Found problems: 85335
2017 Purple Comet Problems, 1
Paul starts at $1$ and counts by threes: $1, 4, 7, 10, ... $. At the same time and at the same speed, Penny counts backwards from $2017$ by fi ves: $2017, 2012, 2007, 2002,...$ . Find the one number that both Paul and Penny count at the same time.
2017 Romania National Olympiad, 1
Prove that the line joining the centroid and the incenter of a non-isosceles triangle is perpendicular to the base if and only if the sum of the other two sides is thrice the base.
1985 IMO Shortlist, 14
A set of $1985$ points is distributed around the circumference of a circle and each of the points is marked with $1$ or $-1$. A point is called “good” if the partial sums that can be formed by starting at that point and proceeding around the circle for any distance in either direction are all strictly positive. Show that if the number of points marked with $-1$ is less than $662$, there must be at least one good point.
1990 Putnam, A5
If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$.
2019-2020 Fall SDPC, 6
Let $ABCD$ be an isosceles trapezoid inscribed in circle $\omega$, such that $AD \| BC$. Point $E$ is chosen on the arc $BC$ of $\omega$ not containing $A$. Let $BC$ and $DE$ intersect at $F$. Show that if $E$ is chosen such that $EB = EC$, the area of $AEF$ is maximized.
1993 All-Russian Olympiad Regional Round, 10.3
Solve in positive numbers the system
$ x_1\plus{}\frac{1}{x_2}\equal{}4, x_2\plus{}\frac{1}{x_3}\equal{}1, x_3\plus{}\frac{1}{x_4}\equal{}4, ..., x_{99}\plus{}\frac{1}{x_{100}}\equal{}4, x_{100}\plus{}\frac{1}{x_1}\equal{}1$
LMT Theme Rounds, 6
How many functions $f:\{1,2,3,4\}\rightarrow \{1,2,3\}$ are surjective?
[i]Proposed by Nathan Ramesh
2010 Oral Moscow Geometry Olympiad, 5
All edges of a regular right pyramid are equal to $1$, and all vertices lie on the side surface of a (infinite) right circular cylinder of radius $R$. Find all possible values of $R$.
2006 AMC 10, 7
Which of the following is equivalent to $ \displaystyle \sqrt {\frac {x}{1 \minus{} \frac {x \minus{} 1}{x}}}$ when $ x < 0$?
$ \textbf{(A) } \minus{} x \qquad \textbf{(B) } x \qquad \textbf{(C) } 1 \qquad \textbf{(D) } \sqrt {\frac x2} \qquad \textbf{(E) } x\sqrt { \minus{} 1}$
2016 May Olympiad, 5
Rosa and Sara play with a triangle $ABC$, right at $B$. Rosa begins by marking two interior points of the hypotenuse $AC$, then Sara marks an interior point of the hypotenuse $AC$ different from those of Rosa. Then, from these three points the perpendiculars to the sides $AB$ and $BC$ are drawn, forming the following figure.
[img]https://cdn.artofproblemsolving.com/attachments/9/9/c964bbacc4a5960bee170865cc43902410e504.png[/img]
Sara wins if the area of the shaded surface is equal to the area of the unshaded surface, in other case wins Rosa. Determine who of the two has a winning strategy.
1985 IMO Longlists, 80
Let $E = \{1, 2, \dots , 16\}$ and let $M$ be the collection of all $4 \times 4$ matrices whose entries are distinct members of $E$. If a matrix $A = (a_{ij} )_{4\times4}$ is chosen randomly from $M$, compute the probability $p(k)$ of $\max_i \min_j a_{ij} = k$ for $k \in E$. Furthermore, determine $l \in E$ such that $p(l) = \max \{p(k) | k \in E \}.$
1992 Brazil National Olympiad, 7
Find all 4-tuples $(a,b,c,n)$ of naturals such that
$n^a + n^b = n^c$
1983 IMO Shortlist, 17
Let $P_1, P_2, \dots , P_n$ be distinct points of the plane, $n \geq 2$. Prove that
\[ \max_{1\leq i<j\leq n} P_iP_j > \frac{\sqrt 3}{2}(\sqrt n -1) \min_{1\leq i<j\leq n} P_iP_j \]
2021 Sharygin Geometry Olympiad, 9.1
Three cevians concur at a point lying inside a triangle. The feet of these cevians divide the sides into six segments, and the lengths of these segments form (in some order) a geometric progression. Prove that the lengths of the cevians also form a geometric progression.
IV Soros Olympiad 1997 - 98 (Russia), 9.8
There is a king in the lower left corner of a chessboard of dimensions $6$ and $6$. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?
2012 Iran MO (3rd Round), 5
We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$.
[i]Solution by Mostafa Eynollahzade and Erfan Salavati[/i]
1984 IMO Shortlist, 15
Angles of a given triangle $ABC$ are all smaller than $120^\circ$. Equilateral triangles $AFB, BDC$ and $CEA$ are constructed in the exterior of $ABC$.
(a) Prove that the lines $AD, BE$, and $CF$ pass through one point $S.$
(b) Prove that $SD + SE + SF = 2(SA + SB + SC).$
1988 AMC 12/AHSME, 26
Suppose that $p$ and $q$ are positive numbers for which \[ \log_{9}(p) = \log_{12}(q) = \log_{16}(p+q) \] What is the value of $\frac{q}{p}$?
$\textbf{(A)}\ \frac{4}{3}\qquad\textbf{(B)}\ \frac{1+\sqrt{3}}{2}\qquad\textbf{(C)}\ \frac{8}{5}\qquad\textbf{(D)}\ \frac{1+\sqrt{5}}{2}\qquad\textbf{(E)}\ \frac{16}{9} $
2011 National Olympiad First Round, 30
For which value of $m$, there is no integer pair $(x,y)$ satisfying the equation $3x^2-10xy-8y^2=m^{19}$?
$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 3$
2015 Iran Team Selection Test, 1
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that
$$P(x)^3+Q(x)^3=x^{12}+1.$$
1993 Denmark MO - Mohr Contest, 2
A rectangular piece of paper has the side lengths $12$ and $15$. A corner is bent about as shown in the figure. Determine the area of the gray triangle.
[img]https://1.bp.blogspot.com/-HCfqWF0p_eA/XzcIhnHS1rI/AAAAAAAAMYg/KfY14frGPXUvF-H6ZVpV4RymlhD_kMs-ACLcBGAsYHQ/s0/1993%2BMohr%2Bp2.png[/img]
2001 IMO Shortlist, 6
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]
2023 Sharygin Geometry Olympiad, 15
Let $ABCD$ be a convex quadrilateral. Points $X$ and $Y$ lie on the extensions beyond $D$ of the sides $CD$ and $AD$ respectively in such a way that $DX = AB$ and $DY = BC$. Similarly points $Z$ and $T$ lie on the extensions beyond $B$ of the sides $CB$ and $AB$ respectively in such a way that $BZ = AD$ and $BT = DC$. Let $M_1$ be the midpoint of $XY$, and $M_2$ be the midpoint of $ZT$. Prove that the lines $DM_1, BM_2$ and $AC$ concur.
2020 Durer Math Competition Finals, 4
Let $ABC$ be a scalene triangle and its incentre $I$. Denote by $F_A$ the intersection of the line $BC$ and the perpendicular to the angle bisector at $A$ through $I$. Let us define points $F_B$ and $F_C$ in a similar manner. Prove that points $F_A, F_B$ and $F_C$ are collinear.
2019 OMMock - Mexico National Olympiad Mock Exam, 4
Find all positive integers $n$ such that it is possible to split the numbers from $1$ to $2n$ in two groups $(a_1,a_2,..,a_n)$, $(b_1,b_2,...,b_n)$ in such a way that $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$.
[i]Proposed by Alef Pineda[/i]