Found problems: 85335
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.
2022 MIG, 4
There exists a real number $n$ such that $4^n=5$. What is the value of $8^n$?
$\textbf{(A) }5\sqrt{5}\qquad\textbf{(B) }10\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }25\sqrt{5}$
2014 India PRMO, 17
For a natural number $b$, let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$?
2014 ELMO Shortlist, 8
Let $a, b, c$ be positive reals with $a^{2014}+b^{2014}+c^{2014}+abc=4$. Prove that
\[ \frac{a^{2013}+b^{2013}-c}{c^{2013}} + \frac{b^{2013}+c^{2013}-a}{a^{2013}} + \frac{c^{2013}+a^{2013}-b}{b^{2013}} \ge a^{2012}+b^{2012}+c^{2012}. \][i]Proposed by David Stoner[/i]
1965 AMC 12/AHSME, 22
If $ a_2 \neq 0$ and $ r$ and $ s$ are the roots of $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} 0$, then the equality $ a_0 \plus{} a_1x \plus{} a_2x^2 \equal{} a_0\left (1 \minus{} \frac {x}{r} \right ) \left (1 \minus{} \frac {x}{s} \right )$ holds:
$ \textbf{(A)}\ \text{for all values of }x, a_0\neq 0$
$ \textbf{(B)}\ \text{for all values of }x$
$ \textbf{(C)}\ \text{only when }x \equal{} 0$
$ \textbf{(D)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s$
$ \textbf{(E)}\ \text{only when }x \equal{} r \text{ or }x \equal{} s, a_0 \neq 0$
1990 Chile National Olympiad, 5
Determine a natural $n$ such that $$996 \le \sum_{k = 1}^{n}\frac{1}{k}$$
1993 Hungary-Israel Binational, 2
Determine all polynomials $f (x)$ with real coeffcients that satisfy
\[f (x^{2}-2x) = f^{2}(x-2)\]
for all $x.$