Found problems: 85335
2023 VN Math Olympiad For High School Students, Problem 10
Given a triangle $ABC$ with [i]Lemoine[/i] point $L.$ Choose points $X,Y,Z$ on the segments $LA,LB,LC,$ respectively such that:$$\angle XBA=\angle YAB,\angle XCA=\angle ZAC.$$
Prove that: $\angle ZBC=\angle YCB.$
2003 AMC 12-AHSME, 18
Let $ x$ and $ y$ be positive integers such that $ 7x^5 \equal{} 11y^{13}$. The minimum possible value of $ x$ has a prime factorization $ a^cb^d$. What is $ a \plus{} b \plus{} c \plus{} d$?
$ \textbf{(A)}\ 30 \qquad \textbf{(B)}\ 31 \qquad \textbf{(C)}\ 32 \qquad \textbf{(D)}\ 33 \qquad \textbf{(E)}\ 34$
2013 India Regional Mathematical Olympiad, 4
Find the number of $10$-tuples $(a_1,a_2,\dots,a_9,a_{10})$ of integers such that $|a_1|\leq 1$ and
\[a_1^2+a_2^2+a_3^2+\cdots+a_{10}^2-a_1a_2-a_2a_3-a_3a_4-\cdots-a_9a_{10}-a_{10}a_1=2.\]
2017 USAMTS Problems, 5
Does there exist a set $S$ consisting of rational numbers with the following property:
for every integer $n$ there is a unique nonempty, finite subset of $S$, whose elements sum to $n$?
1994 Baltic Way, 17
In a certain kingdom, the king has decided to build $25$ new towns on $13$ uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections.
2010 Iran Team Selection Test, 1
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that
\[f(a+1), f(a+2), \dots, f(a+n)\]
form an arithmetic progression.
2007 Oral Moscow Geometry Olympiad, 1
The triangle was divided into five triangles similar to it. Is it true that the original triangle is right-angled?
(S. Markelov)
2015 China Team Selection Test, 2
Let $a_1,a_2,a_3, \cdots $ be distinct positive integers, and $0<c<\frac{3}{2}$ . Prove that : There exist infinitely many positive integers $k$, such that $[a_k,a_{k+1}]>ck $.
2022 Belarusian National Olympiad, 11.3
$2021$ points are marked on a circle. $2021$ segments with marked endpoints are drawn. After that one counts the number of different points where some $2$ drawn segments intersect(endpoints of segments do [b]not[/b] count as intersections)
Find the maximum number one can get.
1987 Spain Mathematical Olympiad, 3
A given triangle is divided into $n$ triangles in such a way that any line segment which is a side of a tiling triangle is either a side of another tiling triangle or a side of the given triangle. Let $s$ be the total number of sides and $v$ be the total number of vertices of the tiling triangles (counted without multiplicity).
(a) Show that if $n$ is odd then such divisions are possible, but each of them has the same number $v$ of vertices and the same number $s$ of sides. Express $v$ and $s$ as functions of $n$.
(b) Show that, for $n$ even, no such tiling is possible
2002 Nordic, 4
Eva, Per and Anna play with their pocket calculators. They choose different integers and check, whether or not they are divisible by ${11}$. They only look at nine-digit numbers consisting of all the digits ${1, 2, . . . , 9}$. Anna claims that the probability of such a number to be a multiple of ${11}$ is exactly ${1/11}$. Eva has a different opinion: she thinks the probability is less than ${1/11}$. Per thinks the probability is more than ${1/11}$. Who is correct?
2016 ASDAN Math Tournament, 6
Rectangle $ABCD$ has $AB=20$ and $BC=15$. $2$ circles with diameters $AB$ and $AC$ intersect again at point $E$. What is the length of $DE$?
2011 National Olympiad First Round, 4
How many subsets, which does not contain two consecutive numbers, are there of the set $\{1,2,\dots ,20\}$ with $8$ elements?
$\textbf{(A)}\ {{13}\choose{8}} \qquad\textbf{(B)}\ {{13}\choose{9}} \qquad\textbf{(C)}\ {{14}\choose{8}} \qquad\textbf{(D)}\ {{14}\choose{9}} \qquad\textbf{(E)}\ {{20}\choose{15}}$
1990 IMO Longlists, 86
Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.
2009 District Round (Round II), 3
$A,B,C$ are the three angles in a triangle such that
$2\sin B\sin (A+B)-\cos A=1$,
$2\sin C\sin (B+C)-\cos B=0$
find the three angles.
1982 IMO Longlists, 16
Let $p(x)$ be a cubic polynomial with integer coefficients with leading coefficient $1$ and with one of its roots equal to the product of the other two. Show that $2p(-1)$ is a multiple of $p(1)+p(-1)-2(1+p(0)).$
2010 Germany Team Selection Test, 3
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
2000 German National Olympiad, 2
For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.
2017 ASDAN Math Tournament, 4
How many $6$-digit positive integers have their digits in nondecreasing order from left to right? Note that $0$ cannot be a leading digit.
1998 Belarus Team Selection Test, 3
Let $1=d_1<d_2<d_3<...<d_k=n$ be all different divisors of positive integer $n$ written in ascending order. Determine all $n$ such that $$d_7^2+d_{10}^2=(n/d_{22})^2.$$
2021 Saudi Arabia IMO TST, 2
Find all positive integers $n$, such that $n$ is a perfect number and $\varphi (n)$ is power of $2$.
[i]Note:a positive integer $n$, is called perfect if the sum of all its positive divisors is equal to $2n$.[/i]
2017 Finnish National High School Mathematics Comp, 4
Let $m$ be a positive integer.
Two players, Axel and Elina play the game HAUKKU ($m$) proceeds as follows:
Axel starts and the players choose integers alternately. Initially, the set of integers is the set of positive divisors of a positive integer $m$ .The player in turn chooses one of the remaining numbers, and removes that number and all of its multiples from the list of selectable numbers. A player who has to choose number $1$, loses. Show that the beginner player, Axel, has a winning strategy in the HAUKKU ($m$) game for all $m \in Z_{+}$.
PS. As member Loppukilpailija noted, it should be written $m>1$, as the statement does not hold for $m = 1$.
2018 Iran MO (3rd Round), 2
Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$)
Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.
2013 Math Prize For Girls Problems, 7
In the figure below, $\triangle ABC$ is an equilateral triangle.
[asy]
import graph;
unitsize(60);
axes("$x$", "$y$", (0, 0), (1.5, 1.5), EndArrow);
real w = sqrt(3) - 1;
pair A = (1, 1);
pair B = (0, w);
pair C = (w, 0);
draw(A -- B -- C -- cycle);
dot(Label("$A(1, 1)$", A, NE), A);
dot(Label("$B$", B, W), B);
dot(Label("$C$", C, S), C);
[/asy]
Point $A$ has coordinates $(1, 1)$, point $B$ is on the positive $y$-axis, and point $C$ is on the positive $x$-axis. What is the area of $\triangle ABC$?
1980 Miklós Schweitzer, 2
Let $ \mathcal{H}$ be the class of all graphs with at most $ 2^{\aleph_0}$ vertices not containing a complete subgraph of size $ \aleph_1$. Show that there is no graph $ H \in \mathcal{H}$ such that every graph in $ \mathcal{H}$ is a subgraph of $ H$.
[i]F. Galvin[/i]