Found problems: 85335
2022 Durer Math Competition Finals, 3
Let $x, y, z$ denote positive real numbers for which $x+y+z = 1$ and $x > yz$, $y > zx$, $z > xy$. Prove that
$$\left(\frac{x - yz}{x + yz}\right)^2+ \left(\frac{y - zx}{y + zx}\right)^2+\left(\frac{z - xy}{z + xy}\right)^2< 1.$$
2016 Taiwan TST Round 2, 2
Let $x,y$ be positive real numbers such that $x+y=1$.
Prove that$\frac{x}{x^2+y^3}+\frac{y}{x^3+y^2}\leq2(\frac{x}{x+y^2}+\frac{y}{x^2+y})$.
2018 Indonesia MO, 4
In a game, Andi and a computer take turns. At the beginning, the computer shows a polynomial $x^2 + mx + n$ where $m,n \in \mathbb{Z}$, such that it doesn't have real roots. Andi then begins the game. On his turn, Andi may change a polynomial in the form $x^2 + ax + b$ into either $x^2 + (a+b)x + b$ or $x^2 + ax + (a+b)$. However, Andi may only choose a polynomial that has real roots. On the computer's turn, it simply switches the coefficient of $x$ and the constant of the polynomial. Andi loses if he can't continue to play. Find all $(m,n)$ such that Andi always loses (in finitely many turns).
2014 Belarus Team Selection Test, 2
Let $x,y,z$ be pairwise distinct real numbers such that $x^2-1/y = y^2 -1/z = z^2 -1/x$.
Given $z^2 -1/x = a$, prove that $(x + y + z)xyz= -a^2$.
(I. Voronovich)
2008 Romania Team Selection Test, 3
Let $ \mathcal{P}$ be a square and let $ n$ be a nonzero positive integer for which we denote by $ f(n)$ the maximum number of elements of a partition of $ \mathcal{P}$ into rectangles such that each line which is parallel to some side of $ \mathcal{P}$ intersects at most $ n$ interiors (of rectangles). Prove that
\[ 3 \cdot 2^{n\minus{}1} \minus{} 2 \le f(n) \le 3^n \minus{} 2.\]
2008 Baltic Way, 11
Consider a subset $A$ of $84$ elements of the set $\{1,\,2,\,\dots,\,169\}$ such that no two elements in the set add up to $169$. Show that $A$ contains a perfect square.
2014 USAJMO, 1
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
2002 SNSB Admission, 3
Classify up to homeomorphism the topological spaces of the support of functions that are real quadratic polynoms of three variables and and irreducible over the set of real numbers.
2002 HKIMO Preliminary Selection Contest, 14
In $\triangle ABC$, $\angle ACB=3\angle BAC$, $BC=5$, $AB=11$. Find $AC$
2011 AMC 12/AHSME, 11
A frog located at $(x,y)$, with both $x$ and $y$ integers, makes successive jumps of length $5$ and always lands on points with integer coordinates. Suppose that the frog starts at $(0,0)$ and ends at $(1,0)$. What is the smallest possible number of jumps the frog makes?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 3 \qquad
\textbf{(C)}\ 4 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ 6 $
2021 Spain Mathematical Olympiad, 6
Let $ABC$ be a triangle with $AB \neq AC$, let $I$ be its incenter, $\gamma$ its inscribed circle and $D$ the midpoint of $BC$. The tangent to $\gamma$ from $D$ different to $BC$ touches $\gamma$ in $E$. Prove that $AE$ and $DI$ are parallel.
2007 Iran Team Selection Test, 3
Find all solutions of the following functional equation: \[f(x^{2}+y+f(y))=2y+f(x)^{2}. \]
2018 MMATHS, 4
A sequence of integers fsng is defined as follows: fix integers $a$, $b$, $c$, and $d$, then set $s_1 = a$, $s_2 = b$, and $$s_n = cs_{n-1} + ds_{n-2}$$ for all $n \ge 3$. Create a second sequence $\{t_n\}$ by defining each $t_n$ to be the remainder when $s_n$ is divided by $2018$ (so we always have $0 \le t_n \le 2017$). Let $N = (2018^2)!$. Prove that $t_N = t_{2N}$ regardless of the choices of $a$, $b$, $c$, and $d$.
2003 China Team Selection Test, 2
Let $x<y$ be positive integers and $P=\frac{x^3-y}{1+xy}$. Find all integer values that $P$ can take.
1991 IMO, 3
An infinite sequence $ \,x_{0},x_{1},x_{2},\ldots \,$ of real numbers is said to be [b]bounded[/b] if there is a constant $ \,C\,$ such that $ \, \vert x_{i} \vert \leq C\,$ for every $ \,i\geq 0$. Given any real number $ \,a > 1,\,$ construct a bounded infinite sequence $ x_{0},x_{1},x_{2},\ldots \,$ such that
\[ \vert x_{i} \minus{} x_{j} \vert \vert i \minus{} j \vert^{a}\geq 1
\]
for every pair of distinct nonnegative integers $ i, j$.
2020 May Olympiad, 4
Maria has a $6 \times 5$ board with some shaded squares, as in the figure. She writes, in some order, the digits $1, 2, 3, 4$ and $5$ in the first row and then completes the board as follows: look at the number written in the shaded box and write the number that occupies the position indicated by the box shaded as the last number in the next row, and repeat the other numbers in the first four squares, following the same order as in the previous row.
For example, if you wrote $2, 3, 4, 1, 5$ in the first row, then since $4$ is in the shaded box, the number that occupies the fourth place $(1)$ is written in the last box of the second row and completes it with the remaining numbers in the order in which. They were. She remains: $2, 3, 4, 5, 1$.
Then, to complete the third row, as in the shaded box is $3$, the number located in the third place $(4)$ writes it in the last box and gets $2, 3, 5, 1, 4$. Following in the same way, he gets the board of the figure.
Show a way to locate the numbers in the first row to get the numbers $2, 4, 5, 1, 3$ in the last row.
2014 Bundeswettbewerb Mathematik, 2
For all positive integers $m$ and $k$ with $m\ge k$, define $a_{m,k}=\binom{m}{k-1}-3^{m-k}$.
Determine all sequences of real numbers $\{x_1, x_2, x_3, \ldots\}$, such that each positive integer $n$ satisfies the equation
\[a_{n,1}x_1+ a_{n,2}x_2+ \cdots + a_{n,n}x_n = 0\]
2006 Iran MO (3rd Round), 5
$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that \[\cos B+\cos C=1\Longleftrightarrow MI=MT\]
2012 NIMO Problems, 3
For positive integers $1 \le n \le 100$, let \[ f(n) = \sum_{i=1}^{100} i\left\lvert i-n \right\rvert. \] Compute $f(54)-f(55)$.
[i]Proposed by Aaron Lin[/i]
2013 Stanford Mathematics Tournament, 3
Robin has obtained a circular pizza with radius $2$. However, being rebellious, instead of slicing the pizza radially, he decides to slice the pizza into $4$ strips of equal width both vertically and horizontally. What is the area of the smallest piece of pizza?
2019 Serbia JBMO TST, 2
If a b c positive reals smaller than 1, prove:
a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)
2015 IMC, 9
An $n \times n$ complex matrix $A$ is called \emph{t-normal} if
$AA^t = A^t A$ where $A^t$ is the transpose of $A$. For each $n$,
determine the maximum dimension of a linear space of complex $n
\times n$ matrices consisting of t-normal matrices.
Proposed by Shachar Carmeli, Weizmann Institute of Science
1983 IMO Longlists, 12
The number $0$ or $1$ is to be assigned to each of the $n$ vertices of a regular polygon. In how many different ways can this be done (if we consider two assignments that can be obtained one from the other through rotation in the plane of the polygon to be identical)?
1967 Swedish Mathematical Competition, 1
$p$ parallel lines are drawn in the plane and $q$ lines perpendicular to them are also drawn. How many rectangles are bounded by the lines?
1966 Putnam, B3
Show that if the series $$\sum_{n=1}^{\infty} \frac{1}{p_n}$$ is convergent, where $p_1,p_2,p_3,\dots, p_n, \dots$ are positive real numbers, then the series $$\sum_{n=1}^{\infty} \frac{n^2}{(p_1+p_2+\dots +p_n)^2}p_n$$ is also convergent.