Found problems: 85335
2000 All-Russian Olympiad, 3
A convex pentagon $ABCDE$ is given in the coordinate plane with all vertices in lattice points. Prove that there must be at least one lattice point in the pentagon determined by the diagonals $AC$, $BD$, $CE$, $DA$, $EB$ or on its boundary.
1966 IMO Longlists, 26
Prove the inequality
[b]a.)[/b] $
\left( a_{1}+a_{2}+...+a_{k}\right) ^{2}\leq k\left(
a_{1}^{2}+a_{2}^{2}+...+a_{k}^{2}\right) , $
where $k\geq 1$ is a natural number and $a_{1},$ $a_{2},$ $...,$ $a_{k}$ are arbitrary real numbers.
[b]b.)[/b] Using the inequality (1), show that if the real numbers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ satisfy the inequality
\[
a_{1}+a_{2}+...+a_{n}\geq \sqrt{\left( n-1\right) \left(
a_{1}^{2}+a_{2}^{2}+...+a_{n}^{2}\right) },
\]
then all of these numbers $a_{1},$ $a_{2},$ $\ldots,$ $a_{n}$ are non-negative.
2007 Indonesia TST, 1
Let $ P$ be a point in triangle $ ABC$, and define $ \alpha,\beta,\gamma$ as follows: \[ \alpha\equal{}\angle BPC\minus{}\angle BAC, \quad \beta\equal{}\angle CPA\minus{}\angle \angle CBA, \quad \gamma\equal{}\angle APB\minus{}\angle ACB.\] Prove that \[ PA\dfrac{\sin \angle BAC}{\sin \alpha}\equal{}PB\dfrac{\sin \angle CBA}{\sin \beta}\equal{}PC\dfrac{\sin \angle ACB}{\sin \gamma}.\]
2006 Pre-Preparation Course Examination, 2
a) Show that you can divide an angle $\theta$ to three equal parts using compass and ruler if and only if the polynomial $4t^3-3t-\cos (\theta)$ is reducible over $\mathbb{Q}(\cos (\theta))$.
b) Is it always possible to divide an angle into five equal parts?
1987 AMC 8, 9
When finding the sum $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}+\frac{1}{7}$, the least common denominator used is
$\text{(A)}\ 120 \qquad \text{(B)}\ 210 \qquad \text{(C)}\ 420 \qquad \text{(D)}\ 840 \qquad \text{(E)}\ 5040$
2014 Contests, 3
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.
(Day 1, 3rd problem
author: Michal RolÃnek)
2002 IMO, 4
Let $n\geq2$ be a positive integer, with divisors $1=d_1<d_2<\,\ldots<d_k=n$. Prove that $d_1d_2+d_2d_3+\,\ldots\,+d_{k-1}d_k$ is always less than $n^2$, and determine when it is a divisor of $n^2$.
2014 Peru Iberoamerican Team Selection Test, P6
Determine the largest positive integer $k$ for which there exists a simple graph $G$ of $2014$ vertices that simultaneously satisfies the following conditions:
$a)$ $G$ does not contain triangles
$b)$ For each $i$ between $1$ and $k$, inclusive, at least one vertex of $G$ has degree $i$
$c)$ No vertex of $G$ has a degree greater than $k$
2017 CentroAmerican, 3
Tita the Frog sits on the number line. She is initially on the integer number $k>1$. If she is sitting on the number $n$, she hops to the number $f(n)+g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distinct integers.
1993 Irish Math Olympiad, 5
$ (a)$ The rectangle $ PQRS$ with $ PQ\equal{}l$ and $ QR\equal{}m$ $ (l,m \in \mathbb{N})$ is divided into $ lm$ unit squares. Prove that the diagonal $ PR$ intersects exactly $ l\plus{}m\minus{}d$ of these squares, where $ d\equal{}(l,m)$.
$ (b)$ A box with edge lengths $ l,m,n \in \mathbb{N}$ is divided into $ lmn$ unit cubes. How many of the cubes does a main diagonal of the box intersect?
2015 Kazakhstan National Olympiad, 5
Find all possible $\{ x_1,x_2,...x_n \}$ permutations of $ \{1,2,...,n \}$ so that when $1\le i \le n-2 $ then we have $x_i < x_{i+2}$ and when $1 \le i \le n-3$ then we have $x_i < x_{i+3}$ . Here $n \ge 4$.
2014 Singapore Senior Math Olympiad, 26
Suppose that $x$ is measured in radians. Find the maximum value of \[\frac{\sin2x+\sin4x+\sin6x}{\cos2x+\cos4x+\cos6x}\] for $0\le x\le \frac{\pi}{16}$
2020-21 KVS IOQM India, 11
The prime numbers $a,b$ and $c$ are such that $a+b^2=4c^2$. Determine the sum of all possible values of $a+b+c$.
2017 Switzerland - Final Round, 4
Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold:
$$pq | n^p + 2$$
$$n + 2 | n^p + q^p.$$
Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.
2012 Stanford Mathematics Tournament, 4
Two different squares are randomly chosen from an $8\times8$ chessboard. What is the probability that two queens placed on the two squares can attack each other? Recall that queens in chess can attack any square in a straight line vertically, horizontally, or diagonally from their current position.
1972 AMC 12/AHSME, 21
[asy]
draw((3,-13)--(21.5,-5)--(19,-18)--(9,-18)--(10,-6)--(23,-14.5)--cycle);
label("A",(3,-13),W);label("C",(21.5,-5),N);label("E",(19,-18),E);label("F",(9,-18),W);label("B",(10,-6),N);label("D",(23,-14.5),E);
//Credit to Zimbalono for the diagram[/asy]
If the sum of the measures in degrees of angles $A,~B,~C,~D,~E$ and $F$ in the figure above is $90n$, then $n$ is equal to
$\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$
1989 IMO Longlists, 7
Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.
1971 IMO Longlists, 5
Consider a sequence of polynomials $P_0(x), P_1(x), P_2(x), \ldots, P_n(x), \ldots$, where $P_0(x) = 2, P_1(x) = x$ and for every $n \geq 1$ the following equality holds:
\[P_{n+1}(x) + P_{n-1}(x) = xP_n(x).\]
Prove that there exist three real numbers $a, b, c$ such that for all $n \geq 1,$
\[(x^2 - 4)[P_n^2(x) - 4] = [aP_{n+1}(x) + bP_n(x) + cP_{n-1}(x)]^2.\]
2002 AMC 10, 22
In how many zeroes does the number $\dfrac{2002!}{(1001!)^2}$ end?
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }200\qquad\textbf{(E) }400$
1986 Bulgaria National Olympiad, Problem 4
Find the smallest integer $n\ge3$ for which there exists an $n$-gon and a point within it such that, if a light bulb is placed at that point, on each side of the polygon there will be a point that is not lightened. Show that for this smallest value of $n$ there always exist two points within the $n$-gon such that the bulbs placed at these points will lighten up the whole perimeter of the $n$-gon.
2004 IMC, 6
For $ n\geq 0$ define the matrices $ A_n$ and $ B_n$ as follows: $ A_0 \equal{} B_0 \equal{} (1)$, and for every $ n>0$ let
\[ A_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & B_{n \minus{} 1} \\
\end{array} \right) \ \textrm{and} \ B_n \equal{} \left( \begin{array}{cc} A_{n \minus{} 1} & A_{n \minus{} 1} \\
A_{n \minus{} 1} & 0 \\
\end{array} \right).
\]
Denote by $ S(M)$ the sum of all the elements of a matrix $ M$. Prove that $ S(A_n^{k \minus{} 1}) \equal{} S(A_k^{n \minus{} 1})$, for all $ n,k\geq 2$.
1993 Abels Math Contest (Norwegian MO), 1b
Given a triangle with sides of lengths $a,b,c$, prove that $\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}< 2$.
1997 All-Russian Olympiad Regional Round, 10.5
Given a set of $100$ different numbers such that if each number in the set is replaced by the sum of the others, the same set will be obtained. Prove that the product of numbers in a set is positive.
2009 Today's Calculation Of Integral, 411
Find the area bounded by $ y\equal{}x^2\minus{}|x^2\minus{}1|\plus{}|2|x|\minus{}2|\plus{}2|x|\minus{}7$ and the $ x$ axis.
2000 Bosnia and Herzegovina Team Selection Test, 2
Let $S$ be a point inside triangle $ABC$ and let lines $AS$, $BS$ and $CS$ intersect sides $BC$, $CA$ and $AB$ in points $X$, $Y$ and $Z$, respectively. Prove that $$\frac{BX\cdot CX}{AX^2}+\frac{CY\cdot AY}{BY^2}+\frac{AZ\cdot BZ}{CZ^2}=\frac{R}{r}-1$$ iff $S$ is incenter of $ABC$