Found problems: 85335
1986 AMC 8, 1
In July 1861, $ 366$ inches of rain fell in Cherrapunji, India. What was the average rainfall in inches per hour during that month?
\[ \textbf{(A)}\ \frac{366}{31 \times 24} \qquad
\textbf{(B)}\ \frac{366 \times 31}{24} \qquad
\textbf{(C)}\ \frac{366 \times 24}{31} \\
\textbf{(D)}\ \frac{31 \times 24}{366} \qquad
\textbf{(E)}\ 366 \times 31 \times 24
\]
2015 BMT Spring, 15
Recall that an icosahedron is a $3$-dimensional solid characterized by its $20$ congruent faces, each of which is an equilateral triangle. Determine the number of rigid rotations that preserve the symmetry of the icosahedron. (Each vertex moves to the location of another vertex.)
2025 Kosovo National Mathematical Olympiad`, P4
For a sequence of integers $a_1 < a_2 < \cdot\cdot\cdot < a_n$, a pair $(a_i,a_j)$ where $1 \leq i < j \leq n$ is said to be [i]balanced[/i] if the number $\frac{a_i+a_j}{2}$ belongs to the sequence. For every natural number $n \geq 3$, find the maximum possible number of balanced pairs in a sequence with $n$ numbers.
2000 Moldova National Olympiad, Problem 3
Consider the sets $A_1=\{1\}$, $A_2=\{2,3,4\}$, $A_3=\{5,6,7,8,9\}$, etc. Let $b_n$ be the arithmetic mean of the smallest and the greatest element in $A_n$. Show that the number $\frac{2000}{b_1-1}+\frac{2000}{b_2-1}+\ldots+\frac{2000}{b_{2000}-1}$ is a prime integer.
2017 India PRMO, 20
Attached below:
2009 AMC 12/AHSME, 22
Parallelogram $ ABCD$ has area $ 1,\!000,\!000$. Vertex $ A$ is at $ (0,0)$ and all other vertices are in the first quadrant. Vertices $ B$ and $ D$ are lattice points on the lines $ y\equal{}x$ and $ y\equal{}kx$ for some integer $ k>1$, respectively. How many such parallelograms are there?
$ \textbf{(A)}\ 49\qquad
\textbf{(B)}\ 720\qquad
\textbf{(C)}\ 784\qquad
\textbf{(D)}\ 2009\qquad
\textbf{(E)}\ 2048$
2006 AMC 10, 1
What is $ ( \minus{} 1)^1 \plus{} ( \minus{} 1)^2 \plus{} \cdots \plus{} ( \minus{} 1)^{2006}$?
$ \textbf{(A) } \minus{} 2006 \qquad \textbf{(B) } \minus{} 1 \qquad \textbf{(C) } 0 \qquad \textbf{(D) } 1 \qquad \textbf{(E) } 2006$
2023 Switzerland - Final Round, 8
Let $n$ be a positive integer. We start with $n$ piles of pebbles, each initially containing a single pebble. One can perform moves of the following form: choose two piles, take an equal number of pebbles from each pile and form a new pile out of these pebbles. Find (in terms of $n$) the smallest number of nonempty piles that one can obtain by performing a finite sequence of moves of this form.
Ukrainian TYM Qualifying - geometry, X.12
Inside the convex polygon $A_1A_2...A_n$ , there is a point $M$ such that $\sum_{k=1}^n \overrightarrow {A_kM} = \overrightarrow{0}$. Prove that $nP\ge 4d$, where $P$ is the perimeter of the polygon, and $d=\sum_{k=1}^n |\overrightarrow {A_kM}|$ . Investigate the question of the achievement of equality in this inequality.
2000 Bulgaria National Olympiad, 3
Let $ p$ be a prime number and let $ a_1,a_2,\ldots,a_{p \minus{} 2}$ be positive integers such that $ p$ doesn't $ a_k$ or $ {a_k}^k \minus{} 1$ for any $ k$. Prove that the product of some of the $ a_i$'s is congruent to $ 2$ modulo $ p$.
2015 HMMT Geometry, 10
Let $\mathcal{G}$ be the set of all points $(x,y)$ in the Cartesian plane such that $0\le y\le 8$ and $$(x-3)^2+31=(y-4)^2+8\sqrt{y(8-y)}.$$ There exists a unique line $\ell$ of [b]negative slope[/b] tangent to $\mathcal{G}$ and passing through the point $(0,4)$. Suppose $\ell$ is tangent to $\mathcal{G}$ at a [b]unique[/b] point $P$. Find the coordinates $(\alpha, \beta)$ of $P$.
2008 District Olympiad, 4
Let $ M$ be the set of those positive integers which are not divisible by $ 3$. The sum of $ 2n$ consecutive elements of $ M$ is $ 300$. Determine $ n$.
2023 Bangladesh Mathematical Olympiad, P2
Let the points $A,B,C$ lie on a line in this order. $AB$ is the diameter of semicircle $\omega_1$, $AC$ is the diameter of semicircle $\omega_2$. Assume both $\omega_1$ and $\omega_2$ lie on the same side of $AC$. $D$ is a point on $\omega_2$ such that $BD\perp AC$. A circle centered at $B$ with radius $BD$ intersects $\omega_1$ at $E$. $F$ is on $AC$ such that $EF\perp AC$. Prove that $BC=BF$.
2010 Stanford Mathematics Tournament, 22
We need not restrict our number system radix to be an integer. Consider the phinary numeral system in which the radix is the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$ and the digits $0$ and $1$ are used. Compute $1010100_{\phi}-.010101_{\phi}$
2010 Today's Calculation Of Integral, 628
(1) Evaluate the following definite integrals.
(a) $\int_0^{\frac{\pi}{2}} \cos ^ 2 x\sin x\ dx$
(b) $\int_0^{\frac{\pi}{2}} (\pi - 2x)\cos x\ dx$
(c) $\int_0^{\frac{\pi}{2}} x\cos ^ 3 x\ dx$
(2) Let $a$ be a positive constant. Find the area of the cross section cut by the plane $z=\sin \theta \ \left(0\leq \theta \leq \frac{\pi}{2}\right)$ of the solid such that
\[x^2+y^2+z^2\leq a^2,\ \ x^2+y^2\leq ax,\ \ z\geq 0\]
, then find the volume of the solid.
[i]1984 Yamanashi Medical University entrance exam[/i]
Please slove the problem without multi integral or arcsine function for Japanese high school students aged 17-18 those who don't study them.
Thanks in advance.
kunny
2004 Nordic, 3
Given a finite sequence $x_{1,1}, x_{2,1}, \dots , x_{n,1}$ of integers $(n\ge 2)$, not all equal, define the sequences $x_{1,k}, \dots , x_{n,k}$ by
\[ x_{i,k+1}=\frac{1}{2}(x_{i,k}+x_{i+1,k})\quad\text{where }x_{n+1,k}=x_{1,k}.\]
Show that if $n$ is odd, then not all $x_{j,k}$ are integers. Is this also true for even $n$?
2008 South East Mathematical Olympiad, 4
Let $n$ be a positive integer. $f(n)$ denotes the number of $n$-digit numbers $\overline{a_1a_2\cdots a_n}$(wave numbers) satisfying the following conditions :
(i) for each $a_i \in\{1,2,3,4\}$, $a_i \not= a_{i+1}$, $i=1,2,\cdots$;
(ii) for $n\ge 3$, $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$ is negative, $i=1,2,\cdots$.
(1) Find the value of $f(10)$;
(2) Determine the remainder of $f(2008)$ upon division by $13$.
2020 Moldova Team Selection Test, 11
Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy
$$f(\sin{x})+f(\cos{x})=2020$$
for any real number $x.$
2012 Bosnia And Herzegovina - Regional Olympiad, 3
Prove tha number $19 \cdot 8^n +17$ is composite for every positive integer $n$
2025 International Zhautykov Olympiad, 1
Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]
2018 AMC 8, 7
The $5$-digit number $\underline{2}$ $\underline{0}$ $\underline{1}$ $\underline{8}$ $\underline{U}$ is divisible by $9$. What is the remainder when this number is divided by $8$?
$\textbf{(A) }1\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7$
2005 International Zhautykov Olympiad, 2
The inner point $ X$ of a quadrilateral is [i]observable[/i] from the side $ YZ$ if the perpendicular to the line $ YZ$ meet it in the colosed interval $ [YZ].$ The inner point of a quadrilateral is a $ k\minus{}$point if it is observable from the exactly $ k$ sides of the quadrilateral. Prove that if a convex quadrilateral has a 1-point then it has a $ k\minus{}$point for each $ k\equal{}2,3,4.$
2022 China Team Selection Test, 5
Show that there exist constants $c$ and $\alpha > \frac{1}{2}$, such that for any positive integer $n$, there is a subset $A$ of $\{1,2,\ldots,n\}$ with cardinality $|A| \ge c \cdot n^\alpha$, and for any $x,y \in A$ with $x \neq y$, the difference $x-y$ is not a perfect square.
2020 Australian Maths Olympiad, 5
Each term of an infinite sequence $a_1 ,a_2 ,a_3 , \dots$ is equal to 0 or 1. For each positive integer $n$,
$$a_n + a_{n+1} \neq a_{n+2} + a_{n+3},\, \text{and}$$
$$a_n + a_{n+1} + a_{n+2} \neq a_{n+3} + a_{n+4} + a_{n+5}.$$
Prove that if $a_1 = 0$, then $a_{2020} = 1$.
2009 Germany Team Selection Test, 1
Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.