Found problems: 85335
2019 AMC 8, 2
Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles $5$ feet, what is the area in square feet of rectangle $ABCD$?
[asy]draw((0,0)--(0,10)--(15,10)--(15,0)--(0,0));
draw((0,5)--(10,5));
draw((10,0)--(10,10));
label("$A$",(0,0),SW);
label("$B$",(15,0),SE);
label("$C$",(15,10),NE);
label("$D$",(0,10),NW);
dot((0,10));
dot((15,0));
dot((15,10));
dot((0,0));
[/asy]
$\textbf{(A) }45\qquad
\textbf{(B) }75\qquad
\textbf{(C) }100\qquad
\textbf{(D) }125\qquad
\textbf{(E) }150\qquad$
1974 IMO Shortlist, 6
Prove that for any n natural, the number \[ \sum \limits_{k=0}^{n} \binom{2n+1}{2k+1} 2^{3k} \]
cannot be divided by $5$.
2010 Contests, 3
Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent.
[i]Pavel Kozhevnikov, Russia[/i]
2012 Federal Competition For Advanced Students, Part 2, 1
Determine the maximum value of $m$, such that the inequality
\[ (a^2+4(b^2+c^2))(b^2+4(a^2+c^2))(c^2+4(a^2+b^2)) \ge m \]
holds for every $a,b,c \in \mathbb{R} \setminus \{0\}$ with $\left|\frac{1}{a}\right|+\left|\frac{1}{b}\right|+\left|\frac{1}{c}\right|\le 3$.
When does equality occur?
2004 District Olympiad, 1
Let $n\geq 2$ and $1 \leq r \leq n$. Consider the set $S_r=(A \in M_n(\mathbb{Z}_2), rankA=r)$. Compute the sum $\sum_{X \in S_r}X$
2011 SEEMOUS, Problem 3
Given vectors $\overline a,\overline b,\overline c\in\mathbb R^n$, show that
$$(\lVert\overline a\rVert\langle\overline b,\overline c\rangle)^2+(\lVert\overline b\rVert\langle\overline a,\overline c\rangle)^2\le\lVert\overline a\rVert\lVert\overline b\rVert(\lVert\overline a\rVert\lVert\overline b\rVert+|\langle\overline a,\overline b\rangle|)\lVert\overline c\rVert^2$$where $\langle\overline x,\overline y\rangle$ denotes the scalar (inner) product of the vectors $\overline x$ and $\overline y$ and $\lVert\overline x\rVert^2=\langle\overline x,\overline x\rangle$.
May Olympiad L2 - geometry, 2018.5
Each point on a circle is colored with one of $10$ colors. Is it true that for any coloring there are $4$ points of the same color that are vertices of a quadrilateral with two parallel sides (an isosceles trapezoid or a rectangle)?
2008 Sharygin Geometry Olympiad, 5
(N.Avilov) Can the surface of a regular tetrahedron be glued over with equal regular hexagons?
2004 Federal Competition For Advanced Students, P2, 5
Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} \\
b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} \\
c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} \\
d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$
2009 Thailand Mathematical Olympiad, 6
Let $\vartriangle ABC$ be a triangle with $AB > AC$, its incircle is tangent to $BC$ at $D$. Let $DE$ be a diameter of the incircle, and let $F$ be the intersection between line $AE$ and side $BC$. Find the ratio between the areas of $\vartriangle DEF$ and $\vartriangle ABC$ in terms of the three side lengths of$\vartriangle ABC$.
2018 Brazil Team Selection Test, 1
Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that
$$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$
If $M>1$, prove that the polynomial
$$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$
has no positive roots.
1965 Swedish Mathematical Competition, 1
The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?
2022 Belarus - Iran Friendly Competition, 4
From a point $S$, which lies outside the circle $\Omega$, tangent lines $SA$ and $SB$ to that circle are drawn. On the chord $AB$ an arbitrary point $K$ is chosen. $SK$ intersects $\Omega$ at points $P$ and $Q$, and chords $RT$ and $UW$ pass through $K$ such that $W, Q$ and $T$ lie in the same half-plane with respect to $AB$. Lines $WR$ and $TU$ intersect chord $AB$ at $C$ and $D$, and $M$ is the midpoint of $PQ$.
Prove that $\angle AMC = \angle BMD$
2018 IMO Shortlist, G3
A circle $\omega$ with radius $1$ is given. A collection $T$ of triangles is called [i]good[/i], if the following conditions hold:
[list=1]
[*] each triangle from $T$ is inscribed in $\omega$;
[*] no two triangles from $T$ have a common interior point.
[/list]
Determine all positive real numbers $t$ such that, for each positive integer $n$, there exists a good collection of $n$ triangles, each of perimeter greater than $t$.
2017 ASDAN Math Tournament, 15
Each face of a regular tetrahedron can be colored one of red, purple, blue, or orange. How many distinct ways can we color the faces of the tetrahedron? Colorings are considered distinct if they cannot reach one another by rotation.
2016 ASDAN Math Tournament, 2
Let $f(x)=ax^3+bx^2+cx+d$ be some cubic polynomial. Given that $f(1)=20$ and $f(-1)=16$, what is $b+d$?
2024 Girls in Mathematics Tournament, 4
Find all the positive integers $a,b,c$ such that $3ab= 2c^2$ and $a^3+b^3+c^3$ is the double of a prime number.
1996 Akdeniz University MO, 3
A $x>2$ real number is given. Bob has got $1997$ labels and writes one of the numbers $"x^0, x^1, x^2 ,\dotsm x^{1995}, x^{1996}"$ each labels such that all labels has distinct numbers. Bob puts some labels to right pocket, some labels to left pocket. Prove that sum of numbers of the right pocket never equal to sum of numbers of the left pocket.
2009 Bosnia And Herzegovina - Regional Olympiad, 4
Let $x$ and $y$ be positive integers such that $\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}$ is integer. Prove that numbers $\frac{x^2-1}{y+1}$ and $\frac{y^2-1}{x+1}$ are integers
2010 Contests, 2
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
1998 Austrian-Polish Competition, 8
In each unit square of an infinite square grid a natural number is written. The polygons of area $n$ with sides going along the gridlines are called [i]admissible[/i], where $n > 2$ is a given natural number. The [i]value [/i] of an admissible polygon is defined as the sum of the numbers inside it. Prove that if the values of any two congruent admissible polygons are equal, then all the numbers written in the unit squares of the grid are equal. (We recall that a symmetric image of polygon $P$ is congruent to $P$.)
2021 Sharygin Geometry Olympiad, 5
Five points are given in the plane. Find the maximum number of similar triangles whose vertices are among those five points.
2025 Ukraine National Mathematical Olympiad, 8.4
What is the maximum number of knights that can be placed on a chessboard of size \(8 \times 8\) such that any knight, after making 1 or 2 arbitrary moves, does not land on a square occupied by another knight?
[i]Proposed by Bogdan Rublov[/i]
2014 ASDAN Math Tournament, 6
Consider $7$ points on a circle. Compute the number of ways there are to draw chords between pairs of points such that two chords never intersect and one point can only belong to one chord. It is acceptable to draw no chords.
2013 All-Russian Olympiad, 3
The head of the Mint wants to release 12 coins denominations (each - a natural number rubles) so that any amount from 1 to 6543 rubles could be paid without having to pass, using no more than 8 coins. Can he do it? (If the payment amount you can use a few coins of the same denomination.)