Olympiad problems

2015 India National Olympiad, 5

Tags: parallelogram , analytic geometry , trigonometry , geometry , vector , quadratic

Let $ABCD$ be a convex quadrilateral.Let diagonals $AC$ and $BD$ intersect at $P$. Let $PE,PF,PG$ and $PH$ are altitudes from $P$ on the side $AB,BC,CD$ and $DA$ respectively. Show that $ABCD$ has a incircle if and only if $\frac{1}{PE}+\frac{1}{PG}=\frac{1}{PF}+\frac{1}{PH}.$

2010 Iran MO (3rd Round), 4

Tags: algebra , group theory , abstract algebra , vector

a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms: i-$\{0\}$. ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$. iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$) b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)

2015 AMC 12/AHSME, 16

Tags: analytic geometry , geometry , vector , 3d geometry , tetrahedron , calculus

Tetrahedron $ABCD$ has $AB=5$, $AC=3$, $BC=4$, $BD=4$, $AD=3$, and $CD=\tfrac{12}5\sqrt2$. What is the volume of the tetrahedron? $\textbf{(A) }3\sqrt2\qquad\textbf{(B) }2\sqrt5\qquad\textbf{(C) }\dfrac{24}5\qquad\textbf{(D) }3\sqrt3\qquad\textbf{(E) }\dfrac{24}5\sqrt2$

1996 IMC, 3

Tags: vector , linear algebra

The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if $A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$. i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors of $A$. ii) Find the maximal number of distinct pairwise commuting involutions on $V$.

2005 Croatia National Olympiad, 4

Tags: vector , geometry

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 $AC.$

2014 AIME Problems, 7

Tags: vector , analytic geometry , trigonometry , derivative , graphing lines , calculus , ratio

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.

2006 District Olympiad, 4

Tags: vector

Let $ABC$ be a triangle with $AB=AC$. Let $D$ be the midpoint of $BC$, $M$ the midpoint of $AD$ and $N$ the foot of the perpendicular from $D$ to $BM$. Prove that $\angle ANC = 90^\circ$.

2012 Putnam, 5

Tags: function , algebra , vector , linear algebra , matrix , domain

Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.

1991 Arnold's Trivium, 87

Tags: derivative , function , calculus , quadratic , vector , linear algebra , matrix

Find the derivatives of the lengths of the semiaxes of the ellipsoid $x^2 + y^2 + z^2 + xy + yz + zx = 1 + \epsilon xy$ with respect to $\epsilon$ at $\epsilon = 0$.

2012 Indonesia TST, 3

Tags: geometry , quadratic , rectangle , vector

The [i]cross[/i] of a convex $n$-gon is the quadratic mean of the lengths between the possible pairs of vertices. For example, the cross of a $3 \times 4$ rectangle is $\sqrt{ \dfrac{3^2 + 3^2 + 4^2 + 4^2 + 5^2 + 5^2}{6} } = \dfrac{5}{3} \sqrt{6}$. Suppose $S$ is a dodecagon ($12$-gon) inscribed in a unit circle. Find the greatest possible cross of $S$.