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Stage 5 · Mixed

10 problems · AIME late · mathsolympiadprep.com

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1. What number does not change when raised to a power? What number retains its absolute value when raised to any positive integer power?

   Number theory Solution and answer checking →

2. Find all solutions to the equation $2017^{x}-2016^{x}=1$.

   Translate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.

   Algebra Solution and answer checking →

3. $\frac{1}{2} n(n+1)$ distinct numbers are randomly arranged in a triangle:

   Let $M_{\mathrm{k}}$ be the maximum number in the $k$-th row (counting from the top), find the probability that $M_{1}<M_{2}<M_{3}<\cdots<M_{\mathrm{n}}$ holds.

   Combinatorics Solution and answer checking →

4. Veronica observes that $81 \cdot 3=243$ and $81 \cdot 4=324$ and wonders how many numbers $m$ there are with $10 \leq m \leq 99$ such that $3 m=A B C$ and $4 m=C A B$, with $A, B$ and $C$ decimal digits (the cases in which one or more of the digits $A, B, C$ are equal to zero are also considered valid).

   1. A1
   2. B2
   3. C3
   4. D4
   5. E6

   Number theory Solution and answer checking →

5. 199 people have registered to participate in a tennis tournament. In the first round, pairs of opponents are selected by lottery. The same process is used to select pairs in the second, third, and all subsequent rounds. After each match, one of the two opponents is eliminated, and whenever the number of participants in the tournament is odd, one of them skips the current round.

   Assume that in each match between two tennis players, a new can of balls is used. How many cans of balls will be needed for the entire tournament?

   Combinatorics Solution and answer checking →

6. Inside square $A B C D$, a point $E$ is chosen so that triangle $D E C$ is equilateral. Find the measure of $\angle A E B$.

   Geometry Solution and answer checking →

7. Given $t$ is a real number. Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$ such that
   $$f(x+t+f(y))=f(f(x))+f(t)+y .$$
   (2014, Croatian Mathematical Olympiad)

   Algebra Solution and answer checking →

8. In a chess tournament, there are an odd number of participants, and each participant plays one game against every other participant. A win earns 1 point, a draw earns 0.5 points, and a loss earns 0 points; it is known that two of the participants scored a total of 8 points, and the average score of the others is an integer. How many participants are there in the tournament?

   Combinatorics Solution and answer checking →

9. In $\triangle A B C$, it is known that $A B=A C, D$ is the midpoint of side $B C$, $B E \perp A C$ at point $E, B E$ intersects $A D$ at point $P$. If $B P=3, P E=1$, then $A E=(\quad)$.

   1. A$\frac{\sqrt{6}}{2}$
   2. B$\sqrt{2}$
   3. C$\sqrt{3}$
   4. D$\sqrt{6}$

   Geometry Solution and answer checking →

10. Six natural numbers are written on the board, such that for any two $a$ and $b$ among them (where $b>a$), $\log _{a} b$ is an integer. What is the smallest value that the maximum of these numbers can take? The answer can be written in the form of a power of a number: $m^{n}$ is denoted as $\mathrm{m}^{\wedge} \mathrm{n}$.

    Number theory Solution and answer checking →

Answer key — Stage 5 · Mixed

Worked solutions for every problem are on the site, one page per problem.

1. $\cos\alpha+i\sin\alpha$ open
2. $1$ open
3. $\frac{2^{n}}{(n+1)!}$ open
4. $C$ open
5. $198$ open
6. $150$ open
7. $f(x)=x$ open
8. $9$ open
9. $\sqrt{2}$ open
10. $4294967296$ open

Problems belong to the competitions that set them and are reproduced from open datasets under their licences; every problem page names its source. Free to copy for classroom use.