Found problems: 85335
2020 Online Math Open Problems, 10
Compute the number of functions $f\colon\{1, \dots, 15\} \to \{1, \dots, 15\}$ such that, for all $x \in \{1, \dots, 15\}$, \[
\frac{f(f(x)) - 2f(x) + x}{15}
\]is an integer.
[i]Proposed by Ankan Bhattacharya[/i]
1974 AMC 12/AHSME, 22
The minimum of $ \sin \frac{A}{2} \minus{} \sqrt3 \cos \frac{A}{2}$ is attained when $ A$ is
$ \textbf{(A)}\ \minus{}180^{\circ} \qquad
\textbf{(B)}\ 60^{\circ} \qquad
\textbf{(C)}\ 120^{\circ} \qquad
\textbf{(D)}\ 0^{\circ} \qquad
\textbf{(E)}\ \text{none of these}$
2010 Today's Calculation Of Integral, 570
Let $ f(x) \equal{} 1 \minus{} \cos x \minus{} x\sin x$.
(1) Show that $ f(x) \equal{} 0$ has a unique solution in $ 0 < x < \pi$.
(2) Let $ J \equal{} \int_0^{\pi} |f(x)|dx$. Denote by $ \alpha$ the solution in (1), express $ J$ in terms of $ \sin \alpha$.
(3) Compare the size of $ J$ defined in (2) with $ \sqrt {2}$.
2003 China Team Selection Test, 1
Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.
1970 IMO, 3
The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$.
[b]a.)[/b] Prove that $0\le b_n<2$.
[b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.
2020 Italy National Olympiad, #5
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions:
1) $f$ is surjective
2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$)
3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$).
Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.
2009 National Olympiad First Round, 18
$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ?
$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$
1991 Brazil National Olympiad, 4
Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).
Cono Sur Shortlist - geometry, 2012.G2
Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.
2002 Korea Junior Math Olympiad, 8
On a long metal stick, $1000$ red marbles are embedded in the stick so the stick is equally partitioned into $1001$ parts by them. $1001$ blue marbles are embedded in the stick too, so the stick is equally partitioned into $1002$ parts by them. If you cut the metal stick equally into $2003$ smaller parts, how many of the smaller parts would contain at least one embedded marble?
1981 Putnam, B2
Determine the minimum value of
$$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$
for all real numbers $1\leq r \leq s \leq t \leq 4.$
1966 AMC 12/AHSME, 32
Let $M$ be the midpoint of side $AB$ of the triangle $ABC$. Let$P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of the triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then
$\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\
\text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\
\text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\
\text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\
\text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P$
Ukrainian TYM Qualifying - geometry, 2019.8
Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that:
a) all three drawn triangles are equal to each other?
b) among them there are at least two equal triangles?
2013 Junior Balkan Team Selection Tests - Romania, 1
Find all pairs of integers $(x,y)$ satisfying the following condition:
[i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$
[/i]
Tournament of Towns
2015 Balkan MO Shortlist, A6
For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid
$$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$
and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$?
(Serbia)
1947 Moscow Mathematical Olympiad, 124
a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected.
b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.
2022 USAMO, 3
Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that for all $x,y\in \mathbb{R}_{>0}$ we have
\[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]
2023 Indonesia TST, 2
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
1973 USAMO, 2
Let $ \{X_n\}$ and $ \{Y_n\}$ denote two sequences of integers defined as follows:
\begin{align*} X_0 \equal{} 1,\ X_1 \equal{} 1,\ X_{n \plus{} 1} \equal{} X_n \plus{} 2X_{n \minus{} 1} \quad (n \equal{} 1,2,3,\ldots), \\
Y_0 \equal{} 1,\ Y_1 \equal{} 7,\ Y_{n \plus{} 1} \equal{} 2Y_n \plus{} 3Y_{n \minus{} 1} \quad (n \equal{} 1,2,3,\ldots).\end{align*}
Prove that, except for the "1", there is no term which occurs in both sequences.
2018 Tournament Of Towns, 4.
Let O be the center of the circumscribed circle of the triangle ABC. Let AH be the altitude in this triangle, and let P be the base of the perpendicular drawn from point A to the line CO. Prove that the line HP passes through the midpoint of the side AB. (6 points)
Egor Bakaev
2024 HMNT, 25
Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$
2001 District Olympiad, 3
Consider an inscriptible polygon $ABCDE$. Let $H_1,H_2,H_3,H_4,H_5$ be the orthocenters of the triangles $ABC,BCD,CDE,DEA,EAB$ and let $M_1,M_2,M_3,M_4,M_5$ be the midpoints of $DE,EA,AB,BC$ and $CD$, respectively. Prove that the lines $H_1M_1,H_2M_2,H_3M_3,H_4M_4,H_5M_5$ have a common point.
[i]Dinu Serbanescu[/i]
2023 Korea National Olympiad, 6
Let $\Omega$ and $O$ be the circumcircle and the circumcenter of an acute triangle $ABC$ $(\overline{AB} < \overline{AC})$. Define $D,E(\neq A)$ be the points such that ray $AO$ intersects $BC$ and $\Omega$. Let the line passing through $D$ and perpendicular to $AB$ intersects $AC$ at $P$ and define $Q$ similarly. Tangents to $\Omega$ on $A,E$ intersects $BC$ at $X,Y$. Prove that $X,Y,P,Q$ lie on a circle.
2008 Turkey MO (2nd round), 2
A circle $ \Gamma$ and a line $ \ell$ is given in a plane such that $ \ell$ doesn't cut $ \Gamma$.Determine the intersection set of the circles has $ [AB]$ as diameter for all pairs of $ \left\{A,B\right\}$ (lie on $ \ell$) and satisfy $ P,Q,R,S \in \Gamma$ such that $ PQ \cap RS\equal{}\left\{A\right\}$ and $ PS \cap QR\equal{}\left\{B\right\}$
2019 AMC 12/AHSME, 2
Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?
$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$