Found problems: 85335
2012 USA TSTST, 8
Let $n$ be a positive integer. Consider a triangular array of nonnegative integers as follows: \[
\begin{array}{rccccccccc}
\text{Row } 1: &&&&& a_{0,1} &&&& \smallskip\\
\text{Row } 2: &&&& a_{0,2} && a_{1,2} &&& \smallskip\\
&&& \vdots && \vdots && \vdots && \smallskip\\
\text{Row } n-1: && a_{0,n-1} && a_{1,n-1} && \cdots && a_{n-2,n-1} & \smallskip\\
\text{Row } n: & a_{0,n} && a_{1,n} && a_{2,n} && \cdots && a_{n-1,n}
\end{array}
\] Call such a triangular array [i]stable[/i] if for every $0 \le i < j < k \le n$ we have \[ a_{i,j} + a_{j,k} \le a_{i,k} \le a_{i,j} + a_{j,k} + 1. \] For $s_1, \ldots s_n$ any nondecreasing sequence of nonnegative integers, prove that there exists a unique stable triangular array such that the sum of all of the entries in row $k$ is equal to $s_k$.
2021 Thailand TST, 2
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$
2010 Austria Beginners' Competition, 2
In a national park there is a group of sequoia trees, all of which have a positive integer age. Their average age is $41$ years. After a $2010$ year old building was destroyed by lightning, the average age drops to $40$ years. How many trees were originally in the group? At most, how many of them were exactly $2010$ years old?
(W. Janous, WRG Ursulinen, Innsbruck)
2024 5th Memorial "Aleksandar Blazhevski-Cane", P4
Let $D$ be a point inside $\triangle ABC$ such that $\angle CDA + \angle CBA = 180^{\circ}.$ The line $CD$ meets the circle $\odot ABC$ at the point $E$ for the second time. Let $G$ be the common point of the circle centered at $C$ with radius $CD$ and the arc $\overset{\LARGE \frown}{AC}$ of $\odot ABC$ which does not contain the point $B$. The circle centered at $A$ with radius $AD$ meets $\odot BCD$ for the second time at $F$.
Prove that the lines $GE, FD, CB$ are concurrent or parallel.
Kvant 2019, M2543
Let $a$ and $b$ be 2019-digit numbers. Exactly 12 digits of $a$ are non-zero: the five leftmost and seven rightmost, and exactly 14 digits of $b$ are non-zero: the five leftmost and nine rightmost. Prove that the largest common divisor of $a$ and $b$ has no more than 14 digits.
[i]Proposed by L. Samoilov[/i]
2014 Contests, 1
On a circle there are $99$ natural numbers. If $a,b$ are any two neighbouring numbers on the circle, then $a-b$ is equal to $1$ or $2$ or $ \frac{a}{b}=2 $. Prove that there exists a natural number on the circle that is divisible by $3$.
[i]S. Berlov[/i]
2017 Gulf Math Olympiad, 2
One country consists of islands $A_1,A_2,\cdots,A_N$,The ministry of transport decided to build some bridges such that anyone will can travel by car from any of the islands $A_1,A_2,\cdots,A_N$ to any another island by one or more of these bridges. For technical reasons the only bridges that can be built is between $A_i$ and $A_{i+1}$ where $i = 1,2,\cdots,N-1$ , and between $A_i$ and $A_N$ where $i<N$.
We say that a plan to build some bridges is good if it is satisfies the above conditions , but when we remove any bridge it will not satisfy this conditions. We assume that there is $a_N$ of good plans. Observe that $a_1 = 1$ (The only good plan is to not build any bridge) , and $a_2 = 1$ (We build one bridge).
1-Prove that $a_3 = 3$
2-Draw at least $5$ different good plans in the case that $N=4$ and the islands are the vertices of a square
3-Compute $a_4$
4-Compute $a_6$
5-Prove that there is a positive integer $i$ such that $1438$ divides $a_i$
2006 Vietnam Team Selection Test, 3
In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points.
A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on.
Find the minimum value of a [i]good[/i] number $m$.
2021 Science ON all problems, 4
Find the least positive integer which is a multiple of $13$ and all its digits are the same.
[i](Adapted from Gazeta Matematică 1/1982, Florin Nicolăită)[/i]
2013 BMT Spring, 1
A time is called [i]reflexive [/i] if its representation on an analog clock would still be permissible if the hour and minute hand were switched. In a given non-leap day ($12:00:00.00$ a.m. to $11:59:59.99$ p.m.), how many times are reflexive?
2017 Harvard-MIT Mathematics Tournament, 7
[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.
1969 Spain Mathematical Olympiad, 4
A circle of radius $R$ is divided into $8$ equal parts. The points of division are denoted successively by $A, B, C, D, E, F , G$ and $H$. Find the area of the square formed by drawing the chords $AF$ , $BE$, $CH$ and $DG$.
2003 JHMMC 8, 21
The surface area and the volume of a cube are numerically equal. Find the cube’s volume.
1996 IMC, 4
Let $a_{1}=1$, $a_{n}=\frac{1}{n} \sum_{k=1}^{n-1}a_{k}a_{n-k}$ for $n\geq 2$. Show that
i) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}<2^{-\frac{1}{2}}$;
ii) $\limsup_{n\to \infty} |a_{n}|^{\frac{1}{n}}\geq \frac{2}{3}$
2009 Tournament Of Towns, 4
Denote by $[n]!$ the product $ 1 \cdot 11 \cdot 111\cdot ... \cdot \underbrace{111...1}_{\text{n ones}}$.($n$ factors in total). Prove that $[n + m]!$ is divisible by $ [n]! \times [m]!$
[i](8 points)[/i]
1999 China Team Selection Test, 3
Let $S = \lbrace 1, 2, \ldots, 15 \rbrace$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:
[b]I.[/b] $|A_i| = 7, i = 1, 2, \ldots, n$;
[b]II.[/b] $|A_i \cap A_j| \leq 3, 1 \leq i < j \leq n$
[b]III.[/b] For any 3-element subset $M$ of $S$, there exists $A_k$ such
that $M \subset A_k$.
Find the smallest possible value of $n$.
2010 Today's Calculation Of Integral, 597
In space given a board shaped the equilateral triangle $PQR$ with vertices $P\left(1,\ \frac 12,\ 0\right),\ Q\left(1,-\frac 12,\ 0\right),\ R\left(\frac 14,\ 0,\ \frac{\sqrt{3}}{4}\right)$. When $S$ is revolved about the $z$-axis, find the volume of the solid generated by the whole points through which $S$ passes.
1984 Tokyo University entrance exam/Science
1994 Tournament Of Towns, (424) 1
Nuts are placed in boxes. The mean value of the number of nuts in a box is $10$, and the mean value of the squares of the numbers of nuts in the boxes is less than $1000$. Prove that at least $10\%$ of the boxes are not empty.
(AY Belov)
2020 USAMTS Problems, 3:
Find, with proof, all positive integers $n$ with the following property: There are only finitely many positive multiples of $n$ which have exactly $n$ positive divisors
2008 China Team Selection Test, 1
Prove that in a plane, arbitrary $ n$ points can be overlapped by discs that the sum of all the diameters is less than $ n$, and the distances between arbitrary two are greater than $ 1$. (where the distances between two discs that have no common points are defined as that the distances between its centers subtract the sum of its radii; the distances between two discs that have common points are zero)
2019 LIMIT Category B, Problem 12
Find the number of rational solutions of the following equations (i.e., rational $x$ and $y$ satisfy the equations)
$$x^2+y^2=2$$$$x^2+y^2=3$$$\textbf{(A)}~2\text{ and }2$
$\textbf{(B)}~2\text{ and }0$
$\textbf{(C)}~2\text{ and infinitely many}$
$\textbf{(D)}~\text{Infinitely many and }0$
2007 Croatia Team Selection Test, 7
Let $a,b,c>0$ such that $a+b+c=1$. Prove: \[\frac{a^{2}}b+\frac{b^{2}}c+\frac{c^{2}}a \ge 3(a^{2}+b^{2}+c^{2}) \]
2013 IPhOO, 1
A block of mass $m$ on a frictionless inclined plane of angle $\theta$ is connected by a cord over a small frictionless, massless pulley to a second block of mass $M$ hanging vertically, as shown. If $M=1.5m$, and the acceleration of the system is $\frac{g}{3}$, where $g$ is the acceleration of gravity, what is $\theta$, in degrees, rounded to the nearest integer?
[asy]size(12cm);
pen p=linewidth(1), dark_grey=gray(0.25), ll_grey=gray(0.90), light_grey=gray(0.75);
pair B = (-1,-1);
pair C = (-1,-7);
pair A = (-13,-7);
path inclined_plane = A--B--C--cycle;
draw(inclined_plane, p);
real r = 1; // for marking angles
draw(arc(A, r, 0, degrees(B-A))); // mark angle
label("$\theta$", A + r/1.337*(dir(C-A)+dir(B-A)), (0,0), fontsize(16pt)); // label angle as theta
draw((C+(-r/2,0))--(C+(-r/2,r/2))--(C+(0,r/2))); // draw right angle
real h = 1.2; // height of box
real w = 1.9; // width of box
path box = (0,0)--(0,h)--(w,h)--(w,0)--cycle; // the box
// box on slope with label
picture box_on_slope;
filldraw(box_on_slope, box, light_grey, black);
label(box_on_slope, "$m$", (w/2,h/2));
pair V = A + rotate(90) * (h/2 * dir(B-A)); // point with distance l/2 from AB
pair T1 = dir(125); // point of tangency with pulley
pair X1 = intersectionpoint(T1--(T1 - rotate(-90)*(2013*dir(T1))), V--(V+B-A)); // construct midpoint of right side of box
draw(T1--X1); // string
add(shift(X1-(w,h/2))*rotate(degrees(B-A), (w,h/2)) * box_on_slope);
// picture for the hanging box
picture hanging_box;
filldraw(hanging_box, box, light_grey, black);
label(hanging_box, "$M$", (w/2,h/2));
pair T2 = (1,0);
pair X2 = (1,-3);
draw(T2--X2); // string
add(shift(X2-(w/2,h)) * hanging_box);
// Draws the actual pulley
filldraw(unitcircle, grey, p); // outer boundary of pulley wheel
filldraw(scale(0.4)*unitcircle, light_grey, p); // inner boundary of pulley wheel
path pulley_body=arc((0,0),0.3,-40,130)--arc((-1,-1),0.5,130,320)--cycle; // defines "arm" of pulley
filldraw(pulley_body, ll_grey, dark_grey+p); // draws the arm
filldraw(scale(0.18)*unitcircle, ll_grey, dark_grey+p); // inner circle of pulley[/asy][i](Proposed by Ahaan Rungta)[/i]
2021 Canadian Mathematical Olympiad Qualification, 7
If $A, B$ and $C$ are real angles such that
$$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$
find
$$\cos (A)+\cos (B)+\cos (C)$$
2005 France Team Selection Test, 2
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).
Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.