Found problems: 85335
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
$\text{(A)}\ \frac{5}{16} \qquad \text{(B)}\ \frac{3}{8} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{5}{8} \qquad \text{(E)}\ \frac{11}{16}$
2010 Serbia National Math Olympiad, 3
Let $a_0$ and $a_n$ be different divisors of a natural number $m$, and $a_0, a_1, \ldots, a_n$ be a sequence of natural numbers such that it satisfies
\[a_{i+1} = |a_i\pm a_{i-1}|\text{ for }0 < i < n\]
If $gcd(a_0,a_1,\ldots, a_n) = 1$, show that there exists a term of the sequence that is smaller than $\sqrt{m}$ .
[i]Proposed by Dusan Djukic[/i]
1996 Turkey Junior National Olympiad, 2
Write out the positive integers consisting of only $1$s, $6$s, and $9$s in ascending order as in: $1,6,9,11,16,\dots$.
a. Find the order of $1996$ in the sequence.
b. Find the $1996$th term in the sequence.
1988 IMO Longlists, 84
A point $ M$ is chosen on the side $ AC$ of the triangle $ ABC$ in such a way that the radii of the circles inscribed in the triangles $ ABM$ and $ BMC$ are equal. Prove that
\[ BM^{2} \equal{} X \cot \left( \frac {B}{2}\right)
\]
where X is the area of triangle $ ABC.$
JOM 2013, 1.
Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.
2021 Romania National Olympiad, 4
Let $n \ge 2$ and matrices $A,B \in M_n(\mathbb{R})$. There exist $x \in \mathbb{R} \backslash \{0,\frac{1}{2}, 1 \}$, such that $ xAB + (1-x)BA = I_n$. Show that $(AB-BA)^n = O_n$.
1991 AIME Problems, 2
Rectangle $ABCD$ has sides $\overline {AB}$ of length 4 and $\overline {CB}$ of length 3. Divide $\overline {AB}$ into 168 congruent segments with points $A=P_0, P_1, \ldots, P_{168}=B$, and divide $\overline {CB}$ into 168 congruent segments with points $C=Q_0, Q_1, \ldots, Q_{168}=B$. For $1 \le k \le 167$, draw the segments $\overline {P_kQ_k}$. Repeat this construction on the sides $\overline {AD}$ and $\overline {CD}$, and then draw the diagonal $\overline {AC}$. Find the sum of the lengths of the 335 parallel segments drawn.
2009 Sharygin Geometry Olympiad, 3
The bisectors of trapezoid's angles form a quadrilateral with perpendicular diagonals. Prove that this trapezoid is isosceles.
2011 Preliminary Round - Switzerland, 4
Given is a circular bus route with $n\geqslant2$ bus stops. The route can be frequented in both directions. The way between two stops is called [i]section[/i] and one of the bus stops is called [i]Zürich[/i]. A bus shall start at Zürich, pass through all the bus stops [b]at least once[/b] and drive along exactly $n+2$ sections before it returns to Zürich in the end. Assuming that the bus can chance directions at each bus stop, how many possible routes are there?
EDIT: Sorry, there was a mistake...corrected now, thanks mavropnevma! :oops:
2021 China Team Selection Test, 6
Proof that there exist constant $\lambda$, so that for any positive integer $m(\ge 2)$, and any lattice triangle $T$ in the Cartesian coordinate plane, if $T$ contains exactly one $m$-lattice point in its interior(not containing boundary), then $T$ has area $\le \lambda m^3$.
PS. lattice triangles are triangles whose vertex are lattice points; $m$-lattice points are lattice points whose both coordinates are divisible by $m$.
1985 Tournament Of Towns, (103) 7
(a)The game of "super- chess" is played on a $30 \times 30$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g. $m$ squares forward and $n$ squares to the right) does not depend on the piece 's starting square . Prove that
(i) A piece cannot cap ture a piece on a given square from more than $20$ starting squares.
(ii) It is possible to arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move.
(b) The game of "super-chess" is played on a $100 \times 100$ board and involves $20$ different pieces. Each piece moves according to its own rules , but cannot move from any square to more than $20$ other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g. $m$ squares forward and $n$ squares to the right) may vary, depending on the piece's position .
Prove that one can arrange all $20$ pieces on the board in such a way that not one of them can capture any of the others in one move.
( A . K . Tolpygo, Kiev)
PS. (a) for Juniors , (b) for Seniors
2018 Harvard-MIT Mathematics Tournament, 7
Rachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When she puts the number $x$ in her right pocket, the number changes to $x^{-1}.$ Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E,$ compute $\left\lfloor\frac{E}{10}\right\rfloor.$
1980 Miklós Schweitzer, 8
Let $ f(x)$ be a nonnegative, integrable function on $ (0,2\pi)$ whose Fourier series is $ f(x)\equal{}a_0\plus{}\sum_{k\equal{}1}^{\infty} a_k \cos (n_k x)$, where none of the positive integers $ n_k$ divides another. Prove that $ |a_k| \leq a_0$.
[i]G. Halasz[/i]