Olympiad problems

1982 IMO Longlists, 39

Let $S$ be the unit circle with center $O$ and let $P_1, P_2,\ldots, P_n$ be points of $S$ such that the sum of vectors $v_i=\stackrel{\longrightarrow}{OP_i}$ is the zero vector. Prove that the inequality $\sum_{i=1}^n XP_i \geq n$ holds for every point $X$.

2001 APMO, 3

Tags: geometry , trigonometry , vector , triangle inequality

Two equal-sized regular $n$-gons intersect to form a $2n$-gon $C$. Prove that the sum of the sides of $C$ which form part of one $n$-gon equals half the perimeter of $C$. [i]Alternative formulation:[/i] Let two equal regular $n$-gons $S$ and $T$ be located in the plane such that their intersection $S\cap T$ is a $2n$-gon (with $n\ge 3$). The sides of the polygon $S$ are coloured in red and the sides of $T$ in blue. Prove that the sum of the lengths of the blue sides of the polygon $S\cap T$ is equal to the sum of the lengths of its red sides.

2007 Bundeswettbewerb Mathematik, 3

Tags: 3d geometry , tetrahedron , vector , geometry

A set $ E$ of points in the 3D space let $ L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $ E.$ Let $ T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $ L(L(T))?$

2023 Romania National Olympiad, 3

Tags: combinatorics , game , vector

Let $n \geq 2$ be a natural number. We consider a $(2n - 1) \times (2n - 1)$ table.Ana and Bob play the following game: starting with Ana, the two of them alternately color the vertices of the unit squares, Ana with red and Bob with blue, in $2n^2$ rounds. Then, starting with Ana, each one forms a vector with origin at a red point and ending at a blue point, resulting in $2n^2$ vectors with distinct origins and endpoints. If the sum of these vectors is zero, Ana wins. Otherwise, Bob wins. Show that Bob has a winning strategy.

2014 PUMaC Combinatorics A, 5

Tags: vector

What is the size of the largest subset $S'$ of $S = \{2^x3^y5^z : 0 \le x,y,z \le 4\}$ such that there are no distinct elements $p,q \in S'$ with $p \mid q$?

2008 Iran Team Selection Test, 4

Tags: function , inequalities , vector , 3d geometry , tetrahedron , geometry

Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i\minus{}P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.

2023 China Second Round, 4

Tags: vector , algebra

if three non-zero vectors on a plane $\vec{a},\vec{b},\vec{c}$ satisfy: (a) $\vec{a}\bot\vec{b}$ (b) $\vec{b}\cdot\vec{c}=2|\vec{a}|$ (c) $\vec{c}\cdot\vec{a}=3|\vec{b}|$ find out the minimum of $|\vec{c}|$

2003 Iran MO (3rd Round), 15

Tags: linear algebra , matrix , inequalities , vector , analytic geometry , modular arithmetic

Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$. ( for example the diffrence of this two row is only in one index 110 100) [i]Edited by Myth[/i]

1991 Arnold's Trivium, 31

Tags: vector

Find the index of the singular point $0$ of the vector field with components \[(x^4+y^4+z^4,x^3y-xy^3,xyz^2)\]

2014 Putnam, 6

Tags: algebra , polynomial , vector , function , putnam matrices

Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$

1985 Greece National Olympiad, 1

Tags: geometry , vector , analytic geometry , area

Inside triangle $ABC$ consider random point $O$. Prove that: $$E_A \overrightarrow{OA}+E_B \overrightarrow{OB}+E_C\overrightarrow{OC}=\overrightarrow{O}$$ where $E_A,E_B,E_C$ the areas of triangle $BOC, COB, AOB$ respectively

2008 IMC, 6

Tags: vector , functional analysis

Let $ \mathcal{H}$ be an infinite-dimensional Hilbert space, let $ d>0$, and suppose that $ S$ is a set of points (not necessarily countable) in $ \mathcal{H}$ such that the distance between any two distinct points in $ S$ is equal to $ d$. Show that there is a point $ y\in\mathcal{H}$ such that \[ \left\{\frac{\sqrt{2}}{d}(x\minus{}y): \ x\in S\right\}\] is an orthonormal system of vectors in $ \mathcal{H}$.

2022 District Olympiad, P4

Tags: geometry , vector

We call a set of $6$ points in the plane [i]splittable[/i] if we if can denote its elements by $A,B,C,D,E$ and $F$ in such a way that $\triangle ABC$ and $\triangle DEF$ have the same centroid. [list=a] [*]Construct a splittable set. [*]Show that any set of $7$ points has a subset of $6$ points which is [i]not[/i] splittable. [/list]

1990 IMO Longlists, 15

Tags: incenter , vector , euler , trigonometry , orthocenter , geometry

Given a triangle $ ABC$. Let $ G$, $ I$, $ H$ be the centroid, the incenter and the orthocenter of triangle $ ABC$, respectively. Prove that $ \angle GIH > 90^{\circ}$.

2012 China Team Selection Test, 2

Tags: function , vector , inequalities

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

1999 AIME Problems, 15

Tags: geometry , 3d geometry , pyramid , tetrahedron , analytic geometry , trigonometry , vector

Consider the paper triangle whose vertices are $(0,0), (34,0),$ and $(16,24).$ The vertices of its midpoint triangle are the midpoints of its sides. A triangular pyramid is formed by folding the triangle along the sides of its midpoint triangle. What is the volume of this pyramid?

1995 Moldova Team Selection Test, 7

Tags: vector

Let $S=\{a_1,a_2,\ldots,a_n\}$ of nenul vectors in a plane. Show that $S{}$ can be partitioned in nenul subsets $B_1, B_2,\ldots, B_m$ with the properties: 1) each vector from $S{}$ is part of only on subset; 2) if $a_i\in B_j$ then the angle between vectors $a_i$ and $c_j$, which is the sum of all vectors from $B_j$ is not greater than $\frac{\pi}{2}$; 3) if $i\neq j$ then the angle between vectors $c_i$ and $c_j$, which is the sum of all vectors from $B_i$ and $B_j$, respectively, is greater than $\frac{\pi}{2}$. What are the possible values of $m$?

2002 AIME Problems, 11

Tags: geometry , 3d geometry , geometric transformation , reflection , vector

Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}.$ The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n},$ where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n.$

2011 USA Team Selection Test, 9

Tags: inequalities , vector , pigeonhole principle , algebra

Determine whether or not there exist two different sets $A,B$, each consisting of at most $2011^2$ positive integers, such that every $x$ with $0 < x < 1$ satisfies the following inequality: \[\left| \sum_{a \in A} x^a - \sum_{b \in B} x^b \right| < (1-x)^{2011}.\]

1998 Flanders Math Olympiad, 2

Tags: geometry , 3d geometry , analytic geometry , vector , trigonometry

Given a cube with edges of length 1, $e$ the midpoint of $[bc]$, and $m$ midpoint of the face $cdc_1d_1$, as on the figure. Find the area of intersection of the cube with the plane through the points $a,m,e$. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=279[/img]

2022 CIIM, 2

Tags: linear algebra , vector , matrix

Let $v \in \mathbb{R}^2$ a vector of length 1 and $A$ a $2 \times 2$ matrix with real entries such that: (i) The vectors $A v, A^2 v$ y $A^3 v$ are also of length 1. (ii) The vector $A^2 v$ isn't equal to $\pm v$ nor to $\pm A v$. Prove that $A^t A=I_2$.

2003 IMO Shortlist, 1

Tags: geometry , 3d geometry , tetrahedron , linear algebra , algebra , vector

Let $a_{ij}$ $i=1,2,3$; $j=1,2,3$ be real numbers such that $a_{ij}$ is positive for $i=j$ and negative for $i\neq j$. Prove the existence of positive real numbers $c_{1}$, $c_{2}$, $c_{3}$ such that the numbers \[a_{11}c_{1}+a_{12}c_{2}+a_{13}c_{3},\qquad a_{21}c_{1}+a_{22}c_{2}+a_{23}c_{3},\qquad a_{31}c_{1}+a_{32}c_{2}+a_{33}c_{3}\] are either all negative, all positive, or all zero. [i]Proposed by Kiran Kedlaya, USA[/i]

2008 IMAC Arhimede, 5

Tags: geometry , parallelogram , vector , trigonometry , geometric transformation , reflection , cyclic quadrilateral

The diagonals of the cyclic quadrilateral $ ABCD$ are intersecting at the point $ E$. $ K$ and $ M$ are the midpoints of $ AB$ and $ CD$, respectively. Let the points $ L$ on $ BC$ and $ N$ on $ AD$ s.t. $ EL\perp BC$ and $ EN\perp AD$.Prove that $ KM\perp LN$.

Oliforum Contest I 2008, 2

Tags: vector , geometry , circumcircle , geometric transformation , reflection , parallelogram , trapezoid

Let $ ABCD$ be a cyclic quadrilateral with $ AB>CD$ and $ BC>AD$. Take points $ X$ and $ Y$ on the sides $ AB$ and $ BC$, respectively, so that $ AX\equal{}CD$ and $ AD\equal{}CY$. Let $ M$ be the midpoint of $ XY$. Prove that $ AMC$ is a right angle.

1962 IMO Shortlist, 3

Tags: geometry , 3d geometry , perimeter , vector , cube

Consider the cube $ABCDA'B'C'D'$ ($ABCD$ and $A'B'C'D'$ are the upper and lower bases, repsectively, and edges $AA', BB', CC', DD'$ are parallel). The point $X$ moves at a constant speed along the perimeter of the square $ABCD$ in the direction $ABCDA$, and the point $Y$ moves at the same rate along the perimiter of the square $B'C'CB$ in the direction $B'C'CBB'$. Points $X$ and $Y$ begin their motion at the same instant from the starting positions $A$ and $B'$, respectively. Determine and draw the locus of the midpionts of the segments $XY$.