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: 10

1963 IMO, 6
Tags: combinatorics , permutations , IMO , IMO 1963
Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

1963 IMO Shortlist, 4
Tags: parameterization , algebra , system of equations , IMO , IMO 1963
Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

1963 IMO Shortlist, 3
Tags: inequalities , geometry , polygon , angles , geometric inequality , IMO , IMO 1963
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

1963 IMO Shortlist, 2
Tags: geometry , 3D geometry , sphere , Locus , Locus problems , IMO , IMO 1963
Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

1963 IMO, 3
Tags: inequalities , geometry , polygon , angles , geometric inequality , IMO , IMO 1963
In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

1963 IMO Shortlist, 6
Tags: combinatorics , permutations , IMO , IMO 1963
Five students $ A, B, C, D, E$ took part in a contest. One prediction was that the contestants would finish in the order $ ABCDE$. This prediction was very poor. In fact, no contestant finished in the position predicted, and no two contestants predicted to finish consecutively actually did so. A second prediction had the contestants finishing in the order $ DAECB$. This prediction was better. Exactly two of the contestants finished in the places predicted, and two disjoint pairs of students predicted to finish consecutively actually did so. Determine the order in which the contestants finished.

1963 IMO Shortlist, 1
Tags: algebra , equation , parametric equation , IMO , IMO 1963
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.

1963 IMO, 2
Tags: geometry , 3D geometry , sphere , Locus , Locus problems , IMO , IMO 1963
Point $A$ and segment $BC$ are given. Determine the locus of points in space which are vertices of right angles with one side passing through $A$, and the other side intersecting segment $BC$.

1963 IMO, 4
Tags: parameterization , algebra , system of equations , IMO , IMO 1963
Find all solutions $x_1, x_2, x_3, x_4, x_5$ of the system \[ x_5+x_2=yx_1 \] \[ x_1+x_3=yx_2 \] \[ x_2+x_4=yx_3 \] \[ x_3+x_5=yx_4 \] \[ x_4+x_1=yx_5 \] where $y$ is a parameter.

1963 IMO, 1
Tags: algebra , equation , parametric equation , IMO , IMO 1963
Find all real roots of the equation \[ \sqrt{x^2-p}+2\sqrt{x^2-1}=x \] where $p$ is a real parameter.