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.

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Found problems: 23

1979 IMO Shortlist, 17
Tags: Triangle , angles , geometry , IMO Shortlist , IMO 1979
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that \[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\] Determine the angles of triangle $PQR.$

1979 IMO Longlists, 37
Tags: logarithms , IMO 1979 , IMO Shortlist
Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

1979 IMO Shortlist, 23
Tags: number theory , Perfect Square , IMO Shortlist , IMO 1979
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

1979 IMO Longlists, 66
Tags: number theory , Perfect Square , IMO Shortlist , IMO 1979
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.

1979 IMO Longlists, 25
Tags: number theory , relatively prime , series summation , Divisibility , prime , IMO , IMO 1979
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

1979 IMO, 1
Tags: number theory , relatively prime , series summation , Divisibility , prime , IMO , IMO 1979
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

1979 IMO Longlists, 71
Tags: geometry , parallelogram , conics , circles , Fixed point , IMO , IMO 1979
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

1979 IMO, 3
Tags: combinatorics , recurrence relation , Linear Recurrences , counting , IMO , IMO 1979
Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

1979 IMO, 3
Tags: geometry , parallelogram , conics , circles , Fixed point , IMO , IMO 1979
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

1979 IMO Shortlist, 25
Tags: trigonometry , geometry , 3D geometry , maximization , IMO , IMO 1979
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

1979 IMO Longlists, 68
Tags: trigonometry , geometry , 3D geometry , maximization , IMO , IMO 1979
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

1979 IMO Shortlist, 14
Tags: logarithms , IMO 1979 , IMO Shortlist
Find all bases of logarithms in which a real positive number can be equal to its logarithm or prove that none exist.

1979 IMO Shortlist, 22
Tags: geometry , parallelogram , conics , circles , Fixed point , IMO , IMO 1979
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

1979 IMO Longlists, 80
Tags: algebra , functional equation , IMO Shortlist , IMO 1979
Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

1979 IMO Longlists, 47
Tags: Triangle , angles , geometry , IMO Shortlist , IMO 1979
Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that \[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\] Determine the angles of triangle $PQR.$

1979 IMO Shortlist, 26
Tags: algebra , functional equation , IMO Shortlist , IMO 1979
Prove that the functional equations \[f(x + y) = f(x) + f(y),\] \[ \text{and} \qquad f(x + y + xy) = f(x) + f(y) + f(xy) \quad (x, y \in \mathbb R)\] are equivalent.

1979 IMO Longlists, 28
Tags: combinatorics , recurrence relation , Linear Recurrences , counting , IMO , IMO 1979
Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

1979 IMO Shortlist, 9
Tags: combinatorics , recurrence relation , Linear Recurrences , counting , IMO , IMO 1979
Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: \[ a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}. \]

1979 IMO, 1
Tags: trigonometry , geometry , 3D geometry , maximization , IMO , IMO 1979
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.

1979 IMO, 2
Tags: 3D geometry , prism , combinatorics , graph theory , Coloring , IMO , IMO 1979
We consider a prism which has the upper and inferior basis the pentagons: $A_{1}A_{2}A_{3}A_{4}A_{5}$ and $B_{1}B_{2}B_{3}B_{4}B_{5}$. Each of the sides of the two pentagons and the segments $A_{i}B_{j}$ with $i,j=1,\ldots$,5 is colored in red or blue. In every triangle which has all sides colored there exists one red side and one blue side. Prove that all the 10 sides of the two basis are colored in the same color.

1979 IMO Longlists, 17
Tags: parametric equation , IMO Shortlist , IMO 1979
Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).

1979 IMO Shortlist, 7
Tags: number theory , relatively prime , series summation , Divisibility , prime , IMO , IMO 1979
If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

1979 IMO Shortlist, 6
Tags: parametric equation , IMO Shortlist , IMO 1979
Find the real values of $p$ for which the equation \[\sqrt{2p+ 1 - x^2} +\sqrt{3x + p + 4} = \sqrt{x^2 + 9x+ 3p + 9}\] in $x$ has exactly two real distinct roots.($\sqrt t $ means the positive square root of $t$).