Found problems: 85335
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 $x^{12}_1+x^{12}_2+\ldots +x^{12}_{1997}$ .
2014 Belarus Team Selection Test, 2
Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that $$\frac{a^2}{(b+c)^3}+\frac{b^2}{(c+a)^3}+\frac{c^2}{(a+b)^3}\geq \frac98$$
2015 China Second Round Olympiad, 2
Let $S=\{A_1,A_2,\ldots ,A_n\}$, where $A_1,A_2,\ldots ,A_n$ are $n$ pairwise distinct finite sets $(n\ge 2)$, such that for any $A_i,A_j\in S$, $A_i\cup A_j\in S$. If $k= \min_{1\le i\le n}|A_i|\ge 2$, prove that there exist $x\in \bigcup_{i=1}^n A_i$, such that $x$ is in at least $\frac{n}{k}$ of the sets $A_1,A_2,\ldots ,A_n$ (Here $|X|$ denotes the number of elements in finite set $X$).
2021 Cyprus JBMO TST, 3
George plays the following game: At every step he can replace a triple of integers $(x,y,z)$ which is written on the blackboard, with any of the following triples:
(i) $(x,z,y)$
(ii) $(-x,y,z)$
(iii) $(x+y,y,2x+y+z)$
(iv) $(x-y,y,y+z-2x)$
Initially, the triple $(1,1,1)$ is written on the blackboard. Determine whether George can, with a sequence of allowed steps, end up at the triple $(2021,2019,2023)$, fully justifying your answer.
1984 Bundeswettbewerb Mathematik, 2
Given is a regular $n$-gon with circumradius $1$. $L$ is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of $L$?
2020 MBMT, 38
Consider $\triangle ABC$ with circumcenter $O$ and $\angle ABC$ obtuse. Construct $A'$ as the reflection of $A$ over $O$, and let $P$ be the intersection of $\overline{A'B}$ and $\overline{AC}$. Let $P'$ be the intersection of the circumcircle of $(OPA)$ with $\overline{AB}$. Given that the circumdiameter of $\triangle ABC$ is $25$, $\overline{AB} = 7$, and $\overline{BC} = 15$, find the length of $PP'$.
[i]Proposed by Kevin Wu[/i]
2018 AMC 10, 21
Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?
$
\textbf{(A) }a=\frac14 \qquad
\textbf{(B) }\frac14 < a < \frac12 \qquad
\textbf{(C) }a>\frac14 \qquad
\textbf{(D) }a=\frac12 \qquad
\textbf{(E) }a>\frac12 \qquad
$
2018 IFYM, Sozopol, 1
Find the number of solutions to the equation:
$6\{x\}^3 + \{x\}^2 + \{x\} + 2x = 2018. $
With {x} we denote the fractional part of the number x.
1998 All-Russian Olympiad, 5
Initially the numbers $19$ and $98$ are written on a board. Every minute, each of the two numbers is either squared or increased by $1$. Is it possible to obtain two equal numbers at some time?
2016 Indonesia TST, 5
For a finite set $A$ of positive integers, a partition of $A$ into two disjoint nonempty subsets $A_1$ and $A_2$ is $\textit{good}$ if the least common multiple of the elements in $A_1$ is equal to the greatest common divisor of the elements in $A_2$. Determine the minimum value of $n$ such that there exists a set of $n$ positive integers with exactly $2015$ good partitions.
2015 PAMO, Problem 5
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$, $k=1,2,...,7$. Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$. What is the most probable sum he can get?
2008 AMC 12/AHSME, 25
A sequence $ (a_1,b_1)$, $ (a_2,b_2)$, $ (a_3,b_3)$, $ \ldots$ of points in the coordinate plane satisfies \[ (a_{n \plus{} 1}, b_{n \plus{} 1}) \equal{} (\sqrt {3}a_n \minus{} b_n, \sqrt {3}b_n \plus{} a_n)\hspace{3ex}\text{for}\hspace{3ex} n \equal{} 1,2,3,\ldots.\] Suppose that $ (a_{100},b_{100}) \equal{} (2,4)$. What is $ a_1 \plus{} b_1$?
$ \textbf{(A)}\\minus{} \frac {1}{2^{97}} \qquad
\textbf{(B)}\\minus{} \frac {1}{2^{99}} \qquad
\textbf{(C)}\ 0 \qquad
\textbf{(D)}\ \frac {1}{2^{98}} \qquad
\textbf{(E)}\ \frac {1}{2^{96}}$
2011 Sharygin Geometry Olympiad, 5
Given triangle $ABC$. The midperpendicular of side $AB$ meets one of the remaining sides at point $C'$. Points $A'$ and $B'$ are defined similarly. Find all triangles $ABC$ such that triangle $A'B'C'$ is regular.