Found problems: 1132
2002 Tournament Of Towns, 1
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2014 Harvard-MIT Mathematics Tournament, 12
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
2005 Vietnam Team Selection Test, 2
Let $p\in \mathbb P,p>3$. Calcute:
a)$S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$ if $ p\equiv 1 \mod 4$
b) $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$ if $p\equiv 1 \mod 8$
2008 Harvard-MIT Mathematics Tournament, 4
([b]4[/b]) Let $ a$, $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$. Determine the pair $ (a,b)$.
2005 Harvard-MIT Mathematics Tournament, 6
A triangular piece of paper of area $1$ is folded along a line parallel to one of the sides and pressed flat. What is the minimum possible area of the resulting figure?
2009 Indonesia MO, 1
Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that
\[ 4n^6 \plus{} n^3 \plus{} 5\]
is divisible by $ 7$.
2012 Dutch BxMO/EGMO TST, 1
Do there exist quadratic polynomials $P(x)$ and $Q(x)$ with real coeffcients such that the polynomial $P(Q(x))$ has precisely the zeros $x = 2, x = 3, x =5$ and $x = 7$?
1985 AIME Problems, 4
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
pair point=( 0.5 , 0.5 );
//label("$A$", A, dir(point--A));
//label("$B$", B, dir(point--B));
//label("$C$", C, dir(point--C));
//label("$D$", D, dir(point--D));
label("$1/n$", (11/12,1), N, fontsize(9));[/asy]
2007 Harvard-MIT Mathematics Tournament, 5
A convex quadrilateral is determined by the points of intersection of the curves $x^4+y^4=100$ and $xy=4$; determine its area.
2010 AIME Problems, 10
Find the number of second-degree polynomials $ f(x)$ with integer coefficients and integer zeros for which $ f(0)\equal{}2010$.
1975 AMC 12/AHSME, 22
If $ p$ and $ q$ are primes and $ x^2 \minus{} px \plus{} q \equal{} 0$ has distinct positive integral roots, then which of the following statements are true?
$ \text{I. The difference of the roots is odd.}$
$ \text{II. At least one root is prime.}$
$ \text{III. } p^2 \minus{} q \text{ is prime.}$
$ \text{IV. } p \plus{} q \text{ is prime.}$
$ \textbf{(A)}\ \text{I only} \qquad
\textbf{(B)}\ \text{II only} \qquad
\textbf{(C)}\ \text{II and III only} \qquad$
$ \textbf{(D)}\ \text{I, II and IV only} \qquad
\textbf{(E)}\ \text{All are true.}$
2007 Irish Math Olympiad, 5
Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)\equal{}x^2\plus{}ax\plus{}b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)\equal{}\frac{g(x)}{h(x)}$ for all real numbers $ x$.
2010 AIME Problems, 9
Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2022 CIIM, 1
Given the function $f(x) = x^2$, the sector of $f$ from $a$ to $b$ is defined as the limited region between the
graph of $y = f(x)$ and the straight line segment that joins the points $(a, f(a))$ and $(b, f(b))$. Define the
increasing sequence $x_0$, $x_1, \cdots$ with $x_0 = 0$ and $x_1 = 1$, such that the area of the sector of $f$ from $x_n$ to $x_{n+1}$ is constant for $n \geq 0$. Determine the value of $x_n$ in function of $n$.
2008 IMO Shortlist, 6
Prove that there are infinitely many positive integers $ n$ such that $ n^{2} \plus{} 1$ has a prime divisor greater than $ 2n \plus{} \sqrt {2n}$.
[i]Author: Kestutis Cesnavicius, Lithuania[/i]
2019 Philippine TST, 3
Determine all ordered triples $(a, b, c)$ of real numbers such that whenever a function $f : \mathbb{R} \to \mathbb{R}$ satisfies $$|f(x) - f(y)| \le a(x - y)^2 + b(x - y) + c$$ for all real numbers $x$ and $y$, then $f$ must be a constant function.
1998 National Olympiad First Round, 16
If $ x^{2} \plus{}y^{2} \plus{}z\equal{}15$, $ x\plus{}y\plus{}z^{2} \equal{}27$ and $ xy\plus{}yz\plus{}zx\equal{}7$, then
$\textbf{(A)}\ 3\le \left|x\plus{}y\plus{}z\right|\le 4 \\ \textbf{(B)}\ 5\le \left|x\plus{}y\plus{}z\right|\le 6 \\ \textbf{(C)}\ 7\le \left|x\plus{}y\plus{}z\right|\le 8 \\ \textbf{(D)}\ 9\le \left|x\plus{}y\plus{}z\right|\le 10 \\ \textbf{(E)}\ \text{None}$
2012 Bosnia Herzegovina Team Selection Test, 3
Prove that for all odd prime numbers $p$ there exist a natural number $m<p$ and integers $x_1, x_2, x_3$ such that:
\[mp=x_1^2+x_2^2+x_3^2.\]
2013 IFYM, Sozopol, 2
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]
1999 Brazil Team Selection Test, Problem 5
(a) If $m, n$ are positive integers such that $2^n-1$ divides $m^2 + 9$, prove
that $n$ is a power of $2$;
(b) If $n$ is a power of $2$, prove that there exists a positive integer $m$ such
that $2^n-1$ divides $m^2 + 9$.
2011 All-Russian Olympiad, 1
A quadratic trinomial $P(x)$ with the $x^2$ coefficient of one is such, that $P(x)$ and $P(P(P(x)))$ share a root. Prove that $P(0)*P(1)=0$.
2001 Pan African, 3
Let $S_1$ be a semicircle with centre $O$ and diameter $AB$.A circle $C_1$ with centre $P$ is drawn, tangent to $S_1$, and tangent to $AB$ at $O$. A semicircle $S_2$ is drawn, with centre $Q$ on $AB$, tangent to $S_1$ and to $C_1$. A circle $C_2$ with centre $R$ is drawn, internally tangent to $S_1$ and externally tangent to $S_2$ and $C_1$. Prove that $OPRQ$ is a rectangle.
2012 Stanford Mathematics Tournament, 1
Compute the minimum possible value of
$(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2$
For real values $x$
1986 USAMO, 3
What is the smallest integer $n$, greater than one, for which the root-mean-square of the first $n$ positive integers is an integer?
$\mathbf{Note.}$ The root-mean-square of $n$ numbers $a_1, a_2, \cdots, a_n$ is defined to be
\[\left[\frac{a_1^2 + a_2^2 + \cdots + a_n^2}n\right]^{1/2}\]
2012 IMO Shortlist, N1
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.
[i]Proposed by Warut Suksompong, Thailand[/i]