Cool math Olympics - 1960

1. Determine all three-digit numbers `N` having the property that `N` is divisible by `11`. and `N / 11` is equal to the sum of the squares of the digits of `N`.

2. For what values of the variable `x` does the following inequality hold: `(4x^{2})/(1 -\sqrt{1 + 2x})^{2} < 2x + 9`.

3. In a given right triangle `ABC` the hypotenuse `BC` of length `a` is divided into `n` equal parts (`n` an odd integer). Let `\alpha` be the acute angle subtending, from `A` that segment which contains the midpoint of the hypotenuse. Let `h` be the length of the altitude to the hypotenuse of the triangle.

Prove: `tan(\alpha) = (4nh) / ((n^{2} - 1) * a)`.

4. Construct triangle `ABC` given `h_{a}, h_{b}` (the altitudes from `A` and `B`) and `m_{a}`, the median from vertex `A`.

5. Consider the cube `ABCDA^{1}B^{1}C^{1}D^{1}` (with face `ABCD` directly above face `A^{1}B^{1}C^{1}D^{1}`).

(a) Find the locus of the midpoints of segments `XY` where `X` is any point of `AC` and `Y` is any point of `B^{1}D^{1}`.

(b) Find the locus of points `Z` which lie on the segments `XY` of part (a) with `ZY = 2XZ`.

6. Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let `V1` be the volume of the cone and `V2` the volume of the cylinder.

(a) Prove that `V1 != V2`.

(b) Find the smallest number `k` for which `V1 = kV2`, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.

7. An isosceles trapezoid with bases `a` and `c` and altitude `h` is given.

(a) On the axis of symmetry of this trapezoid, find all points `P` such that both legs of the trapezoid subtend right angles at `P`:

(b) Calculate the distance of `P` from either base.

(c) Determine under what conditions such points `P` actually exist. (Discuss various cases that might arise.)

1. Determine all three-digit numbers `N` having the property that `N` is divisible by `11`. and `N / 11` is equal to the sum of the squares of the digits of `N`.

2. For what values of the variable `x` does the following inequality hold: `(4x^{2})/(1 -\sqrt{1 + 2x})^{2} < 2x + 9`.

3. In a given right triangle `ABC` the hypotenuse `BC` of length `a` is divided into `n` equal parts (`n` an odd integer). Let `\alpha` be the acute angle subtending, from `A` that segment which contains the midpoint of the hypotenuse. Let `h` be the length of the altitude to the hypotenuse of the triangle.

Prove: `tan(\alpha) = (4nh) / ((n^{2} - 1) * a)`.

4. Construct triangle `ABC` given `h_{a}, h_{b}` (the altitudes from `A` and `B`) and `m_{a}`, the median from vertex `A`.

5. Consider the cube `ABCDA^{1}B^{1}C^{1}D^{1}` (with face `ABCD` directly above face `A^{1}B^{1}C^{1}D^{1}`).

(a) Find the locus of the midpoints of segments `XY` where `X` is any point of `AC` and `Y` is any point of `B^{1}D^{1}`.

(b) Find the locus of points `Z` which lie on the segments `XY` of part (a) with `ZY = 2XZ`.

6. Consider a cone of revolution with an inscribed sphere tangent to the base of the cone. A cylinder is circumscribed about this sphere so that one of its bases lies in the base of the cone. Let `V1` be the volume of the cone and `V2` the volume of the cylinder.

(a) Prove that `V1 != V2`.

(b) Find the smallest number `k` for which `V1 = kV2`, for this case, construct the angle subtended by a diameter of the base of the cone at the vertex of the cone.

7. An isosceles trapezoid with bases `a` and `c` and altitude `h` is given.

(a) On the axis of symmetry of this trapezoid, find all points `P` such that both legs of the trapezoid subtend right angles at `P`:

(b) Calculate the distance of `P` from either base.

(c) Determine under what conditions such points `P` actually exist. (Discuss various cases that might arise.)

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