Found problems: 85335
2020 Korean MO winter camp, #5
$\square ABCD$ is a quadrilateral with $\angle A=2\angle C <90^\circ$. $I$ is the incenter of $\triangle BAD$, and the line passing $I$ and perpendicular to $AI$ meets rays $CB$ and $CD$ at $E,F$ respectively. Denote $O$ as the circumcenter of $\triangle CEF$. The line passing $E$ and perpendicular to $OE$ meets ray $OF$ at $Q$, and the line passing $F$ and perpendicular to $OF$ meets ray $OE$ at $P$. Prove that the circle with diameter $PQ$ is tangent to the circumcircle of $\triangle BCD$.
LMT Team Rounds 2021+, B21
A Haiku is a Japanese poem of seventeen syllables, in three lines of five, seven, and five.
Take five good haikus
Scramble their lines randomly
What are the chances
That you end up with
Five completely good haikus
(With five, seven, five)?
Your answer will be
m over n where m,n
Are numbers such that
m,n positive
Integers where gcd
Of m,n is 1.
Take this answer and
Add the numerator and
Denominator.
[i]Proposed by Jeff Lin[/i]
2015 AMC 12/AHSME, 17
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
$\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$
2019 Yasinsky Geometry Olympiad, p2
An isosceles triangle $ABC$ ($AB = AC$) with an incircle of radius $r$ is given. We know that the point $M$ of the intersection of the medians of the triangle $ABC$ lies on this circle. Find the distance from the vertex $A$ to the point of intersection of the bisectrix of the triangle $ABC$.
(Grigory Filippovsky)
2011 Princeton University Math Competition, A3 / B6
Shirley has a magical machine. If she inputs a positive even integer $n$, the machine will output $n/2$, but if she inputs a positive odd integer $m$, the machine will output $m+3$. The machine keeps going by automatically using its output as a new input, stopping immediately before it obtains a number already processed. Shirley wants to create the longest possible output sequence possible with initial input at most $100$. What number should she input?
2017 Iran MO (3rd round), 2
Let $P(z)=a_d z^d+\dots+ a_1z+a_0$ be a polynomial with complex coefficients. The $reverse$ of $P$ is defined by
$$P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}$$
(a) Prove that
$$P^*(z)=z^d \overline{ P\left( \frac{1}{\overline{z}} \right) } $$
(b) Let $m$ be a positive integer and let $q(z)$ be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of $q(z)$ lie inside or on the unit circle. Prove that all roots of the polynomial
$$Q(z)=z^m q(z)+ q^*(z)$$
lie on the unit circle.
2009 District Round (Round II), 4
in an acute triangle $ABC$,$D$ is a point on $BC$,let $Q$ be the intersection of $AD$ and the median of $ABC$from $C$,$P$ is a point on $AD$,distinct from $Q$.the circumcircle of $CPD$ intersects $CQ$ at $C$ and $K$.prove that the circumcircle of $AKP$ passes through a fixed point differ from $A$.
2016 LMT, 22
Albert rolls a fair six-sided die thirteen times. For each time he rolls a number that is strictly greater than the previous number he rolled, he gains a point, where his first roll does not gain him a point. Find the expected number of points that Albert receives.
[i]Proposed by Nathan Ramesh
2017 Argentina National Olympiad, 3
Let $ABC$ be a triangle of perimeter $100$ and $I$ be the point of intersection of its bisectors. Let $M$ be the midpoint of side $BC$. The line parallel to $AB$ drawn by$ I$ cuts the median $AM$ at point $P$ so that $\frac{AP}{PM} =\frac73$. Find the length of side $AB$.
2018 Belarusian National Olympiad, 11.2
The altitudes $AA_1$, $BB_1$ and $CC_1$ are drawn in the acute triangle $ABC$. The bisector of the angle $AA_1C$ intersects the segments $CC_1$ and $CA$ at $E$ and $D$ respectively. The bisector of the angle $AA_1B$ intersects the segments $BB_1$ and $BA$ at $F$ and $G$ respectively. The circumcircles of the triangles $FA_1D$ and $EA_1G$ intersect at $A_1$ and $X$.
Prove that $\angle BXC=90^{\circ}$.
2021 Oral Moscow Geometry Olympiad, 5
Let $ABC$ be a triangle, $I$ and $O$ be its incenter and circumcenter respectively. $A'$ is symmetric to $O$ with respect to line $AI$. Points $B'$ and $C'$ are defined similarly. Prove that the nine-point centers of triangles $ABC$ and $A'B'C'$ coincide.
2020 JBMO TST of France, 1
Players A and B play a game. They are given a box with $n=>1$ candies. A starts first. On a move, if in the box there are $k$ candies, the player chooses positive integer $l$ so that $l<=k$ and $(l, k) =1$, and eats $l$ candies from the box. The player who eats the last candy wins. Who has winning strategy, in terms of $n$.
1999 Spain Mathematical Olympiad, 6
A plane is divided into $N$ regions by three families of parallel lines. No three lines pass through the same point. What is the smallest number of lines needed so that $N > 1999$?
2021 Polish MO Finals, 1
Let $p_i$ for $i=1,2,..., k$ be a sequence of smallest consecutive prime numbers ($p_1=2$, $p_2=3$, $p_3=3$ etc. ). Let $N=p_1\cdot p_2 \cdot ... \cdot p_k$. Prove that in a set $\{ 1,2,...,N \}$ there exist exactly $\frac{N}{2}$ numbers which are divisible by odd number of primes $p_i$.
[hide=example]For $k=2$ $p_1=2$, $p_2=3$, $N=6$. So in set $\{ 1,2,3,4,5,6 \}$ we can find $3$ number satisfying thesis: $2$, $3$ and $4$. ($1$ and $5$ are not divisible by $2$ or $3$, and $6$ is divisible by both of them so by even number of primes )[/hide]
2016 AMC 8, 20
The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$?
$\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$
2023 Lusophon Mathematical Olympiad, 4
A positive integer with 3 digits $\overline{ABC}$ is $Lusophon$ if $\overline{ABC}+\overline{CBA}$ is a perfect square. Find all $Lusophon$ numbers.
1983 IMO Shortlist, 4
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
2009 Princeton University Math Competition, 3
A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$, $BC = 2$. The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$. How large, in degrees, is $\angle ABM$?
[asy]
size(180); pathpen = rgb(0,0,0.6)+linewidth(1); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6) + linewidth(0.7) + linetype("4 4"), dr = rgb(0.8,0,0), dg = rgb(0,0.6,0), db = rgb(0,0,0.6)+linewidth(1);
pair A=(0,1), B=(0,0), C=(2,0), D=(2,1), E=A/2, F=(2,.5), M=(1/3^.5,1), N=reflect(B,M)*A;
D(B--M--D("N",N,NE)--B--D("C",C,SE)--D("D",D,NE)--M); D(D("M",M,plain.N)--D("A",A,NW)--D("B",B,SW),dd); D(D("E",E,W)--D("F",F,plain.E),dd);
[/asy]
2008 Purple Comet Problems, 9
One container of paint is exactly enough to cover the inside of an old rectangle which is three times as long as it is wide. If we make a new rectangle by shortening the old rectangle by $18$ feet and widening it by $8$ feet as shown below, one container of paint is also exactly enough to cover the inside of the new rectangle. Find the length in feet of the perimeter of the new rectangle.
[asy]
size(250);
defaultpen(linewidth(0.8));
draw((-2,0)--(-2,5)--(13,5)--(13,0)--cycle^^(16,-1)--(16,6)--(27,6)--(27,-1)--cycle^^(9,5)--(9,0)^^(16,4)--(27,4));
path rect1=(13,5)--(13,0)--(9,0)--(9,5)--cycle,rect2=(16,6)--(16,4)--(27,4)--(27,6)--cycle;
fill(rect1,lightgray);
fill(rect2,lightgray);
draw(rect1^^rect2);
[/asy]
2018 Iran MO (1st Round), 1
In a village with a population of $1000$, two hundred people have been infected by a disease. A diagnostic test can be done to check whether a person is infected, but the result could be erroneous. That is, there is a $5\%$ probability that the test result of an infected person shows that they are not infected and a $5\%$ probability that the test result of a healthy person shows that they are infected. We randomly choose someone from the population of this village and take the diagnostic test from him. What is the probability that the test result declares that person is infected?
2021 Saudi Arabia Training Tests, 4
Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.
DMM Individual Rounds, 1998 Tie
[b]p1A[/b] Positive reals $x$, $y$, and $z$ are such that $x/y +y/x = 7$ and $y/z +z/y = 7$. There are two possible values for $z/x + x/z;$ find the greater value.
[b]p1B[/b] Real values $x$ and $y$ are such that $x+y = 2$ and $x^3+y^3 = 3$. Find $x^2+y^2$.
[b]p2[/b] Set $A = \{5, 6, 8, 13, 20, 22, 33, 42\}$. Let $\sum S$ denote the sum of the members of $S$; then $\sum A = 149$. Find the number of (not necessarily proper) subsets $B$ of $A$ for which $\sum B \ge 75$.
[b]p3[/b] $99$ dots are evenly spaced around a circle. Call two of these dots ”close” if they have $0$, $1$, or $2$ dots between them on the circle. We wish to color all $99$ dots so that any two dots which are close are colored differently. How many such colorings are possible using no more than $4$ different colors?
[b]p4[/b] Given a $9 \times 9$ grid of points, count the number of nondegenerate squares that can be drawn whose vertices are in the grid and whose center is the middle point of the grid.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
DMM Team Rounds, 1999
[b]p1.[/b] The least prime factor of $a$ is $3$, the least prime factor of $b$ is $7$. Find the least prime factor of $a + b$.
[b]p2.[/b] In a Cartesian coordinate system, the two tangent lines from $P = (39, 52)$ meet the circle defined by $x^2 + y^2 = 625$ at points $Q$ and $R$. Find the length $QR$.
[b]p3.[/b] For a positive integer $n$, there is a sequence $(a_0, a_1, a_2,..., a_n)$ of real values such that $a_0 = 11$ and $(a_k + a_{k+1}) (a_k - a_{k+1}) = 5$ for every $k$ with $0 \le k \le n-1$. Find the maximum possible value of $n$. (Be careful that your answer isn’t off by one!)
[b]p4.[/b] Persons $A$ and $B$ stand at point $P$ on line $\ell$. Point $Q$ lies at a distance of $10$ from point $P$ in the direction perpendicular to $\ell$. Both persons intially face towards $Q$. Person $A$ walks forward and to the left at an angle of $25^o$ with $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the right, and continues walking. Person $B$ walks forward and to the right at an angle of $55^o$ with line $\ell$, when he is again at a distance of $10$ from point $Q$, he stops, turns $90^o$ to the left, and continues walking. Their paths cross at point $R$. Find the distance $PR$.
[b]p5.[/b] Compute
$$\frac{lcm (1,2, 3,..., 200)}{lcm (102, 103, 104, ..., 200)}.$$
[b]p6.[/b] There is a unique real value $A$ such that for all $x$ with $1 < x < 3$ and $x \ne 2$, $$\left| \frac{A}{x^2-x - 2} +\frac{1}{x^2 - 6x + 8} \right|< 1999.$$
Compute $A$.
[b]p7.[/b] Nine poles of height $1, 2,..., 9$ are placed in a line in random order. A pole is called [i]dominant [/i] if it is taller than the pole immediately to the left of it, or if it is the pole farthest to the left. Count the number of possible orderings in which there are exactly $2$ dominant poles.
[b]p8.[/b] $\tan (11x) = \tan (34^o)$ and $\tan (19x) = \tan (21^o)$. Compute $\tan (5x)$.
PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
2010 Peru Iberoamerican Team Selection Test, P6
On an $n$ × $n$ board, the set of all squares that are located on or below the main diagonal of the board is called the$n-ladder$. For example, the following figure shows a $3-ladder$:
[asy]
draw((0,0)--(0,3));
draw((0,0)--(3,0));
draw((0,1)--(3,1));
draw((1,0)--(1,3));
draw((0,2)--(2,2));
draw((2,0)--(2,2));
draw((0,3)--(1,3));
draw((3,0)--(3,1));
[/asy]
In how many ways can a $99-ladder$ be divided into some rectangles, which have their sides on grid lines, in such a way that all the rectangles have distinct areas?
2019 Bangladesh Mathematical Olympiad, 7
Given three cocentric circles $\omega_1$,$\omega_2$,$\omega_3$ with radius $r_1,r_2,r_3$ such that $r_1+r_3\geq {2r_2}$.Constrat a line that intersects $\omega_1$,$\omega_2$,$\omega_3$ at $A,B,C$ respectively such that $AB=BC$.