Found problems: 85335
2010 Today's Calculation Of Integral, 566
In the coordinate space, consider the cubic with vertices $ O(0,\ 0,\ 0),\ A(1,\ 0,\ 0),\ B(1,\ 1,\ 0),\ C(0,\ 1,\ 0),\ D(0,\ 0,\ 1),\ E(1,\ 0,\ 1),\ F(1,\ 1,\ 1),\ G(0,\ 1,\ 1)$. Find the volume of the solid generated by revolution of the cubic around the diagonal $ OF$ as the axis of rotation.
2009 AMC 10, 3
Which of the following is equal to $ 1\plus{}\frac{1}{1\plus{}\frac{1}{1\plus{}1}}$?
$ \textbf{(A)}\ \frac{5}{4} \qquad
\textbf{(B)}\ \frac{3}{2} \qquad
\textbf{(C)}\ \frac{5}{3} \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
2014-2015 SDML (High School), 1
How many ways are there to color the vertices of a square green, red, or blue so that no two adjacent vertices have the same color? (Two colorings are considered different even if one coloring can be rotated to product the other coloring.)
IV Soros Olympiad 1997 - 98 (Russia), 9.3
Several machines were working in the workshop. After reconstruction, the number of machines decreased, and the percentage by which the number of machines decreased turned out to be equal to the number of remaining machines. What was the smallest number of machines that could have been in the workshop before the reconstruction?
2009 Today's Calculation Of Integral, 412
Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$.
(1) Find $ I_0,\ I_{\minus{}1},\ I_2$.
(2) Find $ I_1$.
(3) Express $ I_{n\plus{}2}$ in terms of $ I_n$.
(4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$.
(5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results.
You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.
1980 VTRMC, 8
Let $z=x+iy$ be a complex number with $x$ and $y$ rational and with $|z| = 1.$
(a) Find two such complex numbers.
(b) Show that $|z^{2n}-1|=2|\sin n\theta|,$ where $z=e^{i\theta}.$
(c) Show that $|z^2n -1|$ is rational for every $n.$
1999 National Olympiad First Round, 24
Polynomial $ f\left(x\right)$ satisfies $ \left(x \minus{} 1\right)f\left(x \plus{} 1\right) \minus{} \left(x \plus{} 2\right)f\left(x\right) \equal{} 0$ for every $ x\in \Re$. If $ f\left(2\right) \equal{} 6$, $ f\left({\tfrac{3}{2}} \right) \equal{} ?$
$\textbf{(A)}\ -6 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ \frac {3}{2} \qquad\textbf{(D)}\ \frac {15}{8} \qquad\textbf{(E)}\ \text{None}$
1978 IMO Longlists, 31
Let the polynomials
\[P(x) = x^n + a_{n-1}x^{n-1 }+ \cdots + a_1x + a_0,\]
\[Q(x) = x^m + b_{m-1}x^{m-1} + \cdots + b_1x + b_0,\]
be given satisfying the identity $P(x)^2 = (x^2 - 1)Q(x)^2 + 1$. Prove the identity
\[P'(x) = nQ(x).\]
2012 Indonesia TST, 4
Given a non-zero integer $y$ and a positive integer $n$. If $x_1, x_2, \ldots, x_n \in \mathbb{Z} - \{0, 1\}$ and $z \in \mathbb{Z}^+$ satisfy $(x_1x_2 \ldots x_n)^2y \le 2^{2(n+1)}$ and $x_1x_2 \ldots x_ny = z + 1$, prove that there is a prime among $x_1, x_2, \ldots, x_n, z$.
[color=blue]It appears that the problem statement is incorrect; suppose $y = 5, n = 2$, then $x_1 = x_2 = -1$ and $z = 4$. They all satisfy the problem's conditions, but none of $x_1, x_2, z$ is a prime. What should the problem be, or did I misinterpret the problem badly?[/color]
1979 IMO Longlists, 69
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
2019 Online Math Open Problems, 24
We define the binary operation $\times$ on elements of $\mathbb{Z}^2$ as \[(a,b)\times(c,d)=(ac+bd,ad+bc)\] for all integers $a,b,c,$ and $d$. Compute the number of ordered six-tuples $(a_1,a_2,a_3,a_4,a_5,a_6)$ of integers such that \[[[[[(1,a_1)\times (2,a_2)]\times (3,a_3)]\times (4,a_4)]\times (5,a_5)]\times (6,a_6)=(350,280).\]
[i]Proposed by Michael Ren and James Lin[/i]
2000 Harvard-MIT Mathematics Tournament, 40
Let $\phi(n)$ denote the number of positive integers less than or equal to $n$ and relatively prime to $n$. Find all natural numbers $n$ and primes $p$ such that $\phi(n)=\phi(np)$.
2000 Harvard-MIT Mathematics Tournament, 4
On an $n$ by $n$ chessboard, numbers are written on each square so that the number in a square is the average of the numbers on the adjacent squares. Show that all the numbers are the same.
1986 China National Olympiad, 6
Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.
2016 Korea USCM, 5
For $f(x) = \cos\left(\frac{3\sqrt{3}\pi}{8}(x-x^3 ) \right)$, find the value of
$$\lim_{t\to\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} + \lim_{t\to-\infty} \left( \int_0^1 f(x)^t dx \right)^\frac{1}{t} $$
2013 IFYM, Sozopol, 3
Determine all pairs $(p, q)$ of prime numbers such that $p^p + q^q + 1$ is divisible by $pq.$
2014 Sharygin Geometry Olympiad, 24
A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ in such a way that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. Prove that if $PK$ and $QL$ are complanar then the touching point of the sphere with the base lies on $BD$.
2024 Princeton University Math Competition, A6 / B8
Let $\triangle ABC$ be a triangle with $AB = 10.$ Let $D$ be a point on the opposite side of line $AC$ as $B$ so that $\triangle ACD$ is directly similar to $\triangle ABC$ (i.e. $\angle ACD = \angle ABC,$ etc). Let $M$ be the midpoint of $AD.$ Given that $A$ is the centroid of triangle $\triangle BCM,$ compute $BC^2.$
.
2009 Tournament Of Towns, 3
Are there positive integers $a; b; c$ and $d$ such that $a^3 + b^3 + c^3 + d^3 =100^{100}$ ?
[i](4 points)[/i]
2018 India IMO Training Camp, 2
In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.
Russian TST 2018, P2
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
2014 Sharygin Geometry Olympiad, 17
Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.
2025 China Team Selection Test, 3
Let $n, k, l$ be positive integers satisfying $n \ge 3$, $l \le n - 2, l - k \le \frac{n-3}{2}$. Suppose that $a_1, a_2, \dots, a_k$ are integers chosen from $\{1, 2, \dots, n\}$ such that the set of remainders of the subset sums over all subsets of $a_i$ when divided by $n$ is exactly $\{1, 2, \dots, l\}$. Show that \[ a_1 + a_2 + \dots + a_k = l. \]
2025 Belarusian National Olympiad, 9.2
Snow White and seven dwarfs live in their house in the forest. During several days some dwarfs worked in the diamond mine, while others were collecting mushrooms. Each dwarf each day was doing only one type of job. It is known that in any two consecutive days there are exactly three dwarfs which did both types of job. Also, for any two days at least one dwarf did both types of job.
What is maximum amount of days which this situation could last?
[i]M. Karpuk[/i]
1990 AMC 8, 8
A dress originally priced at 80 dollars was put on sale for $25\%$ off. If $10\%$ tax was added to the sale price, then the total selling price (in dollars) of the dress was
$ \text{(A)}\ \text{45 dollars}\qquad\text{(B)}\ \text{52 dollars}\qquad\text{(C)}\ \text{54 dollars}\qquad\text{(D)}\ \text{66 dollars}\qquad\text{(E)}\ \text{68 dollars} $