Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG = \angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$. Prove that the line $AI$, the line perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.

Given a convex $n$-gon with pairwise (mutually) non-parallel sides and a point inside it. Prove that there are not more than $n$ straight lines coming through that point and halving the area of the $n$-gon.

Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$

Let $ f: \mathbb{R}^2 \rightarrow \mathbb{R} $ be a function such that for every regular $ n $-gon $ A_1A_2 \ldots A_n $ we have $ f(A_1)+f(A_2)+\cdots +f(A_n)=0 $. Prove that $ f(x)=0 $ for all reals $ x $.

$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.

Let \(ABC\) be a triangle. Let \(B'\) denote the reflection of \(b\) in the internal angle bisector \(l\) of \(\angle A\).Show that the circumcentre of the triangle \(CB'I\) lies on the line \(l\) where \(I\) is the incentre of \(ABC\).

For any positive integer $k$ consider the sequence $$a_n=\sqrt{k+\sqrt{k+\dots+\sqrt k}},$$ where there are $n$ square-root signs on the right-hand side. (a) Show that the sequence converges, for every fixed integer $k\ge 1$. (b) Find $k$ such that the limit is an integer. Furthermore, prove that if $k$ is odd, then the limit is irrational.

In a three-dimensional Euclidean space, by $\overrightarrow{u_1}$ , $\overrightarrow{u_2}$ , $\overrightarrow{u_3}$ are denoted the three orthogonal unit vectors on the $x, y$, and $z$ axes, respectively. a) Prove that the point $P(t) = (1-t)\overrightarrow{u_1} +(2-3t)\overrightarrow{u_2} +(2t-1)\overrightarrow{u_3}$ , where $t$ takes all real values, describes a straight line (which we will denote by $L$). b) What describes the point $Q(t) = (1-t^2)\overrightarrow{u_1} +(2-3t^2)\overrightarrow{u_2} +(2t^2 -1)\overrightarrow{u_3}$ if $t$ takes all the real values? c) Find a vector parallel to $L$. d) For what values of $t$ is the point $P(t)$ on the plane $2x+ 3y + 2z +1 = 0$? e) Find the Cartesian equation of the plane parallel to the previous one and containing the point $Q(3)$. f) Find the Cartesian equation of the plane perpendicular to $L$ that contains the point $Q(2)$.

Define the function $f:\mathbb N_{>1}\to\mathbb N_{>1}$ such that $f(x)$ is the greatest prime factor of $x$. A sequence of positive integers $\{a_n\}$ satisfies $a_1=M>1$ and $$a_{n+1}=\begin{cases}a_n-f(a_n)&\text{if }a_n\text{ is composite.}\\a_n+k&\text{otherwise.}\end{cases}$$ Show that for any positive integers $M,k$, the sequence $\{a_n\}$ is bounded. (TAN768092100853)

Let $ABCD$ be a trapezoid such that side $[AB]$ and side $[CD]$ are perpendicular to side $[BC]$. Let $E$ be a point on side $[BC]$ such that $\triangle AED$ is equilateral. If $|AB|=7$ and $|CD|=5$, what is the area of trapezoid $ABCD$? $ \textbf{(A)}\ 27\sqrt{3} \qquad\textbf{(B)}\ 42 \qquad\textbf{(C)}\ 24\sqrt{3} \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 36 $

In a group of people, every two people have exactly one friend in common. Prove that there is a person who is a friend of everyone else.

The numbers $1,2,\dots,10$ are written on a board. Every minute, one can select three numbers $a$, $b$, $c$ on the board, erase them, and write $\sqrt{a^2+b^2+c^2}$ in their place. This process continues until no more numbers can be erased. What is the largest possible number that can remain on the board at this point? [i]Proposed by Evan Chen[/i]

Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold. (Here we denote $XY$ the length of the line segment $XY$.)

The median $CM$ is drawn in the triangle $ABC$ intersecting bisector angle $BL$ at point $O$. Ray $AO$ intersects side $BC$ at point $K$, beyond point $K$ draw the segment $KT = KC$. On the ray $BC$ beyond point $C$ draw a segment $CN = BK$. Prove that is a quadrilateral $ABTN$ is cyclic if and only if $AB = AK$. (Vladislav Yurashev)

Consider the sequence $<{a_n}>$ of natural numbers such that {i} $a_n$ is a square numver for all $n$ ; (ii) $a_{n+1} - a_n$ is either a prime or a square of a prime for each $n$. Show that $<a_n>$ is a finite sequence. Determine the longest such sequence.

In this figure the radius of the circle is equal to the altitude of the equilateral triangle $ABC$. The circle is made to roll along the side $AB$, remaining tangent to it at a variable point $T$ and intersecting lines $AC$ and $BC$ in variable points $M$ and $N$, respectively. Let $n$ be the number of degrees in arc $MTN$. Then $n$, for all permissible positions of the circle: $\textbf{(A) }\text{varies from }30^{\circ}\text{ to }90^{\circ}$ $\textbf{(B) }\text{varies from }30^{\circ}\text{ to }60^{\circ}$ $\textbf{(C) }\text{varies from }60^{\circ}\text{ to }90^{\circ}$ $\textbf{(D) }\text{remains constant at }30^{\circ}$ $\textbf{(E) }\text{remains constant at }60^{\circ}$ [asy] pair A = (0,0), B = (1,0), C = dir(60), T = (2/3,0); pair M = intersectionpoint(A--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)), N = intersectionpoint(B--C,Circle((2/3,sqrt(3)/2),sqrt(3)/2)); draw((0,0)--(1,0)--dir(60)--cycle); draw(Circle((2/3,sqrt(3)/2),sqrt(3)/2)); label("$A$",A,dir(210)); label("$B$",B,dir(-30)); label("$C$",C,dir(90)); label("$M$",M,dir(190)); label("$N$",N,dir(75)); label("$T$",T,dir(-90)); //Credit to bobthesmartypants for the diagram [/asy]

Find all solutions (real and complex) for $x,y,z$, given that: \[ x+y+z = 6 \\ x^2+y^2+z^2 = 12 \\ x^3+y^3+z^3 = 24 \]

Let $ABC$ be a triangle with $\angle BAC = 90^o$. Let $D$, $E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$.

The Fibonacci Sequence $ 1,1,2,3,5,8,13,21,\ldots$ starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in the units position of a number in the Fibonacci Sequence? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$

A circle $\Gamma$ is tangent to the side $BC$ of a triangle $ABC$ at $X$ and tangent to the side $AC$ at $Y$. A point $P$ is taken on the side $AB$. Let $XP$ and $YP$ intersect $\Gamma$ at $K$ and $L$ for the second time, $AK$ and $BL$ intersect $\Gamma$ at $R$ and $S$ for the second time. Prove that $XR$ and $YS$ intersect on $AB$.

$ABCD$ is a quadrilateral such that $\angle ACB=\angle ACD$. $T$ is inside of $ABCD$ such that $\angle ADC-\angle ATB=\angle BAC$ and $\angle ABC-\angle ATD=\angle CAD$. Prove that $\angle BAT=\angle DAC$.

Let $AA',BB', CC'$ be the bisectors of the angles of a triangle $ABC \ (A' \in BC, B' \in CA, C' \in AB)$. Prove that each of the lines $A'B', B'C', C'A'$ intersects the incircle in two points.

Consider $ \triangle ABC$ and points $ M \in (AB)$, $ N \in (BC)$, $ P \in (CA)$, $ R \in (MN)$, $ S \in (NP)$, $ T \in (PM)$ such that $ \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k$ and $ \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k$ for some $ k \in (0, 1)$. Prove that $ \triangle STR \sim \triangle ABC$ and, furthermore, determine $ k$ for which the minimum of $ [STR]$ is attained.

The altitudes of a parallelogram are greater than $1$. Does this yield that the unit square may be covered by this parallelogram?

At an elementary school, the students in third grade, fourth grade, and fifth grade run an average of 12, 15, and 10 minutes per day, respectively. There are twice as many third graders as fourth graders, and twice as many fourth graders as fifth graders. What is the average number of minutes run per day by these students? $ \textbf{(A)}\ 12 \qquad \textbf{(B)}\ \frac{37}{3} \qquad \textbf{(C)}\ \frac{88}{7}\qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$