Incircle of a triangle $ABC$ touches $AB$ and $AC$ at $D$ and $E$, respectively. Point $J$ is the excenter of $A$. Points $M$ and $N$ are midpoints of $JD$ and $JE$. Lines $BM$ and $CN$ cross at point $P$. Prove that $P$ lies on the circumcircle of $ABC$.

The triangle $ABC$ ($AB > BC$) is inscribed in the circle $\Omega$. On the sides $AB$ and $BC$, the points $M$ and $N$ are chosen, respectively, so that $AM = CN$, The lines $MN$ and $AC$ intersect at point $K$. Let $P$ be the center of the inscribed circle of triangle $AMK$, and $Q$ the center of the excircle of the triangle $CNK$ tangent to side $CN$. Prove that the midpoint of the arc $ABC$ of the circle $\Omega$ is equidistant from the $P$ and $Q$.

If distinct digits $D,E,L,M,Q$ (between $0$ and $9$ inclusive) satisfy \begin{tabular}{c@{\,}c@{\,}c@{\,}c} & & $E$ & $L$ \\ + & $M$ & $E$ & $M$ \\\hline & $Q$ & $E$ & $D$ \\ \end{tabular} what is the maximum possible value of the three digit integer $QED$? [i]2019 CCA Math Bonanza Individual Round #6[/i]

Let $ABC$ be an acute triangle, $M$ be the midpoint of $BC$ and $P$ be a point on line segment $AM$. Lines $BP$ and $CP$ meet the circumcircle of $ABC$ again at $X$ and $Y$ , respectively, and sides $AC$ at $D$ and $AB$ at $E$, respectively. Prove that the circumcircles of $AXD$ and $AYE$ have a common point $T \ne A$ on line $AM$.

Let $n$ be an even positive integer. Show that there is a permutation $\left(x_{1},x_{2},\ldots,x_{n}\right)$ of $\left(1,\,2,\,\ldots,n\right)$ such that for every $i\in\left\{1,\ 2,\ ...,\ n\right\}$, the number $x_{i+1}$ is one of the numbers $2x_{i}$, $2x_{i}-1$, $2x_{i}-n$, $2x_{i}-n-1$. Hereby, we use the cyclic subscript convention, so that $x_{n+1}$ means $x_{1}$.

Let $\alpha$ be the angle between two lines containing the diagonals of a regular $1996$-gon, and let $\beta\not= 0$ be another such angle. Prove that $\frac{\alpha}{\beta}$ is a rational number.

There are $20$ pupils in the Backwoods school. Any two of them have a common grandfather. Prove that there exist $14$ pupils all of whom have a common grandfather. (AV Shapovalov)

We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$, there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.) [i]Proposed by Hesam Rajabzadeh[/i]

Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.

Seventeen football fans were planning to go to Korea to watch the World Cup football match. They selected 17 matches. The conditions of the admission tickets they booked were such that - One person should book at most one admission ticket for one match; - At most one match was same in the tickets booked by every two persons; - There was one person who booked six tickets. How many tickets did those football fans book at most?

The National Marriage Council wishes to invite $n$ couples to form 17 discussion groups under the following conditions: (1) All members of a group must be of the same sex; i.e. they are either all male or all female. (2) The difference in the size of any two groups is 0 or 1. (3) All groups have at least 1 member. (4) Each person must belong to one and only one group. Find all values of $n$, $n \leq 1996$, for which this is possible. Justify your answer.

Let $x_1$ be a given positive integer. A sequence $\{x_n\}_ {n\geq 1}$ of positive integers is such that $x_n$, for $n \geq 2$, is obtained from $x_ {n-1}$ by adding some nonzero digit of $x_ {n-1}$. Prove that a) the sequence contains an even term; b) the sequence contains infinitely many even terms.

Let $k\ge 0$ be an integer and suppose the integers $a_1,a_2,\dots,a_n$ give at least $2k$ different residues upon division by $(n+k)$. Show that there are some $a_i$ whose sum is divisible by $n+k$.

[u]Round 1[/u] [b]p1.[/b] If $2m = 200 cm$ and $m \ne 0$, find $c$. [b]p2.[/b] A right triangle has two sides of lengths $3$ and $4$. Find the smallest possible length of the third side. [b]p3.[/b] Given that $20(x + 17) = 17(x + 20)$, determine the value of $x$. [u]Round 2[/u] [b]p4.[/b] According to the Egyptian Metropolitan Culinary Community, food service is delayed on $\frac23$ of flights departing from Cairo airport. On average, if flights with delayed food service have twice as many passengers per flight as those without, what is the probability that a passenger departing from Cairo airport experiences delayed food service? [b]p5.[/b] In a positive geometric sequence $\{a_n\}$, $a_1 = 9$, $a_9 = 25$. Find the integer $k$ such that $a_k = 15$ [b]p6.[/b] In the Delicate, Elegant, and Exotic Music Organization, pianist Hans is selling two types of owers with different prices (per ower): magnolias and myosotis. His friend Alice originally plans to buy a bunch containing $50\%$ more magnolias than myosotis for $\$50$, but then she realizes that if she buys $50\%$ less magnolias and $50\%$ more myosotis than her original plan, she would still need to pay the same amount of money. If instead she buys $50\%$ more magnolias and $50\%$ less myosotis than her original plan, then how much, in dollars, would she need to pay? [u]Round 3[/u] [b]p7.[/b] In square $ABCD$, point $P$ lies on side $AB$ such that $AP = 3$,$BP = 7$. Points $Q,R, S$ lie on sides $BC,CD,DA$ respectively such that $PQ = PR = PS = AB$. Find the area of quadrilateral $PQRS$. [b]p8.[/b] Kristy is thinking of a number $n < 10^4$ and she says that $143$ is one of its divisors. What is the smallest number greater than $143$ that could divide $n$? [b]p9.[/b] A positive integer $n$ is called [i]special [/i] if the product of the $n$ smallest prime numbers is divisible by the sum of the $n$ smallest prime numbers. Find the sum of the three smallest special numbers. [u]Round 4[/u] [b]p10.[/b] In the diagram below, all adjacent points connected with a segment are unit distance apart. Find the number of squares whose vertices are among the points in the diagram and whose sides coincide with the drawn segments. [img]https://cdn.artofproblemsolving.com/attachments/b/a/923e4d2d44e436ccec90661648967908306fea.png[/img] [b]p11.[/b] Geyang tells Junze that he is thinking of a positive integer. Geyang gives Junze the following clues: $\bullet$ My number has three distinct odd digits. $\bullet$ It is divisible by each of its three digits, as well as their sum. What is the sum of all possible values of Geyang's number? [b]p12.[/b] Regular octagon $ABCDEFGH$ has center $O$ and side length $2$. A circle passes through $A,B$, and $O$. What is the area of the part of the circle that lies outside of the octagon? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h2936505p26278645]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider skew lines $PP'$, $QQ'$ and points $X$, $Y$ lying on them, respectively. Initially, we have $X=P$, $Y=Q$. Both points $X$, $Y$ start moving simultaneously along the rays $PP'$, $QQ'$ with the speeds $c_1$, $c_2$, respectively. Show that midpoint $Z$ of segment $XY$ always lies on a fixed ray $RR'$, where $R$ is midpoint of $PQ$.

The side-lengths of two equilaterals $ABC$ and $KLM$ are $1$ and $1/4$, respectively. And triangle $KLM$ located inside triangle $ABC$. Denote by $\Sigma$ the sum of the distances from $A$ to lines $KL, LM$ and $MK$. Find the location of triangle $KLM$ when $\Sigma$ is maximal.

SEEMOUS 2016 COMPETITION PROBLEMS

Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is its area? (A)$20\sqrt{3}$ (B)$20\sqrt{2}$ (C)$25\sqrt{3}$ (D)None of these

We are given an infinite deck of cards, each with a real number on it. For every real number $x$, there is exactly one card in the deck that has $x$ written on it. Now two players draw disjoint sets $A$ and $B$ of $100$ cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions: 1. The winner only depends on the relative order of the $200$ cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner. 2. If we write the elements of both sets in increasing order as $A =\{ a_1 , a_2 , \ldots, a_{100} \}$ and $B= \{ b_1 , b_2 , \ldots , b_{100} \}$, and $a_i > b_i$ for all $i$, then $A$ beats $B$. 3. If three players draw three disjoint sets $A, B, C$ from the deck, $A$ beats $B$ and $B$ beats $C$ then $A$ also beats $C$. How many ways are there to define such a rule? Here, we consider two rules as different if there exist two sets $A$ and $B$ such that $A$ beats $B$ according to one rule, but $B$ beats $A$ according to the other. [i]Proposed by Ilya Bogdanov, Russia[/i]

For how many positive integers $ m$ does there exist at least one positive integer $ n$ such that $ m\cdot n \le m \plus{} n$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}$ infinitely many

Consider an acute triangle $ABC$ and its altitudes from $A$ ,$B$ that intersect the respective sides at $D ,E$. Let us call the point of intersection of the altitudes $H$. Construct the circle with center $H$ and radius $HE$. From $C$ draw a tangent line to the circle at point $P$. With center $B$ and radius $BE$ draw another circle and from $C$ another tangent line is drawn to this circle in the point $Q$. Prove that the points $D, P$, and $Q$ are collinear.

Compute the area of the polygon formed by connecting the roots of \[x^{10} + x^9 + x^8 + x^6 + x^5 + x^4 + x^2 + x + 1\] graphed in the complex plane with line segments in counterclockwise order.

Let $\mathbb{N}$ be the set of positive integers. Across all function $f:\mathbb{N}\to\mathbb{N}$ such that $$mn+1\text{ divides } f(m)f(n)+1$$ for all positive integers $m$ and $n$, determine all possible values of $f(101).$

On planet $X31$ there are only two types of tickets, however the system is not so bad because there are only fifteen full prices that cannot be paid exactly (you pay more and receive change). If $18$ is one of those prices that cannot be paid exactly, find the value of each type of bill.

Let $H$ be orthocenter and $O$ circumcenter of an acuted angled triangle $ABC$. $D$ and $E$ are feets of perpendiculars from $A$ and $B$ on $BC$ and $AC$ respectively. Let $OD$ and $OE$ intersect $BE$ and $AD$ in $K$ and $L$, respectively. Let $X$ be intersection of circumcircles of $HKD$ and $HLE$ different than $H$, and $M$ is midpoint of $AB$. Prove that $K, L, M$ are collinear iff $X$ is circumcenter of $EOD$.