Given acute triangle $ABC$ with circumcenter $O$ and the center of nine-point circle $N$. Point $N_1$ are given such that $\angle NAB = \angle N_1AC$ and $\angle NBC = \angle N_1BA$. Perpendicular bisector of segment $OA$ intersects the line $BC$ at $A_1$. Analogously define $B_1$ and $C_1$. Show that all three points $A_1,B_1,C_1$ are collinear at a line that is perpendicular to $ON_1$.

In the adjoining figure, the triangle $ABC$ is a right triangle with $\angle BCA=90^\circ$. Median $CM$ is perpendicular to median $BN$, and side $BC=s$. The length of $BN$ is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10));real r=54.72; pair B=origin, C=dir(r), A=intersectionpoint(B--(9,0), C--C+4*dir(r-90)), M=midpoint(B--A), N=midpoint(A--C), P=intersectionpoint(B--N, C--M); draw(M--C--A--B--C^^B--N); pair point=P; markscalefactor=0.005; draw(rightanglemark(C,P,B)); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$M$", M, S); label("$N$", N, dir(C--A)*dir(90)); label("$s$", B--C, NW);[/asy] $\textbf {(A) } s\sqrt 2 \qquad \textbf {(B) } \frac 32s\sqrt2 \qquad \textbf {(C) } 2s\sqrt2 \qquad \textbf {(D) } \frac 12s\sqrt5 \qquad \textbf {(E) } \frac 12s\sqrt6$

In an acute-angled triangle $ABC$ the angle at $B$ is greater than $45^\circ$. Points $D,E, F$ are the feet of the altitudes from $A,B,C$ respectively, and $K$ is the point on segment $AF$ such that $\angle DKF = \angle KEF$. (a) Show that such a point $K$ always exists. (b) Prove that $KD^2 = FD^2 + AF \cdot BF$.

For a positive real number $t$, in the coordiante space, consider 4 points $O(0,\ 0,\ 0),\ A(t,\ 0,\ 0),\ B(0,\ 1,\ 0),\ C(0,\ 0,\ 1)$. Let $r$ be the radius of the sphere $P$ which is inscribed to all faces of the tetrahedron $OABC$. When $t$ moves, find the maximum value of $\frac{\text{vol[P]}}{\text{vol[OABC]}}.$

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

In triangle $[ABC]$, the bisector in $C$ and the altitude passing through $B$ intersect at point $D$. Point $E$ is the symmetric of point $D$ wrt $BC$ and lies on the circle circumscribed to the triangle $[ABC]$. Prove that the triangle is $[ABC]$ isosceles.

Let $\Gamma$ be the circumcircle of an acute triangle $ABC.$ The perpendicular to $AB$ from $C$ meets $AB$ at $D$ and $\Gamma$ again at $E.$ The bisector of angle $C$ meets $AB$ at $F$ and $\Gamma$ again at $G.$ The line $GD$ meets $\Gamma$ again at $H$ and the line $HF$ meets $\Gamma$ again at $I.$ Prove that $AI = EB.$

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$. Points $X$ and $Y$ lie on sides $AB$ and $CD$, respectively. Suppose the circumcircles of $ADX$ and $BCY$ meet line $XY$ again at $P$ and $Q$, respectively. Show that $OP=OQ$. [i]Holden Mui[/i]

$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$. A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear. [i]V. Astakhov[/i]

Let $G$ be the intersection of medians of $\triangle ABC$ and $I$ be the incenter of $\triangle ABC$. If $|AB|=c$, $|AC|=b$ and $GI \perp BC$, what is $|BC|$? $ \textbf{(A)}\ \dfrac{b+c}2 \qquad\textbf{(B)}\ \dfrac{b+c}{3} \qquad\textbf{(C)}\ \dfrac{\sqrt{b^2+c^2}}{2} \qquad\textbf{(D)}\ \dfrac{\sqrt{b^2+c^2}}{3\sqrt 2} \qquad\textbf{(E)}\ \text{None of the preceding} $

In triangle $ABC$ with centroid $G$ and circumcircle $\omega$, line $\overline{AG}$ intersects $BC$ at $D$ and $\omega$ at $P$. Given that $GD =DP = 3$, and $GC = 4$, find $AB^2$. [i]Proposed by Muztaba Syed[/i]

Prove that if the point $ P $ runs through a circle inscribed in the triangle $ ABC $, then the value of the expression $ a \cdot PA^2 + b \cdot PB^2 + c \cdot PC^2 $ is constant ($ a, b, c $ are the lengths of the sides opposite the vertices $ A, B, C $, respectively).

A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.

Let $A,B,C$ denote intersection points of diagonals $A_1A_4$ and $A_2A_5$, $A_1A_6$ and $A_2A_7$, $A_1A_9$ and $A_2A_{10}$ of the regular decagon $A_1A_2...A_{10}$ respectively Find the angles of the triangle $ABC$

In an acute non-isosceles triangle $ABC$, the inscribed circle touches side $BC$ at point $T, Q$ is the midpoint of altitude $AK$, $P$ is the orthocenter of the triangle formed by the bisectors of angles $B$ and $C$ and line $AK$. Prove that the points $P, Q$ and $T$ lie on the same line. (D. Prokopenko)

Three circles pass through a point $P$, and the second points of their intersection $A, B, C$ lie on a straight line. Let $A_1 B_1, C_1$ be the second meets of lines $AP, BP, CP$ with the corresponding circles. Let $C_2$ be the intersections of lines $AB_1$ and $BA_1$. Let $A_2, B_2$ be defined similarly. Prove that the triangles $A_1B_1C_1$ and $A_2B_2C_2$ are equal,

Given are $4004$ distinct points, which lie in the interior of a convex polygon of area $1$. Prove that there exists a convex polygon of area $\frac{1}{2003}$, included in the given polygon, such that it does not contain any of the given points in its interior.

There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.

$ABCD$ is parallelogram. Line $l$ is perpendicular to $BC$ at $B$. Two circles passes through $D,C$, such that $l$ is tangent in points $P$ and $Q$. $M$ - midpoint $AB$. Prove that $\angle DMP=\angle DMQ$

In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $E$. If $AB=BE=5$, $EC=CD=7$, and $BC=11$, compute $AE$.

The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^{\circ}$ around the point $(1,5)$ and then reflected about the line $y=-x$. The image of $P$ after these two transformations is at $(-6,3)$. What is $b-a$? $\textbf{(A) }1 \qquad \textbf{(B) }3 \qquad \textbf{(C) }5 \qquad \textbf{(D) }7 \qquad \textbf{(E) }9$

[b]p1.[/b] Compute $45\times 45 - 6$. [b]p2.[/b] Consecutive integers have nice properties. For example, $3$, $4$, $5$ are three consecutive integers, and $8$, $9$, $10$ are three consecutive integers also. If the sum of three consecutive integers is $24$, what is the smallest of the three numbers? [b]p3.[/b] How many positive integers less than $25$ are either multiples of $2$ or multiples of $3$? [b]p4.[/b] Charlotte has $5$ positive integers. Charlotte tells you that the mean, median, and unique mode of his five numbers are all equal to $10$. What is the largest possible value of the one of Charlotte's numbers? [b]p5.[/b] Mr. Meeseeks starts with a single coin. Every day, Mr. Meeseeks goes to a magical coin converter where he can either exchange $1$ coin for $5$ coins or exchange $5$ coins for $3$ coins. What is the least number of days Mr. Meeseeks needs to end with $15$ coins? [b]p6.[/b] Twelve years ago, Violet's age was twice her sister Holo's age. In $7$ years, Holo's age will be $13$ more than a third of Violet's age. $3$ years ago, Violet and Holo's cousin Rindo's age was the sum of their ages. How old is Rindo? [b]p7.[/b] In a $2 \times 3$ rectangle composed of $6$ unit squares, let $S$ be the set of all points $P$ in the rectangle such that a unit circle centered at $P$ covers some point in exactly $3$ of the unit squares. Find the area of the region $S$. For example, the diagram below shows a valid unit circle in a $2 \times 3$ rectangle. [img]https://cdn.artofproblemsolving.com/attachments/d/9/b6e00306886249898c2bdb13f5206ced37d345.png[/img] [b]p8.[/b] What are the last four digits of $2^{1000}$? [b]p9.[/b] There is a point $X$ in the center of a $2 \times 2 \times 2$ box. Find the volume of the region of points that are closer to $X$ than to any of the vertices of the box. [b]p10.[/b] Evaluate $\sqrt{37 \cdot 41 \cdot 113 \cdot 290 - 4319^2}$ [b]p11.[/b] (Estimation) A number is abundant if the sum of all its divisors is greater than twice the number. One such number is $12$, because $1+2+3+4+6+12 = 28 > 24$: How many abundant positive integers less than $20190$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The triangle $ABC$ is inscribed in circle $\Gamma$. The points X, Y, Z are the midpoints of the arcs $BC$, $CA$ and $AB$ respectively in $\Gamma$ (those that do not contain the third vertex, in each case). The intersection points of the sides of the triangles $\vartriangle ABC$ and $\vartriangle XY Z$ form the hexagon $DEFGHK$. Prove that the diagonals $DG$, $EH$ and $FK$ are concurrent

Let $ABC$ be an acute triangle with altitude $AD$ ($D \in BC$). The line through $C$ parallel to $AB$ meets the perpendicular bisector of $AD$ at $G$. Show that $AC = BC$ if and only if $\angle AGC = 90^{\circ}$.

$ABCDE$ is a convex pentagon with $BD=CD=AC$, and $B$, $C$, $D$, $E$ are concyclic. If $\angle BAC+\angle AED=180^{\circ}$ and $\angle DCA=\angle BDE$, prove that $AB=DE$ or $AB=2AE$.