Olympiad problems

Mathley 2014-15, 3

A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$. Let $E, F$ be the projections of $CA, AB$. Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$. Prove that $P$ is the orthocenter of triangle $ABC$. Do Thanh Son, High School of Natural Sciences, National University, Hanoi

VMEO II 2005, 9

Tags: chessboard , combinatorics , queen

On a board with $64$ ($8 \times 8$) squares, find a way to arrange $9$ queens and $ 1$ king so that every queen cannot capture another queen.

2016 AMC 10, 25

Tags: floor function

Let $f(x)=\sum_{k=2}^{10}(\lfloor kx \rfloor -k \lfloor x \rfloor)$, where $\lfloor r \rfloor$ denotes the greatest integer less than or equal to $r$. How many distinct values does $f(x)$ assume for $x \ge 0$? $\textbf{(A)}\ 32\qquad\textbf{(B)}\ 36\qquad\textbf{(C)}\ 45\qquad\textbf{(D)}\ 46\qquad\textbf{(E)}\ \text{infinitely many}$

2015 VTRMC, Problem 1

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Find all n such that $n^{4}+6n^{3}+11n^{2}+3n+31$ is a perfect square.

2017 Sharygin Geometry Olympiad, P11

Tags: geometry , circumcircle

A finite number of points is marked on the plane. Each three of them are not collinear. A circle is circumscribed around each triangle with marked vertices. Is it possible that all centers of these circles are also marked? [i]Proposed by A.Tolesnikov[/i]

1993 Greece National Olympiad, 15

Tags: number theory , relatively prime , pythagorean theorem , geometry

Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $m/n$, where $m$ and $n$ are relatively prime integers. Find $m + n$

1955 AMC 12/AHSME, 42

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If $ a$, $ b$, and $ c$ are positive integers, the radicals $ \sqrt{a\plus{}\frac{b}{c}}$ and $ a\sqrt{\frac{b}{c}}$ are equal when and only when: $ \textbf{(A)}\ a\equal{}b\equal{}c\equal{}1 \qquad \textbf{(B)}\ a\equal{}b\text{ and }c\equal{}a\equal{}1 \qquad \textbf{(C)}\ c\equal{}\frac{b(a^2\minus{}1)}{2} \\ \textbf{(D)}\ a\equal{}b \text{ and }c\text{ is any value} \qquad \textbf{(E)}\ a\equal{}b \text{ and }c\equal{}a\minus{}1$

1959 AMC 12/AHSME, 8

Tags: quadratic , parabola , analytic geometry , optimizing quadratics , algebra , conic

The value of $x^2-6x+13$ can never be less than: $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 4.5 \qquad\textbf{(C)}\ 5\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 13 $

2000 Greece National Olympiad, 1

Tags: rectangle , geometry

Consider a rectangle $ABCD$ with $AB = a$ and $AD = b.$ Let $l$ be a line through $O,$ the center of the rectangle, that cuts $AD$ in $E$ such that $AE/ED = 1/2$. Let $M$ be any point on $l,$ interior to the rectangle. Find the necessary and suﬃcient condition on $a$ and $b$ that the four distances from M to lines $AD, AB, DC, BC$ in this order form an arithmetic progression.

2018 Balkan MO Shortlist, G3

Tags: geometry , geometric inequality , inequalities , semiperimeter , maximum value , minimum

Let $P$ be an interior point of triangle $ABC$. Let $a,b,c$ be the sidelengths of triangle $ABC$ and let $p$ be it's semiperimeter. Find the maximum possible value of $$ \min\left(\frac{PA}{p-a},\frac{PB}{p-b},\frac{PC}{p-c}\right)$$ taking into consideration all possible choices of triangle $ABC$ and of point $P$. by Elton Bojaxhiu, Albania

2001 Spain Mathematical Olympiad, Problem 5

Tags: geometry

A quadrilateral $ABCD$ is inscribed in a circle of radius 1 whose diameter is $AB$. If the quadrilateral $ABCD$ has an incircle, prove that $CD \leq 2 \sqrt{5} - 2$.

2003 Poland - Second Round, 3

Tags: algebra , polynomial , polynomial equation

Let $W(x) = x^4 - 3x^3 + 5x^2 - 9x$ be a polynomial. Determine all pairs of different integers $a$, $b$ satisfying the equation $W(a) = W(b)$.

1968 Putnam, B6

Tags: topology , compact set , real analysis

Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that (1) All elements of $A_n$ are rational. (2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.

2015 JBMO Shortlist, 4

Tags: geometry

Let $ABC$ be an acute triangle.The lines $l_1$ and $l_2$ are perpendicular to $AB$ at the points $A$ and $B$, respectively.The perpendicular lines from the midpoint $M$ of $AB$ to the lines $AC$ and $BC$ intersect $l_1$ and $l_2$ at the points $E$ and $F$, respectively.If $D$ is the intersection point of the lines $EF$ and $MC$, prove that \[\angle ADB = \angle EMF.\]

2014 Contests, 1

Tags: number theory

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

2024 Mathematical Talent Reward Programme, 6

Tags: algebra

Find the maximum possible length of a sequence consisting of non-zero integers, in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.

2020 BMT Fall, 6

Tags: geometry

A tetrahedron has four congruent faces, each of which is a triangle with side lengths $6$, $5$, and $5$. If the volume of the tetrahedron is $V$ , compute $V^2$ .

1997 All-Russian Olympiad Regional Round, 9.1

Tags: geometry , combinatorics , combinatorial geometry , acute

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.

2018 District Olympiad, 2

Tags: algebra , inequalities

Let $a,b,c \in [1, \infty)$. Prove that: $$\frac{a\sqrt{b}}{a+b}+\frac{b\sqrt{c}}{b+c}+\frac{c\sqrt{b}}{c+a}+\frac32 \le a+b+c$$

1996 Abels Math Contest (Norwegian MO), 1

Tags: locus , geometry

Let $S$ be a circle with center $C$ and radius $r$, and let $P \ne C$ be an arbitrary point. A line $\ell$ through $P$ intersects the circle in $X$ and $Y$. Let $Z$ be the midpoint of $XY$. Prove that the points $Z$, as $\ell$ varies, describe a circle. Find the center and radius of this circle.

2004 Purple Comet Problems, 22

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Two circles have radii $15$ and $95$. If the two external tangents to the circles intersect at $60$ degrees, how far apart are the centers of the circles?

1966 Putnam, B1

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Let a convex polygon $P$ be contained in a square of side one. Show that the sum of the sides of $P$ is less than or equal to $4$.

2008 Sharygin Geometry Olympiad, 19

Tags: ratio , parallelogram , geometry

(V.Protasov, 10-11) Given parallelogram $ ABCD$ such that $ AB \equal{} a$, $ AD \equal{} b$. The first circle has its center at vertex $ A$ and passes through $ D$, and the second circle has its center at $ C$ and passes through $ D$. A circle with center $ B$ meets the first circle at points $ M_1$, $ N_1$, and the second circle at points $ M_2$, $ N_2$. Determine the ratio $ M_1N_1/M_2N_2$.

1977 AMC 12/AHSME, 1

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If $y = 2x$ and $z = 2y$, then $x + y + z$ equals \[ \text{(A)}\ x \qquad \text{(B)}\ 3x \qquad \text{(C)}\ 5x \qquad \text{(D)}\ 7x \qquad \text{(E)}\ 9x \]

2020 Princeton University Math Competition, A8

Tags: combinatorics

Let $f(k)$ denote the number of triples $(a, b, c)$ of positive integers satisfying $a + b + c = 2020$ with $(k - 1)$ not dividing $a, k$ not dividing $b$, and $(k + 1)$ not dividing $c$. Find the product of all integers $k$ in the range 3 \le k \le 20 such that $(k + 1)$ divides $f(k)$.