Found problems: 85335
2017 Peru IMO TST, 16
Let $n$ and $k$ be positive integers. A simple graph $G$ does not contain any cycle whose length be an odd number greater than $1$ and less than $ 2k + 1$. If $G$ has at most $n + \frac{(k-1) (n-1) (n+2)}{2}$ vertices, prove that the vertices of $G$ can be painted with $n$ colors in such a way that any edge of $G$ has its ends of different colors.
2020 Brazil Team Selection Test, 4
Let $n$ be an odd positive integer. Some of the unit squares of an $n\times n$ unit-square board are colored green. It turns out that a chess king can travel from any green unit square to any other green unit squares by a finite series of moves that visit only green unit squares along the way. Prove that it can always do so in at most $\tfrac{1}{2}(n^2-1)$ moves. (In one move, a chess king can travel from one unit square to another if and only if the two unit squares share either a corner or a side.)
[i]Proposed by Nikolai Beluhov[/i]
2013 Vietnam Team Selection Test, 1
The $ABCD$ is a cyclic quadrilateral with no parallel sides inscribed in circle $(O, R)$. Let $E$ be the intersection of two diagonals and the angle bisector of $AEB$ cut the lines $AB, BC, CD, DA$ at $M, N, P, Q$ respectively .
a) Prove that the circles $(AQM), (BMN), (CNP), (DPQ)$ are passing through a point. Call that point $K$.
b) Denote $min \,\{AC, BD\} = m$. Prove that $OK \le \dfrac{2R^2}{\sqrt{4R^2-m^2}}$.
1995 Rioplatense Mathematical Olympiad, Level 3, 2
In a circle of center $O$ and radius $r$, a triangle $ABC$ of orthocenter $H$ is inscribed. It is considered a triangle $A'B'C'$ whose sides have by length the measurements of the segments $AB, CH$ and $2r$. Determine the triangle $ABC$ so that the area of the triangle $A'B'C'$ is maximum.
2024 Taiwan Mathematics Olympiad, 4
Suppose $O$ is the circumcenter of $\Delta ABC$, and $E, F$ are points on segments $CA$ and $AB$ respectively with $E, F \neq A$. Let $P$ be a point such that $PB = PF$ and $PC = PE$.
Let $OP$ intersect $CA$ and $AB$ at points $Q$ and $R$ respectively. Let the line passing through $P$ and perpendicular to $EF$ intersect $CA$ and $AB$ at points $S$ and $T$ respectively. Prove that points $Q, R, S$, and $T$ are concyclic.
[i]Proposed by Li4 and usjl[/i]
Brazil L2 Finals (OBM) - geometry, 2001.6
An altitude of a convex quadrilateral is a line through the midpoint of a side perpendicular to the opposite side. Show that the four altitudes are concurrent iff the quadrilateral is cyclic.
1974 Czech and Slovak Olympiad III A, 2
Let a triangle $ABC$ be given. For any point $X$ of the triangle denote $m(X)=\min\{XA,XB,XC\}.$ Find all points $X$ (of triangle $ABC$) such that $m(X)$ is maximal.
1986 Traian Lălescu, 2.1
Find the real values $ m\in\mathbb{R} $ such that all solutions of the equation
$$ 1=2mx(2x-1)(2x-2)(2x-3) $$
are real.
1975 All Soviet Union Mathematical Olympiad, 209
Denote the midpoints of the convex hexagon $A_1A_2A_3A_4A_5A_6$ diagonals $A_6A_2$, $A_1A_3$, $A_2A_4$, $A_3A_5$, $A_4A_6$, $A_5A_1$ as $B_1, B_2, B_3, B_4, B_5, B_6$ respectively. Prove that if the hexagon $B_1B_2B_3B_4B_5B_6$ is convex, than its area equals to the quarter of the initial hexagon.
2009 Harvard-MIT Mathematics Tournament, 10
Given a rearrangement of the numbers from $1$ to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a < b$) or decreasing (if $b < a$). How many rearrangements of the numbers from $1$ to $n$ have exactly two increasing pairs of consecutive elements? Express your answer in terms of $n$.
2000 Junior Balkan Team Selection Tests - Romania, 3
Let be a real number $ a. $ For any real number $ p $ and natural number $ k, $ let be the set
$$ A_k(p)=\{ px\in\mathbb{Z}\mid k=\lceil x \rceil \} . $$
Find all real numbers $ b $ such that $ \# A_n(a)=\# A_n(b) , $ for any natural number $ n. $
$ \# $ [i]denotes the cardinal.[/i]
[i]Eugen Păltânea[/i]
1978 AMC 12/AHSME, 26
[asy]
import cse5;
size(180);
real a=4, b=3;
pathpen=black;
pair A=(a,0), B=(0,b), C=(0,0);
D(MP("A",A)--MP("B",B,N)--MP("C",C,SW)--cycle);
pair X=IP(B--A,(0,0)--(b,a));
D(CP((X+C)/2,C));
D(MP("R",IP(CP((X+C)/2,C),B--C),NW)--MP("Q",IP(CP((X+C)/2,C),A--C+(0.1,0))));
//Credit to chezbgone2 for the diagram[/asy]
In $\triangle ABC$, $AB = 10~ AC = 8$ and $BC = 6$. Circle $P$ is the circle with smallest radius which passes through $C$ and is tangent to $AB$. Let $Q$ and $R$ be the points of intersection, distinct from $C$ , of circle $P$ with sides $AC$ and $BC$, respectively. The length of segment $QR$ is
$\textbf{(A) }4.75\qquad\textbf{(B) }4.8\qquad\textbf{(C) }5\qquad\textbf{(D) }4\sqrt{2}\qquad \textbf{(E) }3\sqrt{3}$
2006 Lithuania National Olympiad, 3
Show that if $a+b+c=0$ then $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})=9$.
1984 IMO Longlists, 44
Let $a,b,c$ be positive numbers with $\sqrt{a}+\sqrt{b}+\sqrt{c}= \frac{\sqrt{3}}{2}$
Prove that the system of equations
\[\sqrt{y-a}+\sqrt{z-a}=1\]
\[\sqrt{z-b}+\sqrt{x-b}=1\]
\[\sqrt{x-c}+\sqrt{y-c}=1\]
has exactly one solution $(x,y,z)$ in real numbers.
It was proposed by Poland. Have fun! :lol:
2021 Albanians Cup in Mathematics, 3
Let $\mathcal{S}$ be a set consisting of $n \ge 3$ positive integers, none of which is a sum of two other distinct members of $\mathcal{S}$. Prove that the elements of $\mathcal{S}$ may be ordered as $a_1, a_2, \dots, a_n$ so that $a_i$ does not divide $a_{i - 1} + a_{i + 1}$ for all $i = 2, 3, \dots, n - 1$.
1999 Federal Competition For Advanced Students, Part 2, 1
Ninety-nine points are given on one of the diagonals of a unit square. Prove that there is at most one vertex of the square such that the average squared distance from a given point to the vertex is less than or equal to $1/2$.
2014 Oral Moscow Geometry Olympiad, 1
In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.
2007 CentroAmerican, 1
The Central American Olympiad is an annual competition. The ninth Olympiad is held in 2007. Find all the positive integers $n$ such that $n$ divides the number of the year in which the $n$-th Olympiad takes place.
2001 South africa National Olympiad, 3
For a certain real number $x$, the differences between $x^{1919}$, $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.
2018 Sharygin Geometry Olympiad, 8
Consider a fixed regular $n$-gon of unit side. When a second regular $n$-gon of unit size rolls around the first one, one of its vertices successively pinpoints the vertices of a closed broken line $\kappa$ as in the figure.
[asy]
int n=9;
draw(polygon(n));
for (int i = 0; i<n;++i) {
draw(reflect(dir(360*i/n + 90), dir(360*(i+1)/n + 90))*polygon(n), dashed+linewidth(0.4));
draw(reflect(dir(360*i/n + 90),dir(360*(i+1)/n + 90))*(0,1)--reflect(dir(360*(i-1)/n + 90),dir(360*i/n + 90))*(0,1), linewidth(1.2));
}
[/asy]
Let $A$ be the area of a regular $n$-gon of unit side, and let $B$ be the area of a regular $n$-gon of unit circumradius. Prove that the area enclosed by $\kappa$ equals $6A-2B$.
2018 ELMO Shortlist, 2
We say that a positive integer $n$ is $m$[i]-expressible[/i] if it is possible to get $n$ from some $m$ digits and the six operations $+,-,\times,\div$, exponentiation $^\wedge$, and concatenation $\oplus$. For example, $5625$ is $3$-expressible (in two ways): both $5\oplus (5^\wedge 4)$ and $(7\oplus 5)^\wedge 2$ yield $5625$.
Does there exist a positive integer $N$ such that all positive integers with $N$ digits are $(N-1)$-expressible?
[i]Proposed by Krit Boonsiriseth[/i]
2010 Today's Calculation Of Integral, 531
(1) Let $ f(x)$ be a continuous function defined on $ [a,\ b]$, it is known that there exists some $ c$ such that
\[ \int_a^b f(x)\ dx \equal{} (b \minus{} a)f(c)\ (a < c < b)\]
Explain the fact by using graph. Note that you don't need to prove the statement.
(2) Let $ f(x) \equal{} a_0 \plus{} a_1x \plus{} a_2x^2 \plus{} \cdots\cdots \plus{} a_nx^n$,
Prove that there exists $ \theta$ such that
\[ f(\sin \theta) \equal{} a_0 \plus{} \frac {a_1}{2} \plus{} \frac {a_3}{3} \plus{} \cdots\cdots \plus{} \frac {a_n}{n \plus{} 1},\ 0 < \theta < \frac {\pi}{2}.\]
1974 IMO Shortlist, 9
Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that
\[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]
2009 Postal Coaching, 5
Define a sequence $<x_n>$ by $x_1 = 1, x_2 = x, x_{n+2} = xx_{n+1} + nx_n, n \ge 1$.
Consider the polynomial $P_n(x) = x_{n-1}x_{n+1} - x_n^2$, for each $n \ge 2$.
Prove or disprove that the coefficients of $P_n(x)$ are all non-negative, except for the constant term when $n$ is odd.
2010 Contests, 1
Given that $a,b,c > 0$ and $a + b + c = 1$. Prove that $\sqrt {\frac{{ab}}{{ab + c}}} + \sqrt {\frac{{bc}}{{bc + a}}} + \sqrt {\frac{{ca}}{{ca + b}}} \leqslant \frac{3}{2}$.