Found problems: 85335
2007 Bulgaria Team Selection Test, 3
Let $I$ be the center of the incircle of non-isosceles triangle $ABC,A_{1}=AI\cap BC$ and $B_{1}=BI\cap AC.$ Let $l_{a}$ be the line through $A_{1}$ which is parallel to $AC$ and $l_{b}$ be the line through $B_{1}$ parallel to $BC.$ Let $l_{a}\cap CI=A_{2}$ and $l_{b}\cap CI=B_{2}.$ Also $N=AA_{2}\cap BB_{2}$ and $M$ is the midpoint of $AB.$ If $CN\parallel IM$ find $\frac{CN}{IM}$.
MOAA Gunga Bowls, 2021.15
Let $a,b,c,d$ be the four roots of the polynomial
\[x^4+3x^3-x^2+x-2.\]
Given that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}=\frac{1}{2}$ and $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{d^2}=-\frac{3}{4}$, the value of
\[\frac{1}{a^3}+\frac{1}{b^3}+\frac{1}{c^3}+\frac{1}{d^3}\]
can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
1996 Romania National Olympiad, 2
Let $ABCD$ a tetrahedron and $M$ a variable point on the face $BCD$. The line perpendicular to $(BCD)$ in $M$ . intersects the planes$ (ABC)$, $(ACD)$, and $(ADB)$ in $M_1$, $M_2$, and $M_3$. Show that the sum $MM_1 + MM_2 + MM_3$ is constant if and only if the perpendicular dropped from $A$ to $(BCD)$ passes through the centroid of triangle $BCD$.
2003 Hong kong National Olympiad, 1
Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$
2010 Harvard-MIT Mathematics Tournament, 9
Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.
JOM 2013, 5.
Consider a triangle $ABC$ with height $AH$ and $H$ on $BC$. Let $\gamma_1$ and $\gamma_2$ be the circles with diameter $BH,CH$ respectively, and let their centers be $O_1$ and $O_2$. Points $X,Y$ lie on $\gamma_1,\gamma_2$ respectively such that $AX,AY$ are tangent to each circle and $X,Y,H$ are all distinct. $P$ is a point such that $PO_1$ is perpendicular to $BX$ and $PO_2$ is perpendicular to $CY$.
Prove that the circumcircles of $PXY$ and $AO_1O_2$ are tangent to each other.
2016 Harvard-MIT Mathematics Tournament, 8
For each positive integer $n$ and non-negative integer $k$, define $W(n,k)$ recursively by
\[
W(n,k) =
\begin{cases}
n^n & k = 0 \\
W(W(n,k-1), k-1) & k > 0.
\end{cases}
\]
Find the last three digits in the decimal representation of $W(555,2)$.
2009 Germany Team Selection Test, 1
For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i \equal{} 1} (A_i \plus{} k)$ is a power for each natural number $ k.$
1982 IMO Shortlist, 19
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
2013 India Regional Mathematical Olympiad, 2
In a triangle $ABC$, $AD$ is the altitude from $A$, and $H$ is the orthocentre. Let $K$ be the centre of the circle passing through $D$ and tangent to $BH$ at $H$. Prove that the line $DK$ bisects $AC$.
2013 NIMO Problems, 1
At ARML, Santa is asked to give rubber duckies to $2013$ students, one for each student. The students are conveniently numbered $1,2,\cdots,2013$, and for any integers $1 \le m < n \le 2013$, students $m$ and $n$ are friends if and only if $0 \le n-2m \le 1$.
Santa has only four different colors of duckies, but because he wants each student to feel special, he decides to give duckies of different colors to any two students who are either friends or who share a common friend. Let $N$ denote the number of ways in which he can select a color for each student. Find the remainder when $N$ is divided by $1000$.
[i]Proposed by Lewis Chen[/i]
2023 Malaysian IMO Training Camp, 3
Let $ABC$ be an acute triangle with $AB\neq AC$. Let $D, E, F$ be the midpoints of the sides $BC$, $CA$, and $AB$ respectively, and $M, N$ be the midpoints of minor arc $BC$ not containing $A$ and major arc $BAC$ respectively. Suppose $W, X, Y, Z$ are the incenter, $D$-excenter, $E$-excenter, and $F$-excenter of triangle $DEF$ respectively.
Prove that the circumcircles of the triangles $ABC$, $WNX$, $YMZ$ meet at a common point.
[i]Proposed by Ivan Chan Kai Chin[/i]
2019 Estonia Team Selection Test, 3
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.
1981 Swedish Mathematical Competition, 3
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
2012 Czech-Polish-Slovak Junior Match, 4
A rhombus $ABCD$ is given with $\angle BAD = 60^o$ . Point $P$ lies inside the rhombus such that $BP = 1$, $DP = 2$, $CP = 3$. Determine the length of the segment $AP$.
2000 AMC 10, 24
Let $f$ be a function for which $f\left(\frac x3\right)=x^2+x+1$. Find the sum of all values of $z$ for which $f(3z)=7$.
$\text{(A)}\ -\frac13\qquad\text{(B)}\ -\frac19 \qquad\text{(C)}\ 0 \qquad\text{(D)}\ \frac59 \qquad\text{(E)}\ \frac53$
2003 Iran MO (3rd Round), 5
Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$.
If not, then what is $S$ congruent to $\mod p$ ?
2018 Canadian Open Math Challenge, C1
Source: 2018 Canadian Open Math Challenge Part C Problem 1
-----
At Math-$e^e$-Mart, cans of cat food are arranged in an pentagonal pyramid of 15 layers high, with 1 can in the top layer, 5 cans in the second layer, 12 cans in the third layer, 22 cans in the fourth layer etc, so that the $k^{\text{th}}$ layer is a pentagon with $k$ cans on each side.
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNC9lLzA0NTc0MmM2OGUzMWIyYmE1OGJmZWQzMGNjMGY1NTVmNDExZjU2LnBuZw==&rn=YzFhLlBORw==[/img][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYS9hLzA1YWJlYmE1ODBjMzYwZDFkYWQyOWQ1YTFhOTkzN2IyNzJlN2NmLnBuZw==&rn=YzFiLlBORw==[/img][/center]
$\text{(a)}$ How many cans are on the bottom, $15^{\text{th}}$,
[color=transparent](A.)[/color]layer of this pyramid?
$\text{(b)}$ The pentagonal pyramid is rearranged into a prism consisting of 15 identical layers.
[color=transparent](B.)[/color]How many cans are on the bottom layer of the prism?
$\text{(c)}$ A triangular prism consist of indentical layers, each of which has a shape of a triangle.
[color=transparent](C.)[/color](the number of cans in a triangular layer is one of the triangular numbers: 1,3,6,10,...)
[color=transparent](C.)[/color]For example, a prism could be composed of the following layers:
[center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMi85L2NlZmE2M2Y3ODhiN2UzMTRkYzIxY2MzNjFmMDJkYmE0ZTJhMTcwLnBuZw==&rn=YzFjLlBORw==[/img][/center]
Prove that a pentagonal pyramid of cans with any number of layers $l\ge 2$ can be rearranged (without a deficit or leftover) into a triangluar prism of cans with the same number of layers $l$.
2019 Brazil Team Selection Test, 4
Let $p \geq 7$ be a prime number and $$S = \bigg\{jp+1 : 1 \leq j \leq \frac{p-5}{2}\bigg\}.$$ Prove that at least one element of $S$ can be written as $x^2+y^2$, where $x, y$ are integers.
2023 SG Originals, Q3
Let $n$ be a positive integer. There are $n$ islands with $n-1$ bridges connecting them such that one can travel from any island to another. One afternoon, a fire breaks out in one of the islands. Every morning, it spreads to all neighbouring islands. (Two islands are neighbours if they are connected by a bridge.) To control the spread, one bridge is destroyed every night until the fire has nowhere to spread the next day. Let $X$ be the minimum possible number of bridges one has to destroy before the fire stops spreading. Find the maximum possible value of $X$ over all possible configurations of bridges and island where the fire starts at.
2016 CCA Math Bonanza, I1
Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$
[i]2016 CCA Math Bonanza Individual Round #1[/i]
2006 IberoAmerican Olympiad For University Students, 7
Consider the multiplicative group $A=\{z\in\mathbb{C}|z^{2006^k}=1, 0<k\in\mathbb{Z}\}$ of all the roots of unity of degree $2006^k$ for all positive integers $k$.
Find the number of homomorphisms $f:A\to A$ that satisfy $f(f(x))=f(x)$ for all elements $x\in A$.
Kyiv City MO Juniors 2003+ geometry, 2017.9.5
Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle.
(Danilo Hilko)
1982 Tournament Of Towns, (031) 5
The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.
2024 Harvard-MIT Mathematics Tournament, 10
Suppose point $P$ is inside quadrilateral $ABCD$ such that
$\angle{PAB} = \angle{PDA}, \angle{PAD} = \angle{PDC}, \angle{PBA} = \angle{PCB}, \angle{PBC} = \angle{PCD}.$
If $PA = 4, PB = 5,$ and $PC = 10$, compute the perimeter of $ABCD$.