Found problems: 126
2013 NIMO Problems, 4
Let $a,b,c$ be the answers to problems $4$, $5$, and $6$, respectively. In $\triangle ABC$, the measures of $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, $c$ in degrees, respectively. Let $D$ and $E$ be points on segments $AB$ and $AC$ with $\frac{AD}{BD} = \frac{AE}{CE} = 2013$. A point $P$ is selected in the interior of $\triangle ADE$, with barycentric coordinates $(x,y,z)$ with respect to $\triangle ABC$ (here $x+y+z=1$). Lines $BP$ and $CP$ meet line $DE$ at $B_1$ and $C_1$, respectively. Suppose that the radical axis of the circumcircles of $\triangle PDC_1$ and $\triangle PEB_1$ pass through point $A$. Find $100x$.
[i]Proposed by Evan Chen[/i]
2009 China Team Selection Test, 1
Let $ ABC$ be a triangle. Point $ D$ lies on its sideline $ BC$ such that $ \angle CAD \equal{} \angle CBA.$ Circle $ (O)$ passing through $ B,D$ intersects $ AB,AD$ at $ E,F$, respectively. $ BF$ meets $ DE$ at $ G$.Denote by$ M$ the midpoint of $ AG.$ Show that $ CM\perp AO.$
2005 Gheorghe Vranceanu, 2
Let be a twice-differentiable function $ f:(0,\infty )\longrightarrow\mathbb{R} $ that admits a polynomial function of degree $ 1 $ or $ 2, $ namely, $ \alpha :(0,\infty )\longrightarrow\mathbb{R} $ as its asymptote. Prove the following propositions:
[b]a)[/b] $ f''>0\implies f-\alpha >0 $
[b]b)[/b] $ \text{supp} f''=(0,\infty )\wedge f-\alpha >0\implies f''=0 $
2021 AMC 12/AHSME Fall, 6
As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. What is the degree measure of $\angle AFE?$
[asy]
size(6cm);
pair A = (0,10);
label("$A$", A, N);
pair B = (0,0);
label("$B$", B, S);
pair C = (10,0);
label("$C$", C, S);
pair D = (10,10);
label("$D$", D, SW);
pair EE = (15,11.8);
label("$E$", EE, N);
pair F = (3,10);
label("$F$", F, N);
filldraw(D--arc(D,2.5,270,380)--cycle,lightgray);
dot(A^^B^^C^^D^^EE^^F);
draw(A--B--C--D--cycle);
draw(D--EE--F--cycle);
label("$110^\circ$", (15,9), SW);
[/asy]
$\textbf{(A) }160\qquad\textbf{(B) }164\qquad\textbf{(C) }166\qquad\textbf{(D) }170\qquad\textbf{(E) }174$
2013 ELMO Shortlist, 5
Let $\omega_1$ and $\omega_2$ be two orthogonal circles, and let the center of $\omega_1$ be $O$. Diameter $AB$ of $\omega_1$ is selected so that $B$ lies strictly inside $\omega_2$. The two circles tangent to $\omega_2$, passing through $O$ and $A$, touch $\omega_2$ at $F$ and $G$. Prove that $FGOB$ is cyclic.
[i]Proposed by Eric Chen[/i]
2014 AMC 8, 11
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad \textbf{(E) }10$
1975 AMC 12/AHSME, 20
In the adjoining figure triangle $ ABC$ is such that $ AB \equal{} 4$ and $ AC \equal{} 8$. If $ M$ is the midpoint of $ BC$ and $ AM \equal{} 3$, what is the length of $ BC$?
$ \textbf{(A)}\ 2\sqrt{26} \qquad
\textbf{(B)}\ 2\sqrt{31} \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 4\plus{}2\sqrt{13} \qquad$
$ \textbf{(E)}\ \text{not enough information given to solve the problem}$
[asy]draw((0,0)--(2.8284,2)--(8,0)--cycle);
draw((2.8284,2)--(4,0));
label("A",(2.8284,2),N);
label("B",(0,0),S);
label("C",(8,0),S);
label("M",(4,0),S);[/asy]
2014 ELMO Shortlist, 2
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\omega$. Let $AB \cap CD = E$, $AD \cap BC = F$. Let $\omega_1, \omega_2$ be the circumcircles of $AEF, CEF$, respectively. Let $\omega \cap \omega_1 = G$, $\omega \cap \omega_2 = H$. Show that $AC, BD, GH$ are concurrent.
[i]Proposed by Yang Liu[/i]
2013 AIME Problems, 9
A paper equilateral triangle $ABC$ has side length $12$. The paper triangle is folded so that vertex $A$ touches a point on side $\overline{BC}$ a distance $9$ from point $B$. The length of the line segment along which the triangle is folded can be written as $\frac{m\sqrt{p}}{n}$, where $m$, $n$, and $p$ are positive integers, $m$ and $n$ are relatively prime, and $p$ is not divisible by the square of any prime. Find $m+n+p$.
[asy]
import cse5;
size(12cm);
pen tpen = defaultpen + 1.337;
real a = 39/5.0;
real b = 39/7.0;
pair B = MP("B", (0,0), dir(200));
pair A = MP("A", (9,0), dir(-80));
pair C = MP("C", (12,0), dir(-20));
pair K = (6,10.392);
pair M = (a*B+(12-a)*K) / 12;
pair N = (b*C+(12-b)*K) / 12;
draw(B--M--N--C--cycle, tpen);
draw(M--A--N--cycle);
fill(M--A--N--cycle, mediumgrey);
pair shift = (-20.13, 0);
pair B1 = MP("B", B+shift, dir(200));
pair A1 = MP("A", K+shift, dir(90));
pair C1 = MP("C", C+shift, dir(-20));
draw(A1--B1--C1--cycle, tpen);[/asy]
2012 Brazil Team Selection Test, 4
Let $ ABC $ be an acute triangle. Denote by $ D $ the foot of the perpendicular line drawn from the point $ A $ to the side $ BC $, by $M$ the midpoint of $ BC $, and by $ H $ the orthocenter of $ ABC $. Let $ E $ be the point of intersection of the circumcircle $ \Gamma $ of the triangle $ ABC $ and the half line $ MH $, and $ F $ be the point of intersection (other than $E$) of the line $ ED $ and the circle $ \Gamma $. Prove that $ \tfrac{BF}{CF} = \tfrac{AB}{AC} $ must hold.
(Here we denote $XY$ the length of the line segment $XY$.)
2014 AMC 12/AHSME, 20
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
$\textbf{(A) }6\sqrt 3+3\qquad
\textbf{(B) }\dfrac{27}2\qquad
\textbf{(C) }8\sqrt 3\qquad
\textbf{(D) }14\qquad
\textbf{(E) }3\sqrt 3+9\qquad$
2017 Romania National Olympiad, 4
A function $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ has the property that $ \lim_{x\to\infty } \frac{1}{x^2}\int_0^x f(t)dt=1. $
[b]a)[/b] Give an example of what $ f $ could be if it's continuous and $ f/\text{id.} $ doesn't have a limit at $ \infty . $
[b]b)[/b] Prove that if $ f $ is nondecreasing then $ f/\text{id.} $ has a limit at $ \infty , $ and determine it.
2014 Online Math Open Problems, 23
Let $\Gamma_1$ and $\Gamma_2$ be circles in the plane with centers $O_1$ and $O_2$ and radii $13$ and $10$, respectively. Assume $O_1O_2=2$. Fix a circle $\Omega$ with radius $2$, internally tangent to $\Gamma_1$ at $P$ and externally tangent to $\Gamma_2$ at $Q$ . Let $\omega$ be a second variable circle internally tangent to $\Gamma_1$ at $X$ and externally tangent to $\Gamma_2$ at $Y$. Line $PQ$ meets $\Gamma_2$ again at $R$, line $XY$ meets $\Gamma_2$ again at $Z$, and lines $PZ$ and $XR$ meet at $M$.
As $\omega$ varies, the locus of point $M$ encloses a region of area $\tfrac{p}{q} \pi$, where $p$ and $q$ are relatively prime positive integers. Compute $p+q$.
[i]Proposed by Michael Kural[/i]
2011 AMC 8, 19
How many rectangles are in this figure?
[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(0,0);
B=(20,0);
C=(20,20);
D=(0,20);
draw(A--B--C--D--cycle);
E=(-10,-5);
F=(13,-5);
G=(13,5);
H=(-10,5);
draw(E--F--G--H--cycle);
I=(10,-20);
J=(18,-20);
K=(18,13);
L=(10,13);
draw(I--J--K--L--cycle);[/asy]
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $
2014 ELMO Shortlist, 1
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
[i]Proposed by Sammy Luo[/i]
2005 IberoAmerican Olympiad For University Students, 4
A variable tangent $t$ to the circle $C_1$, of radius $r_1$, intersects the circle $C_2$, of radius $r_2$ in $A$ and $B$. The tangents to $C_2$ through $A$ and $B$ intersect in $P$.
Find, as a function of $r_1$ and $r_2$, the distance between the centers of $C_1$ and $C_2$ such that the locus of $P$ when $t$ varies is contained in an equilateral hyperbola.
[b]Note[/b]: A hyperbola is said to be [i]equilateral[/i] if its asymptotes are perpendicular.
2018 Canadian Open Math Challenge, C1
Source: 2018 Canadian Open Math Challenge Part C Problem 1
-----
At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center]
$\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$,
[color=transparent](A.)[/color]layer of this pyramid?
$\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers.
[color=transparent](B.)[/color]How many cans are on the bottom layer of the prism?
$\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle.
[color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...)
[color=transparent](C.)[/color]For example, a prism could be composed of the following layers:
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center]
Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.
1979 AMC 12/AHSME, 30
[asy]
/*Using regular asymptote, this diagram would take 30 min to make. Using cse5, this takes 5 minutes. Conclusion? CSE5 IS THE BEST PACKAGE EVER CREATED!!!!*/
size(100);
import cse5;
pathpen=black;
anglefontpen=black;
pointpen=black;
anglepen=black;
dotfactor=3;
pair A=(0,0),B=(0.5,0.5*sqrt(3)),C=(3,0),D=(1.7,0),EE;
EE=(B+C)/2;
D(MP("$A$",A,W)--MP("$B$",B,N)--MP("$C$",C,E)--cycle);
D(MP("$E$",EE,N)--MP("$D$",D,S));
D(D);D(EE);
MA("80^\circ",8,D,EE,C,0.1);
MA("20^\circ",8,EE,C,D,0.3,2,shift(1,3)*C);
draw(arc(shift(-0.1,0.05)*C,0.25,100,180),arrow =ArcArrow());
MA("100^\circ",8,A,B,C,0.1,0);
MA("60^\circ",8,C,A,B,0.1,0);
//Credit to TheMaskedMagician for the diagram
[/asy]
In $\triangle ABC$, $E$ is the midpoint of side $BC$ and $D$ is on side $AC$. If the length of $AC$ is $1$ and $\measuredangle BAC = 60^\circ$, $\measuredangle ABC = 100^\circ$, $\measuredangle ACB = 20^\circ$ and $\measuredangle DEC = 80^\circ$, then the area of $\triangle ABC$ plus twice the area of $\triangle CDE$ equals
$\textbf{(A) }\frac{1}{4}\cos 10^\circ\qquad\textbf{(B) }\frac{\sqrt{3}}{8}\qquad\textbf{(C) }\frac{1}{4}\cos 40^\circ\qquad\textbf{(D) }\frac{1}{4}\cos 50^\circ\qquad\textbf{(E) }\frac{1}{8}$
2013 Sharygin Geometry Olympiad, 10
The incircle of triangle $ABC$ touches the side $AB$ at point $C'$; the incircle of triangle $ACC'$ touches the sides $AB$ and $AC$ at points $C_1, B_1$; the incircle of triangle $BCC'$ touches the sides $AB$ and $BC$ at points $C_2$, $A_2$. Prove that the lines $B_1C_1$, $A_2C_2$, and $CC'$ concur.
1983 IMO Longlists, 74
In a plane we are given two distinct points $A,B$ and two lines $a, b$ passing through $B$ and $A$ respectively $(a \ni B, b \ni A)$ such that the line $AB$ is equally inclined to a and b. Find the locus of points $M$ in the plane such that the product of distances from $M$ to $A$ and a equals the product of distances from $M$ to $B$ and $b$ (i.e., $MA \cdot MA' = MB \cdot MB'$, where $A'$ and $B'$ are the feet of the perpendiculars from $M$ to $a$ and $b$ respectively).
2008 AIME Problems, 14
Let $ \overline{AB}$ be a diameter of circle $ \omega$. Extend $ \overline{AB}$ through $ A$ to $ C$. Point $ T$ lies on $ \omega$ so that line $ CT$ is tangent to $ \omega$. Point $ P$ is the foot of the perpendicular from $ A$ to line $ CT$. Suppose $ AB \equal{} 18$, and let $ m$ denote the maximum possible length of segment $ BP$. Find $ m^{2}$.
2014 ELMO Shortlist, 1
Let $ABC$ be a triangle with symmedian point $K$. Select a point $A_1$ on line $BC$ such that the lines $AB$, $AC$, $A_1K$ and $BC$ are the sides of a cyclic quadrilateral. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
[i]Proposed by Sammy Luo[/i]
2014 AIME Problems, 13
On square $ABCD,$ points $E,F,G,$ and $H$ lie on sides $\overline{AB},\overline{BC},\overline{CD},$ and $\overline{DA},$ respectively, so that $\overline{EG} \perp \overline{FH}$ and $EG=FH = 34.$ Segments $\overline{EG}$ and $\overline{FH}$ intersect at a point $P,$ and the areas of the quadrilaterals $AEPH, BFPE, CGPF,$ and $DHPG$ are in the ratio $269:275:405:411.$ Find the area of square $ABCD$.
[asy]
size(200);
defaultpen(linewidth(0.8)+fontsize(10.6));
pair A = (0,sqrt(850));
pair B = (0,0);
pair C = (sqrt(850),0);
pair D = (sqrt(850),sqrt(850));
draw(A--B--C--D--cycle);
dotfactor = 3;
dot("$A$",A,dir(135));
dot("$B$",B,dir(215));
dot("$C$",C,dir(305));
dot("$D$",D,dir(45));
pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850));
pair F = ((2sqrt(850)+sqrt(306)+7)/6,0);
dot("$H$",H,dir(90));
dot("$F$",F,dir(270));
draw(H--F);
pair E = (0,(sqrt(850)-6)/2);
pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2);
dot("$E$",E,dir(180));
dot("$G$",G,dir(0));
draw(E--G);
pair P = extension(H,F,E,G);
dot("$P$",P,dir(60));
label("$w$", (H+E)/2,fontsize(15));
label("$x$", (E+F)/2,fontsize(15));
label("$y$", (G+F)/2,fontsize(15));
label("$z$", (H+G)/2,fontsize(15));
label("$w:x:y:z=269:275:405:411$",(sqrt(850)/2,-4.5),fontsize(11));
[/asy]
2003 Bulgaria Team Selection Test, 5
Let $ABCD$ be a circumscribed quadrilateral and let $P$ be the orthogonal projection of its in center on $AC$.
Prove that $\angle {APB}=\angle {APD}$
2013 Iran Team Selection Test, 17
In triangle $ABC$, $AD$ and $AH$ are the angle bisector and the altitude of vertex $A$, respectively. The perpendicular bisector of $AD$, intersects the semicircles with diameters $AB$ and $AC$ which are drawn outside triangle $ABC$ in $X$ and $Y$, respectively. Prove that the quadrilateral $XYDH$ is concyclic.
[i]Proposed by Mahan Malihi[/i]