In a triangle $ABC$, $AB>AC$. The foot of the altitude from $A$ to $BC$ is $D$, the intersection of bisector of $B$ and $AD$ is $K$, the foot of the altitude from $B$ to $CK$ is $M$ and let $BM$ and $AK$ intersect at point $N$. The line through $N$ parallel to $DM$ intersects $AC$ at $T$. Prove that $BM$ is the bisector of angle $\widehat{TBC}$.

The triangles $BCD$ and $ACE$ are externally constructed to sides $BC$ and $CA$ of a triangle $ABC$ such that $AE = BD$ and $\angle BDC+\angle AEC = 180^\circ$. Let $F$ be a point on segment $AB$ such that ${AF\over FB}={CD\over CE}$. Prove that ${DE\over CD+CE}={EF\over BC}={FD\over AC}$.

Given a circle and its center $O$, a point $A$ inside the circle and a distance $h$, construct a triangle $BAC$ with $\angle BAC = 90^\circ$, $B$ and $C$ on the circle and the altitude from $A$ length $h$.

Given a triangle $ ABC$, let $ r$ be the external bisector of $ \angle ABC$. $ P$ and $ Q$ are the feet of the perpendiculars from $ A$ and $ C$ to $ r$. If $ CP \cap BA \equal{} M$ and $ AQ \cap BC\equal{}N$, show that $ MN$, $ r$ and $ AC$ concur.

Point $A$ lies on one ray and points $B,C$ lie on the other ray of an angle with the vertex at $O$ such that $B$ lies between $O$ and $C$. Let $O_1$ be the incenter of $\triangle OAB$ and $O_2$ be the center of the excircle of $\triangle OAC$ touching side $AC$. Prove that if $O_1A = O_2A$, then the triangle $ABC$ is isosceles.

Regular octagon $A_1A_2A_3A_4A_5A_6A_7A_8$ is inscribed in a circle of area $1$. Point $P$ lies inside the circle so that the region bounded by $\overline{PA_1}$, $\overline{PA_2}$, and the minor arc $\widehat{A_1A_2}$ of the circle has area $\tfrac17$, while the region bounded by $\overline{PA_3}$, $\overline{PA_4}$, and the minor arc $\widehat{A_3A_4}$ of the circle has area $\tfrac 19$. There is a positive integer $n$ such that the area of the region bounded by $\overline{PA_6}$, $\overline{PA_7}$, and the minor arc $\widehat{A_6A_7}$ is equal to $\tfrac18 - \tfrac{\sqrt 2}n$. Find $n$.

Let $ABC$ be a triangle and $I$ its incenter. The lines $BI$ and $CI$ intersect the circumcircle of $ABC$ again at $M$ and $N$, respectively. Let $C_1$ and $C_2$ be the circumferences of diameters $NI$ and $MI$, respectively. The circle $C_1$ intersects $AB$ at $P$ and $Q$, and the circle $C_2$ intersects $AC$ at $R$ and $S$. Show that $P$, $Q$, $R$ and $S$ are concyclic.

Let $ABCD$ be a quadrilateral inscribed in a circle $(O)$. Let $(O_1), (O_2), (O_3), (O_4)$ be the circles going through $(A,B), (B,C),(C,D),(D,A)$. Let $X, Y,Z, T$ be the second intersection of the pairs of the circles: $(O_1)$ and $(O_2), (O_2)$ and $(O_3), (O_3)$ and $(O_4), (O_4)$ and $(O_1)$. (a) Prove that $X, Y,Z, T$ are on the same circle of radius $I$. (b) Prove that the midpoints of the line segments $O_1O_3,O_2O_4,OI$ are collinear. Nguyễn Văn Linh

Let $ABC$ be a triangle. For any point $X$ on $BC$, let $AX$ meet the circumcircle of $ABC$ in $X'$. Prove or disprove: $XX'$ has maximum length if and only if $AX$ lies between the median and the internal angle bisector from $A$.

A trapezium has parallel sides of length equal to $a$ and $b$ ($a <b$), and the distance between the parallel sides is the altitude $h$. The extensions of the non-parallel lines intersect at a point that is a vertex of two triangles that have as sides the parallel sides of the trapezium. Express the areas of the triangles as functions of $a,b$ and $h$.

Let $S$ be a great circle with pole $P$. On any great circle through $P$, two points $A$ and $B$ are chosen equidistant from $P$. For any [i] spherical triangle [/i] $ABC$ (the sides are great circles ares), where $C$ is on $S$, prove that the great circle are $CP$ is the angle bisector of angle $C$. [b] Note. [/b] A great circle on a sphere is one whose center is the center of the sphere. A pole of the great circle $S$ is a point $P$ on the sphere such that the diameter through $P$ is perpendicular to the plane of $S$.

Let $A_1,B_1,C_1$ be the midpoints of the sides $BC,CA,AB$ respectively of a triangle $ABC$. Points $K$ on segment $C_1A_1$ and $L$ on segment $A_1B_1$ are taken such that $$\frac{C_1K}{KA_1}=\frac{BC+AC}{AC+AB}\enspace\enspace\text{and}\enspace\enspace\frac{A_1L}{LB_1}=\frac{AC+AB}{BC+AB}.$$If $BK$ and $CL$ meet at $S$, prove that $\angle C_1A_1S=\angle B_1A_1S$.

[b]p1.[/b] Give a fake proof that $0 = 1$ on the back of this page. The most convincing answer to this question at this test site will receive a point. [b]p2.[/b] It is often said that once you assume something false, anything can be derived from it. You may assume for this question that $0 = 1$, but you can only use other statements if they are generally accepted as true or if your prove them from this assumption and other generally acceptable mathematical statements. With this in mind, on the back of this page prove that every number is the same number. [b]p3.[/b] Suppose you write out all integers between $1$ and $1000$ inclusive. (The list would look something like $1$, $2$, $3$, $...$ , $10$, $11$, $...$ , $999$, $1000$.) Which digit occurs least frequently? [b]p4.[/b] Pick a real number between $0$ and $1$ inclusive. If your response is $r$ and the standard deviation of all responses at this site to this question is $\sigma$, you will receive $r(1 - (r - \sigma)^2)$ points. [b]p5.[/b] Find the sum of all possible values of $x$ that satisfy $243^{x+1} = 81^{x^2+2x}$. [b]p6.[/b] How many times during the day are the hour and minute hands of a clock aligned? [b]p7.[/b] A group of $N + 1$ students are at a math competition. All of them are wearing a single hat on their head. $N$ of the hats are red; one is blue. Anyone wearing a red hat can steal the blue hat, but in the process that person’s red hat disappears. In fact, someone can only steal the blue hat if they are wearing a red hat. After stealing it, they would wear the blue hat. Everyone prefers the blue hat over a red hat, but they would rather have a red hat than no hat at all. Assuming that everyone is perfectly rational, find the largest prime $N$ such that nobody will ever steal the blue hat. [b]p8.[/b] On the back of this page, prove there is no function f$(x)$ for which there exists a (finite degree) polynomial $p(x)$ such that $f(x) = p(x)(x + 3) + 8$ and $f(3x) = 2f(x)$. [b]p9.[/b] Given a cyclic quadrilateral $YALE$ with $Y A = 2$, $AL = 10$, $LE = 11$, $EY = 5$, what is the area of $YALE$? [b]p10.[/b] About how many pencils are made in the U.S. every year? If your answer to this question is $p$, and our (good) estimate is $\rho$, then you will receive $\max(0, 1 -\frac 12 | \log_{10}(p) - \log_{10}(\rho)|)$ points. [b]p11.[/b] The largest prime factor of $520, 302, 325$ has $5$ digits. What is this prime factor? [b]p12.[/b] The previous question was on the individual round from last year. It was one of the least frequently correctly answered questions. The first step to solving the problem and spotting the pattern is to divide $520, 302, 325$ by an appropriate integer. Unfortunately, when solving the problem many people divide it by $n$ instead, and then they fail to see the pattern. What is $n$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be an acute triangle and let $D$ be the foot of the $A$-bisector. Moreover, let $M$ be the midpoint of $AD$. The circle $\omega_1$ with diameter $AC$ meets $BM$ at $E$, while the circle $\omega_2$ with diameter $AB$ meets $CM$ at $F$. Assume that $E$ and $F$ lie inside $ABC$. Prove that $B$, $E$, $F$, $C$ are concyclic.

An acute triangle $ABC$ has side lenghths $a$, $b$, $c$ such that $a$, $b$, $c$ forms an arithmetic sequence. Given that the area of triangle $ABC$ is an integer, what is the smallest value of its perimeter? [i]2017 CCA Math Bonanza Lightning Round #3.3[/i]

In a triangle $ABC$, $D$ and $E$ lies on $AB$ and $AC$ such that $DE$ is parallel to $BC$. There exists point $P$ in the interior of $BDEC$ such that \[ \angle BPD = \angle CPE = 90^{\circ} \]Prove that the line $AP$ passes through the circumcenter of triangles $EPD$ and $BPC$.

This figure is cut out from a sheet of paper. Folding the sides upwards along the dashed lines, one gets a (non-equilateral) pyramid with a square base. Calculate the area of the base. [img]https://1.bp.blogspot.com/-lPpfHqfMMRY/XzcBIiF-n2I/AAAAAAAAMW8/nPs_mLe5C8srcxNz45Wg-_SqHlRAsAmigCLcBGAsYHQ/s0/2005%2BMohr%2Bp1.png[/img]

Determine all values of $n$ such that it is possible to divide a triangle in $n$ smaller triangles such that there are not three collinear vertices and such that each vertex belongs to the same number of segments.

Let $ABC$ be an acute triangle with circumcircle $\omega$. Let $\ell$ be the tangent line to $\omega$ at $A$. Let $X$ and $Y$ be the projections of $B$ onto lines $\ell$ and $AC$, respectively. Let $H$ be the orthocenter of $BXY$. Let $CH$ intersect $\ell$ at $D$. Prove that $BA$ bisects angle $CBD$.

A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.

The incircle of triangle $ABC$ touches its sides in points $A', B',C'$ . It is known that the orthocenters of triangles $ABC$ and $A' B'C'$ coincide. Is triangle $ABC$ regular?

Let $ABC$ be a non-obtuse triangle satisfying $AB>AC$ and $\angle B=45^{\circ}$. The circumcentre $O$ and incentre $I$ of triangle $ABC$ are such that $\sqrt{2}\ OI=AB-AC$. Find the value of $\sin A$.

On the sides of a triangle $ABC$ are constructed similar triangles $ABD,BCE,CAF$ with $k=AD/DB=BE/EC=CF/FA$ and $\alpha=\angle ADB=\angle BEC=\angle CFA$. Prove that the midpoints of the segments $AC,BC,CD$ and $EF$ form a parallelogram with an angle $\alpha$ and two sides whose ratio is $k$.

Let $A_0A_1A_2$ be a scalene triangle. Find the locus of the centres of the equilateral triangles $X_0X_1X_2$ , such that $A_k$ lies on the line $X_{k+1}X_{k+2}$ for each $k=0,1,2$ (with indices taken modulo $3$).

The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$. [img]https://cdn.artofproblemsolving.com/attachments/7/5/d90827028f426f2d2772f7d7b875eea4909211.png[/img]