Found problems: 85335
2017 Morocco TST-, 4
Two circles $ G_1$ and $ G_2$ intersect at two points $ M$ and $ N$. Let $ AB$ be the line tangent to these circles at $ A$ and $ B$, respectively, so that $ M$ lies closer to $ AB$ than $ N$. Let $ CD$ be the line parallel to $ AB$ and passing through the point $ M$, with $ C$ on $ G_1$ and $ D$ on $ G_2$. Lines $ AC$ and $ BD$ meet at $ E$; lines $ AN$ and $ CD$ meet at $ P$; lines $ BN$ and $ CD$ meet at $ Q$. Show that $ EP \equal{} EQ$.
2014 Contests, 1
Prove that for positive reals $a$,$b$,$c$ so that $a+b+c+abc=4$, \[\left (1+\dfrac{a}{b}+ca \right )\left (1+\dfrac{b}{c}+ab \right)\left (1+\dfrac{c}{a}+bc \right) \ge 27\] holds.
2014 Sharygin Geometry Olympiad, 3
An acute angle $A$ and a point $E$ inside it are given. Construct points $B, C$ on the sides of the angle such that $E$ is the center of the Euler circle of triangle $ABC$.
(E. Diomidov)
2011 District Olympiad, 4
Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have:
\[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\]
for all $x,y\in [0,1]$.
2022 Utah Mathematical Olympiad, 3
Find all sequences $a_1, a_2, a_3, \dots$ of real numbers such that for all positive integers $m,n\ge 1$, we have
\begin{align*}
a_{m+n} &= a_m+a_n - mn \text{ and} \\
a_{mn} &= m^2a_n + n^2a_m + 2a_ma_n. \\
\end{align*}
2022 Irish Math Olympiad, 5
5. Let $\triangle$ABC be a triangle with circumcentre [i]O[/i]. The perpendicular line from [i]O[/i] to [i]BC[/i] intersects line [i]BC[/i] at [i]M[/i] and line [i]AC[/i] at [i]P[/i], and the perpendicular line from [i]O[/i] to [i]AC[/i] intersects line [i]AC[/i] at [i]N[/i] and line [i]BC[/i] at [i]Q[/i]. Let [i]D[/i] be the intersection point of lines [i]PQ[/i] and [i]MN[/i]. construct the parallelogram [i]PCQJ[/i] with [i]PJ[/i] || [i]CQ[/i] and [i]QJ[/i] || [i]CP[/i].
Prove the following:
a) The points [i]A[/i], [i]B[/i], [i]O[/i], [i]P[/i], [i]Q[/i], [i]J[/i] are all on the same circle.
b) line [i]OD[/i] is perpendicular to line [i]CJ[/i].
2019 Auckland Mathematical Olympiad, 2
Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.
2014 Romania Team Selection Test, 1
Let $\triangle ABC$ be an acute triangle of circumcentre $O$. Let the tangents to the circumcircle of $\triangle ABC$ in points $B$ and $C$ meet at point $P$. The circle of centre $P$ and radius $PB=PC$ meets the internal angle bisector of $\angle BAC$ inside $\triangle ABC$ at point $S$, and $OS \cap BC = D$. The projections of $S$ on $AC$ and $AB$ respectively are $E$ and $F$. Prove that $AD$, $BE$ and $CF$ are concurrent.
[i]Author: Cosmin Pohoata[/i]
2018 India National Olympiad, 4
Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.
2021 Junior Balkan Team Selection Tests - Romania, P1
Let $a,b,c>0$ be real numbers with the property that $a+b+c=1$. Prove that \[\frac{1}{a+bc}+\frac{1}{b+ca}+\frac{1}{c+ab}\geq\frac{7}{1+abc}.\]
2023 MIG, 13
Five cards numbered $1,2,3,4,$ and $5$ are given to Paige, Quincy, Ronald, Selena, and Terrence. Paige, Quincy, and Ronald have the following conversation:
[list=disc]
[*]Paige: My number is between is between Selena's number and Quincy's number.
[*]Quincy: My number is between Ronald's number and Terrence's number.
[*]Ronald: My number is between Paige's number and Quincy's number.
[/list]
Who received the card numbered $3$?
$\textbf{(A) } \text{Paige}\qquad\textbf{(B) } \text{Quincy}\qquad\textbf{(C) } \text{Ronald}\qquad\textbf{(D) } \text{Selena}\qquad\textbf{(E) } \text{Terrence}$
2011 District Olympiad, 1
a) Prove that $\{x+y\}-\{y\}$ can only be equal to $\{x\}$ or $\{x\}-1$ for any $x,y\in \mathbb{R}$.
b) Let $\alpha\in \mathbb{R}\backslash \mathbb{Q}$. Denote $a_n=\{n\alpha\}$ for all $n\in \mathbb{N}^*$ and define the sequence $(x_n)_{n\ge 1}$ by
\[x_n=(a_2-a_1)(a_3-a_2)\cdot \ldots \cdot (a_{n+1}-a_n)\]
Prove that the sequence $(x_n)_{n\ge 1}$ is convergent and find it's limit.
2022 Abelkonkurransen Finale, 3
Nils has an $M \times N$ board where $M$ and $N$ are positive integers, and a tile shaped as shown below. What is the smallest number of squares that Nils must color, so that it is impossible to place the tile on the board without covering a colored square? The tile can be freely rotated and mirrored, but it must completely cover four squares.
[asy]
usepackage("tikz");
label("%
\begin{tikzpicture}
\draw[step=1cm,color=black] (0,0) grid (2,1);
\draw[step=1cm,color=black] (1,1) grid (3,2);
\fill [yellow] (0,0) rectangle (2,1);
\fill [yellow] (1,1) rectangle (3,2);
\draw[step=1cm,color=black] (0,0) grid (2,1);
\draw[step=1cm,color=black] (1,1) grid (3,2);
\end{tikzpicture}
");
[/asy]
2013 Canada National Olympiad, 2
The sequence $a_1, a_2, \dots, a_n$ consists of the numbers $1, 2, \dots, n$ in some order. For which positive integers $n$ is it possible that the $n+1$ numbers $0, a_1, a_1+a_2, a_1+a_2+a_3,\dots, a_1 + a_2 +\cdots + a_n$ all have di fferent remainders when divided by $n + 1$?
2005 District Olympiad, 4
Let $\{a_k\}_{k\geq 1}$ be a sequence of non-negative integers, such that $a_k \geq a_{2k} + a_{2k+1}$, for all $k\geq 1$.
a) Prove that for all positive integers $n\geq 1$ there exist $n$ consecutive terms equal with 0 in the sequence $\{a_k\}_k$;
b) State an example of sequence with the property in the hypothesis which contains an infinite number of non-zero terms.
1996 AMC 12/AHSME, 16
A fair standard six-sided dice is tossed three times. Given that the sum of the first two tosses equal the third, what is the probability that at least one $2$ is tossed?
$\displaystyle \textbf{(A)} \ \frac{1}{6} \qquad \textbf{(B)} \ \frac{91}{216} \qquad \textbf{(C)} \ \frac{1}{2} \qquad \textbf{(D)} \ \frac{8}{15} \qquad \textbf{(E)} \ \frac{7}{12}$
2017 Singapore Junior Math Olympiad, 1
A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a, b,a<b$, compute the ratio $a/b$. Prove that sum of these ratios is at least $1$.
2019 Sharygin Geometry Olympiad, 1
A trapezoid with bases $AB$ and $CD$ is inscribed into a circle centered at $O$. Let $AP$ and $AQ$ be the tangents from $A$ to the circumcircle of triangle $CDO$. Prove that the circumcircle of triangle $APQ$ passes through the midpoint of $AB$.
2016 BMT Spring, 8
A regular unit $7$-simplex is a polytope in $7$-dimensional space with $8$ vertices that are all exactly a distance of $ 1$ apart. (It is the $7$-dimensional analogue to the triangle and the tetrahedron.) In this $7$-dimensional space, there exists a point that is equidistant from all $8$ vertices, at a distance $d$. Determine $d$.
2009 Bundeswettbewerb Mathematik, 1
Determine all possible digits $z$ for which
$\underbrace{9...9}_{100}z\underbrace{0...0}_{100}9$ is a square number.
2017 LMT, Max Area
The goal of this problem is to show that the maximum area of a polygon with a fixed number of sides and a fixed perimeter is achieved by a regular polygon.
(a) Prove that the polygon with maximum area must be convex. (Hint: If any angle is concave, show that the polygon’s area can be increased.)
(b) Prove that if two adjacent sides have different lengths, the area of the polygon can be increased without changing the perimeter.
(c) Prove that the polygon with maximum area is equilateral, that is, has all the same side lengths.
It is true that when given all four side lengths in order of a quadrilateral, the maximum area is achieved in the unique configuration in which the quadrilateral is cyclic, that is, it can be inscribed in a circle.
(d) Prove that in an equilateral polygon, if any two adjacent angles are different then the area of the polygon
can be increased without changing the perimeter.
(e) Prove that the polygon of maximum area must be equiangular, or have all angles equal.
(f ) Prove that the polygon of maximum area is a regular polygon.
PS. You had better use hide for answers.
2012 Moldova Team Selection Test, 7
Let $C(O_1),C(O_2)$ be two externally tangent circles at point $P$. A line $t$ is tangent to $C(O_1)$ in point $R$ and intersects $C(O_2)$ in points $A,B$ such that $A$ is closer to $R$ than $B$ is. The line $AO_1$ intersects the perpendicular to $t$ in $B$ at point $C$, the line $PC$ intersects $AB$ in $Q$.
Prove that $QO_1$ passes through the midpoint of $BC$.
2008 Costa Rica - Final Round, 6
Let $ O$ be the circumcircle of a $ \Delta ABC$ and let $ I$ be its incenter, for a point $ P$ of the plane let $ f(P)$ be the point obtained by reflecting $ P'$ by the midpoint of $ OI$, with $ P'$ the homothety of $ P$ with center $ O$ and ratio $ \frac{R}{r}$ with $ r$ the inradii and $ R$ the circumradii,(understand it by $ \frac{OP}{OP'}\equal{}\frac{R}{r}$). Let $ A_1$, $ B_1$ and $ C_1$ the midpoints of $ BC$, $ AC$ and $ AB$, respectively. Show that the rays $ A_1f(A)$, $ B_1f(B)$ and $ C_1f(C)$ concur on the incircle.
2008 Tuymaada Olympiad, 2
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?
[i]Author: L. Emelyanov[/i]
[hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$
may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]
1966 Czech and Slovak Olympiad III A, 4
Two triangles $ABC,ABD$ (with the common side $c=AB$) are given in space. Triangle $ABC$ is right with hypotenuse $AB$, $ABD$ is equilateral. Denote $\varphi$ the dihedral angle between planes $ABC,ABD$.
1) Determine the length of $CD$ in terms of $a=BC,b=CA,c$ and $\varphi$.
2) Determine all values of $\varphi$ such that the tetrahedron $ABCD$ has four sides of the same length.