$BL $ is the bisector of an isosceles triangle $ABC $. A point $D $ is chosen on the Base $BC $ and a point $E $ is chosen on the lateral side $AB $ so that $AE=\frac {1}{2}AL=CD $. Prove that $LE=LD $. Tuymaada 2017 Q5 Juniors

Let $ABC$ be a triangle with $\angle A = 90$ and let $D$ be the orthogonal projection of $A$ onto $BC$. The midpoints of $AD$ and $AC$ are called $E$ and $F$, respectively. Let $M$ be the circumcentre of $BEF$. Prove that $AC$ and $ BM$ are parallel.

A cube with side length $100cm$ is filled with water and has a hole through which the water drains into a cylinder of radius $100cm.$ If the water level in the cube is falling at a rate of $1 \frac{cm}{s} ,$ how fast is the water level in the cylinder rising?

Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)

Given a triangle $ABC$, with right $\angle A$, we know: the point $T$ of tangency of the circumference inscribed in $ABC$ with the hypotenuse $BC$, the point $D$ of intersection of the angle bisector of $\angle B$ with side AC and the point E of intersection of the angle bisector of $\angle C$ with side $AB$ . Describe a construction with ruler and compass for points $A$, $B$, and $C$. Justify.

Let $ABCD$ be a convex quadrilateral with the line $CD$ being tangent to the circle on diameter $AB$. Prove that the line $AB$ is tangent to the circle on diameter $CD$ if and only if the lines $BC$ and $AD$ are parallel.

All vertices of a convex polyhedron with 100 edges are cut off by some planes. The planes do not intersect either inside or on the surface of the polyhedron. For this new polyhedron find a) the number of vertices; [i](1 point)[/i] b) the number of edges. [i](2 points)[/i]

Two circles $\omega_1$ and $\omega_2$ meeting at point $A$ and a line $a$ are given. Let $BC$ be an arbitrary chord of $\omega_2$ parallel to $a$, and $E$, $F$ be the second common points of $AB$ and $AC$ respectively with $\omega_1$. Find the locus of common points of lines $BC$ and $EF$.

In the $x$-$y$ plane, for the origin $ O$, given an isosceles triangle $ OAB$ with $ AO \equal{} AB$ such that $ A$ is on the first quadrant and $ B$ is on the $ x$ axis. Denote the area by $ s$. Find the area of the common part of the traingle and the region expressed by the inequality $ xy\leq 1$ to give the area as the function of $ s$.

Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and[list][*]if the card is black, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance 1 away from the vertex;[*]if the card is white, she simultaneously replaces the number on each vertex by the sum of the three numbers on vertices that are distance $\sqrt2$ away from the vertex.[/list]When Tessa finishes drawing all cards of the deck, what is the maximum possible value of a number that is on the cube?

In a triangle $ABC$ with circumcircle $(O)$ suppose that $A$-altitude cut $(O)$ at $D$. Let altitude of $B,C$ cut $AC,AB$ at $E,F$. $H$ is orthocenter and $T$ is midpoint of $AH$. Parallel line with $EF$ passes through $T$ cut $AB,AC$ at $X,Y$. Prove that $\angle XDF = \angle YDE$.

Let circles $C_1$ and $C_2$ be internally tangent at point $P$, with $C_1$ being the smaller circle. Consider a line passing through $P$ which intersects $C_1$ at $Q$ and $C_2$ at $R$. Let the line tangent to $C_2$ at $R$ and the line perpendicular to $\overline{PR}$ passing through $Q$ intersect at a point $S$ outside both circles. Given that $SR = 5$, $RQ = 3$, and $QP = 2$, compute the radius of $C_2$.

[b]p13.[/b] Emma has the five letters: $A, B, C, D, E$. How many ways can she rearrange the letters into words? Note that the order of words matter, ie $ABC DE$ and $DE ABC$ are different. [b]p14.[/b] Seven students are doing a holiday gift exchange. Each student writes their name on a slip of paper and places it into a hat. Then, each student draws a name from the hat to determine who they will buy a gift for. What is the probability that no student draws himself/herself? [b]p15.[/b] We model a fidget spinner as shown below (include diagram) with a series of arcs on circles of radii $1$. What is the area swept out by the fidget spinner as it’s turned $60^o$ ? [img]https://cdn.artofproblemsolving.com/attachments/9/8/db27ffce2af68d27eee5903c9f09a36c2a6edf.png[/img] [b]p16.[/b] Let $a,b,c$ be the sides of a triangle such that $gcd(a, b) = 3528$, $gcd(b, c) = 1008$, $gcd(a, c) = 504$. Find the value of $a * b * c$. Write your answer as a prime factorization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be an arbitrary point on the side $BC$ of triangle $ABC$ and let $D$ be the tangency point between the incircle of the triangle $ABC$ and the side $BC$. If $Q$ and $R$ are respectively the incenters in the triangles $ABP$ and $ACP$, prove that $\angle QDR$ is a right angle. Prove that the triangle $QDR$ is isosceles if and only if $P$ is the foot of the altitude from $A$ in the triangle $ABC$.

Maria Ivanovna drew on the blackboard a right triangle $ABC$ with a right angle $B$. Three students looked at her and said: $\bullet$ Yura said: "The hypotenuse of this triangle is $10$ cm." $\bullet$ Roma said: "The altitude drawn from the vertex $B$ on the side $AC$ is $6$ cm." $\bullet$ Seva said: "The area of the triangle $ABC$ is $25$ cm$^2$." Determine which of the students was mistaken if it is known that there is exactly one such person.

Show that it is possible to circumscribe a circle of radius $R$ about, and inscribe a circle of radius $r$ in some triangle with one angle equal to $a$, if and only if $$\frac{2R}{r} \ge \dfrac{1}{ \sin \frac{a}{2} \left(1- \sin \frac{a}{2} \right)}$$

A square with sides of length $6$ has the same area as a rectangle with a length of $9$. What is the width of the rectangle? $\textbf{(A) } 2\qquad\textbf{(B) } \frac73\qquad\textbf{(C) } 3\qquad\textbf{(D) } \frac{10}{3}\qquad\textbf{(E) } 4$

Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points. (a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with $n$ chosen points.

Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.

Let $\Delta ABC$ be an acute-angled triangle with $AB < AC$, $H$ be a point on $BC$ such that $AH\ \bot BC$ and $G$ be the centroid of $\Delta ABC$. Let $P,Q$ be the point of tangency of the inscribed circle of $\Delta ABC$ with $AB,AC$, correspondingly. Define $M,N$ to be the midpoint of $PB,QC$, correspondingly. Let $D,E$ be points on the inscribed circle of $\Delta ABC$ such that $\angle BDH + \angle ABC = 180^{\circ}$, $\angle CEH + \angle ACB = 180^{\circ}$. Prove that lines $MD,NE,HG$ share a common point.

We have an empty equilateral triangle with length of a side $l$. We put the triangle, horizontally, over a sphere of radius $r$. Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ and $r$)?

The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.

In $\triangle ABC$, $X, Y$, are points on BC such that $BX=XY=YC$, $M , N$ are points on $AC$ such that $AM=MN=NC$. $BM$ and $BN$ intersect $AY$ at $S$ and $R$ and respectively. If the area of $\triangle ABC$ is $1$, find the area of $SMNR$.