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Is there a triangle that can be cut into $7$ congruent triangles? (no)
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Is there a triangle that can be cut into $7$ congruent triangles?

  • Is there a dense subset of a plane having only rational distances between its points?
  • What is the least $V$ such that any convex body of unit volume can be fit into a tetrahedron of volume $V$? It is known that $V \ge 9/2$ and conjectured that $V = 9/2$.
  • What is the least $S$ (if any) such that any subset of a plane of area $S$ contains $3$ vertices of a triangle of unit area?
  • Is there an upper bound of quotients in the continued fraction representation of $\sqrt[3]{2}=[ 1; 3, 1, 5, 1, 1, \dots]$?
  • Is Hilbert's tenth problem for Diophantine equations of power $3$ decidable?
  • Is Hilbert's tenth problem for Diophantine equations in rational numbers decidable?

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    • Is there a triangle that can be cut into $7$ congruent triangles?
    • Is there a dense subset of a plane having only rational distances between its points?
    • What is the least $V$ such that any convex body of unit volume can be fit into a tetrahedron of volume $V$? It is known that $V \ge 9/2$ and conjectured that $V = 9/2$.
    • What is the least $S$ (if any) such that any subset of a plane of area $S$ contains $3$ vertices of a triangle of unit area?
    • Is there an upper bound of quotients in the continued fraction representation of $\sqrt[3]{2}=[ 1; 3, 1, 5, 1, 1, \dots]$?
    • Is Hilbert's tenth problem for Diophantine equations of power $3$ decidable?
    • Is Hilbert's tenth problem for Diophantine equations in rational numbers decidable?
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