Found problems: 85335
2008 Harvard-MIT Mathematics Tournament, 4
The function $ f$ satisfies \[f(x) \plus{} f(2x \plus{} y) \plus{} 5xy \equal{} f(3x \minus{} y) \plus{} 2x^2 \plus{} 1\] for all real numbers $ x$, $ y$. Determine the value of $ f(10)$.
2007 Greece National Olympiad, 2
Let $a,b,c$ be sides of a triangle, show that
\[\frac{(c+a-b)^{4}}{a(a+b-c)}+\frac{(a+b-c)^{4}}{b(b+c-a)}+\frac{(b+c-a)^{4}}{c(c+a-b)}\geq ab+bc+ca.\]
1981 IMO, 2
Take $r$ such that $1\le r\le n$, and consider all subsets of $r$ elements of the set $\{1,2,\ldots,n\}$. Each subset has a smallest element. Let $F(n,r)$ be the arithmetic mean of these smallest elements. Prove that: \[ F(n,r)={n+1\over r+1}. \]
2016 Czech-Polish-Slovak Junior Match, 2
Let $x$ and $y$ be real numbers such that $x^2 + y^2 - 1 < xy$. Prove that $x + y - |x - y| < 2$.
Slovakia
2012 Tournament of Towns, 4
In a triangle $ABC$ two points, $C_1$ and $A_1$ are marked on the sides $AB$ and $BC$ respectively (the points do not coincide with the vertices). Let $K$ be the midpoint of $A_1C_1$ and $I$ be the incentre of the triangle $ABC$. Given that the quadrilateral $A_1BC_1I$ is cyclic, prove that the angle $AKC$ is obtuse.
2000 Chile National Olympiad, 4
Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $
2023 AMC 10, 3
A $3-4-5$ right triangle is inscribed in circle $A$, and a $5-12-13$ right triangle is inscribed in circle $B$. What is the ratio of the area of circle $A$ to the area of circle $B$?
$\textbf{(A)}~\frac{9}{25}\qquad\textbf{(B)}~\frac{1}{9}\qquad\textbf{(C)}~\frac{1}{5}\qquad\textbf{(D)}~\frac{25}{169}\qquad\textbf{(E)}~\frac{4}{25}$
1998 Iran MO (3rd Round), 2
Let $ABCDEF$ be a convex hexagon such that $AB = BC, CD = DE$ and $EF = FA$. Prove that
\[\frac{AB}{BE}+\frac{CD}{AD}+\frac{EF}{CF} \geq \frac{3}{2}.\]
2002 Cono Sur Olympiad, 1
Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and Julián do the same, emptying the ballot box. Pedro informs the teacher that he has picked three consecutive numbers whose product is $5$
times the sum. Ana informs that she has no prime number, but two consecutive and that the product of these three numbers is $4$ times the sum of them. What numbers did Julian remove?
2018 Taiwan TST Round 3, 6
For any finite sets $X$ and $Y$ of positive integers, denote by $f_X(k)$ the $k^{\text{th}}$ smallest positive integer not in $X$, and let $$X*Y=X\cup \{ f_X(y):y\in Y\}.$$Let $A$ be a set of $a>0$ positive integers and let $B$ be a set of $b>0$ positive integers. Prove that if $A*B=B*A$, then $$\underbrace{A*(A*\cdots (A*(A*A))\cdots )}_{\text{ A appears $b$ times}}=\underbrace{B*(B*\cdots (B*(B*B))\cdots )}_{\text{ B appears $a$ times}}.$$
[i]Proposed by Alex Zhai, United States[/i]
1995 IMO Shortlist, 6
Let $ n$ be an integer,$ n \geq 3.$ Let $ x_1, x_2, \ldots, x_n$ be real numbers such that $ x_i < x_{i\plus{}1}$ for $ 1 \leq i \leq n \minus{} 1$. Prove that
\[ \frac{n(n\minus{}1)}{2} \sum_{i < j} x_ix_j > \left(\sum^{n\minus{}1}_{i\equal{}1} (n\minus{}i)\cdot x_i \right) \cdot \left(\sum^{n}_{j\equal{}2} (j\minus{}1) \cdot x_j \right)\]
2017 Saudi Arabia BMO TST, 3
We put four numbers $1,2, 3,4$ around a circle in order. One starts at the number $1$ and every step, he moves to an adjacent number on either side. How many ways he can move such that sum of the numbers he visits in his path (including the starting number) is equal to $21$?
2021 Centroamerican and Caribbean Math Olympiad, 1
An ordered triple $(p, q, r)$ of prime numbers is called [i]parcera[/i] if $p$ divides $q^2-4$, $q$ divides $r^2-4$ and $r$ divides $p^2-4$. Find all parcera triples.
2009 Costa Rica - Final Round, 5
Suppose the polynomial $ x^{n} \plus{} a_{n \minus{} 1}x^{n \minus{} 1} \plus{} ... \plus{} a_{1} \plus{} a_{0}$ can be factorized as $ (x \plus{} r_{1})(x \plus{} r_{2})...(x \plus{} r_{n})$, with $ r_{1}, r_{2}, ..., r_{n}$ real numbers.
Show that $ (n \minus{} 1)a_{n \minus{} 1}^{2}\geq\ 2na_{n \minus{} 2}$
1997 Putnam, 1
For all reals $x$ define $\{x\}$ to be the difference between $x$ and the closest integer to $x$. For each positive integer $n$ evaluate :
\[ S_n=\sum_{m=1}^{6n-1}\min \left(\left\{\frac{m}{6n}\right\},\left\{\frac{m}{3n}\right\}\right) \]
2002 IMO Shortlist, 2
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
2015 South East Mathematical Olympiad, 4
Given $8$ pairwise distinct positive integers $a_1,a_2,…,a_8$ such that the greatest common divisor of any three of them is equal to $1$. Show that there exists positive integer $n\geq 8$ and $n$ pairwise distinct positive integers $m_1,m_2,…,m_n$ with the greatest common divisor of all $n$ numbers equal to $1$ such that for any positive integers $1\leq p<q<r\leq n$, there exists positive integers $1\leq i<j\leq 8$ that $a_ia_j\mid m_p+m_q+m_r$.
1991 Canada National Olympiad, 4
Can ten distinct numbers $a_1, a_2, b_1, b_2, b_3, c_1, c_2, d_1, d_2, d_3$ be chosen from $\{0, 1, 2, \ldots, 14\}$, so that the $14$ differences $|a_1 - b_1|$, $|a_1 - b_2|$, $|a_1 - b_3|$, $|a_2 - b_1|$, $|a_2 - b_2|$, $|a_2 - b_3|$, $|c_1 - d_1|$, $|c_1 - d_2|$, $|c_1 - d_3|$, $|c_2 - d_1|$, $|c_2 - d_2|$, $|c_2 - d_3|$, $|a_1 - c_1|$, and $|a_2 - c_2|$ are all distinct?
1987 IMO Longlists, 23
A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a distance $d$ from it. By distance is meant the shortest distance measured over the curved survace of the lampshade. Prove that the area of the lampshade is $d^2(2\theta + \sqrt 3)$ where $\cot \frac {\theta}{2} = \frac{3}{\theta}.$
2021 German National Olympiad, 5
a) Determine the largest real number $A$ with the following property: For all non-negative real numbers $x,y,z$, one has
\[\frac{1+yz}{1+x^2}+\frac{1+zx}{1+y^2}+\frac{1+xy}{1+z^2} \ge A.\]
b) For this real number $A$, find all triples $(x,y,z)$ of non-negative real numbers for which equality holds in the above inequality.
2025 Chile TST IMO-Cono, 3
Let \( a, b, c, d \) be real numbers such that \( abcd = 1 \), and
\[
a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0.
\]
Prove that one of the numbers \( ab, ac \) or \( ad \) is equal to \( -1 \).
2013 BMT Spring, 1
A rectangle with sides $a$ and $b$ has an area of $24$ and a diagonal of length $11$. Find the perimeter of this rectangle.
2014 AMC 10, 4
Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana?
$ \textbf {(A) } \frac{3}{2} \qquad \textbf {(B) } \frac{5}{3} \qquad \textbf {(C) } \frac{7}{4} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \frac{13}{4}$
2010 Stanford Mathematics Tournament, 14
A series of lockers, numbered 1 through 100, are all initially closed. Student 1 goes through and opens every locker. Student 3 goes through and "flips" every 3rd locker ("flipping") a locker means changing its state: if the locker is open he closes it, and if the locker is closed he opens it). Thus, Student 3 will close the third locker, open the sixth, close the ninth. . . . Student 5 then goes through and "flips"every 5th locker. This process continues with all students with odd numbers $n<100$ going through and "flipping" every $n$th locker.
How many lockers are open after this process?
2018 Brazil Undergrad MO, 13
A continuous function $ f: \mathbb {R} \to \mathbb {R} $ satisfies $ f (x) f (f (x)) = 1 $ for every real $ x $ and $ f (2020) = 2019 $ . What is the value of $ f (2018) $?