Found problems: 85335
2015 Albania JBMO TST, 5
Let $x$ and $y$ be positive real numbers with $x + y =1 $. Prove that
$$\frac{(3x-1)^2}{x}+ \frac{(3y-1)^2}{y} \ge1.$$ For which $x$ and $y$ equality holds?
(K. Czakler, GRG 21, Vienna)
1966 IMO Longlists, 51
Consider $n$ students with numbers $1, 2, \ldots, n$ standing in the order $1, 2, \ldots, n.$ Upon a command, any of the students either remains on his place or switches his place with another student. (Actually, if student $A$ switches his place with student $B,$ then $B$ cannot switch his place with any other student $C$ any more until the next command comes.)
Is it possible to arrange the students in the order $n,1, 2, \ldots, n-1$ after two commands ?
2016 India PRMO, 9
Let $a$ and $b$ be the roots of the equation $x^2 + x - 3 = 0$. Find the value of the expression $4 b^2 -a^3$.
2023 Auckland Mathematical Olympiad, 2
Triangle $ABC$ of area $1$ is given. Point $A'$ lies on the extension of side $BC$ beyond point $C$ with $BC = CA'$. Point $B'$ lies on extension of side $CA$ beyond $A$ and $CA = AB'$. $C'$ lies on extension of $AB$ beyond $B$ with $AB = BC'$. Find the area of triangle $A'B'C'$.
2009 Mathcenter Contest, 5
Let $a$ and $b$ be real numbers, where $a \not= 0$ and $a \not= b$ and all the roots of the equation $ax^{3}-x^{2}+bx-1 = 0$ is a real and positive number. Find the smallest possible value of $P = \dfrac{5a^{2}-3ab+2}{a^{2}(b-a)}$.
[i](Heir of Ramanujan)[/i]
2019 Jozsef Wildt International Math Competition, W. 20
[list=1]
[*] Let $G$ be a $(4, 4)$ unoriented graph, 2-regulate, containing a cycle with the length 3. Find the characteristic polynomial $P_G (\lambda)$ , its spectrum $Spec (G)$ and draw the graph $G$.
[*] Let $G'$ be another 2-regulate graph, having its characteristic polynomial $P_{G'} (\lambda) = \lambda^4 - 4\lambda^2 + \alpha, \alpha \in \mathbb{R}$. Find the spectrum $Spec(G')$ and draw the graph $G'$.
[*] Are the graphs $G$ and $G'$ cospectral or isomorphic?
[/list]
2011 India National Olympiad, 6
Find all functions $f:\mathbb{R}\to \mathbb R$ satisfying
\[f(x+y)f(x-y)=\left(f(x)+f(y)\right)^2-4x^2f(y),\]
For all $x,y\in\mathbb R$.
2004 Harvard-MIT Mathematics Tournament, 8
A freight train leaves the town of Jenkinsville at $1:00$ PM traveling due east at constant speed. Jim, a hobo, sneaks onto the train and falls asleep. At the same time, Julie leaves Jenkinsville on her bicycle, traveling along a straight road in a northeasterly direction (but not due northeast) at $10$ miles per hour. At $1:12$ PM, Jim rolls over in his sleep and falls from the train onto the side of the tracks. He wakes up and immediately begins walking at $3:5$ miles per hour directly towards the road on which Julie is riding. Jim reaches the road at $2:12$ PM, just as Julie is riding by. What is the speed of the train in miles per hour?
2007 ITest, 2
Find the value of $a+b$ given that $(a,b)$ is a solution to the system \begin{align*}3a+7b&=1977,\\5a+b&=2007.\end{align*}
$\begin{array}{c@{\hspace{14em}}c@{\hspace{14em}}c} \textbf{(A) }488&\textbf{(B) }498&\end{array}$
2017 Online Math Open Problems, 8
A permutation of $\{1, 2, 3, \dots, 16\}$ is called \emph{blocksum-simple} if there exists an integer $n$ such that the sum of any $4$ consecutive numbers in the permutation is either $n$ or $n+1$. How many blocksum-simple permutations are there?
[i]Proposed by Yannick Yao[/i]
2007 ITest, 57
Let $T=\text{TNFTPP}$. How many positive integers are within $T$ of exactly $\lfloor \sqrt T\rfloor$ perfect squares? (Note: $0^2=0$ is considered a perfect square.)
2024 China Team Selection Test, 16
$m>1$ is an integer such that $[2m-\sqrt{m}+1, 2m]$ contains a prime. Prove that for any pairwise distinct positive integers $a_1$, $a_2$, $\dots$, $a_m$, there is always $1\leq i,j\leq m$ such that $\frac{a_i}{(a_i, a_j)}\geq m$.
2022 Girls in Math at Yale, 7
Given that six-digit positive integer $\overline{ABCDEF}$ has distinct digits $A,$ $B,$ $C,$ $D,$ $E,$ $F$ between $1$ and $8$, inclusive, and that it is divisible by $99$, find the maximum possible value of $\overline{ABCDEF}$.
[i]Proposed by Andrew Milas[/i]
2023 OMpD, 4
Are there integers $m, n \geq 2$ such that the following property is always true?
$$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$
2002 AMC 10, 21
Let $f$ be a real-valued function such that \[f(x)+2f\left(\dfrac{2002}x\right)=3x\] for all $x>0$. Find $f(2)$.
$\textbf{(A) }1000\qquad\textbf{(B) }2000\qquad\textbf{(C) }3000\qquad\textbf{(D) }4000\qquad\textbf{(E) }6000$
Russian TST 2017, P1
Prove that $\sqrt{a_1}+\sqrt{a_2}+\cdots+\sqrt{a_{119}}$ is an integer, where \[a_n=2-\frac{1}{n^2+\sqrt{n^4+1/4}}.\]
2025 India STEMS Category A, 1
Alice and Bob play a game. Initially, they write the pair $(1012,1012)$ on the board. They alternate their turns with Alice going first. In each turn the player can turn the pair $(a,b)$ to either $(a-2, b+1), (a+1, b-2)$ or $(a-1, b)$ as long as the resulting pair has only nonnegative values. The game terminates, when there is no legal move possible. Alice wins if the game terminates at $(0,0)$ and Bob wins if the game terminates at $(0,1)$. Determine who has the winning strategy?
[i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]
2006 AMC 10, 20
In rectangle $ ABCD$, we have $ A \equal{} (6, \minus{} 22)$, $ B \equal{} (2006,178)$, and $ D \equal{} (8,y)$, for some integer $ y$. What is the area of rectangle $ ABCD$?
$ \textbf{(A) } 4000 \qquad \textbf{(B) } 4040 \qquad \textbf{(C) } 4400 \qquad \textbf{(D) } 40,000 \qquad \textbf{(E) } 40,400$
2000 Estonia National Olympiad, 4
Let us define the sequences $a_1, a_2, a_3,...$ and $b_1, b_2, b_3,...$. with the following conditions
$a_1 = 3, b_1 = 1$ and $a_{n +1} =\frac{a_n^2+b_n^2}{2}$ and $b_{n + 1}= a_n \cdot b_n$ for each $n = 1, 2,...$.
Find all different prime factors οf the number $a_{2000} + b_{2000}$.
2014 Contests, 2
Find all continuous function $f:\mathbb{R}^{\geq 0}\rightarrow \mathbb{R}^{\geq 0}$ such that :
\[f(xf(y))+f(f(y)) = f(x)f(y)+2 \: \: \forall x,y\in \mathbb{R}^{\geq 0}\]
[i]Proposed by Mohammad Ahmadi[/i]
2011 Akdeniz University MO, 2
Let $a$ and $b$ is roots of the $x^2-6x+1$ equation.
[b]a[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $a^n+b^n$ is a integer.
[b]b[/b]) Show that, for all $n \in{\mathbb Z^+}$ , $5$ isn't divide $a^n+b^n$
2017-2018 SDML (Middle School), 12
If $n$ is an integer such that $2 \leq n \leq 2017$, for how many values of $n$ is $\left(1 + \frac{1}{2}\right)\left(1 + \frac{1}{3}\right)\cdots\left(1 + \frac{1}{n}\right)$ equal to a positive integer?
$\mathrm{(A) \ } 0 \qquad \mathrm{(B) \ } 1 \qquad \mathrm {(C) \ } 1007 \qquad \mathrm{(D) \ } 1008 \qquad \mathrm{(E) \ } 2016$
2012 National Olympiad First Round, 24
There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.
These checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checkers and all of the checkers between the two. At least how many moves are required to make all checkers same color?
$ \textbf{(A)}\ 1006 \qquad \textbf{(B)}\ 1204 \qquad \textbf{(C)}\ 1340 \qquad \textbf{(D)}\ 2011 \qquad \textbf{(E)}\ \text{None}$
1979 AMC 12/AHSME, 22
Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$.
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }9\qquad\textbf{(E) }\text{infinitely many}$
2016 CHMMC (Fall), 6
For any nonempty set of integers $X$, define the function $$f(X) = \frac{(-1)^{|X|}}{ \left(\prod_{k\in X} k \right)^2}$$ where $|X|$ denotes the number of elements in $X$.
Consider the set $S = \{2, 3, . . . , 13\}$ . Note that $1$ is not an element of $S$.
Compute $$\sum_{T\subseteq S, T \ne \emptyset} f(T).$$
where the sum is taken over all nonempty subsets $T$ of $S$.