Found problems: 85335
Let $P$ be a convex n-gon in the plane with vertices labeled $V_1,...,V_n$ in counterclockwise order. A point $Q$ not outside $P$ is called a balancing point of $P$ if, when the triangles the blue and green regions are the same. Suppose $P$ has exactly one balancing point/ Show that the balancing point must be a vertex of $P$
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.
(Nigeria)
A $\textit{shuffle}$ is a permutation of the integers $1,2,3,4,5$. More formally, a shuffle is a function $f:\{1,2,3,4,5\}\rightarrow\{1,2,3,4,5\}$ such that if $i\neq j$ then $f(i)\neq f(j)$. For example, $12345\mapsto23154$ denotes a shuffle $f$ so that $f(1)=2$, $f(2)=3$, $f(3)=1$, $f(4)=5$, and $f(5)=4$. A shuffle can be repeated some number of times to obtain another shuffle. For example, if $f$ is the shuffle $12345\mapsto23154$ from above, then repeating $f$ twice gives the shuffle $g(x)=f(f(x))$ which is $12345\mapsto31245$. How many shuffles are there that, when repeated $6$ times, give the shuffle $12345\mapsto12345$?
Let $A=\{1,2,3,\ldots,40\}$. Find the least positive integer $k$ for which it is possible to partition $A$ into $k$ disjoint subsets with the property that if $a,b,c$ (not necessarily distinct) are in the same subset, then $a\ne b+c$.
Consider the sequence $ a_n\equal{}\sqrt[3]{n^3\plus{}3n^2\plus{}2n\plus{}1}\plus{}a\sqrt[5]{n^5\plus{}5n^4\plus{}1}\plus{}\frac{ln(e^{n^2}\plus{}n\plus{}2)}{n\plus{}2}\plus{}b$. Find $ a,b \in \mathbb{R}$ such that $ \displaystyle\lim_{n\to\infty}a_n\equal{}5$.
Let $ABC$ be an acute scalene triangle. Incircle of $ABC$ touches $BC,CA,AB$ at $D,E,F$ respectively. Let $X,Y,Z$ be feet the altitudes of from $A,B,C$ to the sides $BC,CA,AB$ respectively. Let $A',B',C'$ be the reflections of $X,Y,Z$ in $EF,FD,DE$ respectively. Prove that triangles $ABC$ and $A'B'C'$ are similar.
Let $ABC$ be a triangle such that $AB = 13$, $BC = 14$, $CA = 15$ and let $E$, $F$ be the feet of the altitudes from $B$ and $C$, respectively.
Let the circumcircle of triangle $AEF$ be $\omega$.
We draw three lines, tangent to the circumcircle of triangle $AEF$ at $A$, $E$, and $F$.
Compute the area of the triangle these three lines determine.
Given $\triangle ABC$, let $I,O,\Gamma$ denote its incenter, circumcenter and circumcircle respecitvely. Let $AI$ intersect $\Gamma$ at $M(\neq A)$. Circle $\omega$ is tangent to $AB$, $AC$ and $\Gamma$ internally at $T$ (i.e. the mixtilinear incircle opposite $A$). Let the tangents at $A$ and $T$ to $\Gamma$ meet at $P$, and let $PI$ and $TM$ intersect at $Q$. Prove that $QA$ and $MO$ intersect at a point on $\Gamma$.
A line passing through the incenter $I$ of the triangle $ABC$ intersect its incircle at $D$ and $E$ and its circumcircle at $F$ and $G$, in such a way that the point $D$ lies between $I$ and $F$. Prove that:
$DF \cdot EG \geq r^{2}$.
Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]
$13.$ At a party, each man danced with exactly four women and each woman danced with exactly three men. Nine men attended the party. How many women attended the party $?$
In triangle $ABC$ we inscribe a square such that one of the sides of the square lies on the side $AC$, and the other two vertices lie on sides $AB$ and $BC$. Furthermore we know that $AC = 5$, $BC = 4$ and $AB = 3$. This square cuts out three smaller triangles from $\vartriangle ABC$. Express the sum of reciprocals of the inradii of these three small triangles as a fraction $p/q$ in lowest terms (i.e. with $p$ and $q$ coprime). What is $p + q$?
[b]a)[/b] Let $ u $ be a polynom in $ \mathbb{Q}[X] . $ Prove that the function $ E_u:\mathbb{Q}[X]\longrightarrow\mathbb{Q}[X] $ defined as $ E_u(P)=P(u) $ is an endomorphism.
[b]b)[/b] Let $ E $ be an injective endomorphism of $ \mathbb{Q} [X] . $ Show that there exists a nonconstant polynom $ v $ in $ \mathbb{Q}[X] $ such that $ E(P)=P(v) , $ for any $ P $ in $ \mathbb{Q}[X] . $
[b]c)[/b] Let $ A $ be an automorphism of $ \mathbb{Q}[X] . $ Demonstrate that there is a nonzero constant polynom $ w $ in $ \mathbb{Q}[X] $ which has the property that $ A(P)=P(w) , $ for any $ P $ in $ \mathbb{Q}[X] . $
[i]Marcel Țena[/i]
$(MON 5)$ Find the radius of the circle circumscribed about the isosceles triangle whose sides are the solutions of the equation $x^2 - ax + b = 0$.
Find the smallest natural number $n$ for which there exists a natural number $x$ such that
$$(x+1)^3 + (x + 2)^3 + (x + 3)^3 + (x + 4)^3 = (x + n)^3.$$
Find the greatest constant $\lambda$ such that for any doubly stochastic matrix of order 100, we can pick $150$ entries such that if the other $9850$ entries were replaced by $0$, the sum of entries in each row and each column is at least $\lambda$.
Note: A doubly stochastic matrix of order $n$ is a $n\times n$ matrix, all entries are nonnegative reals, and the sum of entries in each row and column is equal to 1.
Find all pairs $(x,y)$ of real numbers that satisfy,
\begin{align*}
x^2+y^2+x+y &= xy(x+y)-\frac{10}{27}\\
|xy| & \leq \frac{25}{9}.
\end{align*}
Given the equation $x^2+ax+1=0$, determine:
a) The interval of possible values for $a$ where the solutions to the previous equation are not real.
b) The loci of the roots of the polynomial, when $a$ is in the previous interval.
Find all triples $(x, y, z)$ of integers such that
\[\frac{1}{x^2}+\frac{2}{y^2}+\frac{3}{z^2} =\frac 23\]
Alice and Bob play a game. Initially, they write the pair $(1012,1012)$ on the board. They alternate their turns with Alice going first. In each turn the player can turn the pair $(a,b)$ to either $(a-2, b+1), (a+1, b-2)$ or $(a-1, b)$ as long as the resulting pair has only nonnegative values. The game terminates, when there is no legal move possible. Alice wins if the game terminates at $(0,0)$ and Bob wins if the game terminates at $(0,1)$. Determine who has the winning strategy?
[i]Proposed by Shashank Ingalagavi and Krutarth Shah[/i]
* Given several numbers each of which is less than $1951$ and the least common multiple of any two of which is greater than $1951$. Prove that the sum of their reciprocals is less than $2$.
Let $\vartriangle ABC$ be a triangle with integer side lengths such that $BC = 2016$. Let $G$ be the centroid of $\vartriangle ABC$ and $I$ be the incenter of $\vartriangle ABC$. If the area of $\vartriangle BGC$ equals the area of $\vartriangle BIC$, find the largest possible length of $AB$.
Construct a scalene triangle such that
\[a(\tan B - \tan C) = b(\tan A - \tan C)\]
Prove that there exist subfields of $\mathbb R$ that are
a) non-measurable and
b) of measure zero and continuum cardinality.
(translated by Miklós Maróti)
Let $MATH$ be a trapezoid with $MA=AT=TH=5$ and $MH=11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$.