Found problems: 85335
2017 ASDAN Math Tournament, 10
Triangle $ABC$ is inscribed in circle $\gamma_1$ with radius $r_1$. Let $\gamma_2$ (with radius $r_2$) be the circle internally tangent to $\gamma_1$ at $A$ and tangent to $BC$ at $D$. Let $I$ be the incenter of $ABC$, and $P$ and $Q$ be the intersection of $\gamma_2$ with $AB$ and $AC$ respectively. Given that $P$, $I$, and $Q$ are collinear, $AI=25$, and the circumradius of triangle $BIC$ is $24$, compute the ratio of the radii $\tfrac{r_2}{r_1}$.
1982 Miklós Schweitzer, 6
For every positive $ \alpha$, natural number $ n$, and at most $ \alpha n$ points $ x_i$, construct a trigonometric polynomial $ P(x)$ of degree at most $ n$ for which \[ P(x_i) \leq 1, \; \int_0^{2 \pi} P(x)dx=0,\ \; \textrm{and}\ \; \max P(x) > cn\ ,\] where the constant $ c$ depends only on $ \alpha$.
[i]G. Halasz[/i]
2013 ISI Entrance Examination, 5
Let $AD$ be a diameter of a circle of radius $r,$ and let $B,C$ be points on the circle such that $AB=BC=\frac r2$ and $A\neq C.$ Find the ratio $\frac{CD}{r}.$
2012 Czech And Slovak Olympiad IIIA, 1
Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime
1990 IMO Longlists, 94
Given integer $n > 1$ and real number $t \geq 1$. $P$ is a parallelogram with four vertices $(0, 0), (0, t), (tF_{2n+1}, tF_{2n}), (tF_{2n+1}, tF_{2n} + t)$. Here, ${F_n}$ is the $n$-th term of Fibonacci sequence defined by $F_0 = 0, F_1 = 1$ and $F_{m+1} = F_m + F_{m-1}$. Let $L$ be the number of integral points (whose coordinates are integers) interior to $P$, and $M$ be the area of $P$, which is $t^2F_{2n+1}.$
[b][i]i)[/i][/b] Prove that for any integral point $(a, b)$, there exists a unique pair of integers $(j, k)$ such that$ j(F_{n+1}, F_n) + k(F_n, F_{n-1}) = (a, b)$, that is,$ jF_{n+1} + kF_n = a$ and $jF_n + kF_{n-1} = b.$
[i][b]ii)[/b][/i] Using [i][b]i)[/b][/i] or not, prove that $|\sqrt L-\sqrt M| \leq \sqrt 2.$
2020 AIME Problems, 1
In $\triangle ABC$ with $AB=AC$, point $D$ lies strictly between $A$ and $C$ on side $\overline{AC}$, and point $E$ lies strictly between $A$ and $B$ on side $\overline{AB}$ such that $AE=ED=DB=BC$. The degree measure of $\angle ABC$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2001 Baltic Way, 11
The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have
\[f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) \]
Determine all possible values of $f(2001)$.
2018 ELMO Shortlist, 2
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
2011 National Olympiad First Round, 3
How many positive integer $n$ are there satisfying the inequality $1+\sqrt{n^2-9n+20} > \sqrt{n^2-7n+12}$ ?
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{None}$
2022/2023 Tournament of Towns, P3
Consider two concentric circles $\Omega$ and $\omega$. Chord $AD$ of the circle $\Omega$ is tangent to $\omega$. Inside the minor disk segment $AD$ of $\Omega$, an arbitrary point $P{}$ is selected. The tangent lines drawn from the point $P{}$ to the circle $\omega$ intersect the major arc $AD$ of the circle $\Omega$ at points $B{}$ and $C{}$. The line segments $BD$ and $AC$ intersect at the point $Q{}$. Prove that the line segment $PQ$ passes through the midpoint of line segment $AD$.
[i]Note.[/i] A circle together with its interior is called a disk, and a chord $XY$ of the circle divides the disk into disk segments, a minor disk segment $XY$ (the one of smaller area) and a major disk segment $XY$.
2021 Austrian MO National Competition, 6
Let $p$ be an odd prime number and $M$ a set derived from $\frac{p^2 + 1}{2}$ square numbers.
Investigate whether $p$ elements can be selected from this set whose arithmetic mean is an integer.
(Walther Janous)
2018 Korea Junior Math Olympiad, 8
For every set $S$ with $n(\ge3)$ distinct integers, show that there exists a function $f:\{1,2,\dots,n\}\rightarrow S$ satisfying the following two conditions.
(i) $\{ f(1),f(2),\dots,f(n)\} = S$
(ii) $2f(j)\neq f(i)+f(k)$ for all $1\le i<j<k\le n$.
2017 Dutch IMO TST, 2
let $a_1,a_2,...a_n$ a sequence of real numbers such that $a_1+....+a_n=0$.
define $b_i=a_1+a_2+....a_i$ for all $1 \leq i \leq n$ .suppose $b_i(a_{j+1}-a_{i+1}) \geq 0$ for all $1 \leq i \leq j \leq n-1$.
Show that $$\max_{1 \leq l \leq n} |a_l| \geq \max_{1 \leq m \leq n} |b_m|$$
2017 Mathematical Talent Reward Programme, SAQ: P 4
An irreducible polynomial is a not-constant polynomial that cannot be factored into product of two non-constant polynomials. Consider the following statements :-
[b]Statement 1 :[/b] $p(x)$ be any monic irreducible polynomial with integer coefficients and degree $\geq 4$. Then $p(n)$ is a prime for at least one natural number $n$
[b]Statement 2 :[/b] $n^2+1$ is prime for infinitely many values of natural number $n$
Show that if [b]Statement 1[/b] is true then [b]Statement 2[/b] is also true
2022 MIG, 15
There exists a fraction $x$ that satisfies $ \sqrt{x^2+5} - x = \tfrac{1}{3}$. What is the sum of the numerator and denominator of this fraction?
$\textbf{(A) }8\qquad\textbf{(B) }21\qquad\textbf{(C) }25\qquad\textbf{(D) }32\qquad\textbf{(E) }34$
2004 Thailand Mathematical Olympiad, 4
Find all real solutions $x$ to the equation $$x =\sqrt{x -\frac{1}{x}} +\sqrt{1 -\frac{1}{x}}$$
MathLinks Contest 4th, 4.3
Given is a graph $G$. An [i]empty [/i] subgraph of $G$ is a subgraph of $G$ with no edges between its vertices. An edge of $G$ is called [i]important [/i] if and only if the removal of this edge will increase the size of the maximal empty subgraph.
Suppose that two important edges in $G$ have a common endpoint. Prove there exists a cycle of odd length in $G$.
2001 Grosman Memorial Mathematical Olympiad, 6
(a) Find a pair of integers (x,y) such that $15x^2 +y^2 = 2^{2000}$
(b) Does there exist a pair of integers $(x,y)$ such that $15x^2 + y^2 = 2^{2000}$ and $x$ is odd?
2021 Peru Cono Sur TST., P1
Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers.
Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.
2014 Contests, 3
There are $ n$ students; each student knows exactly $d $ girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $ of integers .
2015 Greece National Olympiad, 2
Let $P(x)=ax^3+(b-a)x^2-(c+b)x+c$ and $Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c$ be polynomials of $x$ with $a,b,c$ non-zero real numbers and $b>0$.If $P(x)$ has three distinct real roots $x_0,x_1,x_2$ which are also roots of $Q(x)$ then:
A)Prove that $abc>28$,
B)If $a,b,c$ are non-zero integers with $b>0$,find all their possible values.
2023 South Africa National Olympiad, 3
Consider $2$ positive integers $a,b$ such that $a+2b=2020$.
(a) Determine the largest possible value of the greatest common divisor of $a$ and $b$.
(b) Determine the smallest possible value of the least common multiple of $a$ and $b$.
2002 AMC 12/AHSME, 4
Let $ n$ be a positive integer such that $ \tfrac{1}{2}\plus{}\tfrac{1}{3}\plus{}\tfrac{1}{7}\plus{}\tfrac{1}{n}$ is an integer. Which of the following statements is [b]not[/b] true?
$ \textbf{(A)}\ 2\text{ divides }n \qquad
\textbf{(B)}\ 3\text{ divides }n \qquad
\textbf{(C)}\ 6\text{ divides }n \qquad
\textbf{(D)}\ 7\text{ divides }n \\
\textbf{(E)}\ n>84$
2016 ASDAN Math Tournament, 7
What is
$$\sum_{n=1996}^{2016}\lfloor\sqrt{n}\rfloor?$$
Russian TST 2015, P1
Let $n>4$ be a natural number. Prove that \[\sum_{k=2}^n\sqrt[k]{\frac{k}{k-1}}<n.\]