Found problems: 85335
2004 Tuymaada Olympiad, 2
In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines?
[i]Proposed by A. Golovanov[/i]
2010 Vietnam Team Selection Test, 1
Let $a,b,c$ be positive integers which satisfy the condition: $16(a+b+c)\geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.
Prove that
\[\sum_{cyc} \left( \frac{1}{a+b+\sqrt{2a+2c}} \right)^{3}\leq \frac{8}{9}\]
2024 Macedonian TST, Problem 6
Let \(a,b\) be positive integers such that \(a+1\), \(b+1\), and \(ab\) are perfect squares. Prove that $\gcd(a,b)+1$ is also a perfect square.
2003 India National Olympiad, 6
Each lottery ticket has a 9-digit numbers, which uses only the digits $1$, $2$, $3$. Each ticket is colored [color=red]red[/color],[color=blue] blue [/color]or [color=green]green[/color]. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket $122222222$ is red, and ticket $222222222$ is [color=green]green.[/color] What color is ticket $123123123$ ?
2024 ELMO Shortlist, G2
Let $ABC$ be a triangle. Suppose that $D$, $E$, and $F$ are points on segments $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$ respectively such that triangles $AEF$, $BFD$, and $CDE$ have equal inradii. Prove that the sum of the inradii of $\triangle AEF$ and $\triangle DEF$ is equal to the inradius of $\triangle ABC$.
[i]Aprameya Tripathy[/i]
1993 AMC 12/AHSME, 2
In $\triangle ABC$, $\angle A=55^{\circ}$, $\angle C=75^{\circ}$, $D$ is on side $\overline{AB}$ and $E$ is on side $\overline{BC}$. If $DB=BE$, then $\angle BED=$
[asy]
size((100));
draw((0,0)--(10,0)--(8,10)--cycle);
draw((4,5)--(9.2,4));
dot((0,0));
dot((10,0));
dot((8,10));
dot((4,5));
dot((9.2,4));
label("A", (0,0), SW);
label("B", (8,10), N);
label("C", (10,0), SE);
label("D", (4,5), NW);
label("E", (9.2,4), E);
label("$55^{\circ}$", (.5,0), NE);
label("$75^{\circ}$", (9.8,0), NW);
[/asy]
$ \textbf{(A)}\ 50^{\circ} \qquad\textbf{(B)}\ 55^{\circ} \qquad\textbf{(C)}\ 60^{\circ} \qquad\textbf{(D)}\ 65^{\circ} \qquad\textbf{(E)}\ 70^{\circ} $
1987 National High School Mathematics League, 1
For any given positive integer $n$, $n^6+3a$ is a perfect cube, where $a$ is a positive integer. Then
$\text{(A)}$There is no such $a$.
$\text{(B)}$There are infinitely many such $a$.
$\text{(C)}$There is finitely many such $a$.
$\text{(D)}$None of $\text{(A)(B)(C)}$ is correct.
2011 Saudi Arabia IMO TST, 2
Consider the set $S= \{(a + b)^7 - a^7 - b^7 : a,b \in Z\}$. Find the greatest common divisor of all members in $S$.
2007 Tournament Of Towns, 1
Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.
2023 Belarus Team Selection Test, 1.3
Let $Q$ be a set of prime numbers, not necessarily finite. For a positive integer $n$ consider its prime factorization: define $p(n)$ to be the sum of all the exponents and $q(n)$ to be the sum of the exponents corresponding only to primes in $Q$. A positive integer $n$ is called [i]special[/i] if $p(n)+p(n+1)$ and $q(n)+q(n+1)$ are both even integers. Prove that there is a constant $c>0$ independent of the set $Q$ such that for any positive integer $N>100$, the number of special integers in $[1,N]$ is at least $cN$.
(For example, if $Q=\{3,7\}$, then $p(42)=3$, $q(42)=2$, $p(63)=3$, $q(63)=3$, $p(2022)=3$, $q(2022)=1$.)
2006 QEDMO 3rd, 10
Define a sequence $\left( a_{n}\right) _{n\in\mathbb{N}}$ by $a_{1}=a_{2}=a_{3}=1$ and $a_{n+1}=\dfrac{a_{n}^{2}+a_{n-1}^{2}}{a_{n-2}}$ for every integer $n\geq3$. Show that all elements $a_{i}$ of this sequence are integers.
(L. J. Mordell and apparently Dana Scott, see also http://oeis.org/A064098)
2013 China Team Selection Test, 1
The quadrilateral $ABCD$ is inscribed in circle $\omega$. $F$ is the intersection point of $AC$ and $BD$. $BA$ and $CD$ meet at $E$. Let the projection of $F$ on $AB$ and $CD$ be $G$ and $H$, respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$, respectively. If the circumcircle of $\triangle MNG$ only meets segment $BF$ at $P$, and the circumcircle of $\triangle MNH$ only meets segment $CF$ at $Q$, prove that $PQ$ is parallel to $BC$.
2019 BMT Spring, 6
Define $ f(n) = \dfrac{n^2 + n}{2} $. Compute the number of positive integers $ n $ such that $ f(n) \leq 1000 $ and $ f(n) $ is the product of two prime numbers.
2016 EGMO, 2
Let $ABCD$ be a cyclic quadrilateral, and let diagonals $AC$ and $BD$ intersect at $X$.Let $C_1,D_1$ and $M$ be the midpoints of segments $CX,DX$ and $CD$, respecctively. Lines $AD_1$ and $BC_1$ intersect at $Y$, and line $MY$ intersects diagonals $AC$ and $BD$ at different points $E$ and $F$, respectively. Prove that line $XY$ is tangent to the circle through $E,F$ and $X$.
1980 USAMO, 3
Let $F_r=x^r\sin{rA}+y^r\sin{rB}+z^r\sin{rC}$, where $x,y,z,A,B,C$ are real and $A+B+C$ is an integral multiple of $\pi$. Prove that if $F_1=F_2=0$, then $F_r=0$ for all positive integral $r$.
2006 Germany Team Selection Test, 2
In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit.
[b]a)[/b] Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas.
[b]b)[/b] Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?
2013 Princeton University Math Competition, 3
Consider the shape formed from taking equilateral triangle $ABC$ with side length $6$ and tracing out the arc $BC$ with center $A$. Set the shape down on line $l$ so that segment $AB$ is perpendicular to $l$, and $B$ touches $l$. Beginning from arc $BC$ touching $l$, we roll $ABC$ along $l$ until both points $A$ and $C$ are on the line. The area traced out by the roll can be written in the form $n\pi$, where $n$ is an integer. Find $n$.
1999 Korea Junior Math Olympiad, 8
For $S_n=\{1, 2, ..., n\}$, find the maximum value of $m$ that makes the following proposition true.
[b]Proposition[/b]
There exists $m$ different subsets of $S$, say $A_1, A_2, ..., A_m$, such that for every $i, j=1, 2, ..., m$, the set $A_i \cup A_j$ is not $S$.
2023 Harvard-MIT Mathematics Tournament, 16
The graph of the equation $x+y=\lfloor x^2+y^2 \rfloor$ consists of several line segments. Compute the sum of their lengths.
2020 Tournament Of Towns, 1
Consider two parabolas $y = x^2$ and $y = x^2 - 1$. Let $U$ be the set of points between the parabolas (including the points on the parabolas themselves). Does $U$ contain a line segment of length greater than $10^6$ ?
Alexey Tolpygo
1990 AMC 12/AHSME, 27
Which of these triples could [u]not[/u] be the lengths of the three altitudes of a triangle?
$ \textbf{(A)}\ 1,\sqrt{3},2 \qquad\textbf{(B)}\ 3,4,5 \qquad\textbf{(C)}\ 5,12,13 \qquad\textbf{(D)}\ 7,8,\sqrt{113} \qquad\textbf{(E)}\ 8,15,17 $
1991 Arnold's Trivium, 95
Decompose the space of homogeneous polynomials of degree $5$ in $(x, y, z)$ into irreducible subspaces invariant with respect to the rotation group $SO(3)$.
2021 IMO Shortlist, C7
Consider a checkered $3m\times 3m$ square, where $m$ is an integer greater than $1.$ A frog sits on the lower left corner cell $S$ and wants to get to the upper right corner cell $F.$ The frog can hop from any cell to either the next cell to the right or the next cell upwards.
Some cells can be [i]sticky[/i], and the frog gets trapped once it hops on such a cell. A set $X$ of cells is called [i]blocking[/i] if the frog cannot reach $F$ from $S$ when all the cells of $X$ are sticky. A blocking set is [i] minimal[/i] if it does not contain a smaller blocking set.[list=a][*]Prove that there exists a minimal blocking set containing at least $3m^2-3m$ cells.
[*]Prove that every minimal blocking set containing at most $3m^2$ cells.
2019 Cono Sur Olympiad, 6
Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.
2005 India National Olympiad, 3
Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say $\alpha$. Prove that
$a)$ $\alpha$ is real and negative;
$b)$ the remaining third quadratic equation has non-real roots.