Found problems: 85335
2015 AMC 12/AHSME, 9
A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets $2$ marbles of the same color?
$\textbf{(A) }\dfrac1{10}\qquad\textbf{(B) }\dfrac16\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac13\qquad\textbf{(E) }\dfrac12$
2019 Brazil Team Selection Test, 2
Let $m$ be a fixed positive integer. The infinite sequence $\{a_n\}_{n\geq 1}$ is defined in the following way: $a_1$ is a positive integer, and for every integer $n\geq 1$ we have
$$a_{n+1} = \begin{cases}a_n^2+2^m & \text{if } a_n< 2^m \\ a_n/2 &\text{if } a_n\geq 2^m\end{cases}$$
For each $m$, determine all possible values of $a_1$ such that every term in the sequence is an integer.
2019 Thailand TST, 1
Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.
2021 Science ON grade V, 3
A nonnegative integer $n$ is said to be $\textit{squarish}$ is it satisfies the following conditions:
$\textbf{(i)}$ it contains no digits of $9$;
$\textbf{(ii)}$ no matter how we increase exactly one of its digits with $1$, the outcome is a square.
Find all squarish nonnegative integers.
$\textit{(Vlad Robu)}$
1981 Polish MO Finals, 2
In a triangle $ABC$, the perpendicular bisectors of sides $AB$ and $AC$ intersect $BC$ at $X$ and $Y$. Prove that $BC = XY$ if and only if $\tan B\tan C = 3$ or $\tan B\tan C = -1$.
2020-21 IOQM India, 7
Let $\triangle ABC$ be a triangle with $AB=AC$. Let $D$ be a point on the segment $BC$ such that $BD= 48 \frac{1}{61}$ and $DC=61$. Let $E$ be a point on $AD$ such that $CE$ is perpendicular to $AD$ and $DE=11$. Find $AE$.
2023 Czech and Slovak Olympiad III A., 2
Let $n$ be a positive integer, where $n \geq 3$ and let $a_1, a_2, ..., a_n$ be the lengths of sides of some $n$-gon. Prove that
$$a_1 + a_2 + ... + a_n \geq \sqrt{2 \cdot (a_1^2 + a_2^2 + ... + a_n^2)} $$
2013 India IMO Training Camp, 1
Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order.
We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.
2020 Italy National Olympiad, #3
Let $a_1, a_2, \dots, a_{2020}$ and $b_1, b_2, \dots, b_{2020}$ be real numbers(not necessarily distinct). Suppose that the set of positive integers $n$ for which the following equation:
$|a_1|x-b_1|+a_2|x-b_2|+\dots+a_{2020}|x-b_{2020}||=n$ (1) has exactly two real solutions, is a finite set. Prove that the set of positive integers $n$ for which the equation (1) has at least one real solution, is also a finite set.
1990 Bundeswettbewerb Mathematik, 2
Let $A(n)$ be the least possible number of distinct points in the plane with the following property: For every $k = 1,2,...,n$ there is a line containing precisely $k$ of these points. Show that $A(n) =\left[\frac{n+1}{2}\right] \left[\frac{n+2}{2}\right]$
1993 Turkey MO (2nd round), 2
I centered incircle of triangle $ABC$ $(m(\hat{B})=90^\circ)$ touches $\left[AB\right], \left[BC\right], \left[AC\right]$ respectively at $F, D, E$. $\left[CI\right]\cap\left[EF\right]={L}$ and $\left[DL\right]\cap\left[AB\right]=N$. Prove that $\left[AI\right]=\left[ND\right]$.
2012 German National Olympiad, 5
Let $a,b$ be the lengths of two nonadjacent edges of a tetrahedron with inradius $r$. Prove that \[r<\frac{ab}{2(a+b)}.\]
2019 IMC, 9
Determine all positive integers $n$ for which there exist $n\times n$ real invertible matrices $A$ and $B$ that satisfy $AB-BA=B^2A$.
[i]Proposed by Karen Keryan, Yerevan State University & American University of Armenia, Yerevan[/i]
1900 Eotvos Mathematical Competition, 2
Construct a triangle $ABC$, given the length $c$ of its side $AB$, the radius $r$ of its inscribed circle, and the radius $r_c$ of its ex-circle tangent to the side $AB$ and the extensions of $BC$ and $CA$.
2005 MOP Homework, 4
Let $ABCD$ be a convex quadrilateral and let $K$, $L$, $M$, $N$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Let $NL$ and $KM$ meet at point $T$. Show that $8[DNTM] < [ABCD] < 8[DNTM]$, where $[P]$ denotes area of $P$.
2016 Bosnia And Herzegovina - Regional Olympiad, 2
Let $ABC$ be an isosceles triangle such that $\angle BAC = 100^{\circ}$. Let $D$ be an intersection point of angle bisector of $\angle ABC$ and side $AC$, prove that $AD+DB=BC$
1996 South africa National Olympiad, 3
The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.
2011 QEDMO 9th, 9
In a very long corridor there is an infinite number of cabinets, which start with $1,2,3,...$ numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set $A \subseteq \{1, 2,3,...\}$ . In ascending order, the QED people now cause chaos: the person with number $a \in A$ visits the cabinet with the numbers $a,2a,3a,...$ opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from $A$ open.
2005 Oral Moscow Geometry Olympiad, 4
A sphere can be inscribed into a pyramid, the base of which is a parallelogram. Prove that the sums of the areas of its opposite side faces are equal.
(M. Volchkevich)
1975 Vietnam National Olympiad, 2
Solve this equation
$\frac{y^{3}+m^{3}}{\left ( y+m \right )^{3}}+\frac{y^{3}+n^{3}}{\left ( y+n \right )^{3}}+\frac{y^{3}+p^{3}}{\left ( y+p \right )^{3}}-\frac{3}{2}+\frac{3}{2}.\frac{y-m}{y+m}.\frac{y-n}{y+n}.\frac{y-p}{y+p}=0$
2010 Math Prize For Girls Problems, 15
Compute the value of the sum
\begin{align*}
\frac{1}{1 + \tan^3 0^\circ}
&+ \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ}
+ \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\
&+ \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ}
+ \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ}
\, .
\end{align*}
2006 MOP Homework, 1
There are $2005$ players in a chess tournament played a game. Every pair of players played a game against each other. At the end of the tournament, it turned out that if two players $A$ and $B$ drew the game between them, then every other player either lost to $A$ or to $B$. Suppose that there are at least two draws in the tournament. Prove that all players can be lined up in a single file from left to right in the such a way that every play won the game against the person immediately to his right.
2022 Novosibirsk Oral Olympiad in Geometry, 3
Three angle bisectors were drawn in a triangle, and it turned out that the angles between them are $50^o$, $60^o$ and $70^o$. Find the angles of the original triangle.
2002 India IMO Training Camp, 17
Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.