Found problems: 85335
2002 Stanford Mathematics Tournament, 3
An equilateral triangle has has sides $1$ inch long. An ant walks around the triangle maintaining a distance of $1$ inch from the triangle at all times. How far does the ant walk?
2022 Francophone Mathematical Olympiad, 1
find all the integer $n\geq1$ such that $\lfloor\sqrt{n}\rfloor \mid n$
2015 Azerbaijan JBMO TST, 4
Prove that there are not intgers $a$ and $b$ with conditions,
i) $16a-9b$ is a prime number.
ii) $ab$ is a perfect square.
iii) $a+b$ is also perfect square.
2020 Ukrainian Geometry Olympiad - April, 3
The angle $POQ$ is given ($OP$ and $OQ$ are rays). Let $M$ and $N$ be points inside the angle $POQ$ such that $\angle POM = \angle QON$ and $\angle POM < \angle PON$. Consider two circles: one touches the rays $OP$ and $ON$, the other touches the rays $OM$ and $OQ$. Denote by $B$ and $C$ the points of their intersection. Prove that $\angle POC = \angle QOB$.
2023 OlimphÃada, 2
Let $ABCD$ be a quadrilateral circumscribed around a circle $\omega$ with center $I$. Assume $P$ and $Q$ are distinct points and are isogonal conjugates such that $P, Q$, and $I$ are collinear. Show that $ABCD$ is a kite, that is, it has two disjoint pairs of consecutive equal sides.
2009 Today's Calculation Of Integral, 469
Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.
2023 Princeton University Math Competition, A4 / B6
The set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the set of real numbers between $x$ and $y,$ where $x<y.$ If $x+y$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m,n,$ find $m+n.$
2016 China Team Selection Test, 2
In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?
2023 Taiwan TST Round 3, 6
Given triangle $ABC$ with $A$-excenter $I_A$, the foot of the perpendicular from $I_A$ to $BC$ is $D$. Let the midpoint of segment $I_AD$ be $M$, $T$ lies on arc $BC$(not containing $A$) satisfying $\angle BAT=\angle DAC$, $I_AT$ intersects the circumcircle of $ABC$ at $S\neq T$. If $SM$ and $BC$ intersect at $X$, the perpendicular bisector of $AD$ intersects $AC,AB$ at $Y,Z$ respectively, prove that $AX,BY,CZ$ are concurrent.
2008 Moldova Team Selection Test, 3
Let $ \Gamma(I,r)$ and $ \Gamma(O,R)$ denote the incircle and circumcircle, respectively, of a triangle $ ABC$. Consider all the triangels $ A_iB_iC_i$ which are simultaneously inscribed in $ \Gamma(O,R)$ and circumscribed to $ \Gamma(I,r)$. Prove that the centroids of these triangles are concyclic.
2014 Tuymaada Olympiad, 3
Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality
\[\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. \]
[i](N. Alexandrov)[/i]
2014 Balkan MO Shortlist, A4
$\boxed{A4}$Let $m_1,m_2,m_3,n_1,n_2$ and $n_3$ be positive real numbers such that
\[(m_1-n_1)(m_2-n_2)(m_3-n_3)=m_1m_2m_3-n_1n_2n_3\]
Prove that
\[(m_1+n_1)(m_2+n_2)(m_3+n_3)\geq8m_1m_2m_3\]
1999 China National Olympiad, 2
Let $a$ be a real number. Let $(f_n(x))_{n\ge 0}$ be a sequence of polynomials such that $f_0(x)=1$ and $f_{n+1}(x)=xf_n(x)+f_n(ax)$ for all non-negative integers $n$.
a) Prove that $f_n(x)=x^nf_n\left(x^{-1}\right)$ for all non-negative integers $n$.
b) Find an explicit expression for $f_n(x)$.
2008 Costa Rica - Final Round, 2
Let $ ABC$ be a triangle and let $ P$ be a point on the angle bisector $ AD$, with $ D$ on $ BC$. Let $ E$, $ F$ and $ G$ be the intersections of $ AP$, $ BP$ and $ CP$ with the circumcircle of the triangle, respectively. Let $ H$ be the intersection of $ EF$ and $ AC$, and let $ I$ be the intersection of $ EG$ and $ AB$. Determine the geometric place of the intersection of $ BH$ and $ CI$ when $ P$ varies.
1985 Polish MO Finals, 1
Find the largest $k$ such that for every positive integer $n$ we can find at least $k$ numbers in the set $\{n+1, n+2, ... , n+16\}$ which are coprime with $n(n+17)$.
2019 Belarus Team Selection Test, 1.3
Given the equation
$$
a^b\cdot b^c=c^a
$$
in positive integers $a$, $b$, and $c$.
[i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well.
[i](ii)[/i] Solve the equation under the assumption $b\ge a$.
[i](iii)[/i] Prove that the equation has infinitely many solutions.
[i](I. Voronovich)[/i]
2008 AMC 8, 23
In square $ABCE$, $AF=2FE$ and $CD=2DE$. What is the ratio of the area of $\triangle BFD$ to the area of square $ABCE$?
[asy]
size((100));
draw((0,0)--(9,0)--(9,9)--(0,9)--cycle);
draw((3,0)--(9,9)--(0,3)--cycle);
dot((3,0));
dot((0,3));
dot((9,9));
dot((0,0));
dot((9,0));
dot((0,9));
label("$A$", (0,9), NW);
label("$B$", (9,9), NE);
label("$C$", (9,0), SE);
label("$D$", (3,0), S);
label("$E$", (0,0), SW);
label("$F$", (0,3), W);
[/asy]
$ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{5}{18}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{7}{20} $
2023 Pan-African, 5
Let $a, b$ be reals with $a \neq 0$ and let $$P(x)=ax^4-4ax^3+(5a+b)x^2-4bx+b.$$ Show that all roots of $P(x)$ are real and positive if and only if $a=b$.
2010 Iran MO (3rd Round), 4
a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms:
i-$\{0\}$.
ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$.
iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$)
b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)
2005 Portugal MO, 6
Prove that there is a unique function $f: N\to N$, that verifies $$f(a + b)f(a - b) = f(a^2)$$, for any $a, b\in N$ such that $a > b$.
2020 AMC 10, 4
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?
$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$
2021 Romania EGMO TST, P1
Let $(a_n)_{n\geq 1}$ be a sequence for real numbers given by $a_1=1/2$ and for each positive integer $n$
\[ a_{n+1}=\frac{a_n^2}{a_n^2-a_n+1}. \]
Prove that for every positive integer $n$ we have $a_1+a_2+\cdots + a_n<1$.
2015 Romania National Olympiad, 4
Let be a finite set $ A $ of real numbers, and define the sets $ S_{\pm }=\{ x\pm y| x,y\in A \} . $
Show that $ \left| A \right|\cdot\left| S_{-} \right| \le \left| S_{+} \right|^2 . $
2022 Centroamerican and Caribbean Math Olympiad, 4
Let $A_1A_2A_3A_4$ be a rectangle and let $S_1,S_2,S_3,S_4$ four circumferences inside of the rectangle such that $S_k$ and $S_{k+1}$ are tangent to each other and tangent to the side $A_kA_{k+1}$ for $k=1,2,3,4$, where $A_5=A_1$ and $S_5=S_1$. Prove that $A_1A_2A_3A_4$ is a square.
1997 Brazil Team Selection Test, Problem 5
Consider an infinite strip, divided into unit squares. A finite number of nuts is placed in some of these squares. In a step, we choose a square $A$ which has more than one nut and take one of them and put it on the square on the right, take another nut (from $A$) and put it on the square on the left. The procedure ends when all squares has at most one nut. Prove that, given the initial configuration, any procedure one takes will end after the same number of steps and with the same final configuration.