# Tagged Questions

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### Arithmetic Mean and Geometric Mean Question, Guidance Needed

I am super new to olympiad-style math which focuses on a lot of inequalities, and tough problems which highschool students do not go over. I'm in 9th grade, and am trying to get into all of this stuff ...
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### Ratio of angles in a right triangle

P.S: I only want a hint,not the whole solution. BdMO 2009 Problem 5 Secondary In triangle ABC, $\angle A = 90$. M is the midpoint of BC. Choose D on AC such that AD = AM. The circumcircles ...
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### An equilateral triangle formed using points of tangency

P.S:I am looking for a hint and not the whole solution. BdMO 2012 nationals secondary: The vertices of a right triangle $ABC$ inscribed in a circle divide the circumference into three arcs. The ...
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### An angular inequality

In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on ...
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### Find all such functions defined on the space

$f:\mathbb{R}^3\to \mathbb{R}^{\ast}$ is such that for any non-degenerate tetrahedron $ABCD$ with $O$ the center of the inscribed sphere, we have : $$f(O)=f(A)f(B)f(C)f(D)$$ Prove that $f(X)=1$ for ...
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### A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
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Let $\triangle ABC$ be any triangle. Suppose the angle bisector of $\angle BAC$ intersects $BC$ at $D$. Let $\Gamma$ be a circle tangent to $BC$ at $D$ and so that $A$ belongs to the circumference ...
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### Geometry Math Olympiad Question [closed]

In the diagram below, AD is perpendicular to AC and $∠BAD = ∠DAE = 12^\circ$. If $AB + AE = BC$, find $∠ABC$. The above is the diagram. I came across this question in a Math Olympiad ...
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### Parallelogram constructed through medians

Bdmo In $\Delta ABC$, Medians AD and CF intersect at P.Let Q be any point on AC.Construct QM and QN parallel to AD and CF respectively.Now the line joining M and N intersects CF and T and AD at ...
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### Angle Manipulation Contest Math Problem

The problem is as follows: In triangle $ABC$, $BC=2$. Point $D$ is on $\overline{AC}$ such that $AD=1$ and $CD=2$. If $m\angle BDC=2m\angle A$, compute $\sin A$. I tried several ways of making ...
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### Math Olympiad Geometry Question: Similar Triangles

In the diagram below, △ABC and △CDE are two right-angled triangles with AC = 24, CE =7 and ∠ ACB = ∠ CED. Find the length of the line segment AE. The above is the diagram. I came ...
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### 3D Geometry Contest Math Problem

The problem is as follows: Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon H with side length 1, and so that the vertices are not lying in the ...
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### Inscribed Hexagon Geometry Contest Problem

The problem was as follows: Regular hexagon $HEXAGN$ is inscribed in the circle $O$, and $R$ is a point on minor arc $HN$ of circle $O$. If $RE=10$ and $RG=8$, then $RN$ can be expressed in the form ...
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### Equivalent Planes?

The three planes $x=y$, $y=z$, $x=z$ cut the unit cube $0\le x\le1$, $0\le y\le1$, $0\le z\le1$ into $n$ pieces. Find $n$. My question is this: what does $x=y$, $y=z$, $x=z$ mean? If all of the ...
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### Bisector of angle formed at the orthocentre passes through the circumcentre

BdMO 2012 In an acute angled triangle $ABC$, $\angle A= 60$. We have to prove that the bisector of one of the angles formed by the altitudes drawn from $B$ and $C$ passes through the center of the ...
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### Inscribed Angles in Two Cyclic Quadrilaterals

This problem is driving me crazy. It's from Andreescu's Mathematical Olympiad Challenges: Let $AB$ be a chord in a circle and $P$ a point on the circle. Let $Q$ be the projection of $P$ onto $AB$ ...
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### 2014 USAMO Problem :with Points Collinear iff Sum is Constant

Prove that there exists an infinite set of points $$\dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots$$ in the plane with the following property: For any three distinct ...
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### Ratio of area of triangle to that formed by its medians

What is the ratio of the area of a triangle $ABC$ to the area of the triangle whose sides are equal in length to the medians of triangle $ABC$? I see an obvious method of brute-force wherein I can ...
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### Areas in a rectangle

Suppose $P,Q, R$, and $S$ are the midpoints of the sides $AB, BC, CD$, and $DA$, respectively of rectangle $ABCD$. If the area of the rectangle is $\delta$, then prove that the area of the figure ...
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### Orthocentre of triangle and related ratio

$ABC$ is a triangle with $AB = 13$, $BC = 14$ and $CA = 15$. $AD$ and $BE$ are the altitudes from $A$ to $B$ to $BC$ and $AC$ respectively. $H$ is the point of intersection of $AD$ and $BE$. Then the ...
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### How many different right triangles are possible with the shorter side of odd length?

I was trying to solve this problem but unable to figure it out completely. I thing number of was odd integer $n$ can be the side of right triangle is number of factor of $\frac{n^2}{2}$. Can some one ...
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### Given perimeter of triangle and one side, find other two sides

In triangle ABC, all three sides have integer lengths. If AB = 21, the perimeter is 54, and the area is a positive integer, what are the lengths of BC and AC? I tried using Heron's Formula, but I ...
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### Squares constructed externally on the sides of a triangle and concurrent lines

On the sides $BC, CA$ and $AB$ of the triangle $ABC$ we construct externally the squares $BCDE, ACFG$ and $ABHI$. Denote $A', B'$ and $C'$ the intersectiond points of the lines $BF$ and $CH$, $AD$ ...
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### Tricky Triangle Area Problem

This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that ...
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### Euclidean Geometry problem: prove that $C'$ is the midpoint of $A'B'$.

The tangents to a circumference centered at $O$, passing through an exterior point $C$, meet the circumference at the points $A$ and $B$. Let $S$ be an arbitrary point on the circumference. The ...
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### Maximize the inradius given the base and the area of the triangle

BdMO 2013 Secondary: A triangle has base of length 8 and area 12. What is the radius of the largest circle that can be inscribed in this triangle? Let $A,r,s$ denote the area,inradius and ...
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### Locus of the centres of equilateral triangles (contest problem)

Given a triangle $A_0A_1A_2$ determine the locus of the centres of the equilateral triangles $X_0X_1X_2$ satisfying the condition that each of the lines $X_kX_{k+1}$, $k=0,1,2$ passes through ...
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### A geometry problem on power of points

An acute triangle $ABC$ is inscribed in a circumference of center $O$. Its heights are $AD$, $BE$ and $CF$. The line $EF$ intersects the circumference at two points, $P$ and $Q$. (a) Prove ...
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### How prove $G,H,T$ are collinear.

Question: Circle $O_{1}$ and $O_{2}$ are internally tangent at point $T$. $AB$ and $CD$ are tangents of circle $O_{1}$, the angle bisectors of Angle $\angle ADB$ and $\angle CBD$ intersects at ...
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### A geometry problem on cyclic quadrilaterals

The problem: Let $M$ be the point of intersection between the diagonals of a cyclic quadrilateral $ABCD$, where $\angle AMB$ is acute. The isosceles triangle $BCK$, whose base is $BC$, is ...
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### Distance between two points in the plane

my teacher asked in the class today the following question: There exists an infinite set M of points in the plane with the property that any three points are non-collinear and such that the distance ...
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### Competition math geometry question

The perimeter of triangle ABC is $36$, and its area is $36$. Compute $\tan\frac{A}2 \tan\frac{B}2 \tan\frac{C}2$. I found that the answer is $1/9$, but I was not able to find a reason for this. Could ...
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I recently took the AIME, and the following question was one I was not able to answer: On square $ABCD$, points $E,F,G,$ and $H$ lie on sides $\overline{AB}$,$\overline{BC}$,$\overline{CD}$, and ...
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### Inequality in triangle involving side lenghs, medians and area

A, B and C are the vertices of a triangle. Denote $m_a$, $m_b$ and $m_c$ the medians from A, B and C. Prove the inequality: $$\sum_{cyc}{a^2bcm_a}\geq\sum_{cyc}{cS(a^2+b^2)}$$where a, b and c are the ...
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### Finding the Rate of distance between hands of clock

First, I think I don't understand the problem which asks about the greatest rate of change in distance between the tips of the hands of clocks. Does it mean where the increasing of distance is the ...
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### An inequality for sides of a triangle

Let $a, b, c$ be sides of a triangle and $ab+bc+ca=1$. Show $$(a+1)(b+1)(c+1)<4$$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4$. I am ...
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### I am looking for a proof of the “ begonia theorem”.

Let $D$, $E$, $F$ be points on respective (extended) sides $\overleftrightarrow{BC}$, $\overleftrightarrow{CA}$, $\overleftrightarrow{AB}$ of $\triangle ABC$, such that $\overleftrightarrow{AD}$, ...
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### Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
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### Maximizing the perimeter of a triangle inside a square

BdMO 2014: We have a square $ABCD$ of side length 5.We take a point $E$ on $AD$ and $F$ on $AB$ so that $\angle FCE=45^\circ$. What can be the maximum perimeter of $\triangle AEF$? I can ...
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### The second triangle?

BdMO National 2013 Junior Q. 2 Two isosceles triangles are possible with 120 square unit area of each and length of edges are integers. Such one is with 17, 17 and 16 unit edges. Determine the ...
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### Contest geometry problem

$|AM|=|CM|$ $\angle BCA = 15^{\circ}$ $\angle CBM = \angle ABH$ $\angle BHC = 90^\circ$ Find $|AC|$ The solution states that $\overline{BM}$ is the isogonal conjugate of $\overline{BH}$ but I ...
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### $7$ points inside a circle at equal distances

BdMO 2014 There are $7$ points on a circle.Any 2 consecutive points are at equal distance from one another.How many acute angled triangles can you form taking any 3 of these points? I believe ...
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### Separating points on a plane

BdMO 2011 There are $25$ points on a plane, no three of which lie on a line. Find the minimum number of lines needed to separate them from one another. Can we assume that the points lie on a ...
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### What is the meaning of $(x^2+y^2)^n$? Is this an already known geometric object?

We all know that $x^2+y^2=r^2$ is a circle. What does $(x^2+y^2)^2$ signify? In general, what is $(x^2+y^2)^n$?
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In every polyhedron there is at least one pair of faces with the same number of sides. Solution: Let $N$ be the greatest number of sides in a face of a given polyhedron. Then the number of ...
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### Ratios in a rhombus

NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...
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### Separating $3n$ points on the plane by a line

I am trying to solve a problem in geometry (a contest-type question), and I wondering if the following result is true. (If it is true, then it makes life much easier!) Suppose there are $3n$ ...
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### Competition Math Geometry Problem

Note: I am paraphrasing this problem Consider a quadrilateral with 3 sides of equal length, and one longer side. This quadrilateral also has equal diagonals, both of which are equal in length to the ...
A non-planar pentagon in $\mathbb{R}^3$ has equal sides and four right angles. What are the possible values for the fifth angle? My attempt It was quite easy to find an example for 60 $^\circ$: ...