I think this has to be Tunnel in the Sky; the bit about a pentagon of stars is so spot on:
"I knew the Southern Cross before I could read. These aren't our stars, Rod; I know. There is a pentagon of bright stars above where the sun sets; there is nothing like that to be seen from Earth. And besides, anybody would recognise Luna, if it was there."
Posted notice of the upcoming test:
The bulletin board outside lecture hall 1712-A of Patrick Henry High School showed a flashing red light. Rod Walker pushed his way into a knot of students and tried to see what the special notice had to say. He received an elbow in the stomach, accompanied by: "Hey! Quit shoving!"
"Sorry. Take it easy, Jimmy." Rod locked the elbow in a bone breaker but put no pressure on, craned his neck to look over Jimmy Throxton's head. "What's on the board?"
[...]
Twenty-Four Hour Notice is hereby given of final examination in Solo Survival. Students will present themselves for physical check at 0900 Saturday in the dispensary of Templeton Gate and will start passing through the gate at 1000, using three-minute intervals by lot.
The bit about the chlorine-breathers:
While he watched, the red light flickered out and a blue light replaced it; the door slowly opened and a traveling capsule for a chlorine-breather crawled out.
Rod's sister Helen gives him the blade she used in her test:
She retrieved both knives, stood balancing them, one on each hand. She flipped her own so that the grip was toward Rod. "This is my pet, 'Lady Macbeth.' I carried her on my own solo, Buddy. I want you to carry her on yours."
Chased up a tree by a predator:
Having picked his route, he straightened up and at a fast dogtrot headed for the nearest tree. He went past it, cut left for another tree, passed it and cut right toward the tree he wanted. He was about fifteen meters from it when something charged him.
He closed the gap with instantaneous apportation which would have done credit to a Ramsbotham hyperfold. He reached the first branch, ten meters above ground, in what amounted to levitation. From there on he climbed more conventionally, digging the spurs into the tree's smooth bark and setting his feet more comfortably on branches when they began to be close enough together to form a ladder.
About twenty meters above ground he stopped and looked down. The branches interfered and it was darker under the trees than it had been out in the open; nevertheless he could see, prowling around the tree, the denizen that had favored him with attention.
Rod tried to get a better view, but the light was failing rapidly. But it looked like... well, if he had not been certain that he was on some uncolonized planet 'way out behind and beyond, he would have said that it was a lion.
Except that it looked eight times as big as any lion ought to look.
Jack (not Rod) found the deceased classmate, who had previously stolen Rod's stuff:
"Do you want to hear this? I ran across this hombre in the bush, just as the jackals were getting to him. I don't know what got him—stobor, maybe; he was pretty well clawed and half eaten. He wasn't one of my class, for his face wasn't marked and I could tell. He was carrying a Thunderbolt and—"
"Wait a minute. A Thunderbolt gun?"
"I said so, didn't I? I guess he tried to use it and had no luck. Anyhow, I took what I could use—this knife and a couple of other things; I'll show you. I left the Thunderbolt; the power pack was exhausted and it was junk."
Linking from the Moon to Johannesburg:
"It's a good one. What the Deacon forgot is that any environment is as unknown as any other if you don't have the slightest idea where you are. So they rigged it so that we could not know. First they shot us to Luna; the Moon gates are always open and that doesn't cost anything extra. Of course that made us think we were in for a long jump. Besides, it confused us; we wouldn't know we were being dumped back into the gravity field we had left- for that was what they did next; they shoved us back on Earth. Where? Africa, I'd say. I think they used the Luna Link to jump us to Witwatersrand Gate outside Johannesburg and there they were all set with a matched-in temporary link to drop us into the bush. Tshaka Memorial Park or some other primitive preserve, on a guess. Everything matches. A wide variety of antelope-type game, carnivores to feed on them—I've seen a couple of lions and—"
A new star in the pentagon:
Framed by the ledge above him and by tree tops across the stream was a pattern of six stars, a lopsided pentagon with a star in its center. The six stars were as bright and unmistakable as the seven stars of Earth's Big Dipper... nor did it take a degree in astrography to know that this constellation had never been seen from Terra.
Rod stared while the hard convictions he had formed fell in ruins. He felt lost and alone. The trees across the way seemed frightening. He turned to Jack, his cocky sophistication gone. "You've convinced me," he said dully. "What do we do now?"
Jack did not answer.
"Well?" Rod insisted. "No good standing here."
"Rod," Jack answered, "that star in the middle of the Pentagon-it wasn't there before."
"Huh? You probably don't remember."
"No, no, I'm sure! Rod, you know what? We're seeing a nova."
