Found problems: 85335
2022 Romania National Olympiad, P3
Let $Z\subset \mathbb{C}$ be a set of $n$ complex numbers, $n\geqslant 2.$ Prove that for any positive integer $m$ satisfying $m\leqslant n/2$ there exists a subset $U$ of $Z$ with $m$ elements such that\[\Bigg|\sum_{z\in U}z\Bigg|\leqslant\Bigg|\sum_{z\in Z\setminus U}z\Bigg|.\][i]Vasile Pop[/i]
2021 Thailand TSTST, 2
Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties.
$\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational.
$\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer.
$\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.
2008 Postal Coaching, 1
For each positive $ x \in \mathbb{R}$, define
$ E(x)=\{[nx]: n\in \mathbb{N}\}$
Find all irrational $ \alpha >1$ with the following property:
If a positive real $ \beta$ satisfies $ E(\beta) \subset E(\alpha)$. then $ \frac{\beta}{\alpha}$ is a natural number.
2021 Kazakhstan National Olympiad, 3
Let $(a_n)$ and $(b_n)$ be sequences of real numbers, such that $a_1 = b_1 = 1$, $a_{n+1} = a_n + \sqrt{a_n}$, $b_{n+1} = b_n + \sqrt[3]{b_n}$ for all positive integers $n$. Prove that there is a positive integer $n$ for which the inequality $a_n \leq b_k < a_{n+1}$ holds for exactly 2021 values of $k$.
2024 Sharygin Geometry Olympiad, 9
Let $ABCD$ ($AD \parallel BC$) be a trapezoid circumscribed around a circle $\omega$, which touches the sides $AB, BC, CD, $ and $AD$ at points $P, Q, R, S$ respectively. The line passing through $P$ and parallel to the bases of the trapezoid meets $QR$ at point $X$. Prove that $AB, QS$ and $DX$ concur.
2006 MOP Homework, 1
Triangle $ABC$ is inscribed in circle $w$. Line $l_{1}$ bisects $\angle BAC$ and meets segments $BC$ and $w$ in $D$ and $M$,respectively. Let $y$ denote the circle centered at $M$ with radius $BM$. Line $l_{2}$ passes through $D$ and meets circle $y$ at $X$ and $Y$. Prove that line $l_{1}$ also bisects $\angle XAY$
2021 ISI Entrance Examination, 8
A pond has been dug at the Indian Statistical Institute as an inverted truncated pyramid with a square base (see figure below). The depth of the pond is 6m. The square at the bottom has side length 2m and the top square has side length 8m. Water is filled in at a rate of $\tfrac{19}{3}$ cubic meters per hour. At what rate is the water level rising exactly $1$ hour after the water started to fill the pond?
[img]https://cdn.artofproblemsolving.com/attachments/0/9/ff8cac4bb4596ec6c030813da7e827e9a09dfd.png[/img]
2020 LIMIT Category 2, 20
Let $\{a_n \}_n$ be a sequence of real numbers such there there are countably infinite distinct subsequences converging to the same point. We call two subsequences distinct if they do not have a common term. Which of the following statements always holds:
(A) $\{a_n \}_n$ is bounded
(B) $\{a_n \}_n$ is unbounded
(C) The set of convergent subsequence $\{a_n \}_n$ is countable
(D) None of these
2020 Moldova Team Selection Test, 9
Let $\Delta ABC$ be an acute triangle and $\Omega$ its circumscribed circle, with diameter $AP$. Points $E$ and $F$ are the orthogonal projections from $B$ on $AC$ and $AP$, points $M$ and $N$ are the midpoints of segments $EF$ and $CP$. Prove that $\angle BMN=90$.
2000 Harvard-MIT Mathematics Tournament, 3
Suppose the positive integers $a,b,c$ satisfy $a^n+b^n=c^n$, where $n$ is a positive integer greater than $1$. Prove that $a,b,c>n$.
(Note: Fermat's Last Theorem may [i]not[/i] be used)
2018 CCA Math Bonanza, L1.4
What is the sum of all distinct values of $x$ that satisfy $x^4-x^3-7x^2+13x-6=0$?
[i]2018 CCA Math Bonanza Lightning Round #1.4[/i]
2008 Greece National Olympiad, 1
A computer generates all pairs of real numbers $x, y \in (0, 1)$ for which the numbers $a = x+my$ and $b = y+mx$ are both integers, where $m$ is a given positive integer. Finding one such pair $(x, y)$ takes $5$ seconds. Find $m$ if the computer needs $595$ seconds to find all possible ordered pairs $(x, y)$.
2017 Purple Comet Problems, 13
Let $ABCDE$ be a pentagon with area $2017$ such that four of its sides $AB, BC, CD$, and $EA$ have integer length. Suppose that $\angle A = \angle B = \angle C = 90^o$, $AB = BC$, and $CD = EA$. The maximum possible perimeter of $ABCDE$ is $a + b \sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a + b + c$.
2010 Balkan MO Shortlist, G2
Consider a cyclic quadrilateral such that the midpoints of its sides form another cyclic quadrilateral. Prove that the area of the smaller circle is less than or equal to half the area of the bigger circle
2020 Puerto Rico Team Selection Test, 3
The side $BC$ of the triangle $ABC$ is extended beyond $C$ to $D$, such that $CD=BC$. The side $CA$ is extended beyond $A$ to $E$, such that $AE=2CA$. Prove that if $AD=BE$, then the triangle $ABC$ is right.
MBMT Team Rounds, 2020.31
Consider the infinite sequence $\{a_i\}$ that extends the pattern
\[1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \dots\]
Formally, $a_i = i-T(i)$ for all $i \geq 1$, where $T(i)$ represents the largest triangular number less than $i$ (triangle numbers are integers of the form $\frac{k(k+1)}2$ for some nonnegative integer $k$). Find the number of indices $i$ such that $a_i = a_{i + 2020}$.
[i]Proposed by Gabriel Wu[/i]
2017 All-Russian Olympiad, 2
$ABCD$ is an isosceles trapezoid with $BC || AD$. A circle $\omega$ passing through $B$ and $C$ intersects the side $AB$ and the diagonal $BD$ at points $X$ and $Y$ respectively. Tangent to $\omega$ at $C$ intersects the line $AD$ at $Z$. Prove that the points $X$, $Y$, and $Z$ are collinear.
1999 Romania National Olympiad, 3
Let $a,b,c \in \mathbb{C}$ and $a \neq 0$. The roots $z_1$ and $z_2$ of the equation $az^2+bz+c=0$ satisfy $|z_1|<1$ and $|z_2|<1$. Prove that the roots $z_3$ and $z_4$ of the equation $$(a+\overline{c})z^2+(b+\overline{b})z+\overline{a}+c=0$$
satisfy $|z_3|=|z_4|=1$
1983 Austrian-Polish Competition, 1
Nonnegative real numbers $a, b,x,y$ satisfy $a^5 + b^5 \le $1 and $x^5 + y^5 \le 1$. Prove that $a^2x^3 + b^2y^3 \le 1$.
Indonesia Regional MO OSP SMA - geometry, 2005.1
The length of the largest side of the cyclic quadrilateral $ABCD$ is $a$, while the radius of the circumcircle of $\vartriangle ACD$ is $1$. Find the smallest possible value for $a$. Which cyclic quadrilateral $ABCD$ gives the value $a$ equal to the smallest value?
1997 Vietnam National Olympiad, 2
Let n be an integer which is greater than 1, not divisible by 1997.
Let $ a_m\equal{}m\plus{}\frac{mn}{1997}$ for all m=1,2,..,1996
$ b_m\equal{}m\plus{}\frac{1997m}{n}$ for all m=1,2,..,n-1
We arrange the terms of two sequence $ (a_i), (b_j)$ in the ascending order to form a new sequence $ c_1\le c_2\le ...\le c_{1995\plus{}n}$
Prove that $ c_{k\plus{}1}\minus{}c_k<2$ for all k=1,2,...,1994+n
2020 Tournament Of Towns, 1
Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ?
Mikhail Evdokimov
2021 Middle European Mathematical Olympiad, 4
Let $n$ be a positive integer. Prove that in a regular $6n$-gon, we can draw $3n$ diagonals with pairwise distinct ends and partition the drawn diagonals into $n$ triplets so that:
[list]
[*] the diagonals in each triplet intersect in one interior point of the polygon and
[*] all these $n$ intersection points are distinct.
[/list]
1991 USAMO, 2
For any nonempty set $\,S\,$ of numbers, let $\,\sigma(S)\,$ and $\,\pi(S)\,$ denote the sum and product, respectively, of the elements of $\,S\,$. Prove that
\[ \sum \frac{\sigma(S)}{\pi(S)} = (n^2 + 2n) - \left(1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{n} \right) (n+1), \]
where ``$\Sigma$'' denotes a sum involving all nonempty subsets $S$ of $\{1,2,3, \ldots,n\}$.
1997 China National Olympiad, 1
Let $x_1,x_2,\ldots ,x_{1997}$ be real numbers satisfying the following conditions:
i) $-\dfrac{1}{\sqrt{3}}\le x_i\le \sqrt{3}$ for $i=1,2,\ldots ,1997$;
ii) $x_1+x_2+\cdots +x_{1997}=-318 \sqrt{3}$ .
Determine (with proof) the maximum value of $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .