportional to the velocity. If it started with a velocity of $1,000 \mathrm{ft}$. per sec. and had a velocity of 900 ft. per sec. when it had travelled $1,200 \mathrm{ft}$., calculate to the nearest hundredth of a second the time it t... | 126 |
Find the shortest distance between the plane $A x+B y+C z+1=0$ and the ellipsoid $x^{2} / a^{2}+y^{2} / b^{2}+z^{2} / c^{2}=1$. (For brevity, let $$ h=1 / \sqrt{A^{2}+B^{2}+C^{2}} \text { and } m=\sqrt{a^{2} A^{2}+b^{2} B^{2}+c^{2} C^{2}} \text {.) } $$ State algebraically the condition that the plane shall lie outside... | 0 |
Let \alpha and \beta be given positive real numbers, with \alpha<\beta. If two points are selected at random from a straight line segment of length \beta, what is the probability that the distance between them is at least \alpha? Please provide the value of the answer in the form simplest fraction by assuming \alpha = ... | 5 |
Find all positive integers that are within $250$ of exactly $15$ perfect squares. Please provide the sum of the endpoints of the intervals that contain these integers. | 1002 |
For which real numbers $c$ is there a straight line that intersects the curve $$ y=x^{4}+9 x^{3}+c x^{2}+9 x+4 $$ in four distinct points? The original answer is in the format $c < \frac{m}{n}$. Please provide the value of $m + n$. | 251 |
Let $I_{m} = \int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x$. For which integers $m, 1 \leqslant m \leqslant 10$, is $I_{m} \neq 0$? Please provide the sum of all such integers $m$. | 22 |
Define a polynomial $f_{n}(x)$, where $n \geqslant 0$ is $f_{0}(x)=1$, $n \geqslant 1$ is $f_{n}(0)=0$, and $n \geqslant 0$ is $$ \frac{d}{d x}\left(f_{n+1}(x)\right)=(n+1) f_{n}(x+1) $$ . Clearly decompose $f_{100}(1)$ into powers of different prime numbers, $a^{b}$, and find the value of $a+b$. | 200 |
Let $C$ be the unit circle $x^{2}+y^{2}=1$. A point $p$ is chosen randomly on the circumference $C$ and another point $q$ is chosen randomly from the interior of $C$ (these points are chosen independently and uniformly over their domains). Let $R$ be the rectangle with sides parallel to the $x$-and $y$-axes with diagon... | 6 |
Evaluate $\int_{0}^{\infty} t^{-1 / 2} e^{-1985\left(t+t^{-1}\right)} d t$. You may assume that $\int_{-\infty}^{\infty} e^{-x^{2}} d x=\sqrt{\pi}$. The original answer is in the form $\frac{\sqrt{\pi}}{\sqrt{m}} e^{-2 \times m}$ . Please provide the value of $m + 1$. | 1986 |
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