Olympiad problems

2021 All-Russian Olympiad, 6

Tags: geometry

Given is a non-isosceles triangle $ABC$ with $\angle ABC=60^{\circ}$, and in its interior, a point $T$ is selected such that $\angle ATC= \angle BTC=\angle BTA=120^{\circ}$. Let $M$ the intersection point of the medians in $ABC$. Let $TM$ intersect $(ATC)$ at $K$. Find $TM/MK$.

2013 Thailand Mathematical Olympiad, 2

Tags: geometry , angle bisector , incircle , max

Let $\vartriangle ABC$ be a triangle with $\angle ABC > \angle BCA \ge 30^o$ . The angle bisectors of $\angle ABC$ and $\angle BCA$ intersect $CA$ and $AB$ at $D$ and $E$ respectively, and $BD$ and $CE$ intersect at $P$. Suppose that $P D = P E$ and the incircle of $\vartriangle ABC$ has unit radius. What is the maximum possible length of $BC$?

2006 IMO Shortlist, 3

Tags: geometry , circumcircle , pentagon

Let $ ABCDE$ be a convex pentagon such that \[ \angle BAC \equal{} \angle CAD \equal{} \angle DAE\qquad \text{and}\qquad \angle ABC \equal{} \angle ACD \equal{} \angle ADE. \]The diagonals $BD$ and $CE$ meet at $P$. Prove that the line $AP$ bisects the side $CD$. [i]Proposed by Zuming Feng, USA[/i]

Kyiv City MO Juniors Round2 2010+ geometry, 2019.7.31

Tags: point , geometry , construction , analytic geometry , coordinate geometry

The teacher drew a coordinate plane on the board and marked some points on this plane. Unfortunately, Vasya's second-grader, who was on duty, erased almost the entire drawing, except for two points $A (1, 2)$ and $B (3,1)$. Will the excellent Andriyko be able to follow these two points to construct the beginning of the coordinate system point $O (0, 0)$? Point A on the board located above and to the left of point $B$.

2004 AMC 12/AHSME, 20

Tags: 3d geometry , probability , geometry

Each face of a cube is painted either red or blue, each with probability $ 1/2$. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color? $ \textbf{(A)}\ \frac14 \qquad \textbf{(B)}\ \frac{5}{16} \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac{7}{16} \qquad \textbf{(E)}\ \frac12$

Denmark (Mohr) - geometry, 1994.1

Tags: geometry , cylinder , 3d geometry , sphere

A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass. [img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]

2002 Balkan MO, 3

Tags: geometry , rectangle , geometric transformation , homothety , reflection , circumcircle , parallelogram

Two circles with different radii intersect in two points $A$ and $B$. Let the common tangents of the two circles be $MN$ and $ST$ such that $M,S$ lie on the first circle, and $N,T$ on the second. Prove that the orthocenters of the triangles $AMN$, $AST$, $BMN$ and $BST$ are the four vertices of a rectangle.

2014 Iranian Geometry Olympiad (junior), P5

Tags: circumcircle , arc midpoint , geometric inequality , geometry

Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$ . Let $M$ denotes the midpoint of the chord $AX$ . Show that $BM +CM > AY$ . by Mahan Tajrobekar

1990 National High School Mathematics League, 15

Tags: geometry , 3d geometry , pyramid , sphere

In pyramid $M-ABCD$, bottom surface $ABCD$ is a square. $MA=MC,MA\perp AB$. If the area of $\triangle AMD$ is $1$, find the maximum value of radius of sphere that can be put inside the pyramid.

2001 Moldova National Olympiad, Problem 7

Tags: geometry

Let $ABCD$ and $AB’C’D’$ be equally oriented squares. Prove that the lines $BB_1,CC_1,DD_1$ are concurrent.

Today's calculation of integrals, 857

Tags: calculus , integration , limit , trigonometry , geometry , geometric transformation , rotation

Let $f(x)=\lim_{n\to\infty} (\cos ^ n x+\sin ^ n x)^{\frac{1}{n}}$ for $0\leq x\leq \frac{\pi}{2}.$ (1) Find $f(x).$ (2) Find the volume of the solid generated by a rotation of the figure bounded by the curve $y=f(x)$ and the line $y=1$ around the $y$-axis.

2014 PUMaC Team, 2

Tags: geometry , circumcircle

Given a Pacman of radius $1$, and mouth opening angle $90^\circ$, what is the largest (circular) pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt b+c$. where $b$ is square free. Find $a+b+c$.

2001 All-Russian Olympiad, 3

Tags: geometry , circumcircle , parallelogram , similar triangles

Let $N$ be a point on the longest side $AC$ of a triangle $ABC$. The perpendicular bisectors of $AN$ and $NC$ intersect $AB$ and $BC$ respectively in $K$ and $M$. Prove that the circumcenter $O$ of $\triangle ABC$ lies on the circumcircle of triangle $KBM$.

2011 Tokio University Entry Examination, 3

Tags: calculus , integration , analytic geometry , geometry , perimeter , limit , logarithm

Let $L$ be a positive constant. For a point $P(t,\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\ v(t))$ the point at which the point reach starting from $P$ proceeds by　distance $L$ in counter-clockwise on the perimeter of a circle passing the point $P$ with center $O$. (1) Find $u(t),\ v(t)$. (2) For real number $a$ with $0<a<1$, find $f(a)=\int_a^1 \sqrt{\{u'(t)\}^2+\{v'(t)\}^2}\ dt$. (3) Find $\lim_{a\rightarrow +0} \frac{f(a)}{\ln a}$. [i]2011 Tokyo University entrance exam/Science, Problem 3[/i]

2005 Postal Coaching, 2

Tags: geometry

Let $< \Gamma _j >$ be a sequnce of concentric circles such that the sequence $< R_j >$ , where $R_j$ denotes the radius of $\Gamma_j$, is increasing and $R_j \longrightarrow \infty$ as $j \longrightarrow \infty$. Let $A_1 B_1 C_1$ be a triangle inscribed in $\Gamma _1$. extend the rays $\vec{A_i B_1} , \vec{B_1 C_1 }, \vec{C_1 A_1}$ to meet $\Gamma_2$ in $B_2, C_2$and $A_2$ respectively and form the triangle $A_2 B_2 C_2$. Continue this process. Show that the sequence of triangles $< A_n B_n C_n >$ tends to an equilateral triangle as $n \longrightarrow \infty$

2010 Portugal MO, 2

Tags: geometry

On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.

2014 Romania Team Selection Test, 1

Tags: circumcircle , trigonometry , angle bisector , projective geometry , geometry

Let $\triangle ABC$ be an acute triangle of circumcentre $O$. Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$. The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$, and $OS \cap BC = D$. The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$. Prove that $AD$, $BE$ and $CF$ are concurrent. [i]Author: Cosmin Pohoata[/i]

2006 China Team Selection Test, 1

Tags: circumcircle , geometry

$H$ is the orthocentre of $\triangle{ABC}$. $D$, $E$, $F$ are on the circumcircle of $\triangle{ABC}$ such that $AD \parallel BE \parallel CF$. $S$, $T$, $U$ are the semetrical points of $D$, $E$, $F$ with respect to $BC$, $CA$, $AB$. Show that $S, T, U, H$ lie on the same circle.

2024 Singapore Senior Math Olympiad, Q1

Tags: geometry

In an acute triangle $ABC$, $AC>AB$, $D$ is the point on $BC$ such that $AD=AB$. Let $\omega_1$ be the circle through $C$ tangent to $AD$ at $D$, and $\omega_2$ the circle through $C$ tangent to $AB$ at $B$. Let $F(\ne C)$ be the second intersection of $\omega_1$ and $\omega_2$. Prove that $F$ lies on $AC$.

2019 Azerbaijan BMO TST, 2

Tags: geometry , perpendicular bisector , orthocenter , perpendicular , perpendicularity , tangent , circumcircle

Let $ABC$ be a triangle inscribed in circle $\Gamma$ with center $O$. Let $H$ be the orthocenter of triangle $ABC$ and let $K$ be the midpoint of $OH$. Tangent of $\Gamma$ at $B$ intersects the perpendicular bisector of $AC$ at $L$. Tangent of $\Gamma$ at $C$ intersects the perpendicular bisector of $AB$ at $M$. Prove that $AK$ and $LM$ are perpendicular. by Michael Sarantis, Greece

2024 ITAMO, 4

Tags: geometry

Let $ABCD$ be a rectangle with $AB<BC$ and circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) and let $Q$ be a point on the arc $CD$ (not containing $A$) such that $BP=CQ$. The circle with diameter $AQ$ intersects $AP$ again in $S$. The perpendicular to $AQ$ through $B$ intersects $AP$ in $X$. (a) Show that $XS=PS$. (b) Show that $AX=DQ$.

May Olympiad L2 - geometry, 2002.3

Tags: right triangle , isosceles , square , geometry

In a triangle $ABC$, right in $A$ and isosceles, let $D$ be a point on the side $AC$ ($A \ne D \ne C$) and $E$ be the point on the extension of $BA$ such that the triangle $ADE$ is isosceles. Let $P$ be the midpoint of segment $BD$, $R$ be the midpoint of the segment $CE$ and $Q$ the intersection point of $ED$ and $BC$. Prove that the quadrilateral $ARQP$ is a square.

2015 Taiwan TST Round 2, 3

Tags: geometry

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

2021 Bundeswettbewerb Mathematik, 4

Tags: combinatorics , graph theory , coloring , graph coloring , edge coloring , geometry

Consider a pyramid with a regular $n$-gon as its base. We colour all the segments connecting two of the vertices of the pyramid except for the sides of the base either red or blue. Show that if $n=9$ then for each such colouring there are three vertices of the pyramid connecting by three segments of the same colour, and that this is not necessarily the case if $n=8$.

2012 District Olympiad, 3

Tags: geometry , pure geometry

A circle that passes through the vertices $ B,C $ of a triangle $ ABC, $ cuts the segments $ AB,AC $ (endpoints excluded) in $ N, $ respectively, $ M. $ Consider the point $ P $ on the segment $ MN $ and $ Q $ on the segment $ BC $ (endpoints excluded on both segments) such that the angles $ \angle BAC,\angle PAQ $ have the same bisector. Show that: [b]a)[/b] $ \frac{PM}{PN} =\frac{QB}{QC} . $ [b]b)[/b] The midpoints of the segments $ BM,CN,PQ $ are collinear.