Found problems: 85335
2008 Iran MO (2nd Round), 2
Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.
2009 AMC 10, 10
Triangle $ ABC$ has a right angle at $ B$. Point $ D$ is the foot of the altitude from $ B$, $ AD\equal{}3$, and $ DC\equal{}4$. What is the area of $ \triangle{ABC}$?
[asy]unitsize(5mm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=4;
pair B=(0,0), C=(sqrt(28),0), A=(0,sqrt(21));
pair D=foot(B,A,C);
pair[] ps={B,C,A,D};
draw(A--B--C--cycle);
draw(B--D);
draw(rightanglemark(B,D,C));
dot(ps);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,NE);
label("$3$",midpoint(A--D),NE);
label("$4$",midpoint(D--C),NE);[/asy]$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 7\sqrt3 \qquad
\textbf{(C)}\ 21 \qquad
\textbf{(D)}\ 14\sqrt3 \qquad
\textbf{(E)}\ 42$
2015 USAMTS Problems, 1
In the grid to the right, the shortest path through unit squares between between the pair of 2's has length 2. Fill in some of the unit squares in the grid so that
(i) exactly half of the squares in each row and column contain a number,
(ii) each of the number 1 through 12 appears exactly twice, and
(iii) for $n=1,2,\cdot\cdot\cdot,12$, the shortest path between the pair of $n$'s has length exactly $n$.
2019 Bundeswettbewerb Mathematik, 4
In the decimal expansion of $\sqrt{2}=1.4142\dots$, Isabelle finds a sequence of $k$ successive zeroes where $k$ is a positive integer.
Show that the first zero of this sequence can occur no earlier than at the $k$-th position after the decimal point.
2008 Canada National Olympiad, 1
$ ABCD$ is a convex quadrilateral for which $ AB$ is the longest side. Points $ M$ and $ N$ are located on sides $ AB$ and $ BC$ respectively, so that each of the segments $ AN$ and $ CM$ divides the quadrilateral into two parts of equal area. Prove that the segment $ MN$ bisects the diagonal $ BD$.
1997 Slovenia Team Selection Test, 4
Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that
$A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.
2013 China Team Selection Test, 1
For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define
\[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\]
Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.
2004 IMO Shortlist, 2
Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible.
[i]Proposed by Horst Sewerin, Germany[/i]
2015 Moldova Team Selection Test, 3
Let $p$ be a fixed odd prime. Find the minimum positive value of $E_{p}(x,y) = \sqrt{2p}-\sqrt{x}-\sqrt{y}$ where $x,y \in \mathbb{Z}_{+}$.
2013 IPhOO, 5
[asy]
import olympiad;
import cse5;
size(5cm);
pointpen = black;
pair A = Drawing((10,17.32));
pair B = Drawing((0,0));
pair C = Drawing((20,0));
draw(A--B--C--cycle);
pair X = 0.85*A + 0.15*B;
pair Y = 0.82*A + 0.18*C;
pair W = (-11,0) + X;
pair Z = (19, 9);
draw(W--X, EndArrow);
draw(X--Y, EndArrow);
draw(Y--Z, EndArrow);
anglepen=black; anglefontpen=black;
MarkAngle("\theta", C,Y,Z, 3);
[/asy]
The cross-section of a prism with index of refraction $1.5$ is an equilateral triangle, as shown above. A ray of light comes in horizontally from air into the prism, and has the opportunity to leave the prism, at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number.
[i](Ahaan Rungta, 5 points)[/i]
2004 AMC 12/AHSME, 21
The graph of $ 2x^2 \plus{} xy \plus{} 3y^2 \minus{} 11x \minus{} 20y \plus{} 40 \equal{} 0$ is an ellipse in the first quadrant of the $ xy$-plane. Let $ a$ and $ b$ be the maximum and minimum values of $ \frac {y}{x}$ over all points $ (x, y)$ on the ellipse. What is the value of $ a \plus{} b$?
$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ \frac72 \qquad \textbf{(D)}\ \frac92 \qquad \textbf{(E)}\ 2\sqrt {14}$
1996 Romania Team Selection Test, 3
Let $ x,y\in \mathbb{R} $. Show that if the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $.
[i]Vasile Pop[/i]
1996 Korea National Olympiad, 8
Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.
2017 Iran MO (3rd round), 1
Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.
1968 IMO Shortlist, 14
A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$ . Find the locus of the center of the circumcircle of triangle $XAY .$
2010 LMT, 3
J has $98$ cheetahs in his pants, some of which are male and the rest of which are female. He realizes that three times the number of male cheetahs in his pants is equal to nine more than twice the number of female cheetahs. How many male cheetahs are in his pants?
1988 AIME Problems, 10
A convex polyhedron has for its faces 12 squares, 8 regular hexagons, and 6 regular octagons. At each vertex of the polyhedron one square, one hexagon, and one octagon meet. How many segments joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an edge or a face?
2013 CIIM, Problem 6
Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$. Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$, there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$
2008 IMC, 4
Let $ \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients, and let $ f(x), g(x) \in\mathbb{Z}[x]$ be nonconstant polynomials such that $ g(x)$ divides $ f(x)$ in $ \mathbb{Z}[x]$. Prove that if the polynomial $ f(x)\minus{}2008$ has at least 81 distinct integer roots, then the degree of $ g(x)$ is greater than 5.
2005 Junior Balkan Team Selection Tests - Romania, 4
Let $a,b,c$ be positive numbers such that $a+b+c \geq \dfrac 1a + \dfrac 1b + \dfrac 1c$. Prove that
\[ a+b+c \geq \frac 3{abc}. \]
2018 AIME Problems, 14
The incircle of $\omega$ of $\triangle ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP=3, PB = 4, AC=8$, and $AQ = \tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
MBMT Team Rounds, 2015 F11 E9
The right triangle below has legs of length $1$ and $2$. Find the sum of the areas of the shaded regions (of which there are infinitely many), given that the regions into which the triangle has been divided are all right triangles.
2015 Korea - Final Round, 2
In a triangle $\triangle ABC$ with incenter $I$, the incircle meets lines $BC, CA, AB$ at $D, E, F$ respectively.
Define the circumcenter of $\triangle IAB$ and $\triangle IAC$ $O_1$ and $O_2$ respectively.
Let the two intersections of the circumcircle of $\triangle ABC$ and line $EF$ be $P, Q$.
Prove that the circumcenter of $\triangle DPQ$ lies on the line $O_1O_2$.
2014 Junior Balkan Team Selection Tests - Moldova, 5
Show that for any natural number $n$, the number $A = [\frac{n + 3}{4}] + [ \frac{n + 5}{4} ] + [\frac{n}{2} ] +n^2 + 3n + 3$ is a perfect square. ($[x]$ denotes the integer part of the real number x.)
2015 Hanoi Open Mathematics Competitions, 2
The last digit of number $2017^{2017} - 2013^{2015}$ is
(A): $2$, (B): $4$, (C): $6$, (D): $8$, (E): None of the above.