Found problems: 85335
2023 USAMTS Problems, 3
We say that three numbers are balanced if either all three numbers are the same, or they are all different. A grid consisting of hexagons is presented in Figure 1. Each hexagon is filled with the number 1, 2, or 3, so that for any three hexagons that are mutually adjacent and oriented with two hexagons on the bottom and one hexagon on the top (as in Figure 3), the three numbers in the hexagons are balanced. Prove that when the grid is filled completely, the three numbers in the three shaded hexagons are balanced.
(An example of a partially filled-in grid is shown in Figure 2. There are other ways of filling in the grid.)
2014 IMO, 1
Let $a_0 < a_1 < a_2 < \dots$ be an infinite sequence of positive integers. Prove that there exists a unique integer $n\geq 1$ such that
\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]
[i]Proposed by Gerhard Wöginger, Austria.[/i]
2008 AMC 12/AHSME, 1
A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM the machine has completed one third of the day's job. At what time will the doughnut machine complete the job?
$ \textbf{(A)}$ 1:50 PM $ \qquad
\textbf{(B)}$ 3:00 PM $ \qquad
\textbf{(C)}$ 3:30 PM $ \qquad
\textbf{(D)}$ 4:30 PM $ \qquad
\textbf{(E)}$ 5:50 PM
1979 IMO Longlists, 53
An infinite increasing sequence of positive integers $n_j (j = 1, 2, \ldots )$ has the property that for a certain $c$,
\[\frac{1}{N}\sum_{n_j\le N} n_j \le c,\]
for every $N >0$.
Prove that there exist finitely many sequences $m^{(i)}_j (i = 1, 2,\ldots, k)$ such
that
\[\{n_1, n_2, \ldots \} =\bigcup_{i=1}^k\{m^{(i)}_1 ,m^{(i)}_2 ,\ldots\}\]
and
\[m^{(i)}_{j+1} > 2m^{(i)}_j (1 \le i \le k, j = 1, 2,\ldots).\]
1982 IMO Longlists, 30
Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$
2020 Korean MO winter camp, #6
Find all strictly increasing sequences $\{a_n\}_{n=0}^\infty$ of positive integers such that for all positive integers $k,m,n$
$$\frac{a_{n+1} +a_{n+2} +\dots +a_{n+k}}{k+m}$$ is not an integer larger than $2020$.
2009 National Olympiad First Round, 27
$ f\left( x \right) \equal{} \frac {x^5}{5x^4 \minus{} 10x^3 \plus{} 10x^2 \minus{} 5x \plus{} 1}$.
$ \sum_{i \equal{} 1}^{2009} f\left( \frac {i}{2009} \right) \equal{} ?$
$\textbf{(A)}\ 1000 \qquad\textbf{(B)}\ 1005 \qquad\textbf{(C)}\ 1010 \qquad\textbf{(D)}\ 2009 \qquad\textbf{(E)}\ 2010$
1978 IMO, 1
In a triangle $ABC$ we have $AB = AC.$ A circle which is internally tangent with the circumscribed circle of the triangle is also tangent to the sides $AB, AC$ in the points $P,$ respectively $Q.$ Prove that the midpoint of $PQ$ is the center of the inscribed circle of the triangle $ABC.$
2023 VN Math Olympiad For High School Students, Problem 9
Prove that: the polynomial$$(x(x+1))^{2^{2023}}+1$$is irreducible in $\mathbb{Q}[x].$
2002 AMC 10, 7
Let $ n$ be a positive integer such that $ \tfrac{1}{2}\plus{}\tfrac{1}{3}\plus{}\tfrac{1}{7}\plus{}\tfrac{1}{n}$ is an integer. Which of the following statements is [b]not[/b] true?
$ \textbf{(A)}\ 2\text{ divides }n \qquad
\textbf{(B)}\ 3\text{ divides }n \qquad
\textbf{(C)}\ 6\text{ divides }n \qquad
\textbf{(D)}\ 7\text{ divides }n \\
\textbf{(E)}\ n>84$
2016 PUMaC Number Theory A, 7
Compute the number of positive integers $n$ between $2017$ and $2017^2$ such that $n^n \equiv 1$ (mod $2017$). ($2017$ is prime.)
2021 Francophone Mathematical Olympiad, 1
Let $R$ and $S$ be the numbers defined by
\[R = \dfrac{1}{2} \times \dfrac{3}{4} \times \dfrac{5}{6} \times \cdots \times \dfrac{223}{224} \text{ and } S = \dfrac{2}{3} \times \dfrac{4}{5} \times \dfrac{6}{7} \times \cdots \times \dfrac{224}{225}.\]Prove that $R < \dfrac{1}{15} < S$.
2013 Math Prize For Girls Problems, 9
Let $A$ and $B$ be distinct positive integers such that each has the same number of positive divisors that 2013 has. Compute the least possible value of $\left| A - B \right|$.
2016 All-Russian Olympiad, 7
In triangle $ABC$,$AB<AC$ and $\omega$ is incirle.The $A$-excircle is tangent to $BC$ at $A^\prime$.Point $X$ lies on $AA^\prime$ such that segment $A^\prime X$ doesn't intersect with $\omega$.The tangents from $X$ to $\omega$ intersect with $BC$ at $Y,Z$.Prove that the sum $XY+XZ$ not depends to point $X$.(Mitrofanov)
2024-25 IOQM India, 22
In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. if $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, Where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.
The Golden Digits 2024, P1
Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$?
[i]Proposed by Pavel Ciurea[/i]
1955 Poland - Second Round, 6
Inside the trihedral angle $ OABC $, whose plane angles $ AOB $, $ BOC $, $ COA $ are equal, a point $ S $ is chosen equidistant from the faces of this angle. Through point $ S $ a plane is drawn that intersects the edges $ OA $, $ OB $, $ OC $ at points $ M $, $ N $, $ P $, respectively. Prove that the sum
$$
\frac{1}{OM} + \frac{1}{ON} + \frac{1}{OP}$$
has a constant value, i.e. independent of the position of the plane $ MNP $.
2022 IMO Shortlist, A8
For a positive integer $n$, an [i]$n$-sequence[/i] is a sequence $(a_0,\ldots,a_n)$ of non-negative integers satisfying the following condition: if $i$ and $j$ are non-negative integers with $i+j \leqslant n$, then $a_i+a_j \leqslant n$ and $a_{a_i+a_j}=a_{i+j}$.
Let $f(n)$ be the number of $n$-sequences. Prove that there exist positive real numbers $c_1$, $c_2$, and $\lambda$ such that \[c_1\lambda^n<f(n)<c_2\lambda^n\] for all positive integers $n$.
2020 Stanford Mathematics Tournament, 3
Three cities that are located on the vertices of an equilateral triangle with side length $100$ units. A missile flies in a straight line in the same plane as the equilateral triangle formed by the three citiies. The radar from City $A$ reported that the closest approach of the missile was $20$ units. The radar from City $B$ reported that the closest approach of the missile was $60$ units. However, the radar for city $C$ malfunctioned and did not report a distance. Find the minimum possible distance for the closest approach of the missile to city $C$.
2014 ELMO Shortlist, 7
Let $ABC$ be a triangle inscribed in circle $\omega$ with center $O$, let $\omega_A$ be its $A$-mixtilinear incircle, $\omega_B$ be its $B$-mixtilinear incircle, $\omega_C$ be its $C$-mixtilinear incircle, and $X$ be the radical center of $\omega_A$, $\omega_B$, $\omega_C$. Let $A'$, $B'$, $C'$ be the points at which $\omega_A$, $\omega_B$, $\omega_C$ are tangent to $\omega$. Prove that $AA'$, $BB'$, $CC'$ and $OX$ are concurrent.
[i]Proposed by Robin Park[/i]
2023 USA TSTST, 2
Let $n\ge m\ge 1$ be integers. Prove that
\[\sum_{k=m}^n \left (\frac 1{k^2}+\frac 1{k^3}\right) \ge m\cdot \left(\sum_{k=m}^n \frac 1{k^2}\right)^2.\]
[i]Raymond Feng and Luke Robitaille[/i]
2014 ELMO Shortlist, 12
Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$.
[i]Proposed by David Stoner[/i]
1974 All Soviet Union Mathematical Olympiad, 189
Given some cards with either "$-1$" or "$+1$" written on the opposite side. You are allowed to choose a triple of cards and ask about the product of the three numbers on the cards. What is the minimal number of questions allowing to determine all the numbers on the cards ...
a) for $30$ cards,
b) for $31$ cards,
c) for $32$ cards.
(You should prove, that you cannot manage with less questions.)
d) Fifty above mentioned cards are lying along the circumference. You are allowed to ask about the product of three consecutive numbers only. You need to determine the product af all the $50$ numbers. What is the minimal number of questions allowing to determine it?
2004 China Team Selection Test, 1
Given integer $ n$ larger than $ 5$, solve the system of equations (assuming $x_i \geq 0$, for $ i=1,2, \dots n$):
\[ \begin{cases} \displaystyle x_1+ \phantom{2^2} x_2+ \phantom{3^2} x_3 + \cdots + \phantom{n^2} x_n &= n+2, \\ x_1 + 2\phantom{^2}x_2 + 3\phantom{^2}x_3 + \cdots + n\phantom{^2}x_n &= 2n+2, \\ x_1 + 2^2x_2 + 3^2 x_3 + \cdots + n^2x_n &= n^2 + n +4, \\ x_1+ 2^3x_2 + 3^3x_3+ \cdots + n^3x_n &= n^3 + n + 8. \end{cases} \]
2016 AMC 10, 2
For what value of $x$ does $10^{x}\cdot 100^{2x}=1000^{5}$?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$