Olympiad problems

2012 Traian Lălescu, 1

Let $a,b,c,\alpha,\beta,\gamma \in\mathbb{R}$ such as $a^2+b^2+c^2 \neq 0 \neq \alpha\beta\gamma$ and $24^{\alpha}\neq 3^{\beta} \neq 2012^{\gamma} \neq 24^{\alpha}$. Prove that the equation \[ a \cdot 24^{\alpha x}+b \cdot 3^{\beta x} + c \cdot 2012^{\gamma x}=0 \] has at most two real solutions.

2023 IRN-SGP-TWN Friendly Math Competition, 6

Tags: functional equation , polynomial , polynomial with integer coeffi , algebra , number theory

$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\] (We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.) [i]Proposed by the4seasons.[/i]

2020 AMC 10, 9

Tags:

A single bench section at a school event can hold either $7$ adults or $11$ children. When $N$ bench sections are connected end to end, an equal number of adults and children seated together will occupy all the bench space. What is the least possible positive integer value of $N?$ $\textbf{(A) } 9 \qquad \textbf{(B) } 18 \qquad \textbf{(C) } 27 \qquad \textbf{(D) } 36 \qquad \textbf{(E) } 77$

Russian TST 2017, P1

Tags: number theory

Let's call a number of the form $x^3+y^2$ with natural $x, y$ [i]successful[/i]. Are there infinitely many natural $m$ such that among the numbers from $m + 1$ to $m + 2016^2$ exactly 2017 are successful?

2013 Romania Team Selection Test, 3

Tags: function , induction , inequalities , algebra

Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If $S$ is a finite set of positive integers such that $\sum\limits_{s\in S}\frac{1}{s}$ is an integer, then $\sum\limits_{s\in S}\frac{1}{f\left( s\right) }$ is also an integer.

2006 Tournament of Towns, 1

Tags: combinatorics , combinatorial geometry

Prove that one can always mark $50$ points inside of any convex $100$-gon, so that each its vertix is on a straight line connecting some two marked points. (4)

2014 AMC 12/AHSME, 16

Tags: algebra , polynomial , system of equations

Let $P$ be a cubic polynomial with $P(0) = k, P(1) = 2k,$ and $P(-1) = 3k$. What is $P(2) + P(-2)$? $ \textbf{(A) }0 \qquad\textbf{(B) }k \qquad\textbf{(C) }6k \qquad\textbf{(D) }7k\qquad\textbf{(E) }14k\qquad $

2020 Purple Comet Problems, 23

Tags: trigonometry

There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x}=\frac{12}{11}$$ and $\sin x + \cos x =\frac{\sqrt{m}}{n}$ , where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$.

1983 AMC 12/AHSME, 24

Tags: perimeter , trigonometry , inradius , sfft , geometry

How many non-congruent right triangles are there such that the perimeter in $\text{cm}$ and the area in $\text{cm}^2$ are numerically equal? $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 4 \qquad \text{(E)} \ \text{infinitely many}$

2005 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , trigonometry

Compute \[ \displaystyle\sum_{k=0}^{\infty} \dfrac {4}{(4k)!}. \]