Let $n$ be a positive integer and let $0\leq k\leq n$ be an integer. Show that there exist $n$ points in the plane with no three on a line such that the points can be divided into two groups satisfying the following properties. $\text{(i)}$ The first group has $k$ points and the distance between any two distinct points in this group is irrational. $\text{(ii)}$ The second group has $n-k$ points and the distance between any two distinct points in this group is an integer. $\text{(iii)}$ The distance between a point in the first group and a point in the second group is irrational.

[b](a)[/b]Prove that every pentagon with integral coordinates has at least two vertices , whose respective coordinates have the same parity. [b](b)[/b]What is the smallest area possible of pentagons with integral coordinates. Albanian National Mathematical Olympiad 2010---12 GRADE Question 3.

In an acute-angled triangle $ABC$, point $I$ is the incenter, $H$ is the orthocenter, $O$ is the center of the circumscribed circle, $T$ and $K$ are the touchpoints of the $A$-excircle and incircle with side $BC$ respectively. It turned out that the segment $TI$ is passing through the point $O$. Prove that $HK$ is the angle bisector of $\angle BHC$. (Matvii Kurskyi)

In $\vartriangle ABC$, $m\angle A = 90^o$, $m\angle B = 45^o$, and $m\angle C = 45^o$. Point $P$ inside $\vartriangle ABC$ satisfies $m \angle BPC =135^o$. Given that $\vartriangle PAC$ is isosceles, the largest possible value of $\tan \angle PAC$ can be expressed as $s+t\sqrt{u}$, where $s$ and $t$ are integers and $u$ is a positive integer not divisible by the square of any prime. Compute $100s + 10t + u$.

[b]p1.[/b] Compute the degree of the least common multiple of the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$,$...$, $x^{10} -1$. [b]p2.[/b] A line in the $xy$ plane is called wholesome if its equation is $y = mx+b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x,y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when y is divided by $7$. The pair $(r, s)$ is called an [i]ingredient[/i] of the line $\ell$. The (unordered) set of all possible ingredients of a wholesome line $\ell$ is called the [i]recipe [/i] of $\ell$. Compute the number of possible recipes of wholesome lines. [b]p3.[/b] Let $\tau (n)$ be the number of distinct positive divisors of $n$. Compute $\sum_{d|15015} \tau (d)$, that is, the sum of $\tau (d)$ for all $d$ such that $d$ divides $15015$. [b]p4.[/b] Suppose $2202010_b - 2202010_3 = 71813265_{10}$. Compute $b$. ($n_b$ denotes the number $n$ written in base $b$.) [b]p5.[/b] Let $x = (3 -\sqrt5)/2$. Compute the exact value of $x^8 + 1/x^8$. [b]p6.[/b] Compute the largest integer that has the same number of digits when written in base $5$ and when written in base $7$. Express your answer in base $10$. [b]p7.[/b] Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$. [b]p8.[/b] Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this. [b]p9.[/b] Let $A$ and $B$ be points in the plane such that $AB = 30$. A circle with integer radius passes through $A$ and $B$. A point $C$ is constructed on the circle such that $AC$ is a diameter of the circle. Compute all possible radii of the circle such that $BC$ is a positive integer. [b]p10.[/b] Each square of a $3\times 3$ grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings. [b]p11.[/b] Compute all positive integers $n$ such that the sum of all positive integers that are less than $n$ and relatively prime to $n$ is equal to $2n$. [b]p12.[/b] The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle $ABC$ has $AB = 10$, $BC = 21$, and $CA = 17$. Let $P$ be a point inside the triangle. Let $x$ be the distance between $P$ and $\overleftrightarrow{BC}$, let $y$ be the distance between $P$ and $\overleftrightarrow{CA}$, and let $z$ be the distance between $P$ and $\overleftrightarrow{AB}$. Compute the largest possible value of the product $xyz$. [b]p13.[/b] Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac13$ , and he passes it on in the same direction with probability $\frac23$ . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac14$ , and he passes it on with probability $\frac34$ . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability $p$, and passes it on with probability $1-p$. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute $p$. [b]p14.[/b] Circle $O$ is in the plane. Circles $A$, $B$, and $C$ are congruent, and are each internally tangent to circle $O$ and externally tangent to each other. Circle $X$ is internally tangent to circle $O$ and externally tangent to circles $A$ and $B$. Circle $X$ has radius $1$. Compute the radius of circle $O$. [img]https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png[/img] [b]p15.[/b] Compute the number of primes $p$ less than 100 such that $p$ divides $n^2 +n+1$ for some integer $n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$.

Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.

In the acute triangle $ABC$ point $I$ is the incenter, $O$ is the circumcenter, while $I_a$ is the excenter opposite the vertex $A$. Point $A'$ is the reflection of $A$ across the line $BC$. Prove that angles $\angle IOI_a$ and $\angle IA'I_a$ are equal.

Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic. [I]Proposed by S. Berlov[/i]

The area of a pizza with radius $4$ inches is $N$ percent larger than the area of a pizza with radius $3$ inches. What is the integer closest to $N$? $\textbf{(A) } 25 \qquad\textbf{(B) } 33 \qquad\textbf{(C) } 44\qquad\textbf{(D) } 66 \qquad\textbf{(E) } 78$

Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$. [i]Proposed by Josef Tkadlec (Czech Republic)[/i]

Equilateral triangle $ABC$ has side length $840$. Point $D$ lies on the same side of line $BC$ as $A$ such that $\overline{BD} \perp \overline{BC}$. The line $\ell$ through $D$ parallel to line $BC$ intersects sides $\overline{AB}$ and $\overline{AC}$ at points $E$ and $F$, respectively. Point $G$ lies on $\ell$ such that $F$ is between $E$ and $G$, $\triangle AFG$ is isosceles, and the ratio of the area of $\triangle AFG$ to the area of $\triangle BED$ is $8:9$. Find $AF$. [asy] pair A,B,C,D,E,F,G; B=origin; A=5*dir(60); C=(5,0); E=0.6*A+0.4*B; F=0.6*A+0.4*C; G=rotate(240,F)*A; D=extension(E,F,B,dir(90)); draw(D--G--A,grey); draw(B--0.5*A+rotate(60,B)*A*0.5,grey); draw(A--B--C--cycle,linewidth(1.5)); dot(A^^B^^C^^D^^E^^F^^G); label("$A$",A,dir(90)); label("$B$",B,dir(225)); label("$C$",C,dir(-45)); label("$D$",D,dir(180)); label("$E$",E,dir(-45)); label("$F$",F,dir(225)); label("$G$",G,dir(0)); label("$\ell$",midpoint(E--F),dir(90)); [/asy]

Let $ABCD$ be a cyclic quadrilateral. The bisectors of angles $BAD$ and $BCD$ intersect in point $K$ such that $K \in BD$. Let $M$ be the midpoint of $BD$. A line passing through point $C$ and parallel to $AD$ intersects $AM$ in point $P$. Prove that triangle $\triangle DPC$ is isosceles.

Let $A$, $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$, let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$. Further, let $E$ be the second point of intersection of the line $AD$ with $k$. Show that for each circle $k$, the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.

Triangle $ABC$ has $\angle A=90^\circ$, side $BC=25$, $AB>AC$, and area $150$. Circle $\omega$ is inscribed in $ABC$, with $M$ its point of tangency on $AC$. Line $BM$ meets $\omega$ a second time at point $L$. Find the length of segment $BL$.

Let $ABC$ be an equilateral triangle. Let $P$ be a point inside of it such that the square root of the distance of $P$ to one of the sides is equal to the sum of the square roots of the distances of $P$ to the other two sides. Find the geometric place of $P$.

Let $ABC$ be a right triangle with $\angle ACB = 90^o$ and M the center of $AB$. Let $G$ br any point on the line $MC$ and $P$ a point on the line $AG$, such that $\angle CPA = \angle BAC$ . Further let $Q$ be a point on the straight line $BG$, such that $\angle BQC = \angle CBA$ . Show that the circles of the triangles $AQG$ and $BPG$ intersect on the segment $AB$.

Consider the triangle $ABC$ and the points $D \in (BC),E \in (CA), F \in (AB)$, such that $\frac{BD}{DC}=\frac{CE}{EA}=\frac{AF}{FB}$. Prove that if the circumcenters of triangles $DEF$ and $ABC$ coincide, then the triangle $ABC$ is equilateral.

Three different points $A$, $B$ and $C$ are placed in the plane. Construct a line $m$ through $C$ so that the product of the distances from $A$ and $B$ to $m$ has the maximal value. Is $m$ unique for every triple $A$, $B$ and $C$? (NB Vassiliev)

Let $\triangle ABC$ be a triangle such that $AB = AC$, and let its circumcircle be $\Gamma$. Let $\omega$ be a circle which is tangent to $AB$ and $AC$ at $B$ and $C$. Point $P$ belongs to $\omega$, and lines $PB$ and $PC$ intersect $\Gamma$ again at $Q$ and $R$. $X$ and $Y$ are points on lines $BR$ and $CQ$ such that $AX = XB$ and $AY = YC$. Show that as $P$ varies on $\omega$, the circumcircle of $\triangle AXY$ passes through a fixed point other than $A$.

Let $ABCD$ be a square, such that the length of its sides are integers. This square is divided in $89$ smaller squares, $88$ squares that have sides with length $1$, and $1$ square that has sides with length $n$, where $n$ is an integer larger than $1$. Find all possible lengths for the sides of $ABCD$.

We say a triple of real numbers $ (a_1,a_2,a_3)$ is [b]better[/b] than another triple $ (b_1,b_2,b_3)$ when exactly two out of the three following inequalities hold: $ a_1 > b_1$, $ a_2 > b_2$, $ a_3 > b_3$. We call a triple of real numbers [b]special[/b] when they are nonnegative and their sum is $ 1$. For which natural numbers $ n$ does there exist a collection $ S$ of special triples, with $ |S| \equal{} n$, such that any special triple is bettered by at least one element of $ S$?

Let $I$ be the incenter of a triangle $ABC$; $A^{\prime}$, $B^{\prime}$, $C^{\prime}$ be the orthocenters of the triangles $BIC$, $AIC$, $AIB$; $M_{a}$, $M_{b}$, $M_{c}$ be the midpoints of $BC$, $CA$, $AB$, and $S_{a}$, $S_{b}$, $S_{c}$ be the midpoints of $AA^{\prime}$, $BB^{\prime}$, $CC^{\prime}$. Prove that $M_{a}S_{a}$, $M_{b}S_{b}$, $M_{c}S_{c}$ concur. Proposed by: S Kuznetsov

Circle $\omega$ lies in the interior of circle $\Omega$, on which a point $X$ moves. The tangents from $X$ to $\omega$ intersect $\Omega$ for the second time at points $A\neq X$ and $B\neq X$. Prove that the lines $AB$ are either all tangent to a fixed circle, or they all pass through a point.