Some Canadian troops are sent (as part of a U.N. peacekeeping force) to a country located on the Earth's equator. At night, when homesickness makes them gaze sleeplessly at the stars, which of the following will be familiar to them (the same at the equator as in Canada):
a. the celestial poles are on the north and south points of the horizon
b. the celestial equator is overhead and passes through the zenith
c. all stars rise and set (none remain in the sky all night long)
d. all-stars are above the horizon exactly half a day
e. none of the above are the same on the equator as in Canada

The earth's rotation is not straight forward as a sphere just rotating. The shape of the earth (better described as geoid; spherical, but a bit flattened at the poles), some gravitational forces on the earth and the existence of seasons mean the earth's rotation is tilted at an angle of 23.5° relative to our orbital plane; the plane of the earth's orbit round the sun.

When the tilted earth is considered, the celestial north and south poles are the north and south poles on the tilted earth's rotational axis.

And the celestial equator is when the equator on the tilted earth, equidistant from the celestial north and south poles is projected into space. A portion of the celestial equator passes through the real equator.

Now, the options one by one,

A) the celestial poles are on the north and south points of the horizon

At the celestial north and south poles, the poles are observable at the north and south points of the horizon.

At the celestial north pole, it is usually located by locating the north star, Polaris, and at the celestial South poles, there are about 4 different methods of locating the poles on the horizon.

But these poles aren't noticeable on the horizon at the equator. The northern one can only be noticed at regions around the celestial north pole.

Hence, the statement A isn't correct.

B) the celestial equator is overhead and passes through the zenith.

Zenith describes the the point in the sky directly above one's head.

At some points on the equator, the celestial equator is truly overhead, but the celestial equator doesn't pass overhead at the north pole, So, this isn't something common to the equator and the north pole.

C) all stars rise and set (none remain in the sky all night long)

Stars remaining in the sky all night long only occurs at the poles (north pole especially). Depending on the seasons, there are times and/or places that have 24 hour sunlight or darkness.

But, this is usually not the case at the equator. At the equator, all stars rise and set all year round. Only small noticeable longer days, shorter nights or longet nights, shorter days are experienced at the equator.

Hence, the statement given isn't true for the north pole and the equator.

D) all-stars are above the horizon exactly half a day.

The paths of the stars are vertical and are cut exactly in half by the horizon at the equator. Each star is up half the time and down half the time.

Unlike at the poles, as explained in the previous statement, they enjoy more varied daytime and night times.

Again, the statement described isn't true for the north pole and the equator.