Found problems: 85335
2003 Turkey MO (2nd round), 3
An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ rows and $ n$ columns , the percentage of $ 1$s on that row or column is not less than $ a$ or greater than $ 100\minus{}a$. Find the largest beautiful number.
2018 AMC 12/AHSME, 10
How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations?
\begin{align*}x+3y&=3\\ \big||x|-|y|\big|&=1\end{align*}
$\textbf{(A) } 1 \qquad
\textbf{(B) } 2 \qquad
\textbf{(C) } 3 \qquad
\textbf{(D) } 4 \qquad
\textbf{(E) } 8 $
2009 Thailand Mathematical Olympiad, 8
Let $a, b, c$ be side lengths of a triangle, and define $s =\frac{a+b+c}{2}$. Prove that
$$\frac{2a(2a-s)}{b + c}+\frac{2b(2b - s)}{c + a}+\frac{2c(2c - s)}{a + b}\ge s.$$
2015 AMC 8, 2
Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?
$\textbf{(A) }\frac{11}{32} \qquad\textbf{(B) }\frac{3}{8} \qquad\textbf{(C) }\frac{13}{32} \qquad\textbf{(D) }\frac{7}{16}\qquad \textbf{(E) }\frac{15}{32}$
[asy]
pair A,B,C,D,E,F,G,H,O,X;
A=dir(45);
B=dir(90);
C=dir(135);
D=dir(180);
E=dir(-135);
F=dir(-90);
G=dir(-45);
H=dir(0);
O=(0,0);
X=midpoint(A--B);
fill(X--B--C--D--E--O--cycle,rgb(0.75,0.75,0.75));
draw(A--B--C--D--E--F--G--H--cycle);
dot("$A$",A,dir(45));
dot("$B$",B,dir(90));
dot("$C$",C,dir(135));
dot("$D$",D,dir(180));
dot("$E$",E,dir(-135));
dot("$F$",F,dir(-90));
dot("$G$",G,dir(-45));
dot("$H$",H,dir(0));
dot("$X$",X,dir(135/2));
dot("$O$",O,dir(0));
draw(E--O--X);
[/asy]
1985 Traian Lălescu, 1.4
Two planes, $ \alpha $ and $ \beta, $ form a dihedral angle of $ 30^{\circ} , $ and their intersection is the line $ d. $ A point $ A $ situated at the exterior of this angle projects itself in $ P\not\in d $ on $ \alpha , $ and in $ Q\not\in d $ on $ \beta $ such that $ AQ<AP. $ Name $ B $ the projection of $ A $ upon $ d. $
[b]a)[/b] Are $ A,B,P,Q, $ coplanar?
[b]b)[/b] Knowing that a perpendicular to $ \beta $ make with $ AB $ an angle of $ 60^{\circ} , $ and $ AB=4, $ find the area of $ BPQ. $
2021 Indonesia TST, A
Given a polynomial $p(x) =Ax^3+x^2-A$ with $A \neq 0$. Show that for every different real number $a,b,c$, at least one of $ap(b)$, $bp(a)$, and $cp(a)$ not equal to 1.
2024 Auckland Mathematical Olympiad, 11
It is known that for quadratic polynomials $P(x)=x^2+ax+b$ and $Q(x)=x^2+cx+d$ the equation $P(Q(x))=Q(P(x))$ does not have real roots. Prove that $b \neq d$.
2012 AMC 8, 10
How many 4-digit numbers greater than 1000 are there that use the four digits of 2012?
$\textbf{(A)}\hspace{.05in}6 \qquad \textbf{(B)}\hspace{.05in}7 \qquad \textbf{(C)}\hspace{.05in}8 \qquad \textbf{(D)}\hspace{.05in}9 \qquad \textbf{(E)}\hspace{.05in}12 $
2023 Sharygin Geometry Olympiad, 18
Restore a bicentral quadrilateral $ABCD$ if the midpoints of the arcs $AB,BC,CD$ of its circumcircle are given.
2004 Alexandru Myller, 1
Let be a nonnegative integer $ n $ and three real numbers $ a,b,c $ satisfying
$$ a^n+c=b^n+a=c^n+b=a+b+c. $$
Show that $ a=b=c. $
[i]Gheorghe Iurea[/i]
2012 China Team Selection Test, 2
Find all integers $k\ge 3$ with the following property: There exist integers $m,n$ such that $1<m<k$, $1<n<k$, $\gcd (m,k)=\gcd (n,k) =1$, $m+n>k$ and $k\mid (m-1)(n-1)$.
2019 Sharygin Geometry Olympiad, 23
In the plane, let $a$, $b$ be two closed broken lines (possibly self-intersecting), and $K$, $L$, $M$, $N$ be four points. The vertices of $a$, $b$ and the points $K$ $L$, $M$, $N$ are in general position (i.e. no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments $KL$ and $MN$ meets $a$ at an even number of points, and each of segments $LM$ and $NK$ meets $a$ at an odd number of points. Conversely, each of segments $KL$ and $MN$ meets $b$ at an odd number of points, and each of segments $LM$ and $NK$ meets $b$ at an even number of points. Prove that $a$ and $b$ intersect.
2019 MIG, 10
$40$ people, numbered $1$ through $40$ counterclockwise, sit around a circular table. They begin playing a game. Each person is initially considered "alive". Starting with person $1$, the first person eliminates the closest "alive" person to their right (so Person $1$ eliminates Person $2$). Then the next "alive" person, moving counterclockwise, eliminates the closest "alive" person to their right (so since Person $2$ is eliminated, Person $3$ eliminates Person $4$). This process continues until there is only $1$ "alive" person remaining. What is the number of the last "alive" person?
[asy]
usepackage("cancel", "makeroom, thicklines");
usepackage("bm");
size(15cm);
picture p;
draw(p, circle((0,0), 5));
for(int i = 0; i < 4; ++i) {
label(p, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
label(p, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80));
}
int n = 20;
for(int i = 0; i <= n; ++i) {
label(p, scale(2)*"$\cdot$", 6 *dir(180 / n * i));
}
draw(p, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow);
add(shift(-20, 0) * p);
draw((-11, 0)--(-8,0), EndArrow);
picture q;
draw(q, circle((0,0), 5));
for(int i = 0; i < 4; ++i) {
label(q, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
if(i != 1) label(q, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80));
}
int n = 20;
for(int i = 0; i <= n; ++i) {
label(q, scale(2)*"$\cdot$", 6 *dir(180 / n * i));
}
draw(q, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow);
for(int i = 0; i < 1; i+=2) {
//label(q, "\bm\xcancel{~}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
label(q, "\xcancel{2}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80));
}
add(q);
draw((9,0)--(12,0), EndArrow);
picture r;
draw(r, circle((0,0), 5));
for(int i = 0; i < 4; ++i) {
if(i % 2 == 1) label(r, "$" + string(40 - i) + "$", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
if(i % 2 != 1) label(r, "$" + string(i + 1) + "$", 5 * dir(20 * i - 80), 2 * dir(20 * i - 80));
}
int n = 20;
for(int i = 0; i <= n; ++i) {
label(r, scale(2)*"$\cdot$", 6 *dir(180 / n * i));
}
draw(r, arc((0,0), 8 * dir(-80), 8 * dir(0)), EndArrow);
for(int i = 0; i < 4; i+=2) {
label(r, "\xcancel{" + string(40 - i) +"}", 5 * dir(-20 * i - 100), 2 * dir(-20 * i - 100));
label(r, "\xcancel{" + string(i + 1) + "}", 5 * dir(20 * (i + 1) - 80), 2 * dir(20 * (i + 1) - 80));
}
add(shift(20, 0) * r);
[/asy]
[center]In the last step here, Person $39$ eliminates Person $40$. Next turn, Person $1$ eliminates the closest person to his right, Person $3$.[/center]
2010 Kosovo National Mathematical Olympiad, 4
Let $(p_1,p_2,..., p_n)$ be a random permutation of the set $\{1,2,...,n)$. If $n$ is odd, prove that the product
$(p_1-1)\cdot (p_2-2)\cdot ...\cdot (p_n-n)$
is an even number.
@below fixed.
2012 Indonesia TST, 3
Given a convex quadrilateral $ABCD$, let $P$ and $Q$ be points on $BC$ and $CD$ respectively such that $\angle BAP = \angle DAQ$. Prove that the triangles $ABP$ and $ADQ$ have the same area if the line connecting their orthocenters is perpendicular to $AC$.
2012 Today's Calculation Of Integral, 778
In the $xyz$ space with the origin $O$, Let $K_1$ be the surface and inner part of the sphere centered on the point $(1,\ 0,\ 0)$ with radius 2 and let $K_2$ be the surface and inner part of the sphere centered on the point $(-1,\ 0,\ 0)$ with radius 2. For three points $P,\ Q,\ R$ in the space, consider points $X,\ Y$ defined by
\[\overrightarrow{OX}=\overrightarrow{OP}+\overrightarrow{OQ},\ \overrightarrow{OY}=\frac 13(\overrightarrow{OP}+\overrightarrow{OQ}+\overrightarrow{OR}).\]
(1) When $P,\ Q$ move every cranny in $K_1,\ K_2$ respectively, find the volume of the solid generated by the whole points of the point $X$.
(2) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1$.
(3) Find the volume of the solid generated by the whole points of the point $R$ for which for any $P$ belonging to $K_1$ and any $Q$ belonging to $K_2$, $Y$ belongs to $K_1\cup K_2$.
2017-IMOC, C3
Alice and Bob play the following game: Initially, there is a $2016\times2016$ "empty" matrix. Taking turns, with Alice playing first, each player chooses a real number and fill it into an empty entry. If the determinant of the last matrix is non-zero, then Alice wins. Otherwise, Bob wins. Who has the winning strategy?
2022 BMT, 12
Let circles $C_1$ and $C_2$ be internally tangent at point $P$, with $C_1$ being the smaller circle. Consider a line passing through $P$ which intersects $C_1$ at $Q$ and $C_2$ at $R$. Let the line tangent to $C_2$ at $R$ and the line perpendicular to $\overline{PR}$ passing through $Q$ intersect at a point $S$ outside both circles. Given that $SR = 5$, $RQ = 3$, and $QP = 2$, compute the radius of $C_2$.
1990 China National Olympiad, 5
Given a finite set $X$, let $f$ be a rule such that $f$ maps every [i]even-element-subset[/i] $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions:
(I) there exists an [i]even-element-subset[/i] $D$ of $X$ such that $f(D)>1990$;
(II) for any two disjoint [i]even-element-subsets [/i]$A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds.
Prove that there exist two subsets $P,Q$ of $X$ satisfying:
(1) $P\cap Q=\emptyset$, $P\cup Q=X$;
(2) for any [i]non-even-element-subset [/i]$S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$;
(3) for any [i]even-element-subset[/i] $T$ of $Q$, we have $f(T)\le 1990$.
2022 Brazil Team Selection Test, 4
Let $d_1, d_2, \ldots, d_n$ be given integers. Show that there exists a graph whose sequence of degrees is $d_1, d_2, \ldots, d_n$ and which contains an perfect matching if, and only if, there exists a graph whose sequence of degrees is $d_2, d_2, \ldots, d_n$ and a graph whose sequence of degrees is $d_1-1, d_2-1, \ldots, d_n-1$.
2018 Saudi Arabia GMO TST, 1
Let $\{x_n\}$ be a sequence defined by $x_1 = 2$ and $x_{n+1} = x_n^2 - x_n + 1$ for $n \ge 1$. Prove that
$$1 -\frac{1}{2^{2^{n-1}}} < \frac{1}{x_1}+\frac{1}{x_2}+ ... +\frac{1}{x_n}< 1 -\frac{1}{2^{2^n}}$$
for all $n$
2010 Malaysia National Olympiad, 1
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12.
2018 Moscow Mathematical Olympiad, 3
$a_1,a_2,...,a_k$ are positive integers and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1$. Prove that equation $$[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n$$ has no more than $a_1*a_2*...*a_k$ postivie integer solutions in $n$.
2006 AMC 12/AHSME, 17
Square $ ABCD$ has side length $ s$, a circle centered at $ E$ has radius $ r$, and $ r$ and $ s$ are both rational. The circle passes through $ D$, and $ D$ lies on $ \overline{BE}$. Point $ F$ lies on the circle, on the same side of $ \overline{BE}$ as $ A$. Segment $ AF$ is tangent to the circle, and $ AF \equal{} \sqrt {9 \plus{} 5\sqrt {2}}$. What is $ r/s$?
[asy]unitsize(6mm);
defaultpen(linewidth(.8pt)+fontsize(10pt));
dotfactor=3;
pair B=(0,0), C=(3,0), D=(3,3), A=(0,3);
pair Ep=(3+5*sqrt(2)/6,3+5*sqrt(2)/6);
pair F=intersectionpoints(Circle(A,sqrt(9+5*sqrt(2))),Circle(Ep,5/3))[0];
pair[] dots={A,B,C,D,Ep,F};
draw(A--F);
draw(Circle(Ep,5/3));
draw(A--B--C--D--cycle);
dot(dots);
label("$A$",A,NW);
label("$B$",B,SW);
label("$C$",C,SE);
label("$D$",D,SW);
label("$E$",Ep,E);
label("$F$",F,NW);[/asy]$ \textbf{(A) } \frac {1}{2}\qquad \textbf{(B) } \frac {5}{9}\qquad \textbf{(C) } \frac {3}{5}\qquad \textbf{(D) } \frac {5}{3}\qquad \textbf{(E) } \frac {9}{5}$
2002 AMC 8, 21
Harold tosses a nickel four times. The probability that he gets at least as many heads as tails is
$\text{(A)}\ \frac{5}{16} \qquad \text{(B)}\ \frac{3}{8} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{5}{8} \qquad \text{(E)}\ \frac{11}{16}$