Found problems: 85335
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.
VII Soros Olympiad 2000 - 01, 9.2
Find $a, b, c, d$ such that for all $x$ the equality $|| x | -1 | = a | x | + b | x-1 | + c | x + 1 | + d$ holds.
1967 IMO Shortlist, 3
The $n$ points $P_1,P_2, \ldots, P_n$ are placed inside or on the boundary of a disk of radius 1 in such a way that the minimum distance $D_n$ between any two of these points has its largest possible value $D_n.$ Calculate $D_n$ for $n = 2$ to 7. and justify your answer.