Olympiad problems

2017 Korea National Olympiad, problem 3

Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.

1998 Bundeswettbewerb Mathematik, 2

Tags: Subsets , combinatorics , Intersection

Prove that there exist $16$ subsets of set $M = \{1,2,...,10000\}$ with the following property: For every $z \in M$ there are eight of these subsets whose intersection is $\{z\}$.

1964 IMO, 5

Tags: geometry , combinatorial geometry , 3D geometry , counting , Intersection , IMO , IMO 1964

Supppose five points in a plane are situated so that no two of the straight lines joining them are parallel, perpendicular, or coincident. From each point perpendiculars are drawn to all the lines joining the other four points. Determine the maxium number of intersections that these perpendiculars can have.

1967 IMO Longlists, 27

Tags: geometry , 3D geometry , polygon , cube , Intersection , IMO Shortlist , IMO Longlist

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

2024 CAPS Match, 1

Tags: number theory , Intersection , Fractions , decimal representation

Determine whether there exist 2024 distinct positive integers satisfying the following: if we consider every possible ratio between two distinct numbers (we include both $a/b$ and $b/a$), we will obtain numbers with finite decimal expansions (after the decimal point) of mutually distinct non-zero lengths.

2017 Korea Junior Math Olympiad, 2

Tags: geometry , circumcircle , Intersection

Let there be a scalene triangle $ABC$, and its incircle hits $BC, CA, AB$ at $D, E, F$. The perpendicular bisector of $BC$ meets the circumcircle of $ABC$ at $P, Q$, where $P$ is on the same side with $A$ with respect to $BC$. Let the line parallel to $AQ$ and passing through $D$ meet $EF$ at $R$. Prove that the intersection between $EF$ and $PQ$ lies on the circumcircle of $BCR$.

2009 IMAC Arhimede, 3

Tags: geometry , convex polygon , Intersection

In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$. Prove that at least one side of the polygon has not intersection points with the lines $MA_i$, $1\le i\le 2n$. (Spain)

2005 Germany Team Selection Test, 3

Tags: geometry , modular arithmetic , combinatorics , circles , Intersection , IMO Shortlist , double counting

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

1961 IMO, 4

Tags: ratio , inequalities , geometry , Triangle , Intersection , IMO , IMO 1961

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

2005 Germany Team Selection Test, 3

Tags: geometry , modular arithmetic , combinatorics , circles , Intersection , IMO Shortlist , double counting

Let ${n}$ and $k$ be positive integers. There are given ${n}$ circles in the plane. Every two of them intersect at two distinct points, and all points of intersection they determine are pairwise distinct (i. e. no three circles have a common point). No three circles have a point in common. Each intersection point must be colored with one of $n$ distinct colors so that each color is used at least once and exactly $k$ distinct colors occur on each circle. Find all values of $n\geq 2$ and $k$ for which such a coloring is possible. [i]Proposed by Horst Sewerin, Germany[/i]

2013 German National Olympiad, 5

Tags: combinatorics , Sets , Intersection

Five people form several commissions to prepare a competition. Here any commission must be nonempty and any two commissions cannot contain the same members. Moreover, any two commissions have at least one common member. There are already $14$ commissions. Prove that at least one additional commission can be formed.

2024 CAPS Match, 4

Tags: geometry , Intersection , quadrilateral , cyclic quadrilateral

Let $ABCD$ be a quadrilateral, such that $AB = BC = CD.$ There are points $X, Y$ on rays $CA, BD,$ respectively, such that $BX = CY.$ Let $P, Q, R, S$ be the midpoints of segments $BX, CY ,$ $XD, YA,$ respectively. Prove that points $P, Q, R, S$ lie on a circle.

1954 Putnam, B4

Tags: Putnam , conics , parabola , Intersection

Given the focus $F$ and the directrix $D$ of a parabola $P$ and a line $L$, describe a euclidean construction for the point or points of intersection of $P$ and $L.$ Be sure to identify the case for which there are no points of intersection.

1969 IMO Longlists, 42

Tags: algebra , combinatorics , Subsets , Intersection , IMO Shortlist , IMO Longlist

$(MON 3)$ Let $A_k (1 \le k \le h)$ be $n-$element sets such that each two of them have a nonempty intersection. Let $A$ be the union of all the sets $A_k,$ and let $B$ be a subset of $A$ such that for each $k (1\le k \le h)$ the intersection of $A_k$ and $B$ consists of exactly two different elements $a_k$ and $b_k$. Find all subsets $X$ of the set $A$ with $r$ elements satisfying the condition that for at least one index $k,$ both elements $a_k$ and $b_k$ belong to $X$.

2010 ISI B.Stat Entrance Exam, 3

Tags: set , set theory , interval , union , Intersection

Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.

1961 IMO Shortlist, 4

Tags: ratio , inequalities , geometry , Triangle , Intersection , IMO , IMO 1961

Consider triangle $P_1P_2P_3$ and a point $p$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers \[ \dfrac{P_1P}{PQ_1}, \dfrac{P_2P}{PQ_2}, \dfrac{P_3P}{PQ_3} \] at least one is $\leq 2$ and at least one is $\geq 2$

1967 IMO Shortlist, 3

Tags: geometry , 3D geometry , polygon , cube , Intersection , IMO Shortlist , IMO Longlist

Which regular polygon can be obtained (and how) by cutting a cube with a plane ?

2004 Germany Team Selection Test, 1

Tags: geometry , circles , Intersection , IMO Shortlist

Let $D_1$, $D_2$, ..., $D_n$ be closed discs in the plane. (A closed disc is the region limited by a circle, taken jointly with this circle.) Suppose that every point in the plane is contained in at most $2003$ discs $D_i$. Prove that there exists a disc $D_k$ which intersects at most $7\cdot 2003 - 1 = 14020$ other discs $D_i$.