Found problems: 25757
2016 AMC 10, 5
A rectangular box has integer side lengths in the ratio $1: 3: 4$. Which of the following could be the volume of the box?
$\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qquad\textbf{(E)}\ 144$
2013 Oral Moscow Geometry Olympiad, 1
Diagonals of a cyclic quadrilateral $ABCD$ intersect at point $O$. The circumscribed circles of triangles $AOB$ and $COD$ intersect at point $M$ on the side $AD$. Prove that the point $O$ is the center of the inscribed circle of the triangle $BMC$.
Ukraine Correspondence MO - geometry, 2006.3
Find the locus of the points of intersection of the altitudes of the triangles inscribed in a given circle.
2011 Indonesia MO, 3
Given an acute triangle $ABC$, let $l_a$ be the line passing $A$ and perpendicular to $AB$, $l_b$ be the line passing $B$ and perpendicular to $BC$, and $l_c$ be the line passing $C$ and perpendicular to $CA$. Let $D$ be the intersection of $l_b$ and $l_c$, $E$ be the intersection of $l_c$ and $l_a$, and $F$ be the intersection of $l_a$ and $l_b$. Prove that the area of the triangle $DEF$ is at least three times of the area of $ABC$.
2005 Poland - Second Round, 2
A rhombus $ABCD$ with $\angle BAD=60^{\circ}$ is given. Points $E$ on side $AB$ and $F$ on side $AD$ are such that $\angle ECF=\angle ABD$. Lines $CE$ and $CF$ respectively meet line $BD$ at $P$ and $Q$. Prove that $\frac{PQ}{EF}=\frac{AB}{BD}$.
2021 Sharygin Geometry Olympiad, 9.1
Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.
2016 China Team Selection Test, 1
$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.
1952 Moscow Mathematical Olympiad, 229
In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .
2005 MOP Homework, 7
A segment of length $2$ is divided into $n$, $n\ge 2$, subintervals. A square is then constructed on each subinterval. Assume that the sum of the areas of all such squares is greater than $1$. Show that under this assumption one can always choose two subintervals with total length greater than $1$.
1979 IMO Longlists, 1
Prove that in the Euclidean plane every regular polygon having an even number of sides can be dissected into lozenges. (A lozenge is a quadrilateral whose four sides are all of equal length).
2024 Israel National Olympiad (Gillis), P6
Quadrilateral $ABCD$ is inscribed in a circle. Let $\omega_A$, $\omega_B$, $\omega_C$, $\omega_D$ be the incircles of triangles $DAB$, $ABC$, $BCD$, $CDA$ respectively. The common external common tangent of $\omega_A$, $\omega_B$, different from line $AB$, meets the external common tangent of $\omega_A$, $\omega_D$, different from $AD$, at point $A'$. Similarly, the external common tangent of $\omega_B$, $\omega_C$ different from $BC$ meets the external common tangent of $\omega_C$, $\omega_D$ different from $CD$ at $C'$.
Prove that $AA'\parallel CC'$.
2005 Harvard-MIT Mathematics Tournament, 10
Let $AB$ be a diameter of a semicircle $\Gamma$. Two circles, $\omega_1$ and $\omega_2$, externally tangent to each other and internally tangent to $\Gamma$, are tangent to the line $AB$ at $P$ and $Q$, respectively, and to semicircular arc $AB$ at $C$ and $D$, respectively, with $AP<AQ$. Suppose $F$ lies on $\Gamma$ such that $ \angle FQB = \angle CQA $ and that $ \angle ABF = 80^\circ $. Find $ \angle PDQ $ in degrees.
2008 Ukraine Team Selection Test, 1
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.
Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.
[i]Author: Farzan Barekat, Canada[/i]
2010 China Team Selection Test, 1
Given acute triangle $ABC$ with $AB>AC$, let $M$ be the midpoint of $BC$. $P$ is a point in triangle $AMC$ such that $\angle MAB=\angle PAC$. Let $O,O_1,O_2$ be the circumcenters of $\triangle ABC,\triangle ABP,\triangle ACP$ respectively. Prove that line $AO$ passes through the midpoint of $O_1 O_2$.
1993 ITAMO, 4
Let $P$ be a point in the plane of a triangle $ABC$, different from its circumcenter. Prove that the triangle whose vertices are the projections of $P$ on the perpendicular bisectors of the sides of $ABC$, is similar to $ABC$.
2025 Greece National Olympiad, 2
Let $ABC$ be an acute triangle and $D$ be a point of side $ BC$. Consider points $E,Z$ on line $AD$ such that $EB \perp AB$ and $ZC \perp AC$, and points $H, T $ on line $BC$ such that $EH \parallel AC$ and $ZT \parallel AB$. Circumcircle of triangle $BHE$ intersects for second time line $AB$ at point $M$ ($M \ne B$) and circumcircle of triangle $CTZ$ intersects for second time line $AC$ at point $N$ ($N \ne C$). Prove that lines $MH$, $NT$ and $AD$ concur.
2006 Switzerland Team Selection Test, 2
Let $D$ be inside $\triangle ABC$ and $E$ on $AD$ different to $D$. Let $\omega_1$ and $\omega_2$ be the circumscribed circles of $\triangle BDE$ and $\triangle CDE$ respectively. $\omega_1$ and $\omega_2$ intersect $BC$ in the interior points $F$ and $G$ respectively. Let $X$ be the intersection between $DG$ and $AB$ and $Y$ the intersection between $DF$ and $AC$. Show that $XY$ is $\|$ to $BC$.
2018 IMO Shortlist, G4
A point $T$ is chosen inside a triangle $ABC$. Let $A_1$, $B_1$, and $C_1$ be the reflections of $T$ in $BC$, $CA$, and $AB$, respectively. Let $\Omega$ be the circumcircle of the triangle $A_1B_1C_1$. The lines $A_1T$, $B_1T$, and $C_1T$ meet $\Omega$ again at $A_2$, $B_2$, and $C_2$, respectively. Prove that the lines $AA_2$, $BB_2$, and $CC_2$ are concurrent on $\Omega$.
[i]Proposed by Mongolia[/i]
2023 Yasinsky Geometry Olympiad, 4
Pick a point $C$ on a semicircle with diameter $AB$. Let $P$ and $Q$ be two points on segment $AB$ such that $AP= AC$ and $BQ= BC$. The point $O$ is the center of the circumscribed circle of triangle $CPQ$ and point $H$ is the orthocenter of triangle $CPQ$ . Prove that for all posible locations of point $C$, the line $OH$ is passing through a fixed point.
(Mykhailo Sydorenko)
2009 ISI B.Stat Entrance Exam, 3
Let $ABC$ be a right-angled triangle with $BC=AC=1$. Let $P$ be any point on $AB$. Draw perpendiculars $PQ$ and $PR$ on $AC$ and $BC$ respectively from $P$. Define $M$ to be the maximum of the areas of $BPR, APQ$ and $PQCR$. Find the minimum possible value of $M$.
2003 JHMMC 8, 14
In rectangle $ABCD$, $AB = 7$ and $AC = 25$. What is its area?
2019 Baltic Way, 11
Let $ABC$ be a triangle with $AB = AC$. Let $M$ be the midpoint of $BC$. Let the circles with diameters $AC$ and $BM$ intersect at points $M$ and $P$. Let $MP$ intersect $AB$ at $Q$. Let $R$ be a point on $AP$ such that $QR \parallel BP$. Prove that $CP$ bisects $\angle RCB$.
2005 Croatia National Olympiad, 4
Let $P$ and $Q$ be points on the sides $BC$ and $CD$ of a convex quadrilateral $ABCD$, respectively, such that $\angle{BAP}=\angle{ DAQ}$. Prove that the triangles $ABP$ and $ADQ$ have equal area if and only if the line joining their orthocenters is perpendicular to $AC.$
2007 ITest, 16
How many lattice points lie within or on the border of the circle defined in the $xy$-plane by the equation $x^2+y^2=100$?
$\textbf{(A) }1\hspace{14em}\textbf{(B) }2\hspace{14em}\textbf{(C) }4$
$\textbf{(D) }5\hspace{14em}\textbf{(E) }41\hspace{13.5em}\textbf{(F) }42$
$\textbf{(G) }69\hspace{13.5em}\textbf{(H) }76\hspace{13.4em}\textbf{(I) }130$
$\textbf{(J) }133\hspace{13.3em}\textbf{(K) }233\hspace{12.8em}\textbf{(L) }311$
$\textbf{(M) }317\hspace{12.7em}\textbf{(N) }420\hspace{12.9em}\textbf{(O) }520$
$\textbf{(P) }2007$
1993 Greece National Olympiad, 14
A rectangle that is inscribed in a larger rectangle (with one vertex on each side) is called [i]unstuck[/i] if it is possible to rotate (however slightly) the smaller rectangle about its center within the confines of the larger. Of all the rectangles that can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form $\sqrt{N}$, for a positive integer $N$. Find $N$.