Found problems: 85335
2015 AMC 12/AHSME, 17
Eight people are sitting around a circular table, each holding a fair coin. All eight people flip their coins and those who flip heads stand while those who flip tails remain seated. What is the probability that no two adjacent people will stand?
$\textbf{(A) }\dfrac{47}{256}\qquad\textbf{(B) }\dfrac{3}{16}\qquad\textbf{(C) }\dfrac{49}{256}\qquad\textbf{(D) }\dfrac{25}{128}\qquad\textbf{(E) }\dfrac{51}{256}$
2019 Purple Comet Problems, 12
Find the number of ordered triples of positive integers $(a, b, c)$, where $a, b,c$ is a strictly increasing arithmetic progression, $a + b + c = 2019$, and there is a triangle with side lengths $a, b$, and $c$.
2020 Online Math Open Problems, 10
Let $w,x,y,$ and $z$ be nonzero complex numbers, and let $n$ be a positive integer. Suppose that the following conditions hold:
[list]
[*] $\frac{1}{w}+\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=3,$
[*] $wx+wy+wz+xy+xz+yz=14,$
[*] $(w+x)^3+(w+y)^3+(w+z)^3+(x+y)^3+(x+z)^3+(y+z)^3=2160, \text{ and}$
[*] $w+x+y+z+i\sqrt{n} \in \mathbb{R}.$
[/list]
Compute $n$.
[i]Proposed by Luke Robitaille[/i]
2010 Junior Balkan Team Selection Tests - Moldova, 4
In the $25$ squares of a $5 \times 5$ square, zeros are initially written. in the every minute Ionel chooses two squares with a common side. If they are written in them the numbers $a$ and $b$, then he writes instead the numbers $a + 1$ and $b + 1$ or $a - 1$ and $b - 1$. Over time he noticed that the sums of the numbers in each line were equal, as well as the sums of the numbers in each column are equal. Prove that this observation was made after an even number of minutes.
2018 Rio de Janeiro Mathematical Olympiad, 5
Let $n$ be an positive integer and $\sigma = (a_1, \dots, a_n)$ a permutation of $\{1, \dots, n\}$. The [i]cadence number[/i] of $\sigma$ is the number of maximal decrescent blocks.
For example, if $n = 6$ and $\sigma = (4, 2, 1, 5, 6, 3)$, then the cadence number of $\sigma$ is $3$, because $\sigma$ has $3$ maximal decrescent blocks: $(4, 2, 1)$, $(5)$ and $(6, 3)$. Note that $(4, 2)$ and $(2, 1)$ are decrescent, but not maximal, because they are already contained in $(4, 2, 1)$.
Compute the sum of the cadence number of every permutation of $\{1, \dots, n\}$.
2023 Thailand Online MO, 6
Let $ABC$ be a triangle. Construct point $X$ such that $BX=BA$ and $X$ and $C$ lies on the same side of line $AB$. Construct point $Y$ such that $CY=CA$ and $Y$ and $B$ lies on different sides of line $AC$. Suppose that triangle $BAX$ and triangle $CAY$ are similar, prove that the circumcenter of triangle $AXY$ lies on the circumcircle of triangle $ABC$.
2003 Baltic Way, 5
The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$
\[(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. \]
1994 Tournament Of Towns, (430) 7
The figure $F$ is the intersection of $N$ circles (they may have different radii). Find the maximal number of curvilinear “sides” which $F$ can have. Curvilinear sides of $F$ are the arcs (of the given circumferences) that constitute the boundary of $F$. (Their ends are the “vertices” of $F$ - the points of intersection of given circumferences that lie on the boundary of $F$.)
(N Brodsky)
1998 Korea - Final Round, 3
Denote by $\phi(n)$ for all $n\in\mathbb{N}$ the number of positive integer smaller than $n$ and relatively prime to $n$. Also, denote by $\omega(n)$ for all $n\in\mathbb{N}$ the number of prime divisors of $n$. Given that $\phi(n)|n-1$ and $\omega(n)\leq 3$. Prove that $n$ is a prime number.
2006 Iran MO (3rd Round), 2
$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.
2021 OMpD, 4
Determine the smallest positive integer $n$ with the following property: on a board $n \times n$, whose squares are painted in checkerboard pattern (that is, for any two squares with a common edge, one of them is black and the other is white), it is possible to place the numbers $1,2,3 , ... , n^2$, a number in each square, so if $B$ is the sum of the numbers written in the white squares and $P$ is the sum of the numbers written in the black squares, then $\frac {B}{P} = \frac{2021}{4321}$.
2014 Saudi Arabia IMO TST, 3
Let $ABC$ be a triangle and let $P$ be a point on $BC$. Points $M$ and $N$ lie on $AB$ and $AC$, respectively such that $MN$ is not parallel to $BC$ and $AMP N$ is a parallelogram. Line $MN$ meets the circumcircle of $ABC$ at $R$ and $S$. Prove that the circumcircle of triangle $RP S$ is tangent to $BC$.
2013 IPhOO, 1
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$. The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$. The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$. The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$, where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$. If the number is rational, let $r=1$.
[i](Ahaan Rungta, 2 points)[/i]
2015 Junior Balkan Team Selection Tests - Moldova, 3
Let $\Omega$ be the circle circumscribed to the triangle $ABC$. Tangents taken to the circle $\Omega$ at points $A$ and $B$ intersects at the point $P$ , and the perpendicular bisector of $ (BC)$ cuts line $AC$ at point $Q$. Prove that lines $BC$ and $PQ$ are parallel.
2005 Today's Calculation Of Integral, 3
Calculate the following indefinite integrals.
[1] $\int \sin x\sin 2x dx$
[2] $\int \frac{e^{2x}}{e^x-1}dx$
[3] $\int \frac{\tan ^2 x}{\cos ^2 x}dx$
[4] $\int \frac{e^x+e^{-x}}{e^x-e^{-x}}dx$
[5] $\int \frac{e^x}{e^x+1}dx$
2023 Azerbaijan Senior NMO, 1
The teacher calculates and writes on the board all the numbers $a^b$ that satisfy the condition $n = a\times b$ for the natural number $n.$ Here $a$ and $b$ are natural numbers. Is there a natural number $n$ such that each of the numbers $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ is the last digit of one of the numbers written by the teacher on the board? Justify your opinion.
2003 Pan African, 1
Let $N_0=\{0, 1, 2 \cdots \}$. Find all functions: $N_0 \to N_0$ such that:
(1) $f(n) < f(n+1)$, all $n \in N_0$;
(2) $f(2)=2$;
(3) $f(mn)=f(m)f(n)$, all $m, n \in N_0$.
PEN D Problems, 23
Let $p$ be an odd prime of the form $p=4n+1$. [list=a][*] Show that $n$ is a quadratic residue $\pmod{p}$. [*] Calculate the value $n^{n}$ $\pmod{p}$. [/list]
2019 Hong Kong TST, 3
Find an integral solution of the equation
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \]
(Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to $u$.)
2001 Moldova National Olympiad, Problem 6
Two sides of a quadrilateral $ABCD$ are parallel. Let $M$ and $N$ be the midpoints of $BC$ and $CD$ respectively, and $P$ be the intersection point of $AN$ and $DM$. Prove that if $AP=4PN$, then $ABCD$ is a parallelogram.
MathLinks Contest 6th, 5.3
Let $ABC$ be a triangle, and let $ABB_2A_3$, $BCC_3B_1$ and $CAA_1C_2$ be squares constructed outside the triangle. Denote with $S$ the area of the triangle $ABC$ and with s the area of the triangle formed by the intersection of the lines $A_1B_1$, $B_2C_2$ and $C_3A_3$. Prove that $s \le (4 - 2\sqrt3)S$.
Gheorghe Țițeica 2025, P2
Let $a,b,c$ be three positive real numbers with $ab+bc+ca=4$. Find the minimum value of the expression $$E(a,b,c)=\frac{a^2+b^2}{ab}+\frac{b^2+c^2}{bc}+\frac{c^2+a^2}{ca}-(a-b)^2.$$
Croatia MO (HMO) - geometry, 2019.3
Given an isosceles triangle $ABC$ such that $|AB|=|AC|$ . Let $M$ be the midpoint of the segment $BC$ and let $P$ be a point other than $A$ such that $PA\parallel BC$. The points $X$ and $Y$ are located respectively on rays $PB$ and $PC$, so that the point $B$ is between $P$ and $X$, the point $C$ is between $P$ and $Y$ and $\angle PXM=\angle PYM$. Prove that the points $A,P,X$ and $Y$ are concyclic.
1982 AMC 12/AHSME, 28
A set of consecutive positive integers beginning with $1$ is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is $35\frac{7}{17}$. What number was erased?
$\textbf{(A) } 6\qquad \textbf{(B) }7 \qquad \textbf{(C) }8 \qquad \textbf{(D) } 9\qquad \textbf{(E) }\text{cannot be determined}$
1987 Federal Competition For Advanced Students, P2, 1
The sides $ a,b$ and the bisector of the included angle $ \gamma$ of a triangle are given. Determine necessary and sufficient conditions for such triangles to be constructible and show how to reconstruct the triangle.