Found problems: 85335
2007 Hanoi Open Mathematics Competitions, 10
What is the smallest possible value of $x^2+2y^2-x-2y-xy$?
2001 Bundeswettbewerb Mathematik, 2
For each $ n \in \mathbb{N}$ we have two numbers $ p_n, q_n$ with the following property: For exactly $ n$ distinct integer numbers $ x$ the number \[ x^2 \plus{} p_n \cdot x \plus{} q_n\] is the square of a natural number. (Note the definition of natural numbers includes the zero here.)
2014 Online Math Open Problems, 29
Let $ABCD$ be a tetrahedron whose six side lengths are all integers, and let $N$ denote the sum of these side lengths. There exists a point $P$ inside $ABCD$ such that the feet from $P$ onto the faces of the tetrahedron are the orthocenter of $\triangle ABC$, centroid of $\triangle BCD$, circumcenter of $\triangle CDA$, and orthocenter of $\triangle DAB$. If $CD = 3$ and $N < 100{,}000$, determine the maximum possible value of $N$.
[i]Proposed by Sammy Luo and Evan Chen[/i]
2009 AIME Problems, 2
There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that
\[ \frac {z}{z \plus{} n} \equal{} 4i.
\]Find $ n$.
2024 Bulgaria National Olympiad, 6
Given is a triangle $ABC$ and a circle $\omega$ with center $I$ that touches $AB, AC$ and meets $BC$ at $X, Y$. The line through $I$ perpendicular to $BC$ meets the line through $A$ parallel to $BC$ at $Z$. Show that the circumcircles of $\triangle XYZ$ and $\triangle ABC$ are tangent to each other.
2015 USAMTS Problems, 5
Let $a_1,a_2,\dots,a_{100}$ be a sequence of integers. Initially, $a_1=1$, $a_2=-1$ and the remaining numbers are $0$. After every second, we perform the following process on the sequence: for $i=1,2,\dots,99$, replace $a_i$ with $a_i+a_{i+1}$, and replace $a_{100}$ with $a_{100}+a_1$. (All of this is done simultaneously, so each new term is the sum of two terms of the sequence from before any replacements.) Show that for any integer $M$, there is some index $i$ and some time $t$ for which $|a_i|>M$ at time $t$.
2024 Bulgarian Autumn Math Competition, 8.3
Find all positive integers $n$, such that: $$a+b+c \mid a^{2n}+b^{2n}+c^{2n}-n(a^2b^2+b^2c^2+c^2a^2)$$ for all pairwise different positive integers $a,b$ and $c$
2022 VN Math Olympiad For High School Students, Problem 7
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $s$ is a positive integer. Prove that:
a) ${F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})$ and ${F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).$
b) $k({2^s}) = {3.2^{s - 1}}.$
2021 Middle European Mathematical Olympiad, 2
Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor)=\lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$
[list=a]
[*] there exists an n-pretty polynomial;
[*] any $n$-pretty polynomial has a degree of at least $\tfrac{2n+1}{3}$.
[/list]
([i]Remark.[/i] For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)
1997 IMC, 1
Let $f\in C^3(\mathbb{R})$ nonnegative function with $f(0)=f'(0)=0, f''(0)>0$. Define $g(x)$ as follows:
\[ \{ \begin{array}{ccc}g(x)= (\frac{\sqrt{f(x)}}{f'(x)})' &\text{for}& x\not=0 \\ g(x)=0 &\text{for}& x=0\end{array} \]
(a) Show that $g$ is bounded in some neighbourhood of $0$.
(b) Is the above true for $f\in C^2(\mathbb{R})$?
PEN I Problems, 19
Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$, $a-d$ is an integer.
2009 Kazakhstan National Olympiad, 6
Let $P(x)$ be polynomial with integer coefficients.
Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$
1979 AMC 12/AHSME, 8
Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$.
$\textbf{(A) }\frac{\pi}{4}\qquad\textbf{(B) }\frac{3\pi}{4}\qquad\textbf{(C) }\pi\qquad\textbf{(D) }\frac{3\pi}{2}\qquad\textbf{(E) }2\pi$
2011 Mathcenter Contest + Longlist, 10
Let $p,q,r\in R $ with $pqr=1$. Prove that $$\left(\frac{1}{1-p}\right)^2+\left(\frac{1}{1-q}\right)^2+\left(\frac{1}{1-r}\right)^2\ge 1$$
[i](Real Matrik)[/i]
1988 Kurschak Competition, 3
Consider the convex lattice quadrilateral $PQRS$ whose diagonals intersect at $E$. Prove that if $\angle P+\angle Q<180^\circ$, then the $\triangle PQE$ contains inside it or on one of its sides a lattice point other than $P$ and $Q$.
2017 China Second Round Olympiad, 2
Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$.
2015 China Team Selection Test, 5
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$
2002 AMC 12/AHSME, 13
What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1,k_2,\ldots,k_n$ for which \[k_1^2+k_2^2+\ldots+k_n^2=2002?\]
$\textbf{(A) }14\qquad\textbf{(B) }15\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
2021 Polish MO Finals, 5
A convex hexagon $ABCDEF$ is given where $ \measuredangle FAB + \measuredangle BCD + \measuredangle DEF = 360^{\circ}$ and $ \measuredangle AEB = \measuredangle ADB$. Suppose the lines $AB$ and $DE$ are not parallel. Prove that the circumcenters of the triangles $ \triangle AFE, \triangle BCD$ and the intersection of the lines $AB$ and $DE$ are collinear.
2011 Miklós Schweitzer, 10
Let $X_0, \xi_{i, j}, \epsilon_k$ (i, j, k ∈ N) be independent, non-negative integer random variables. Suppose that $\xi_{i, j}$ (i, j ∈ N) have the same distribution, $\epsilon_k$ (k ∈ N) also have the same distribution.
$\mathbb{E}(\xi_{1,1})=1$ , $\mathbb{E}(X_0^l)<\infty$ , $\mathbb{E}(\xi_{1,1}^l)<\infty$ , $\mathbb{E}(\epsilon_1^l)<\infty$ for some $l\in\mathbb{N}$
Consider the random variable $X_n := \epsilon_n + \sum_{j=1}^{X_{n-1}} \xi_{n,j}$ (n ∈ N) , where $\sum_{j=1}^0 \xi_{n,j} :=0$
Introduce the sequence $M_n := X_n-X_{n-1}-\mathbb{E}(\epsilon_n)$ (n ∈ N)
Prove that there is a polynomial P of degree $\leq l/2$ such that $\mathbb{E}(M_n^l) = P_l(n)$ (n ∈ N).
2016 SDMO (Middle School), 2
Let $AB$ be a diameter of a circle and let $C$ be a point on $AB$ with $2\cdot AC=BC$. Let $D$ and $E$ be points on the circle such that $DC\perp AB$ and $DE$ is a second diameter. What is the ratio of the area of $\triangle{DCE}$ to the area of $\triangle{ABD}$?
2022 DIME, 8
Given a parallelogram $ABCD$, let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$, the distance from vertex $B$ to $\mathcal{P}$ is $25$, and the distance from vertex $C$ to $\mathcal{P}$ is $36$. Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$.
[i]Proposed by [b]HrishiP[/b][/i]
1985 Bundeswettbewerb Mathematik, 2
Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.
2008 iTest Tournament of Champions, 1
Find $k$ where $2^k$ is the largest power of $2$ that divides the product \[2008\cdot 2009\cdot 2010\cdots 4014.\]
2006 Hong Kong TST., 3
In triangle ABC, the altitude, angle bisector and median from C divide the angle C into four equal angles. Find angle B.