Olympiad problems

1996 India Regional Mathematical Olympiad, 2

Tags:

Find all triples $a,b,c$ of positive integers such that \[ ( 1 + \frac{1}{a} ) ( 1 + \frac{1}{b}) ( 1 + \frac{1}{c} ) = 3. \]

2016 AMC 10, 22

For some positive integer $n$, the number $110n^3$ has $110$ positive integer divisors, including $1$ and the number $110n^3$. How many positive integer divisors does the number $81n^4$ have? $\textbf{(A) }110 \qquad \textbf{(B) } 191 \qquad \textbf{(C) } 261 \qquad \textbf{(D) } 325 \qquad \textbf{(E) } 425$

2019 NMTC Junior, 7

Tags: inequalities , AM-GM

The perimeter of $\triangle ABC$ is $2$ and it's sides are $BC=a, CA=b,AB=c$. Prove that $$abc+\frac{1}{27}\ge ab+bc+ca-1\ge abc. $$

2012 AMC 10, 19

Tags: geometry , rectangle , AMC , AMC 10 , AMC 10 B , analytic geometry , similar triangles

In rectangle $ABCD$, $AB=6$, $AD=30$, and $G$ is the midpoint of $\overline{AD}$. Segment $AB$ is extended $2$ units beyond $B$ to point $E$, and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$. What is the area of $BFDG$? $ \textbf{(A)}\ \frac{133}{2}\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ \frac{135}{2}\qquad\textbf{(D)}\ 68\qquad\textbf{(E)}\ \frac{137}{2}$

2017 Mathematical Talent Reward Programme, MCQ: P 9

Tags: geometry , tangent , circles

From a point $P$ outside of a circle with centre $O$, tangent segments $PA$ and $PB$ are drawn. $\frac{1}{OA^2}+\frac{1}{PA^2}=\frac{1}{16}$ then $AB=$ [list=1] [*] 4 [*] 6 [*] 8 [*] 10 [/list]

2020-21 IOQM India, 6

Tags: number theory , Perfect Square

What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?

2014 India PRMO, 14

Tags: Inequality

One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1/7$-th of the total amount of milk and $2/17$-th of the total amount of coffee. How many people are there in Manjul’s family?

2019 District Olympiad, 2

Tags: geometry , Vectors

Let $H$ be the orthocenter of the acute triangle $ABC.$ In the plane of the triangle $ABC$ we consider a point $X$ such that the triangle $XAH$ is right and isosceles, having the hypotenuse $AH,$ and $B$ and $X$ are on each part of the line $AH.$ Prove that $\overrightarrow{XA}+\overrightarrow{XC}+\overrightarrow{XH}=\overrightarrow{XB}$ if and only if $ \angle BAC=45^{\circ}.$

2019 Canadian Mathematical Olympiad Qualification, 5

Tags: game , game strategy , winning strategy , combinatorics

Let $(m,n,N)$ be a triple of positive integers. Bruce and Duncan play a game on an m\times n array, where the entries are all initially zeroes. The game has the following rules. $\bullet$ The players alternate turns, with Bruce going first. $\bullet$ On Bruce's turn, he picks a row and either adds $1$ to all of the entries in the row or subtracts $1$ from all the entries in the row. $\bullet$ On Duncan's turn, he picks a column and either adds $1$ to all of the entries in the column or subtracts $1$ from all of the entries in the column. $\bullet$ Bruce wins if at some point there is an entry $x$ with $|x|\ge N$. Find all triples $(m, n,N)$ such that no matter how Duncan plays, Bruce has a winning strategy.

2020 LMT Fall, A16

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Two circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively, and intersect at points $M$ and $N$. The radii of $\omega_1$ and $\omega_2$ are $12$ and $15$, respectively, and $O_1O_2 = 18$. A point $X$ is chosen on segment $MN$. Line $O_1X$ intersects $\omega_2$ at points $A$ and $C$, where $A$ is inside $\omega_1$. Similarly, line $O_2X$ intersects $\omega_1$ at points $B$ and $D$, where $B$ is inside $\omega_2$. The perpendicular bisectors of segments $AB$ and $CD$ intersect at point $P$. Given that $PO_1 = 30$, find $PO_2^2$. [i]Proposed by Andrew Zhao[/i]

2023 Germany Team Selection Test, 2

Tags: geometry , IMO Shortlist

Let $ABCD$ be a cyclic quadrilateral. Assume that the points $Q, A, B, P$ are collinear in this order, in such a way that the line $AC$ is tangent to the circle $ADQ$, and the line $BD$ is tangent to the circle $BCP$. Let $M$ and $N$ be the midpoints of segments $BC$ and $AD$, respectively. Prove that the following three lines are concurrent: line $CD$, the tangent of circle $ANQ$ at point $A$, and the tangent to circle $BMP$ at point $B$.

2004 Junior Balkan Team Selection Tests - Romania, 3

Tags: induction , number theory solved , number theory

Let $A$ be a set of positive integers such that a) if $a\in A$, the all the positive divisors of $a$ are also in $A$; b) if $a,b\in A$, with $1<a<b$, then $1+ab \in A$. Prove that if $A$ has at least 3 elements, then $A$ is the set of all positive integers.

2011 Balkan MO Shortlist, G1

Tags: geometry , power of a point , radical axis , geometry proposed

Let $ABCD$ be a convex quadrangle such that $AB=AC=BD$ (vertices are labelled in circular order). The lines $AC$ and $BD$ meet at point $O$, the circles $ABC$ and $ADO$ meet again at point $P$, and the lines $AP$ and $BC$ meet at the point $Q$. Show that the angles $COQ$ and $DOQ$ are equal.

1996 Spain Mathematical Olympiad, 2

Tags: geometry , isosceles , Centroid

Let $G$ be the centroid of a triangle $ABC$. Prove that if $AB+GC = AC+GB$, then the triangle is isosceles

2000 All-Russian Olympiad, 3

Tags: geometry , parallelogram , trigonometry , geometric transformation , reflection , circumcircle , angle bisector

In an acute scalene triangle $ABC$ the bisector of the acute angle between the altitudes $AA_1$ and $CC_1$ meets the sides $AB$ and $BC$ at $P$ and $Q$ respectively. The bisector of the angle $B$ intersects the segment joining the orthocenter of $ABC$ and the midpoint of $AC$ at point $R$. Prove that $P$, $B$, $Q$, $R$ lie on a circle.

2021 CCA Math Bonanza, T2

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Given that real numbers $a$, $b$, and $c$ satisfy $ab=3$, $ac=4$, and $b+c=5$, the value of $bc$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]2021 CCA Math Bonanza Team Round #2[/i]

2004 All-Russian Olympiad Regional Round, 11.3

Tags: algebra , polynomial

Let the polynomial $P(x) = a_nx^n+a_{n-1}x^{n-1}+...+a_0$ has at least one real root and $a_0 \ne 0$. Prove that, consequently crossing out the monomials in the notation $P(x)$ in some order, we can obtain the number $a_0$ from it so that each intermediate polynomial also has at least one real root.

1955 AMC 12/AHSME, 30

Tags: calculus , integration

Each of the equations $ 3x^2\minus{}2\equal{}25$, $ (2x\minus{}1)^2\equal{}(x\minus{}1)^2$, $ \sqrt{x^2\minus{}7}\equal{}\sqrt{x\minus{}1}$ has: $ \textbf{(A)}\ \text{two integral roots} \qquad \textbf{(B)}\ \text{no root greater than 3} \qquad \textbf{(C)}\ \text{no root zero} \\ \textbf{(D)}\ \text{only one root} \qquad \textbf{(E)}\ \text{one negative root and one positive root}$

1979 AMC 12/AHSME, 22

Tags: modular arithmetic , AMC

Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$. $\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }3\qquad\textbf{(D) }9\qquad\textbf{(E) }\text{infinitely many}$

2013 Stars Of Mathematics, 2

Tags: geometry , rectangle

Three points inside a rectangle determine a triangle. A fourth point is taken inside the triangle. i) Prove at least one of the three concave quadrilaterals formed by these four points has perimeter lesser than that of the rectangle. ii) Assuming the three points inside the rectangle are three corners of it, prove at least two of the three concave quadrilaterals formed by these four points have perimeters lesser than that of the rectangle. [i](Dan Schwarz)[/i]

2008 South East Mathematical Olympiad, 3

Tags: geometry , circumcircle , perpendicular bisector , geometry unsolved

In $\triangle ABC$, side $BC>AB$. Point $D$ lies on side $AC$ such that $\angle ABD=\angle CBD$. Points $Q,P$ lie on line $BD$ such that $AQ\bot BD$ and $CP\bot BD$. $M,E$ are the midpoints of side $AC$ and $BC$ respectively. Circle $O$ is the circumcircle of $\triangle PQM$ intersecting side $AC$ at $H$. Prove that $O,H,E,M$ lie on a circle.

2011 IMO Shortlist, 3

Tags: geometry , IMO 2011 , combinatorics , combinatorial geometry , point set , IMO Shortlist , Geoff Smith

Let $\mathcal{S}$ be a finite set of at least two points in the plane. Assume that no three points of $\mathcal S$ are collinear. A [i]windmill[/i] is a process that starts with a line $\ell$ going through a single point $P \in \mathcal S$. The line rotates clockwise about the [i]pivot[/i] $P$ until the first time that the line meets some other point belonging to $\mathcal S$. This point, $Q$, takes over as the new pivot, and the line now rotates clockwise about $Q$, until it next meets a point of $\mathcal S$. This process continues indefinitely. Show that we can choose a point $P$ in $\mathcal S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $\mathcal S$ as a pivot infinitely many times. [i]Proposed by Geoffrey Smith, United Kingdom[/i]

1999 Poland - Second Round, 4

Tags: geometry , equal angles , right angle , Circumcenter

Let $P$ be a point inside a triangle $ABC$ such that $\angle PAB = \angle PCA$ and $\angle PAC = \angle PBA$. If $O \ne P$ is the circumcenter of $\triangle ABC$, prove that $\angle APO$ is right.

2002 Bosnia Herzegovina Team Selection Test, 3

Tags: number theory , greatest common divisor , modular arithmetic , relatively prime , Divisibility Theory

If $n$ is a natural number, prove that the number $(n+1)(n+2)\cdots(n+10)$ is not a perfect square.

1997 Portugal MO, 1

Tags: algebra , number theory

A test has twenty questions. Seven points are awarded for each correct answer, two points are deducted for each incorrect answer and no points are awarded or deducted for each unanswered question. Joana obtained $87$ points. How many questions did she not answer?