Found problems: 85335
2001 SNSB Admission, 5
Find the fundamental group of the topology of $ \text{SL}_2\left(\mathbb{R}\right) $ on $ \mathbb{R}^4. $
2008 Romania National Olympiad, 3
Let $ A$ be a unitary finite ring with $ n$ elements, such that the equation $ x^n\equal{}1$ has a unique solution in $ A$, $ x\equal{}1$. Prove that
a) $ 0$ is the only nilpotent element of $ A$;
b) there exists an integer $ k\geq 2$, such that the equation $ x^k\equal{}x$ has $ n$ solutions in $ A$.
2006 Australia National Olympiad, 2
Let $f$ be a function defined on the positive integers, taking positive integral values, such that
$f(a)f(b) = f(ab)$ for all positive integers $a$ and $b$,
$f(a) < f(b)$ if $a < b$,
$f(3) \geq 7$.
Find the smallest possible value of $f(3)$.
2023 CCA Math Bonanza, L2.4
A hundred people want to take a photo. They can stand in any number of rows from 1 to 100. Let $N$ be the number of possible photos they can take. What is the largest integer $k$ such that $2^k \mid N$?
[i]Lightning 2.4[/i]
1998 All-Russian Olympiad, 3
In scalene $\triangle ABC$, the tangent from the foot of the bisector of $\angle A$ to the incircle of $\triangle ABC$, other than the line $BC$, meets the incircle at point $K_a$. Points $K_b$ and $K_c$ are analogously defined. Prove that the lines connecting $K_a$, $K_b$, $K_c$ with the midpoints of $BC$, $CA$, $AB$, respectively, have a common point on the incircle.
1990 Polish MO Finals, 2
Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that
\[ \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 \]
Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.
2018 AMC 12/AHSME, 5
What is the sum of all possible values of $k$ for which the polynomials $x^2 - 3x + 2$ and $x^2 - 5x + k$ have a root in common?
$
\textbf{(A) }3 \qquad
\textbf{(B) }4 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }10 \qquad
$
2007 Indonesia TST, 1
Call an $n$-gon to be [i]lattice[/i] if its vertices are lattice points. Prove that inside every lattice convex pentagon there exists a lattice point.
2007 Tuymaada Olympiad, 4
Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.
2019 India Regional Mathematical Olympiad, 4
Let $a_1,a_2,\cdots,a_6,a_7$ be seven positive integers. Let $S$ be the set of all numbers of the form $a_i^2+a_j^2$ where $1\leq i<j\leq 7$.
Prove that there exist two elements of $S$ which have the same remainder on dividing by $36$.
1974 AMC 12/AHSME, 30
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $ R$ is the ratio of the lesser part to the greater part, then the value of \[ R^{[R^{(R^2\plus{}R^{\minus{}1})}\plus{}R^{\minus{}1}]}\plus{}R^{\minus{}1}\] is
$ \textbf{(A)}\ 2
\qquad \textbf{(B)}\ 2R
\qquad \textbf{(C)}\ R^{\minus{}1}
\qquad \textbf{(D)}\ 2\plus{}R^{\minus{}1}
\qquad \textbf{(E)}\ 2\plus{}R$
2021 China Second Round, 4
Find the minimum value of $c$ such that for any positive integer $n\ge 4$ and any set $A\subseteq \{1,2,\cdots,n\}$, if $|A| >cn$, there exists a function $f:A\to\{1,-1\}$ satisfying
$$\left| \sum_{a\in A}a\cdot f(a)\right| \le 1.$$
2011 ISI B.Stat Entrance Exam, 1
Let $x_1, x_2, \cdots , x_n$ be positive reals with $x_1+x_2+\cdots+x_n=1$. Then show that
\[\sum_{i=1}^n \frac{x_i}{2-x_i} \ge \frac{n}{2n-1}\]
2008 Princeton University Math Competition, A9
In tetrahedron $ABCD$ with circumradius $2$, $AB = 2$, $CD = \sqrt{7}$, and $\angle ABC = \angle BAD = \frac{\pi}{2}$. Find all possible angles between the planes containing $ABC$ and $ABD$.
1979 Austrian-Polish Competition, 6
A positive integer $n$ and a real number $a$ are given. Find all $n$-tuples $(x_1, ... ,x_n)$ of real numbers that satisfy the system of equations $$\sum_{i=1}^{n} x_i^k= a^k \,\,\,\, for \,\,\,\, k = 1,2, ... ,n$$
1999 USAMTS Problems, 2
The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$. It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$. Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$.
1964 Poland - Second Round, 3
Prove that if three prime numbers form an arithmetic progression whose difference is not divisible by 6, then the smallest of these numbers is $3 $.
2010 Regional Olympiad of Mexico Northeast, 2
Of all the fractions $\frac{x}{y}$ that satisfy $$\frac{41}{2010}<\frac{x}{y}<\frac{1}{49}$$ find the one with the smallest denominator.
2015 HMNT, 9
Consider a $9 \times 9$ grid of squares. Haruki fills each square in this grid with an integer between 1 and 9, inclusive. The grid is called a $\textit{super-sudoku}$ if each of the following three conditions hold:
[list]
[*] Each column in the grid contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once.
[*] Each row in the grid contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once.
[*] Each $3 \times 3$ subsquare in the grid contains each of the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once.
[/list]
How many possible super-sudoku grids are there?
2014 ELMO Shortlist, 12
Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$.
[i]Proposed by David Stoner[/i]
2017 Princeton University Math Competition, A8
Find the minimum value attained by $\sum_{m=1}^{100} \gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.
2011 Sharygin Geometry Olympiad, 5
The touching point of the excircle with the side of a triangle and the base of the altitude to this side are symmetric wrt the base of the corresponding bisector. Prove that this side is equal to one third of the perimeter.
1999 Abels Math Contest (Norwegian MO), 1a
Find a function $f$ such that $f(t^2 +t +1) = t$ for all real $t \ge 0$
2022 Germany Team Selection Test, 3
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2006 China Northern MO, 7
Can we put positive integers $1,2,3, \cdots 64$ into $8 \times 8$ grids such that the sum of the numbers in any $4$ grids that have the form like $T$ ( $3$ on top and $1$ under the middle one on the top, this can be rotate to any direction) can be divided by $5$?