This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 12

1964 IMO, 3
Tags: geometry , perimeter , inradius , area of a triangle , Heron's formula , IMO , IMO 1964
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).

1964 IMO, 6
Tags: geometry , 3D geometry , tetrahedron , analytic geometry , IMO , IMO 1964
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

1964 IMO Shortlist, 2
Tags: inequalities , geometry , trigonometry , triangle inequality , IMO , IMO 1964 , algebra
Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

1964 IMO Shortlist, 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.

1964 IMO Shortlist, 4
Tags: combinatorics , Ramsey Theory , Coloring , graph theory , IMO , IMO 1964 , Extremal combinatorics
Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

1964 IMO, 2
Tags: inequalities , geometry , trigonometry , triangle inequality , IMO , IMO 1964 , algebra
Suppose $a,b,c$ are the sides of a triangle. Prove that \[ a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c) \leq 3abc \]

1964 IMO Shortlist, 1
Tags: number theory proposed , number theory , IMO , IMO 1964
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.

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.

1964 IMO Shortlist, 3
Tags: geometry , perimeter , inradius , area of a triangle , Heron's formula , IMO , IMO 1964
A circle is inscribed in a triangle $ABC$ with sides $a,b,c$. Tangents to the circle parallel to the sides of the triangle are contructe. Each of these tangents cuts off a triagnle from $\triangle ABC$. In each of these triangles, a circle is inscribed. Find the sum of the areas of all four inscribed circles (in terms of $a,b,c$).

1964 IMO, 4
Tags: combinatorics , Ramsey Theory , Coloring , graph theory , IMO , IMO 1964 , Extremal combinatorics
Seventeen people correspond by mail with one another-each one with all the rest. In their letters only three different topics are discussed. each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

1964 IMO Shortlist, 6
Tags: geometry , 3D geometry , tetrahedron , analytic geometry , IMO , IMO 1964
In tetrahedron $ABCD$, vertex $D$ is connected with $D_0$, the centrod if $\triangle ABC$. Line parallel to $DD_0$ are drawn through $A,B$ and $C$. These lines intersect the planes $BCD, CAD$ and $ABD$ in points $A_2, B_1,$ and $C_1$, respectively. Prove that the volume of $ABCD$ is one third the volume of $A_1B_1C_1D_0$. Is the result if point $D_o$ is selected anywhere within $\triangle ABC$?

1964 IMO, 1
Tags: number theory proposed , number theory , IMO , IMO 1964
(a) Find all positive integers $ n$ for which $ 2^n\minus{}1$ is divisible by $ 7$. (b) Prove that there is no positive integer $ n$ for which $ 2^n\plus{}1$ is divisible by $ 7$.