Found problems: 85335
2018 Iran MO (2nd Round), 4
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:
$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$
for all reals $x, y $.
2016 Postal Coaching, 5
Is it possible to define an operation $\star$ on $\mathbb Z$ such that[list=a][*] for any $a, b, c$ in $\mathbb Z, (a \star b) \star c = a \star (b \star c)$ holds;
[*] for any $x, y$ in $\mathbb Z, x \star x \star y = y \star x \star x=y$?[/list]
2022 HMNT, 19
Define the [i]annoyingness[/i] of a permutation of the first $n$ integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence $1,2 \ldots ,n$ appears. For instance, the annoyingness of $3,2,1$ is $3,$ and the annoyingness of $1,3,4,2$ is $2.$
A random permutation of $1,2, \ldots, 2022$ is selected. Compute the expected value of the annoyingness of this permutation.
2021 BMT, 9
Druv has a $33 \times 33$ grid of unit squares, and he wants to color each unit square with exactly one of three distinct colors such that he uses all three colors and the number of unit squares with each color is the same. However, he realizes that there are internal sides, or unit line segments that have exactly one unit square on each side, with these two unit squares having different colors. What is the minimum possible number of such internal sides?
2007 Regional Olympiad of Mexico Center Zone, 6
Certain tickets are numbered as follows: $1, 2, 3, \dots, N $. Exactly half of the tickets have the digit $ 1$ on them. If $N$ is a three-digit number, determine all possible values ​​of $N $.
1997 Czech And Slovak Olympiad IIIA, 3
A tetrahedron $ABCD$ is divided into five polyhedra so that each face of the tetrahedron is a face of (exactly) one polyhedron, and that the intersection of any two of the polyhedra is either a common vertex, a common edge, or a common face. What is the smallest possible sum of the numbers of faces of the five polyhedra?
2022 Kazakhstan National Olympiad, 1
Given a triangle $ABC$ draw the altitudes $AD$, $BE$, $CF$. Take points $P$ and $Q$ on $AB$ and $AC$, respectively such that $PQ \parallel BC$. Draw the circles with diameters $BQ$ and $CP$ and let them intersect at points $R$
and $T$ where $R$ is closer to $A$ than $T$. Draw the altitudes $BN$ and $CM$ in the triangle $BCR$. Prove that $FM$, $EN$ and $AD$ are concurrent.\\
2003 Iran MO (2nd round), 2
$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that:
\[ BT+CT\leq{2R}. \]
1986 China Team Selection Test, 3
Given a positive integer $A$ written in decimal expansion: $(a_{n},a_{n-1}, \ldots, a_{0})$ and let $f(A)$ denote $\sum^{n}_{k=0} 2^{n-k}\cdot a_k$. Define $A_1=f(A), A_2=f(A_1)$. Prove that:
[b]I.[/b] There exists positive integer $k$ for which $A_{k+1}=A_k$.
[b]II.[/b] Find such $A_k$ for $19^{86}.$
2012 International Zhautykov Olympiad, 1
Do there exist integers $m, n$ and a function $f\colon \mathbb R \to \mathbb R$ satisfying simultaneously the following two conditions?
$\bullet$ i) $f(f(x))=2f(x)-x-2$ for any $x \in \mathbb R$;
$\bullet$ ii) $m \leq n$ and $f(m)=n$.
1969 IMO Shortlist, 63
$(SWE 6)$ Prove that there are infinitely many positive integers that cannot be expressed as the sum of squares of three positive integers.
PEN F Problems, 8
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
2002 China Team Selection Test, 3
Find all groups of positive integers $ (a,x,y,n,m)$ that satisfy $ a(x^n \minus{} x^m) \equal{} (ax^m \minus{} 4) y^2$ and $ m \equiv n \pmod{2}$ and $ ax$ is odd.
1988 China Team Selection Test, 2
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
2021 USA TSTST, 8
Let $ABC$ be a scalene triangle. Points $A_1,B_1$ and $C_1$ are chosen on segments $BC,CA$ and $AB$, respectively, such that $\triangle A_1B_1C_1$ and $\triangle ABC$ are similar. Let $A_2$ be the unique point on line $B_1C_1$ such that $AA_2=A_1A_2$. Points $B_2$ and $C_2$ are defined similarly. Prove that $\triangle A_2B_2C_2$ and $\triangle ABC$ are similar.
[i]Fedir Yudin [/i]
1983 IMO Shortlist, 2
Let $n$ be a positive integer. Let $\sigma(n)$ be the sum of the natural divisors $d$ of $n$ (including $1$ and $n$). We say that an integer $m \geq 1$ is [i]superabundant[/i] (P.Erdos, $1944$) if $\forall k \in \{1, 2, \dots , m - 1 \}$, $\frac{\sigma(m)}{m} >\frac{\sigma(k)}{k}.$
Prove that there exists an infinity of [i]superabundant[/i] numbers.
1950 AMC 12/AHSME, 25
The value of $ \log_5 \frac {(125)(625)}{25}$ is equal to:
$\textbf{(A)}\ 725 \qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 3125 \qquad
\textbf{(D)}\ 5 \qquad
\textbf{(E)}\ \text{None of these}$
2019 LIMIT Category B, Problem 2
Let $\mathbb C$ denote the set of all complex numbers. Define
$$A=\{(z,w)|z,w\in\mathbb C\text{ and }|z|=|w|\}$$$$B=\{(z,w)|z,w\in\mathbb C\text{ and }z^2=w^2\}$$$\textbf{(A)}~A=B$
$\textbf{(B)}~A\subset B\text{ and }A\ne B$
$\textbf{(C)}~B\subset A\text{ and }B\ne A$
$\textbf{(D)}~\text{None of the above}$
2019 CMIMC, 3
Points $A(0,0)$ and $B(1,1)$ are located on the parabola $y=x^2$. A third point $C$ is positioned on this parabola between $A$ and $B$ such that $AC=CB=r$. What is $r^2$?
2022-IMOC, C4
Let $N$ be a given positive integer. Consider a permutation of $1,2,3,\cdots,N$, denoted as $p_1,p_2,\cdots,p_N$. For a section $p_l, p_{l+1},\cdots, p_r$, we call it "extreme" if $p_l$ and $p_r$ are the maximum and minimum value of that section. We say a permutation $p_1,p_2,\cdots,p_N$ is "super balanced" if there isn't an "extreme" section with a length at least $3$. For example, $1,4,2,3$ is "super balanced", but $3,1,2,4$ isn't. Please answer the following questions:
1. How many "super balanced" permutations are there?
2. For each integer $M\leq N$. How many "super balanced" permutations are there such that $p_1=M$?
[i]Proposed by ltf0501[/i]
2024 239 Open Mathematical Olympiad, 8
There are $2n$ points on the plane. No three of them lie on the same straight line and no four lie on the same circle. Prove that it is possible to split these points into $n$ pairs and cover each pair of points with a circle containing no other points.
Durer Math Competition CD 1st Round - geometry, 2011.D5
Is it true that in every convex polygon $3$ adjacent vertices can be selected such that their circumcirscribed circle can cover the entire polygon?
2007 AIME Problems, 3
The complex number $z$ is equal to $9+bi$, where $b$ is a positive real number and $i^{2}=-1$. Given that the imaginary parts of $z^{2}$ and $z^{3}$ are equal, find $b$.
2022 Kyiv City MO Round 1, Problem 4
Let's call integer square-free if it's not divisible by $p^2$ for any prime $p$. You are given a square-free integer $n>1$, which has exactly $d$ positive divisors. Find the largest number of its divisors that you can choose, such that $a^2 + ab - n$ isn't a square of an integer for any $a, b$ among chosen divisors.
[i](Proposed by Oleksii Masalitin)[/i]
2015 Saudi Arabia Pre-TST, 2.2
Find all functions $f : R \to R$ that satisfy $f(x + y^2 - f(y)) = f(x)$ for all $x,y \in R$.
(Vo Quoc Ba Can)