The circumcenter $O$, the incenter $I$, and the midpoint $M$ of a diagonal of a bicentral quadrilateral were marked. After this the quadrilateral was erased. Restore it.

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

If $a,b > 0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by = 6$ has area 6, then $ab =$ $\text{(A)} \ 3 \qquad \text{(B)} \ 6 \qquad \text{(C)} \ 12 \qquad \text{(D)} \ 108 \qquad \text{(E)} \ 432$

Two triangles $ABC$ and $A'B'C'$ are on the plane. It is known that each side length of triangle $ABC$ is not less than $a$, and each side length of triangle $A'B'C'$ is not less than $a'$. Prove that we can always choose two points in the two triangles respectively such that the distance between them is not less than $\sqrt{\dfrac{a^2+a'^2}{3}}$.

Triangles $MAB{}$ and $MA_1B_1{}$ are similar and have the same orientation. Prove that the circumcircles of these triangles cointain the intersection point of lines $AA_1{}$ and $BB_1{}$.

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle and $D$ be the foot of the altitude from $C$ to $AB$. If $|CH|=|HD|$ where $H$ is the orthocenter, what is $\tan \widehat {A} \cdot \tan \widehat{B}$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ \sqrt 2 \qquad\textbf{(C)}\ 3/2 \qquad\textbf{(D)}\ \sqrt 3 \qquad\textbf{(E)}\ \text{None of the preceding} $

Let $ABC$ be an equilateral triangle with side length $8$, and let $D$, $E$, and $F$ be points on the sides $BC$, $CA$, and $AB$ respectively. Suppose that $BD = 2$ and $\angle ADE = \angle DEF = 60^\circ$. Calculate the length of segment $AF$.

[u]Round 9[/u] [b]p25.[/b] Let $S$ be the region bounded by the lines $y = x/2$, $y = -x/2$, and $x = 6$. Pick a random point $P = (x, y)$ in $S$ and translate it $3$ units right to $P' = (x + 3, y)$. What is the probability that $P'$ is in $S$? [b]p26.[/b] A triangle with side lengths $17$, $25$, and $28$ has a circle centered at each of its three vertices such that the three circles are mutually externally tangent to each other. What is the combined area of the circles? [b]p27.[/b] Find all ordered pairs $(x, y)$ of integers such that $x^2 - 2x + y^2 - 6y = -9$. [u]Round 10[/u] [b]p28.[/b] In how many ways can the letters in the word $SCHAFKOPF$ be arranged if the two $F$’s cannot be next to each other and the $A$ and the $O$ must be next to each other? [b]p29.[/b] Let a sequence $a_0, a_1, a_2, ...$ be defined by $a_0 = 20$, $a_1 = 11$, $a_2 = 0$, and for all integers $n \ge 3$, $$a_n + a_{n-1 }= a_{n-2} + a_{n-3}.$$ Find the sum $a_0 + a_1 + a_2 + · · · + a_{2010} + a_{2011}$. [b]p30.[/b] Find the sum of all positive integers b such that the base $b$ number $190_b$ is a perfect square. [u]Round 11[/u] [b]p31.[/b] Find all real values of x such that $\sqrt[3]{4x -1} + \sqrt[3]{4x + 1 }= \sqrt[3]{8x}$. [b]p32.[/b] Right triangle $ABC$ has a right angle at B. The angle bisector of $\angle ABC$ is drawn and extended to a point E such that $\angle ECA = \angle ACB$. Let $F$ be the foot of the perpendicular from $E$ to ray $\overrightarrow{BC}$. Given that $AB = 4$, $BC = 2$, and $EF = 8$, find the area of triangle $ACE$. [b]p33.[/b] You are the soul in the southwest corner of a four by four grid of distinct souls in the Fields of Asphodel. You move one square east and at the same time all the other souls move one square north, south, east, or west so that each square is now reoccupied and no two souls switched places directly. How many end results are possible from this move? [u]Round 12[/u] [b]p34.[/b] A [i]Pythagorean [/i] triple is an ordered triple of positive integers $(a, b, c)$ with $a < b < c $and $a^2 + b^2 = c^2$ . A [i]primitive [/i] Pythagorean triple is a Pythagorean triple where all three numbers are relatively prime to each other. Find the number of primitive Pythagorean triples in which all three members are less than $100,000$. If $P$ is the true answer and $A$ is your team’s answer to this problem, your score will be $max \left\{15 -\frac{|A -P|}{500} , 0 \right\}$ , rounded to the nearest integer. [b]p35.[/b] According to the Enable2k North American word list, how many words in the English language contain the letters $L, M, T$ in order but not necessarily together? If $A$ is your team’s answer to this problem and $W$ is the true answer, the score you will receive is $max \left\{15 -100\left| \frac{A}{W}-1\right| , 0 \right\}$, rounded to the nearest integer. [b]p36.[/b] Write down $5$ positive integers less than or equal to $42$. For each of the numbers written, if no other teams put down that number, your team gets $3$ points. Otherwise, you get $0$ points. Any number written that does not satisfy the given requirement automatically gets $0$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h2952214p26434209]here[/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3133709p28395558]here[/url]. Rest Rounds soon. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Inside the equilateral triangle $ABC$, points $P$ and $Q$ are chosen so that the quadrilateral $APQC$ is convex, $AP = PQ = QC$ and $\angle PBQ = 30^o$. Prove that $AQ = BP$.

In triangle $ABC$, given lines $l_{b}$ and $l_{c}$ containing the bisectors of angles $B$ and $C$, and the foot $L_{1}$ of the bisector of angle $A$. Restore triangle $ABC$.

$ABCDE$ is a pentagon whose vertices lie on circle $\omega$ where $\angle DAB=90^{\circ}$. Let $EB$ and $AC$ intersect at $F$, $EC$ meet $BD$ at $G$. $M$ is the midpoint of arc $AB$ on $\omega$, not containing $C$. If $FG\parallel DE\parallel CM$ holds, then what is the value of $\frac{|GE|}{|GD|}$?

Let $ABC$ be an acute triangle with $AB<AC$. Point $M{}$ from the side $(BC)$ is the foot of the bisector from the vertex $A{}$. The perpendicular bisector of the segment $[AM]$ intersects the side $(AC)$ in $E{}$, the side $(AB)$ in $D$ and the line $(BC)$ in $F{}$. Prove that $\frac{DB}{CE}=\frac{FB}{FC}=\left(\frac{AB}{AC}\right)^2$.

In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.

In a garden there is an $L$-shaped fence, see figure. You also have at your disposal two finished straight fence sections that are $13$ m and $14$ m long respectively. From point $A$ you want to delimit a part of the garden with an area of at least $200$ m$^2$ . Is it possible to do this? [img]https://1.bp.blogspot.com/-VLWIImY7HBA/X0yZq5BrkTI/AAAAAAAAMbg/8CyP6DzfZTE5iX01Qab3HVrTmaUQ7PvcwCK4BGAYYCw/s400/sweden%2B16p1.png[/img]

Let $AD$ be the bisector of an acute-angled triangle $ABC$. The circle circumscribed around the triangle $ABD$ intersects the straight line perpendicular to $AD$ that passes through point $B$, at point $E$. Point $O$ is the center of the circumscribed circle of triangle $ABC$. Prove that the points $A, O, E$ lie on the same line.

Let $\triangle{ABC}$ be a non-equilateral, acute triangle with $\angle A=60^\circ$, and let $O$ and $H$ denote the circumcenter and orthocenter of $\triangle{ABC}$, respectively. (a) Prove that line $OH$ intersects both segments $AB$ and $AC$. (b) Line $OH$ intersects segments $AB$ and $AC$ at $P$ and $Q$, respectively. Denote by $s$ and $t$ the respective areas of triangle $APQ$ and quadrilateral $BPQC$. Determine the range of possible values for $s/t$.

Perpendicular bisectors of the sides $BC$ and $AC$ of an acute-angled triangle $ABC$ intersect lines $AC$ and $BC$ at points $M$ and $N$. Let point $C$ move along the circumscribed circle of triangle $ABC$, remaining in the same half-plane relative to $AB$ (while points $A$ and $B$ are fixed). Prove that line $MN$ touches a fixed circle.

The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png[/img]

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

Given a triangle $ABC$, let $D$ and $E$ be points on the side $BC$ such that $\angle BAD = \angle CAE$. If $M$ and $N$ are, respectively, the points of tangency of the incircles of the triangles $ABD$ and $ACE$ with the line $BC$, then show that \[\frac{1}{MB}+\frac{1}{MD}= \frac{1}{NC}+\frac{1}{NE}. \]

There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?

The intersection of two triangles is a hexagon. If this hexagon is removed, six small triangles remain. These six triangles have the same in-radii. Prove the in-radii of the original two triangles are also equal. Spoiler: This is one of the highlights of TT. Also SA3

Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$. If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.