Found problems: 85335
2018 AIME Problems, 7
Triangle $ABC$ has sides $AB=9,BC = 5\sqrt{3},$ and $AC=12$. Points $A=P_0, P_1, P_2, \dots, P_{2450} = B$ are on segment $\overline{AB}$ with $P_k$ between $P_{k-1}$ and $P_{k+1}$ for $k=1,2,\dots,2449$, and points $A=Q_0, Q_1, Q_2, \dots ,Q_{2450} = C$ for $k=1,2,\dots,2449$. Furthermore, each segment $\overline{P_kQ_k}, k=1,2,\dots,2449$, is parallel to $\overline{BC}$. The segments cut the triangle into $2450$ regions, consisting of $2449$ trapezoids and $1$ triangle. Each of the $2450$ regions have the same area. Find the number of segments $\overline{P_kQ_k}, k=1,2 ,\dots,2450$, that have rational length.
2014 Indonesia MO Shortlist, G5
Given a cyclic quadrilateral $ABCD$. Suppose $E, F, G, H$ are respectively the midpoint of the sides $AB, BC, CD, DA$. The line passing through $G$ and perpendicular on $AB$ intersects the line passing through $H$ and perpendicular on $BC$ at point $K$. Prove that $\angle EKF = \angle ABC$.
2009 Princeton University Math Competition, 8
Find the largest positive integer $k$ such that $\phi ( \sigma ( 2^k)) = 2^k$. ($\phi(n)$ denotes the number of positive integers that are smaller than $n$ and relatively prime to $n$, and $\sigma(n)$ denotes the sum of divisors of $n$). As a hint, you are given that $641|2^{32}+1$.
2003 Hong kong National Olympiad, 3
Let $K, L, M, N$ be the midpoints of sides $AB, BC, CD, DA$ of a cyclic quadrilateral $ABCD$. Prove that the orthocentres of triangles $ANK, BKL, CLM, DMN$ are the vertices of a parallelogram.
2017 CHKMO, Q2
Let k be a positive integer. Find the number of non-negative integers n less than or equal to $10^k$ satisfying the following conditions:
(i) n is divisible by 3;
(ii) Each decimal digit of n is one of the digits 2,0,1 or 7.
2017 Iran Team Selection Test, 1
Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$
$$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$
[i]Proposed by Navid Safaei[/i]
2018-2019 SDML (High School), 8
Five consecutive positive integers have the property that the sum of the second, third, and fourth is a perfect square, while the sum of all five is a perfect cube. If $m$ is the first of these five integers, then the minimum possible value of $m$ satisfies
$ \mathrm{(A) \ } m \leq 200 \qquad \mathrm{(B) \ } 200 < m \leq 400 \qquad \mathrm {(C) \ } 400 < m \leq 600 \qquad \mathrm{(D) \ } 600 < m \leq 800 \qquad \mathrm{(E) \ } 800 < m$
2012 Cono Sur Olympiad, 6
6. Consider a triangle $ABC$ with $1 < \frac{AB}{AC} < \frac{3}{2}$. Let $M$ and $N$, respectively, be variable points of the sides $AB$ and $AC$, different from $A$, such that $\frac{MB}{AC} - \frac{NC}{AB} = 1$. Show that circumcircle of triangle $AMN$ pass through a fixed point different from $A$.
2007 QEDMO 4th, 8
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer
2012 Putnam, 5
Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.
2010 Paraguay Mathematical Olympiad, 3
In a triangle $ABC$, let $M$ be the midpoint of $AC$. If $BC = \frac{2}{3} MC$ and $\angle{BMC}=2 \angle{ABM}$, determine $\frac{AM}{AB}$.
2012 Macedonia National Olympiad, 3
Find all functions $f : \mathbb{R} \to \mathbb{Z}$ which satisfy the conditions:
$f(x+y) < f(x) + f(y)$
$f(f(x)) = \lfloor {x} \rfloor + 2$
2016 Romania National Olympiad, 4
In order to study a certain ancient language, some researchers formatted its discovered words into expressions formed by concatenating letters from an alphabet containing only two letters. Along the study, they noticed that any two distinct words whose formatted expressions have an equal number of letters, greater than $ 2, $ differ by at least three letters.
Prove that if their observation holds indeed, then the number of formatted expressions that have $ n\ge 3 $ letters is at most $ \left[ \frac{2^n}{n+1} \right] . $
2004 Irish Math Olympiad, 5
Suppose $p,q$ are distinct primes and $S$ is a subset of $\{1,2,\dots ,p-1\}$. Let $N(S)$ denote the number of solutions to the equation $$\sum_{i=1}^{q}x_i\equiv 0\mod p$$
where $x_i\in S$, $i=1,2,\dots ,q$. Prove that $N(S)$ is a multiple of $q$.
2010 Iran MO (3rd Round), 2
in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$.
a) in what ratio does $EF$ cut the digonals?(13 points)
b) find $\frac{AF}{FD}$.(5 points)
2016 China Team Selection Test, 2
Find the smallest positive number $\lambda $ , such that for any complex numbers ${z_1},{z_2},{z_3}\in\{z\in C\big| |z|<1\}$ ,if $z_1+z_2+z_3=0$, then $$\left|z_1z_2 +z_2z_3+z_3z_1\right|^2+\left|z_1z_2z_3\right|^2 <\lambda .$$
2006 Tournament of Towns, 3
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)
2020 Princeton University Math Competition, 4
Find the number of points $P \in Z^2$ that satisfy the following two conditions:
1) If $Q$ is a point on the circle of radius $\sqrt{2020}$ centered at the origin such that the line $PQ$ is tangent to the circle at $Q$, then $PQ$ has integral length.
2) The x-coordinate of $P$ is $38$.
2010 LMT, 12
$a,b,c,d,e$ are equal to $1,2,3,4,5$ in some order, such that no two of $a,b,c,d,e$ are equal to the same integer. Given that $b \leq d, c \geq a,a \leq e,b \geq e,$ and that $d\neq5,$ determine the value of $a^b+c^d+e.$
2021 Thailand TSTST, 2
Let $f:\mathbb{R}^+\to\mathbb{R}^+$ be such that $$f(x+f(y))^2\geq f(x)\left(f(x+f(y))+f(y)\right)$$ for all $x,y\in\mathbb{R}^+$. Show that $f$ is [i]unbounded[/i], i.e. for each $M\in\mathbb{R}^+$, there exists $x\in\mathbb{R}^+$ such that $f(x)>M$.
2018 Oral Moscow Geometry Olympiad, 5
The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.
1997 China Team Selection Test, 3
There are 1997 pieces of medicine. Three bottles $A, B, C$ can contain at most 1997, 97, 19 pieces of medicine respectively. At first, all 1997 pieces are placed in bottle $A$, and the three bottles are closed. Each piece of medicine can be split into 100 part. When a bottle is opened, all pieces of medicine in that bottle lose a part each. A man wishes to consume all the medicine. However, he can only open each of the bottles at most once each day, consume one piece of medicine, move some pieces between the bottles, and close them. At least how many parts will be lost by the time he finishes consuming all the medicine?
2021 Balkan MO Shortlist, G3
Let $ABC$ be a triangle with $AB<AC$. Let $\omega$ be a circle passing through $B, C$ and assume that $A$ is inside $\omega$. Suppose $X, Y$ lie on $\omega$ such that $\angle BXA=\angle AYC$. Suppose also that $X$ and $C$ lie on opposite sides of the line $AB$ and that $Y$ and $B$ lie on opposite sides of the line $AC$. Show that, as $X, Y$ vary on $\omega$, the line $XY$ passes through a fixed point.
[i]Proposed by Aaron Thomas, UK[/i]
1995 Nordic, 2
Messages are coded using sequences consisting of zeroes and ones only. Only sequences with at most two consecutive ones or zeroes are allowed. (For instance the sequence $011001$ is allowed, but $011101$ is not.) Determine the number of sequences consisting of exactly $12$ numbers.
2019 Greece JBMO TST, 2
Find all pairs of positive integers $(x,n) $ that are solutions of the equation $3 \cdot 2^x +4 =n^2$.