This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

1963 AMC 12/AHSME, 21
Tags:
The expression $x^2-y^2-z^2+2yz+x+y-z$ has: $\textbf{(A)}\ \text{no linear factor with integer coeficients and integer exponents} \qquad$ $ \textbf{(B)}\ \text{the factor }-x+y+z \qquad$ $ \textbf{(C)}\ \text{the factor }x-y-z+1 \qquad$ $ \textbf{(D)}\ \text{the factor }x+y-z+1 \qquad$ $ \textbf{(E)}\ \text{the factor }x-y+z+1$

2015 Princeton University Math Competition, A5
Tags: number theory
Given that there are $24$ primes between $3$ and $100$, inclusive, what is the number of ordered pairs $(p, a)$ with $p$ prime, $3 \le p < 100$, and $1 \le a < p$ such that the sum \[a+a^2+a^3+\cdots+a^{(p-2)!} \]is not divisible by $p$?

2007 Indonesia TST, 2
Tags: function , algebra
Find all functions $ f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying \[ f(f(x \plus{} y)) \equal{} f(x \plus{} y) \plus{} f(x)f(y) \minus{} xy\] for all real numbers $x$ and $y$.

1959 AMC 12/AHSME, 3
Tags: rhombus , rectangle , trapezoid , quadrilateral , geometry
If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification: $ \textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these} $

IV Soros Olympiad 1997 - 98 (Russia), 11.5
Tags: number theory , logarithm
Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.

2021 Federal Competition For Advanced Students, P2, 3
Tags: number theory , diophantine equation , diophantine
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

1991 Bulgaria National Olympiad, Problem 3
Tags: number theory
Prove that for every prime number $p\ge5$, (a) $p^3$ divides $\binom{2p}p-2$; (b) $p^3$ divides $\binom{kp}p-k$ for every natural number $k$.

2023 Regional Competition For Advanced Students, 4
Tags: number theory , gcd , greatest common divisor , diophantine equation
Determine all pairs $(x, y)$ of positive integers such that for $d = gcd(x, y)$ the equation $$xyd = x + y + d^2$$ holds. [i](Walther Janous)[/i]

1976 AMC 12/AHSME, 29
Tags:
Ann and Barbara were comparing their ages and found that Barbara is as old as Ann was when Barbara was as old as Ann had been when Barbara was half as old as Ann is. If the sum of their present ages is $44$ years, then Ann's age is $\textbf{(A) }22\qquad\textbf{(B) }24\qquad\textbf{(C) }25\qquad\textbf{(D) }26\qquad \textbf{(E) }28$

1973 AMC 12/AHSME, 18
Tags: probability , modular arithmetic , number theory , prime number
If $ p \geq 5$ is a prime number, then $ 24$ divides $ p^2 \minus{} 1$ without remainder $ \textbf{(A)}\ \text{never} \qquad \textbf{(B)}\ \text{sometimes only} \qquad \textbf{(C)}\ \text{always} \qquad$ $ \textbf{(D)}\ \text{only if } p \equal{}5 \qquad \textbf{(E)}\ \text{none of these}$