Found problems: 85335
2014 India IMO Training Camp, 1
Let $x$ and $y$ be rational numbers, such that $x^{5}+y^{5}=2x^{2}y^{2}$. Prove that $1-xy$ is the square of a rational number.
1997 China National Olympiad, 1
Consider a cyclic quadrilateral $ABCD$. The extensions of its sides $AB,DC$ meet at the point $P$ and the extensions of its sides $AD,BC$ meet at the point $Q$. Suppose $\quad QE,QF$ are tangents to the circumcircle of quadrilateral $ABCD$ at $E,F$ respectively. Show that $P,E,F$ are collinear.
2019 Serbia National MO, 1
Find all positive integers $n, n>1$ for wich holds :
If $a_1, a_2 ,\dots ,a_k$ are all numbers less than $n$ and relatively prime to $n$ , and holds $a_1<a_2<\dots <a_k $, then none of sums $a_i+a_{i+1}$ for $i=1,2,3,\dots k-1 $ are divisible by $3$.
2013 IMO Shortlist, C1
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
2023 Turkey Junior National Olympiad, 4
Let $x_1,x_2,\dots,x_{31}$ be real numbers. Then find the maximum value can
$$\sum_{i,j=1,2,\dots,31, \; i\neq j}{\lceil x_ix_j \rceil }-30\left(\sum_{i=1,2,\dots,31}{\lfloor x_i^2 \rfloor } \right)$$
achieve.
P.S.: For a real number $x$ we denote the smallest integer that does not subseed $x$ by $\lceil x \rceil$ and the biggest integer that does not exceed $x$ by $\lfloor x \rfloor$. For example $\lceil 2.7 \rceil=3$, $\lfloor 2.7 \rfloor=2$ and $\lfloor 4 \rfloor=\lceil 4 \rceil=4$
1998 AMC 12/AHSME, 7
If $ N > 1$, then ${ \sqrt [3] {N \sqrt [3] {N \sqrt [3] {N}}}} =$
$ \textbf{(A)}\ N^{\frac {1}{27}}\qquad
\textbf{(B)}\ N^{\frac {1}{9}}\qquad
\textbf{(C)}\ N^{\frac {1}{3}}\qquad
\textbf{(D)}\ N^{\frac {13}{27}}\qquad
\textbf{(E)}\ N$
2015 India Regional MathematicaI Olympiad, 2
Determine the number of $3-$digit numbers in base $10$ having at least one $5$ and at most one $3$.
2017 All-Russian Olympiad, 3
There are $n$ positive real numbers on the board $a_1,\ldots, a_n$. Someone wants to write $n$ real numbers $b_1,\ldots,b_n$,such that:
$b_i\geq a_i$
If $b_i \geq b_j$ then $\frac{b_i}{b_j}$ is integer.
Prove that it is possible to write such numbers with the condition $$b_1 \cdots b_n \leq 2^{\frac{n-1}{2}}a_1\cdots a_n.$$
2023 AMC 10, 11
A square of area $2$ is inscribed in a square of area $3$, creating four congruent triangles, as shown below. What is the ratio of the shorter leg to the longer leg in the shaded right triangle?
[asy]
size(200);
defaultpen(linewidth(0.6pt)+fontsize(10pt));
real y = sqrt(3);
pair A,B,C,D,E,F,G,H;
A = (0,0);
B = (0,y);
C = (y,y);
D = (y,0);
E = ((y + 1)/2,y);
F = (y, (y - 1)/2);
G = ((y - 1)/2, 0);
H = (0,(y + 1)/2);
fill(H--B--E--cycle, gray);
draw(A--B--C--D--cycle);
draw(E--F--G--H--cycle);
[/asy]
$\textbf{(A) }\frac15\qquad\textbf{(B) }\frac14\qquad\textbf{(C) }2-\sqrt3\qquad\textbf{(D) }\sqrt3-\sqrt2\qquad\textbf{(E) }\sqrt2-1$
2019 BMT Spring, Tie 1
Compute the probability that a random permutation of the letters in BERKELEY does not have the three E’s all on the same side of the Y.
1986 IMO Longlists, 21
Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$
1984 National High School Mathematics League, 6
If $F(\frac{1-x}{1+x})=x$, then
$\text{(A)}F(-2-x)=-2-F(x)\qquad\text{(B)}F(-x)=F(\frac{1+x}{1-x})$
$\text{(C)}F(\frac{1}{x})=F(x)\qquad\text{(D)}F(F(-x))=-x$
2016 Dutch IMO TST, 3
Let $\vartriangle ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $|BF| = |BE|$ and such that $ED$ is the internal angle bisector of $\angle BEC$. Prove that $|BD|= |EF|$ if and only if $|AF| = |EC|$.
2010 ITAMO, 3
Let $ABCD$ be a convex quadrilateral. such that $\angle CAB = \angle CDA$ and $\angle BCA = \angle ACD$. If $M$ be the midpoint of $AB$, prove that $\angle BCM = \angle DBA$.
2018 Argentina National Olympiad, 4
There is a $50\times 50$ grid board.. Carlos is going to write a number in each box with the following procedure. He first chooses $100$ distinct numbers that we denote $f_1,f_2,f_3,…,f_{50},c_1,c_2,c_3,…,c_{50}$ among which there are exactly $50$ that they are rational. Then he writes in each box ($i,j)$ the number $f_i \cdot c_j$ (the multiplication of $f_i$ by $c_j$). Determine the maximum number of rational numbers that the squares on the board can contain.
1987 Bundeswettbewerb Mathematik, 2
Let $n$ be a positive integer and $M=\{1,2,\ldots, n\}.$ A subset $T\subset M$ is called [i]heavy[/i] if each of its elements is greater or equal than $|T|.$ Let $f(n)$ denote the number of heavy subsets of $M.$ Describe a method for finding $f(n)$ and use it to calculate $f(32).$
1967 All Soviet Union Mathematical Olympiad, 093
Given natural number $k$ with a property "if $n$ is divisible by $k$, than the number, obtained from $n$ by reversing the order of its digits is also divisible by $k$". Prove that the $k$ is a divisor of $99$.
2006 Junior Balkan Team Selection Tests - Romania, 4
For a positive integer $n$ denote $r(n)$ the number having the digits of $n$ in reverse order- for example, $r(2006) = 6002$. Prove that for any positive integers a and b the numbers $4a^2 + r(b)$ and $4b^2 + r(a)$ can not be simultaneously squares.
2024 China Team Selection Test, 19
$n$ is a positive integer. An equilateral triangle of side length $3n$ is split into $9n^2$ unit equilateral triangles, each colored one of red, yellow, blue, such that each color appears $3n^2$ times. We call a trapezoid formed by three unit equilateral triangles as a "standard trapezoid". If a "standard trapezoid" contains all three colors, we call it a "colorful trapezoid". Find the maximum possible number of "colorful trapezoids".
2009 Jozsef Wildt International Math Competition, W. 5
Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.
1972 Swedish Mathematical Competition, 5
Show that
\[
\int\limits_0^1 \frac{1}{(1+x)^n} dx > 1-\frac{1}{n}
\]
for all positive integers $n$.
2016 Romania Team Selection Test, 3
A set $S=\{ s_1,s_2,...,s_k\}$ of positive real numbers is "polygonal" if $k\geq 3$ and there is a non-degenerate planar $k-$gon whose side lengths are exactly $s_1,s_2,...,s_k$; the set $S$ is multipolygonal if in every partition of $S$ into two subsets,each of which has at least three elements, exactly one of these two subsets in polygonal. Fix an integer $n\geq 7$.
(a) Does there exist an $n-$element multipolygonal set, removal of whose maximal element leaves a multipolygonal set?
(b) Is it possible that every $(n-1)-$element subset of an $n-$element set of positive real numbers be multipolygonal?
2013 District Olympiad, 2
Problem 2. A group $\left( G,\cdot \right)$ has the propriety$\left( P \right)$, if, for any
automorphism f for G,there are two automorphisms
g and h in G, so that $f\left( x \right)=g\left( x \right)\cdot h\left( x \right)$, whatever $x\in G$would be. Prove that:
(a) Every group which the property $\left( P \right)$ is comutative.
(b) Every commutative finite group of odd order doesn’t have the $\left( P \right)$ property.
(c) No finite group of order $4n+2,n\in \mathbb{N}$, doesn’t have the $\left( P \right)$property.
(The order of a finite group is the number of elements of that group).
2010 All-Russian Olympiad Regional Round, 11.3
Quadrangle $ABCD$ is inscribed in a circle with diameter $AC$. Points $K$ and $M$ are projections of vertices $A$ and $C$, respectively, onto line $BD$. A line parallel to $BC$ is drawn through point $K$ and intersecting $AC$ at point $P$. Prove that angle $KPM$ is a right angle.
2015 Saudi Arabia JBMO TST, 2
Given is a binary string $0101010101$. On a move Ali changes 0 to 1 or 1 to 0. The following conditions are fulfilled:
a) All the strings obtained are different.
b) All the strings obtained must have at least 5 times 1. Prove that Ali can't obtain more than 555 strings.