Found problems: 85335
2011 HMNT, 1
Five of James’ friends are sitting around a circular table to play a game of Fish. James chooses a place between two of his friends to pull up a chair and sit. Then, the six friends divide themselves into two disjoint teams, with each team consisting of three consecutive players at the table. If the order in which the three members of a team sit does not matter, how many possible (unordered) pairs of teams are possible?
2004 IMO Shortlist, 6
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[
f(x^2+y^2+2f(xy)) = (f(x+y))^2.
\] for all $x,y \in \mathbb{R}$.
2024 PErA, P3
Let $x_1,x_2,\dots, x_n$ be positive real numbers such that $x_1+x_2+\cdots + x_n=1$. Prove that $$\sum_{i=1}^n \frac{\min\{x_{i-1},x_i\}\cdot \max\{x_i,x_{i+1}\}}{x_i}\leq 1,$$ where we denote $x_0=x_n$ and $x_{n+1}=x_1$.
1967 AMC 12/AHSME, 28
Given the two hypotheses: $\text{I}$ Some Mems are not Ens and $\text{II}$ No Ens are Veens. If "some" means "at least one," we can conclude that:
$\textbf{(A)}\ \text{Some Mems are not Veens}\qquad
\textbf{(B)}\ \text{Some Vees are not Mems}\\
\textbf{(C)}\ \text{No Mem is a Vee}\qquad
\textbf{(D)}\ \text{Some Mems are Vees}\\
\textbf{(E)}\ \text{Neither} \; \textbf{(A)} \; \text{nor} \; \textbf{(B)} \; \text{nor} \; \textbf{(C)} \; \text{nor} \; \textbf{(D)} \; \text{is deducible from the given statements}$
2005 AMC 10, 14
Equilateral $ \triangle ABC$ has side length $ 2$, $ M$ is the midpoint of $ \overline{AC}$, and $ C$ is the midpoint of $ \overline{BD}$. What is the area of $ \triangle CDM$?
[asy]size(200);defaultpen(linewidth(.8pt)+fontsize(8pt));
pair B = (0,0);
pair A = 2*dir(60);
pair C = (2,0);
pair D = (4,0);
pair M = midpoint(A--C);
label("$A$",A,NW);label("$B$",B,SW);label("$C$",C, SE);label("$M$",M,NE);label("$D$",D,SE);
draw(A--B--C--cycle);
draw(C--D--M--cycle);[/asy]$ \textbf{(A)}\ \frac {\sqrt {2}}{2}\qquad \textbf{(B)}\ \frac {3}{4}\qquad \textbf{(C)}\ \frac {\sqrt {3}}{2}\qquad \textbf{(D)}\ 1\qquad \textbf{(E)}\ \sqrt {2}$
2015 HMMT Geometry, 6
In triangle $ABC$, $AB=2$, $AC=1+\sqrt{5}$, and $\angle CAB=54^{\circ}$. Suppose $D$ lies on the extension of $AC$ through $C$ such that $CD=\sqrt{5}-1$. If $M$ is the midpoint of $BD$, determine the measure of $\angle ACM$, in degrees.
2018 Switzerland - Final Round, 7
Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$.
Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.
2014 ELMO Shortlist, 2
Define the Fibanocci sequence recursively by $F_1=1$, $F_2=1$ and $F_{i+2} = F_i + F_{i+1}$ for all $i$. Prove that for all integers $b,c>1$, there exists an integer $n$ such that the sum of the digits of $F_n$ when written in base $b$ is greater than $c$.
[i]Proposed by Ryan Alweiss[/i]
2017 AMC 10, 1
What is the value of $2(2(2(2(2(2+1)+1)+1)+1)+1)+1$?
$\textbf{(A) } 70 \qquad
\textbf{(B) } 97 \qquad
\textbf{(C) } 127 \qquad
\textbf{(D) } 159 \qquad
\textbf{(E) } 729 $
2023 Novosibirsk Oral Olympiad in Geometry, 7
Triangle $ABC$ is given with angles $\angle ABC = 60^o$ and $\angle BCA = 100^o$. On the sides AB and AC, the points $D$ and $E$ are chosen, respectively, in such a way that $\angle EDC = 2\angle BCD = 2\angle CAB$. Find the angle $\angle BED$.
2018 Romania National Olympiad, 1
Find the distinct positive integers $a, b, c,d$, such that the following conditions hold:
(1) exactly three of the four numbers are prime numbers;
(2) $a^2 + b^2 + c^2 + d^2 = 2018.$
2008 Bulgaria Team Selection Test, 2
The point $P$ lies inside, or on the boundary of, the triangle $ABC$. Denote by $d_{a}$, $d_{b}$ and $d_{c}$ the distances between $P$ and $BC$, $CA$, and $AB$, respectively. Prove that $\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}$. When does the equality holds?
2005 All-Russian Olympiad, 1
Do there exist a bounded function $f: \mathbb{R}\to\mathbb{R}$ such that $f(1)>0$ and $f(x)$ satisfies an inequality $f^2(x+y)\ge f^2(x)+2f(xy)+f^2(y)$?
2019 Lusophon Mathematical Olympiad, 6
Two players Arnaldo and Betania play alternately, with Arnaldo being the first to play. Initially there are two piles of stones containing $x$ and $y$ stones respectively. In each play, it is possible to perform one of the following operations:
1. Choose two non-empty piles and take one stone from each pile.
2. Choose a pile with an odd amount of stones, take one of their stones and, if possible, split into two piles with the same amount of stones.
The player who cannot perform either of operations 1 and 2 loses.
Determine who has the winning strategy based on $x$ and $y$.
2011 HMNT, 10
Let $G_1G_2G_3$ be a triangle with $G_1G_2 = 7$, $G_2G_3 = 13$, and $G_3G_1 = 15$. Let $G_4$ be a point outside triangle $G_1G_2G_3$ so that ray $\overrightarrow{G_1G_4}$ cuts through the interior of the triangle, $G_3G_4 = G_4G_2$, and $\angle G_3G_1G_4 = 30^o$. Let $G_3G_4$ and $G_1G_2$ meet at $G_5$. Determine the length of segment $G_2G_5$.
1960 AMC 12/AHSME, 16
In the numeration system with base $5$, counting is as follows: $1, 2, 3, 4, 10, 11, 12, 13, 14, 20, ...$ The number whose description in the decimal system is $69$, when described in the base $5$ system, is a number with:
$ \textbf{(A)}\ \text{two consecutive digits} \qquad\textbf{(B)}\ \text{two non-consecutive digits} \qquad$
$\textbf{(C)}\ \text{three consecutive digits} \qquad\textbf{(D)}\ \text{three non-consecutive digits} \qquad$
$\textbf{(E)}\ \text{four digits} $
2020 MBMT, 2
Daniel, Clarence, and Matthew split a \$20.20 dinner bill so that Daniel pays half of what Clarence pays. If Daniel pays \$6.06, what is the ratio of Clarence's pay to Matthew's pay?
[i]Proposed by Henry Ren[/i]
2010 Turkey MO (2nd round), 2
For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a [i]good polynomial.[/i]
Find the number of [i]good polynomials.[/i]
2001 Estonia National Olympiad, 1
The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.
2012 CHMMC Spring, 5
Suppose $S$ is a subset of $\{1, 2, 3, 4, 5, 6, 7\}$. How many different possible values are there for the product of the elements in $S$?
2024 Saint Petersburg Mathematical Olympiad, 3
On the side $BC$ of acute triangle $ABC$ point $P$ was chosen. Point $E$ is symmetric to point $B$ onto line $AP$. Segment $PE$ meets circumcircle of triangle $ABP$ in point $D$. $M$ is midpoint of side $AC$. Prove that $DE+AC>2BM$.
2009 AMC 8, 6
Steve's empty swimming pool will hold $ 24,000$ gallons of water when full. It will be filled by $ 4$ hoses, each of which supplies $ 2.5$ gallons of water per minute. How many hours will it take to fill Steve's pool?
$ \textbf{(A)}\ 40 \qquad
\textbf{(B)}\ 42 \qquad
\textbf{(C)}\ 44 \qquad
\textbf{(D)}\ 46 \qquad
\textbf{(E)}\ 48$
1989 Iran MO (2nd round), 1
[b](a)[/b] Let $n$ be a positive integer, prove that
\[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\]
[b](b)[/b] Find a positive integer $n$ for which
\[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]
BIMO 2022, 5
Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds:
$$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$
[i]Proposed by Anzo Teh Zhao Yang[/i]
2014 Baltic Way, 10
In a country there are $100$ airports. Super-Air operates direct flights between some pairs of airports (in both directions). The [i]traffic[/i] of an airport is the number of airports it has a direct Super-Air connection with. A new company, Concur-Air, establishes a direct flight between two airports if and only if the sum of their traffics is at least $100.$ It turns out that there exists a round-trip of Concur-Air flights that lands in every airport exactly once. Show that then there also exists a round-trip of Super-Air flights that lands in every airport exactly once.