Found problems: 1132
1993 Hungary-Israel Binational, 1
Find all pairs of coprime natural numbers $a$ and $b$ such that the fraction $\frac{a}{b}$ is written in the decimal system as $b.a.$
2018 CMIMC Individual Finals, 3
Determine the number of integers $a$ with $1\leq a\leq 1007$ and the property that both $a$ and $a+1$ are quadratic residues mod $1009$.
1959 AMC 12/AHSME, 27
Which one of the following is [i] not [/i] true for the equation \[ix^2-x+2i=0,\] where $i=\sqrt{-1}$?
$ \textbf{(A)}\ \text{The sum of the roots is 2} \qquad$
$\textbf{(B)}\ \text{The discriminant is 9}\qquad$
$\textbf{(C)}\ \text{The roots are imaginary}\qquad$
$\textbf{(D)}\ \text{The roots can be found using the quadratic formula}\qquad$
$\textbf{(E)}\ \text{The roots can be found by factoring, using imaginary numbers} $
2002 All-Russian Olympiad, 1
The polynomials $P$, $Q$, $R$ with real coefficients, one of which is degree $2$ and two of degree $3$, satisfy the equality $P^2+Q^2=R^2$. Prove that one of the polynomials of degree $3$ has three real roots.
1991 AIME Problems, 14
A hexagon is inscribed in a circle. Five of the sides have length 81 and the sixth, denoted by $\overline{AB}$, has length 31. Find the sum of the lengths of the three diagonals that can be drawn from $A$.
Cono Sur Shortlist - geometry, 2012.G6.6
6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.
2014 Greece Team Selection Test, 1
Let $(x_{n}) \ n\geq 1$ be a sequence of real numbers with $x_{1}=1$ satisfying $2x_{n+1}=3x_{n}+\sqrt{5x_{n}^{2}-4}$
a) Prove that the sequence consists only of natural numbers.
b) Check if there are terms of the sequence divisible by $2011$.
2024 Belarus Team Selection Test, 4.2
Let $f(x)=x^2+bx+c$, where $b,c \in \mathbb{R}$ and $b>0$
Do there exist disjoint sets $A$ and $B$, whose union is $[0,1]$ and $f(A)=B$, where $f(X)=\{f(x), x \in X\}$
[i]D. Zmiaikou[/i]
1999 Romania Team Selection Test, 16
Let $X$ be a set with $n$ elements, and let $A_{1}$, $A_{2}$, ..., $A_{m}$ be subsets of $X$ such that:
1) $|A_{i}|=3$ for every $i\in\left\{1,2,...,m\right\}$;
2) $|A_{i}\cap A_{j}|\leq 1$ for all $i,j\in\left\{1,2,...,m\right\}$ such that $i \neq j$.
Prove that there exists a subset $A$ of $X$ such that $A$ has at least $\left[\sqrt{2n}\right]$ elements, and for every $i\in\left\{1,2,...,m\right\}$, the set $A$ does not contain $A_{i}$.
[i]Alternative formulation.[/i] Let $X$ be a finite set with $n$ elements and $A_{1},A_{2},\ldots, A_{m}$ be three-elements subsets of $X$, such that $|A_{i}\cap A_{j}|\leq 1$, for every $i\neq j$. Prove that there exists $A\subseteq X$ with $|A|\geq \lfloor \sqrt{2n}\rfloor$, such that none of $A_{i}$'s is a subset of $A$.
2012 Finnish National High School Mathematics Competition, 1
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
2013 Online Math Open Problems, 28
Let $n$ denote the product of the first $2013$ primes. Find the sum of all primes $p$ with $20 \le p \le 150$ such that
(i) $\frac{p+1}{2}$ is even but is not a power of $2$, and
(ii) there exist pairwise distinct positive integers $a,b,c$ for which \[ a^n(a-b)(a-c) + b^n(b-c)(b-a) + c^n(c-a)(c-b) \] is divisible by $p$ but not $p^2$.
[i]Proposed by Evan Chen[/i]
2007 Junior Balkan MO, 1
Let $a$ be positive real number such that $a^{3}=6(a+1)$. Prove that the equation $x^{2}+ax+a^{2}-6=0$ has no real solution.
2009 Croatia Team Selection Test, 4
Prove that there are infinite many positive integers $ n$ such that
$ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.
2009 Vietnam Team Selection Test, 1
Let $ a,b,c$ be positive numbers.Find $ k$ such that:
$ (k \plus{} \frac {a}{b \plus{} c})(k \plus{} \frac {b}{c \plus{} a})(k \plus{} \frac {c}{a \plus{} b}) \ge (k \plus{} \frac {1}{2})^3$
1970 Miklós Schweitzer, 4
If $ c$ is a positive integer and $ p$ is an odd prime, what is the smallest residue (in absolute value) of \[ \sum_{n=0}^{\frac{p-1}{2}} \binom{2n}{n}c^n \;(\textrm{mod}\;p\ ) \ ?\]
J. Suranyi
2008 Serbia National Math Olympiad, 5
The sequence $ (a_n)_{n\ge 1}$ is defined by $ a_1 \equal{} 3$, $ a_2 \equal{} 11$ and $ a_n \equal{} 4a_{n\minus{}1}\minus{}a_{n\minus{}2}$, for $ n \ge 3$. Prove that each term of this sequence is of the form $ a^2 \plus{} 2b^2$ for some natural numbers $ a$ and $ b$.
1986 IMO Shortlist, 5
Let $d$ be any positive integer not equal to $2, 5$ or $13$. Show that one can find distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square.
1995 Flanders Math Olympiad, 2
How many values of $x\in\left[ 1,3 \right]$ are there, for which $x^2$ has the same decimal part as $x$?
2000 Moldova National Olympiad, Problem 1
Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a
$\frac a2x^2+bx+c=0$ between $r$ and $s$.
2007 German National Olympiad, 6
For two real numbers a,b the equation: $x^{4}-ax^{3}+6x^{2}-bx+1=0$ has four solutions (not necessarily distinct). Prove that $a^{2}+b^{2}\ge{32}$
2009 Baltic Way, 7
Suppose that for a prime number $p$ and integers $a,b,c$ the following holds:
\[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\]
Prove that $p\mid a,b,c$.
1990 AIME Problems, 4
Find the positive solution to \[ \frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0 \]
1999 India Regional Mathematical Olympiad, 6
Find all solutions in integers $m,n$ of the equation \[ (m-n)^2 = \frac{4mn}{ m+n-1}. \]
1981 AMC 12/AHSME, 29
If $ a > 1$, then the sum of the real solutions of \[\sqrt{a \minus{} \sqrt{a \plus{} x}} \equal{} x\] is equal to
$ \textbf{(A)}\ \sqrt{a} \minus{} 1\qquad
\textbf{(B)}\ \frac{\sqrt{a} \minus{} 1}{2}\qquad
\textbf{(C)}\ \sqrt{a \minus{} 1}\qquad
\textbf{(D)}\ \frac{\sqrt{a \minus{} 1}}{2}\qquad
\textbf{(E)}\ \frac{\sqrt{4a \minus{} 3} \minus{} 1}{2}$
2013 AIME Problems, 8
A hexagon that is inscribed in a circle has side lengths $22$, $22$, $20$, $22$, $22$, and $20$ in that order. The radius of the circle can be written as $p+\sqrt{q}$, where $p$ and $q$ are positive integers. Find $p+q$.