Found problems: 663
Let $ABC$ and $ABD$ be coplanar triangles with equal perimeters. The lines of support of the internal bisectrices of the angles $CAD$ and $CBD$ meet at $P$. Show that the angles $APC$ and $BPD$ are congruent.
Nondegenerate $ \triangle ABC$ has integer side lengths, $ BD$ is an angle bisector, $ AD \equal{} 3$, and $ DC \equal{} 8$. What is the smallest possible value of the perimeter?
$ \textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 33 \qquad
\textbf{(C)}\ 35 \qquad
\textbf{(D)}\ 36 \qquad
\textbf{(E)}\ 37$
Let $n$ be a positive integer. A regular hexagon with side length $n$ is divided into equilateral triangles with side length $1$ by lines parallel to its sides.
Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.
[i]UK - Sahl Khan[/i]
Let $AB$ be diameter of a circle $\omega$ and let $C$ be a point on $\omega$, different from $A$ and $B$. The perpendicular from $C$ intersects $AB$ at $D$ and $\omega$ at $E(\neq C)$. The circle with centre at $C$ and radius $CD$ intersects $\omega$ at $P$ and $Q$. If the perimeter of the triangle $PEQ$ is $24$, find the length of the side $PQ$
Consider 2 concentric circle radii $ R$ and $ r$ ($ R > r$) with centre $ O.$ Fix $ P$ on the small circle and consider the variable chord $ PA$ of the small circle. Points $ B$ and $ C$ lie on the large circle; $ B,P,C$ are collinear and $ BC$ is perpendicular to $ AP.$
[b]i.)[/b] For which values of $ \angle OPA$ is the sum $ BC^2 \plus{} CA^2 \plus{} AB^2$ extremal?
[b]ii.)[/b] What are the possible positions of the midpoints $ U$ of $ BA$ and $ V$ of $ AC$ as $ \angle OPA$ varies?
In $xyz$ space, let $A$ be the solid generated by a rotation of the figure, enclosed by the curve $y=2-2x^2$ and the $x$-axis about the $y$-axis.
(1) When the solid is cut by the plane $x=a\ (|a|\leq 1)$, find the inequality which expresses the figure of the cross-section.
(2) Denote by $L$ the distance between the point $(a,\ 0,\ 0)$ and the point on the perimeter of the cross-section found in (1), find the maximum value of $L$.
(3) Find the volume of the solid by a rotation of the solid $A$ about the $x$-axis.
[i]1987 Sophia University entrance exam/Science and Technology[/i]
For the triangles of fixed perimeter, find the maximum value of the product of the radius of the incircle with the radius of the excircle.
In triangle $ABC$, side $AB$ is $1$. It is known that one of the angle bisectors of triangle $ABC$ is perpendicular to one of its medians, and some other angle bisector is perpendicular to the other median. What can be the perimeter of triangle $ABC$?
In an urban area whose street plan is a grid, a person started walking from an intersection and turned right or left at every intersection he reached until he ended up in the same initial intersection.
[b]a)[/b] Show that the number of intersections (not necessarily distinct) in which he were is equivalent to $ 1 $ modulo $ 4. $
[b]b)[/b] Enunciate and prove a reciprocal statement.
[i]Marius Beceanu[/i]
A house is in the shape of a triangle, perimeter $P$ metres and area $A$ square metres. The garden consists of all the land within 5 metres of the house. How much land do the garden and house together occupy?
What is the area of the regular hexagon with perimeter $60$?
In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is $17$. What is the greatest possible perimeter of the triangle?
In an acute triangle $ ABC$, let $ D$ be a point on $ \left[AC\right]$ and $ E$ be a point on $ \left[AB\right]$ such that $ \angle ADB\equal{}\angle AEC\equal{}90{}^\circ$. If perimeter of triangle $ AED$ is 9, circumradius of $ AED$ is $ \frac{9}{5}$ and perimeter of triangle $ ABC$ is 15, then $ \left|BC\right|$ is
$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ \frac{24}{5} \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 8 \qquad\textbf{(E)}\ \frac{48}{5}$
Let $ ABCD $ be a regular tetahedron and $ M,N $ be middlepoints for $ AD, $ respectively, $ BC. $ Through a point $ P $ that is on segment $ MN, $ passes a plane perpendicular on $ MN, $ and meets the sides $ AB,AC,CD,BD $ of the tetahedron at $ E,F,G, $ respectively, $ H. $
[b]a)[/b] Prove that the perimeter of the quadrilateral $ EFGH $ doesn't depend on $ P. $
[b]b)[/b] Determine the maximum area of $ EFGH $ (depending on a side of the tetahedron).
Let $PQRS$ be a square piece of paper. $P$ is folded onto $R$ and then $Q$ is folded onto $S$. The area of the resulting figure is 9 square inches. Find the perimeter of square $PQRS$.
[asy]
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
label("$P$",(0,2),SE);
label("$Q$",(2,2),SW);
label("$R$",(2,0),NW);
label("$S$",(0,0),NE);[/asy]
$ \text{(A)}\ 9\qquad\text{(B)}\ 16\qquad\text{(C)}\ 18\qquad\text{(D)}\ 24\qquad\text{(E)}\ 36 $
A rectangle is divided by $10$ horizontal and $10$ vertical lines into $121$ rectangular cells. If $111$ of them have integer perimeters, prove that they all have integer perimeters.
In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters?
$\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$
Let $ABC$ be a triangle with $AB = 130$, $BC = 140$, $CA = 150$. Let $G$, $H$, $I$, $O$, $N$, $K$, $L$ be the centroid, orthocenter, incenter, circumenter, nine-point center, the symmedian point, and the de Longchamps point. Let $D$, $E$, $F$ be the feet of the altitudes of $A$, $B$, $C$ on the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$. Let $X$, $Y$, $Z$ be the $A$, $B$, $C$ excenters and let $U$, $V$, $W$ denote the midpoints of $\overline{IX}$, $\overline{IY}$, $\overline{IZ}$ (i.e. the midpoints of the arcs of $(ABC)$.) Let $R$, $S$, $T$ denote the isogonal conjugates of the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $P$ and $Q$ denote the images of $G$ and $H$ under an inversion around the circumcircle of $ABC$ followed by a dilation at $O$ with factor $\frac 12$, and denote by $M$ the midpoint of $\overline{PQ}$. Then let $J$ be a point such that $JKLM$ is a parallelogram. Find the perimeter of the convex hull of the self-intersecting $17$-gon $LETSTRADEBITCOINS$ to the nearest integer. A diagram has been included but may not be to scale.
[asy]
size(6cm);
import olympiad;
import cse5;
pair A = dir(110);
pair B = dir(210);
pair C = dir(330);
pair D = foot(A,B,C);
pair E = foot(B,C,A);
pair F = foot(C,A,B);
pair G = centroid(A,B,C);
pair H = orthocenter(A,B,C);
pair I = incenter(A,B,C);
pair isocon(pair targ) {
return extension(A,2*foot(targ,I,A)-targ,
C,2*foot(targ,I,C)-targ);
}
pair O = circumcenter(A,B,C);
pair K = isocon(G);
pair N = midpoint(O--H);
pair U = extension(O,midpoint(B--C),A,I);
pair V = extension(O,midpoint(C--A),B,I);
pair W = extension(O,midpoint(A--B),C,I);
pair X = -I + 2*U;
pair Y = -I + 2*V;
pair Z = -I + 2*W;
pair R = isocon(midpoint(A--D));
pair S = isocon(midpoint(B--E));
pair T = isocon(midpoint(C--F));
pair L = 2*H-O;
pair P = 0.5/conj(G);
pair Q = 0.5/conj(H);
pair M = midpoint(P--Q);
pair J = K+M-L;
draw(A--B--C--cycle);
void draw_cevians(pair target) {
draw(A--extension(A,target,B,C));
draw(B--extension(B,target,C,A));
draw(C--extension(C,target,A,B));
}
draw_cevians(H);
draw_cevians(G);
draw_cevians(I);
draw(unitcircle);
draw(circumcircle(D,E,F));
draw(O--P);
draw(O--Q);
draw(P--Q);
draw(CP(X,foot(X,B,C)));
draw(CP(Y,foot(Y,C,A)));
draw(CP(Z,foot(Z,A,B)));
draw(J--K--L--M);
draw(X--Y--Z--cycle);
draw(A--X);
draw(B--Y);
draw(C--Z);
draw(A--foot(X,A,B));
draw(A--foot(X,A,C));
draw(B--foot(Y,B,C));
draw(B--foot(Y,B,A));
draw(C--foot(Z,C,A));
draw(C--foot(Z,C,B));
pen p = black;
dot(A, p);
dot(B, p);
dot(C, p);
dot(D, p);
dot(E, p);
dot(F, p);
dot(G, p);
dot(H, p);
dot(I, p);
dot(J, p);
dot(K, p);
dot(L, p);
dot(M, p);
dot(N, p);
dot(O, p);
dot(P, p);
dot(Q, p);
dot(R, p);
dot(S, p);
dot(T, p);
dot(U, p);
dot(V, p);
dot(W, p);
dot(X, p);
dot(Y, p);
dot(Z, p);
[/asy]
The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is
$\text{(A)} \pi \qquad \text{(B)} \frac{2}{\pi} \qquad \text{(C)} 1 \qquad \text{(D)} \frac{1}{2} \qquad \text{(E)} \frac{4}{\pi} + 2$
$(BUL 3)$ One hundred convex polygons are placed on a square with edge of length $38 cm.$ The area of each of the polygons is smaller than $\pi cm^2,$ and the perimeter of each of the polygons is smaller than $2\pi cm.$ Prove that there exists a disk with radius $1$ in the square that does not intersect any of the polygons.
$ABC$ is a triangle. Find a point $X$ on $BC$ such that :
area $ABX$ / area $ACX$ = perimeter $ABX$ / perimeter $ACX$.
In triangle $ABC$, let points $M$ and $N$ be the midpoints of sides $AB$ and $BC$ respectively. It is known that the perimeter of the triangle $MBN$ is $12$ cm, and the perimeter of the quadrilateral $AMNC$ is $20$ cm. Find the length of the segment $MN$.
$\bullet$ In how many ways can triangles be formed whose sides are integers greater than $50$ and less than $100$?
$\bullet$ In how many of these triangles is the perimeter divisible by $3$?
Find all values of $n$ such that there exists a rectangle with integer side lengths, perimeter $n$, and area $2n$.
Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$?
$ \textbf{(A)}\ 4\sqrt3 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 4\sqrt5 \qquad
\textbf{(D)}\ 4\sqrt7 \qquad
\textbf{(E)}\ 12$