Found problems: 85335
Each of the integers from $1$ to $n$ is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, $A,B,$ and $C,$ take turns in the order $A,B,C,A,\dots$ choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered $1.$
Show that for each of the three players, there are arbitrarily large values of $n$ for which that player has the highest probability among the three players of winning the game.
On the planet Brick, which has the shape of a rectangular parallelepiped with edges of $1$ km,$ 2$ km and $4$ km, the Little Prince built a house in the center of the largest face. What is the distance from the house to the most remote point on the planet? (The distance between two points on the surface of a planet is defined as the length of the shortest path along the surface connecting these points.)
Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$.
Prove that lines $AD, PM$, and $BC$ are concurrent.
A circle through the vertices $A$ and $B$ of $\triangle ABC$ intersects segments $AC$ and $BC$ at points $P$ and $Q$ respectively. If $AQ=AC$, $\angle BAQ=\angle CBP$ and $AB=(\sqrt{3}+1)PQ$, find the measures of the angles of $\triangle ABC$.
Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here.
Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.
Lunasa, Merlin, and Lyrica each has an instrument. We know the following about the prices of their instruments:
(a) If we raise the price of Lunasa's violin by $50\%$ and decrease the price of Merlin's trumpet by $50\%$, the violin will be $\$50$ more expensive than the trumpet;
(b) If we raise the price of Merlin's trumpet by $50\%$ and decrease the price of Lyrica's piano by $50\%$, the trumpet will be $\$50$ more expensive than the piano.
Given these conditions only, there exist integers $m$ and $n$ such that if we raise the price of Lunasa's violin by $m\%$ and decrease the price of Lyrica's piano by $m\%$, the violin must be exactly $\$n$ more expensive than the piano. Find $100m+n$.
[i]Proposed by Yannick Yao[/i]
Each of the $n^2$ cells of an $n \times n$ grid is colored either black or white. Let $a_i$ denote the number of white cells in the $i$-th row, and let $b_i$ denote the number of black cells in the $i$-th column. Determine the maximum value of $\sum_{i=1}^n a_ib_i$ over all coloring schemes of the grid.
[i]Proposed by Alex Zhai[/i]
Let $ABC$ be a triangle and let $D\in (BC)$ be the foot of the $A$- altitude. The circle $w$ with the diameter $[AD]$
meet again the lines $AB$ , $AC$ in the points $K\in (AB)$ , $L\in (AC)$ respectively. Denote the meetpoint $M$
of the tangents to the circle $w$ in the points $K$ , $L$ . Prove that the ray $[AM$ is the $A$-median in $\triangle ABC$ ([b][u]Serbia[/u][/b]).
Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.
Let $a, b$, and $c$ be integers simultaneously satisfying the equations $4abc + a + b + c = 2018$ and $ab + bc + ca = -507$. Find $|a| + |b|+ |c|$.
For each positive integer $k$, let $S_k$ denote the increasing arithmetic sequence of integers whose first term is $1$ and whose common difference is $k$. For example, $S_3$ is the sequence $1,4,7,10,...$. For how many values of $k$ does $S_k$ contain the term $2005$?
$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$
If 1,2, and 3 are solutions to the equation $ x^4 \plus{} ax^2 \plus{} bx \plus{} c \equal{} 0,$ then $ a\plus{}c$ equals
A. -12
B. 24
C. 35
D. -61
E. -63
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.
[i]Proposed by David Monk, United Kingdom[/i]
Two branches of the hyperbola $xy=1$ are $C_1,C_2$ ($C_1$ in Quadrant I, $C_2$ in Quadrant III). Three apexes of regular triangle $PQR$ are on the hyperbola.
[b](a)[/b] $P,Q,R$ cannot be on the same branch.
[b](b)[/b] $P(-1,-1)$ is a point on $C_2$, if $Q,R$ are on $C_1$, find their coordinates.
The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.
The solutions to the equation $(z+6)^8=81$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled $A,B,$ and $C$. What is the least possible area of $\triangle ABC?$
$\textbf{(A) } \frac{1}{6}\sqrt{6} \qquad \textbf{(B) } \frac{3}{2}\sqrt{2}-\frac{3}{2} \qquad \textbf{(C) } 2\sqrt3-3\sqrt2 \qquad \textbf{(D) } \frac{1}{2}\sqrt{2} \qquad \textbf{(E) } \sqrt 3-1$
Let $\Gamma_1$ and $\Gamma_2$ be two circles intersecting at $P$ and $Q$. The common tangent, closer to $P$, of $\Gamma_1$ and $\Gamma_2$ touches $\Gamma_1$ at $A$ and $\Gamma_2$ at $B$. The tangent of $\Gamma_1$ at $P$ meets $\Gamma_2$ at $C$, which is different from $P$, and the extension of $AP$ meets $BC$ at $R$.
Prove that the circumcircle of triangle $PQR$ is tangent to $BP$ and $BR$.
Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$.
[i]Marian Andronache[/i]
Find all functions $f: \mathbb{Q} \mapsto \mathbb{C}$ satisfying
(i) For any $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$, $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$.
(ii) $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$ for all $x \in \mathbb{Q}$.
Let $n(\ge 2)$ be an integer. $2n^2$ contestants participate in a Chinese chess competition, where any two contestant play exactly once. There may be draws. It is known that
(1)If A wins B and B wins C, then A wins C.
(2)there are at most $\frac{n^3}{16}$ draws.
Proof that it is possible to choose $n^2$ contestants and label them $P_{ij}(1\le i,j\le n)$, so that for any $i,j,i',j'\in \{1,2,...,n\}$, if $i<i'$, then $P_{ij}$ wins $P_{i'j'}$.
Let $\triangle ABC$ be a triangle with a right angle $\angle ABC$. Let $D$ be the midpoint of $\overline{BC}$, let $E$ be the midpoint of $\overline{AC}$, and let $F$ be the midpoint of $\overline{AB}$. Let $G$ be the midpoint of $\overline{EC}$. One of the angles of $\triangle DFG$ is a right angle. What is the least possible value of $\frac{BC}{AG}$?
Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.
A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.
In triangle $ABC$ with $\angle ABC = 60^{\circ}$ and $5AB = 4BC$, points $D$ and $E$ are the feet of the altitudes from $B$ and $C$, respectively. $M$ is the midpoint of $BD$ and the circumcircle of triangle $BMC$ meets line $AC$ again at $N$. Lines $BN$ and $CM$ meet at $P$. Prove that $\angle EDP = 90^{\circ}$.