Found problems: 85335
2020 Kazakhstan National Olympiad, 4
The incircle of the triangle $ ABC $ touches the sides of $ AB, BC, CA $ at points $ C_0, A_0, B_0 $, respectively. Let the point $ M $ be the midpoint of the segment connecting the vertex $ C_0 $ with the intersection point of the altitudes of the triangle $ A_0B_0C_0 $, point $ N $ be the midpoint of the arc $ ACB $ of the circumscribed circle of the triangle $ ABC $. Prove that line $ MN $ passes through the center of incircle of triangle $ ABC $.
2006 AMC 10, 2
Define $ x\otimes y \equal{} x^3 \minus{} y$. What is $ h\otimes (h\otimes h)$?
$ \textbf{(A) } \minus{} h\qquad \textbf{(B) } 0\qquad \textbf{(C) } h\qquad \textbf{(D) } 2h\qquad \textbf{(E) } h^3$
2017 China Second Round Olympiad, 1
Given an isocleos triangle $ABC$ with equal sides $AB=AC$ and incenter $I$.Let $\Gamma_1$be the circle centered at $A$ with radius $AB$,$\Gamma_2$ be the circle centered at $I$ with radius $BI$.A circle $\Gamma_3$ passing through $B,I$ intersects $\Gamma_1$,$\Gamma_2$ again at $P,Q$ (different from $B$) respectively.Let $R$ be the intersection of $PI$ and $BQ$.Show that $BR \perp CR$.
2020 BMT Fall, 6
Jack writes whole numbers starting from $ 1$ and skips all numbers that contain either a $2$ or $9$. What is the $100$th number that Jack writes down?
2015 Lusophon Mathematical Olympiad, 4
Let $a$ be a real number, such that $a\ne 0, a\ne 1, a\ne -1$ and $m,n,p,q$ be natural numbers .
Prove that if $a^m+a^n=a^p+a^q$ and $a^{3m}+a^{3n}=a^{3p}+a^{3q}$ , then $m \cdot n = p \cdot q$.
2012 Kyoto University Entry Examination, 3
When real numbers $x,\ y$ moves in the constraint with $x^2+xy+y^2=6.$
Find the range of $x^2y+xy^2-x^2-2xy-y^2+x+y.$
30 points
2017 Korea - Final Round, 2
For a positive integer $n$, $(a_0, a_1, \cdots , a_n)$ is a $n+1$-tuple with integer entries.
For all $k=0, 1, \cdots , n$, we denote $b_k$ as the number of $k$s in $(a_0, a_1, \cdots ,a_n)$.
For all $k = 0,1, \cdots , n$, we denote $c_k$ as the number of $k$s in $(b_0, b_1, \cdots ,b_n)$.
Find all $(a_0, a_1, \cdots ,a_n)$ which satisfies $a_0 = c_0$, $a_1=c_1$, $\cdots$, $a_n=c_n$.
1949 Putnam, B4
Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion
$$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$
are positive integers.
2013 IPhOO, 8
A right-triangulated prism made of benzene sits on a table. The hypotenuse makes an angle of $30^\circ$ with the horizontal table. An incoming ray of light hits the hypotenuse horizontally, and leaves the prism from the vertical leg at an acute angle of $ \gamma $ with respect to the vertical leg. Find $\gamma$, in degrees, to the nearest integer. The index of refraction of benzene is $1.50$.
[i](Proposed by Ahaan Rungta)[/i]
2013 National Olympiad First Round, 16
$16$ white and $4$ red balls that are not identical are distributed randomly into $4$ boxes which contain at most $5$ balls. What is the probability that each box contains exactly $1$ red ball?
$
\textbf{(A)}\ \dfrac{5}{64}
\qquad\textbf{(B)}\ \dfrac{1}{8}
\qquad\textbf{(C)}\ \dfrac{4^4}{\binom{16}{4}}
\qquad\textbf{(D)}\ \dfrac{5^4}{\binom{20}{4}}
\qquad\textbf{(E)}\ \dfrac{3}{32}
$
2016 Novosibirsk Oral Olympiad in Geometry, 5
In the parallelogram $CMNP$ extend the bisectors of angles $MCN$ and $PCN$ and intersect with extensions of sides PN and $MN$ at points $A$ and $B$, respectively. Prove that the bisector of the original angle $C$ of the the parallelogram is perpendicular to $AB$.
[img]https://cdn.artofproblemsolving.com/attachments/f/3/fde8ef133758e06b1faf8bdd815056173f9233.png[/img]
2006 Stanford Mathematics Tournament, 14
Determine the area of the region defined by [i]x[/i]²+[i]y[/i]²≤[i]π[/i]² and [i]y[/i] ≥ sin [i]x[/i].
2008 Germany Team Selection Test, 2
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD \equal{} \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP \equal{} \angle DAQ$.
[i]Author: Vyacheslav Yasinskiy, Ukraine[/i]
2009 JBMO TST - Macedonia, 4
In every $1\times1$ cell of a rectangle board a natural number is written. In one step it is allowed the numbers written in every cell of arbitrary chosen row, to be doubled, or the numbers written in the cells of the arbitrary chosen column to be decreased by 1. Will after final number of steps all the numbers on the board be $0$?
2024 Switzerland Team Selection Test, 12
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$,
\[
f^{bf(a)}(a+1)=(a+1)f(b).
\]
2004 Chile National Olympiad, 4
Take the number $2^{2004}$ and calculate the sum $S$ of all its digits. Then the sum of all the digits of $S$ is calculated to obtain $R$. Next, the sum of all the digits of $R$is calculated and so on until a single digit number is reached. Find it. (For example if we take $2^7=128$, we find that $S=11,R=2$. So in this case of $2^7$ the searched digit will be $2$).
2015 Iran Team Selection Test, 3
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.
Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
1998 AIME Problems, 11
Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?
2018 Canadian Open Math Challenge, B1
Source: 2018 Canadian Open Math Challenge Part B Problem 1
-----
Let $(1+\sqrt2)^5 = a+b\sqrt2$, where $a$ and $b$ are positive integers. Determine the value of $a+b.$
2011 Vietnam National Olympiad, 1
Define the sequence of integers $\langle a_n\rangle$ as;
\[a_0=1, \quad a_1=-1, \quad \text{ and } \quad a_n=6a_{n-1}+5a_{n-2} \quad \forall n\geq 2.\]
Prove that $a_{2012}-2010$ is divisible by $2011.$
2000 Spain Mathematical Olympiad, 2
Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most $1.$
2019-IMOC, C1
Given a natural number $n$, if the tuple $(x_1,x_2,\ldots,x_k)$ satisfies
$$2\mid x_1,x_2,\ldots,x_k$$
$$x_1+x_2+\ldots+x_k=n$$
then we say that it's an [i]even partition[/i]. We define [i]odd partition[/i] in a similar way. Determine all $n$ such that the number of even partitions is equal to the number of odd partitions.
2004 Romania National Olympiad, 4
Let $\mathcal K$ be a field of characteristic $p$, $p \equiv 1 \left( \bmod 4 \right)$.
(a) Prove that $-1$ is the square of an element from $\mathcal K.$
(b) Prove that any element $\neq 0$ from $\mathcal K$ can be written as the sum of three squares, each $\neq 0$, of elements from $\mathcal K$.
(c) Can $0$ be written in the same way?
[i]Marian Andronache[/i]
1992 Balkan MO, 2
Prove that for all positive integers $n$ the following inequality takes place \[ (2n^2+3n+1)^n \geq 6^n (n!)^2 . \]
[i]Cyprus[/i]
2014 Taiwan TST Round 1, 1
Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.