(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this angle (There was an error in published condition of this problem).

Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac mn$, where m and n are relatively prime positive integers. Find $m + n$.

Let $ABC$ be a triangle with $AB < AC$. Let $D$ and $E$ be on the lines $CA$ and $BA$, respectively, such that $CD = AB$, $BE = AC$, and $A$, $D$ and $E$ lie on the same side of $BC$. Let $I$ be the incenter of triangle $ABC$, and let $H$ be the orthocenter of triangle $BCI$. Show that $D$, $E$, and $H$ are collinear.

A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal. [i](2 points)[/i]

Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that $$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$

Angle bisectors from vertices $B$ and $C$ and the perpendicular bisector of side $BC$ are drawn in a non-isosceles triangle $ABC$. Next, three points of pairwise intersection of these three lines were marked (remembering which point is which), and the triangle itself was erased. Restore it according to the marked points using a compass and ruler. (Yu. Blinkov)

At a time $t = 0$, a navy ship is at a point $O$, while an enemy ship is at a point $A$ cruising with speed $v$ perpendicular to $OA = a$. The speed and direction of the enemy ship do not change. The strategy of the navy ship is to travel with constant speed $u$ at a angle $0 < \phi < \pi /2$ to the line $OA$. 1) Let $\phi$ be chosen. What is the minimum distance between the two ships? Under what conditions will the distance vanish? 2) If the distance does not vanish, what is the choice of $\phi$ to minimize the distance? What are directions of the two ships when their distance is minimum?

Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.

An acute triangle $ABC$ has $AB$ as one of its longest sides. The incircle of $ABC$ has center $I$ and radius $r$. Line $CI$ meets the circumcircle of $ABC$ at $D$. Let $E$ be a point on the minor arc $BC$ of the circumcircle of $ABC$ with $\angle ABE > \angle BAD$ and $E\notin \{B,C\}$. Line $AB$ meets $DE$ at $F$ and line $AD$ meets $BE$ at $G$. Let $P$ be a point inside triangle $AGE$ with $\angle APE=\angle AFE$ and $P\neq F$. Let $X$ be a point on side $AE$ with $XP\parallel EG$ and let $S$ be a point on side $EG$ with $PS\parallel AE$. Suppose $XS$ and $GP$ meet on the circumcircle of $AGE$. Determine the possible positions of $E$ as well as the minimum value of $\frac{BE}{r}$.

Consider the closed polygonal discs $P_1$, $P_2$, $P_3$ with the property that for any three points $A\in P_1$, $B\in P_2$, $C\in P_3$, we have $[\triangle ABC]\le 1$. (Here $[X]$ denotes the area of polygon $X$.) (a) Prove that $\min\{[P_1],[P_2],[P_3]\}<4$. (b) Give an example of polygons $P_1,P_2,P_3$ with the above property such that $[P_1]>4$ and $[P_2]>4$.

In an acute traingle $ABC$ with $AB< BC$ let $BH_b$ be its altitude, and let $O$ be the circumcenter. A line through $H_b$ parallel to $CO$ meets $BO$ at $X$. Prove that $X$ and the midpoints of $AB$ and $AC$ are collinear.

[b]8.1[/b] Four circles are placed on planes so that each one touches the other two externally. Prove that the points of tangency lie on one circle. [img]https://cdn.artofproblemsolving.com/attachments/9/8/883a82fb568954b09a4499a955372e2492dbb8.png[/img] [b]8.2[/b]. Let the integers $a$ and $b$ be represented as $x^2-5y^2$, where $x$ and $y$ are integer numbers. Prove that the number $ab$ can also be presented in this form. [b]8.3[/b] Solve the equation $x(x + d)(x + 2d)(x + 3d) = a$. [b]8.4 / 9.1[/b] Let $a+b+c=1$, $m+n+p=1 $. Prove that $$-1 \le am + bn + cp \le 1 $$ [b]8.5[/b] Inscribe a triangle with the largest area in a semicircle. [b]8.6[/b] Three circles of the same radius intersect at one point. Prove that the other three points intersections lie on a circle of the same radius. [img]https://cdn.artofproblemsolving.com/attachments/4/7/014952f2dcf0349d54b07230e45a42c242a49d.png[/img] [b]8.7[/b] Find the circle of smallest radius that contains a given triangle. [b]8.8 / 9.2[/b] Given a polynomial $$x^{2n} +a_1x^{2n-2} + a_2x^{2n-4} + ... + a_{n-1}x^2 + a_n,$$ which is divisible by $ x-1$. Prove that it is divisible by $x^2-1$. [b]8.9[/b] Prove that for any prime number $p$ other than $2$ and from $5$, there is a natural number $k$ such that only ones are involved in the decimal notation of the number $pk$.. PS. You should use hide for answers.Collected [url=https://artofproblemsolving.com/community/c3983459_1962_leningrad_math_olympiad]here[/url].

In a non equilateral triangle $ABC$ the sides $a,b,c$ form an arithmetic progression. Let $I$ be the incentre and $O$ the circumcentre of the triangle $ABC$. Prove that (1) $IO$ is perpendicular to $BI$; (2) If $BI$ meets $AC$ in $K$, and $D$, $E$ are the midpoints of $BC$, $BA$ respectively then $I$ is the circumcentre of triangle $DKE$.

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.

The radius of the circumscribed circle of an acute-angled triangle is $23$ and the radius of its Inscribed circle is $9$. Common external tangents to its ex-circles, other than straight lines containing the sides of the original triangle, form a triangle. Find the radius of its inscribed circle.

Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB \parallel A'B' $, $BC \parallel B'C'$ and $CA \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.

All vertices of triangle $ABC$ lie inside square $K$. Prove that if all of them are reflected symmetrically with respect to the point of intersection of the medians of triangle $ABC$, then at least one of the resulting three points will be inside $K$.

Let $ ABCD$ be a quadrilateral inscribed in a circle. The diagonal $ BD$ bisects $ AC$. If $ AB = 10$, $ AD = 12$ and $ DC = 11$, find $ BC$.

Pyramid $EARLY$ has rectangular base $EARL$ and apex $Y$, and all of its edges are of integer length. The four edges from the apex have lengths $1, 4, 7, 8$ (in no particular order), and EY is perpendicular to $YR$. Find the area of rectangle $EARL$.

A right prism \(ABCA_1B_1C_1\) is given. It is known that triangles \(A_1BC\), \(AB_1C\), \(ABC_1\), and \(ABC\) are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle \(ABC\), lie on the same sphere.

Find all positive values of $a$, for which there is a number $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points. Also prove that for every such a there exists $b$ such that the parabola $y=ax^2-b$ intersects the unit circle at four distinct points whose $x$-coordinates form an arithmetic progression.

In a triangle $ABC$ , we have a point $O$ on $BC$ . Now show that there exists a line $l$ such that $l||AO$ and $l$ divides the triangle $ABC$ into two halves of equal area .

Convex quadrilaterals \(ABCD\), \(A_1B_1C_1D_1\), and \(A_2B_2C_2D_2\) are similar with vertices in order. Points \(A\), \(A_1\), \(B_2\), \(B\) are collinear in order, points \(B\), \(B_1\), \(C_2\), \(C\) are collinear in order, points \(C\), \(C_1\), \(D_2\), \(D\) are collinear in order, and points \(D\), \(D_1\), \(A_2\), \(A\) are collinear in order. Diagonals \(AC\) and \(BD\) intersect at \(P\), diagonals \(A_1C_1\) and \(B_1D_1\) intersect at \(P_1\), and diagonals \(A_2C_2\) and \(B_2D_2\) intersect at \(P_2\). Prove that points \(P\), \(P_1\), and \(P_2\) are collinear. [i]Proposed by Holden Mui[/i]

In $\triangle ABC$, point $M$ is the middle point of $AC$. $MD//AB$ and meet the tangent of $A$ to $\odot(ABC)$ at point $D$. Point $E$ is in $AD$ and point $A$ is the middle point of $DE$. $\{P\}=\odot(ABE)\cap AC,\{Q\}=\odot(ADP)\cap DM$. Prove that $\angle QCB=\angle BAC$. [url=https://imgtu.com/i/4pZ7Zj][img]https://z3.ax1x.com/2021/09/12/4pZ7Zj.jpg[/img][/url]