Found problems: 85335
KoMaL A Problems 2019/2020, A. 761
Let $n\ge3$ be a positive integer. We say that a set $S$ of positive integers is good if $|S|=n$, no element of S is a multiple of n, and the sum of all elements of $S$ is not a multiple of $n$ either. Find, in terms of $n$, the least positive integer $d$ for which there exists a good set $S$ such that there are exactly d nonempty subsets of $S$ the sum of whose elements is a multiple of $n$.
Proposed by Aleksandar Makelov, Burgas, Bulgaria and Nikolai Beluhov, Stara Zagora, Bulgaria
1979 IMO Longlists, 11
Prove that a pyramid $A_1A_2 \ldots A_{2k+1}S$ with equal lateral edges and equal space angles between adjacent lateral walls is regular.
2012 Vietnam Team Selection Test, 3
Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition:
For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.
VI Soros Olympiad 1999 - 2000 (Russia), 8.1
Let $p,q,r$ be prime numbers such that $2p>q$, $q > 2r$ and $q>p+r$. Prove that $p+q+r\ge 20$.
2010 Tournament Of Towns, 6
A broken line consists of $31$ segments. It has no self intersections, and its start and end points are distinct. All segments are extended to become straight lines. Find the least possible number of straight lines.
2025 AMC 8, 22
A classroom has a row of $35$ coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least $1$ coat and at least $1$ empty hook. How many different numbers of coats can satisfy Paulina's pattern?
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 7\qquad \textbf{(E)}\ 9$\\
(need visuals)
2006 Purple Comet Problems, 14
Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$. Find the sum of all the possible values of $n$ in these ordered pairs.
2022 Harvard-MIT Mathematics Tournament, 7
Let $S = \{(x, y) \in Z^2 | 0 \le x \le 11, 0\le y \le 9\}$. Compute the number of sequences $(s_0, s_1, . . . , s_n)$ of elements in $S$ (for any positive integer $n \ge 2$) that satisfy the following conditions:
$\bullet$ $s_0 = (0, 0)$ and $s_1 = (1, 0)$,
$\bullet$ $s_0, s_1, . . . , s_n$ are distinct,
$\bullet$ for all integers $2 \le i \le n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^o$ or $180^o$ in the
clockwise direction.
2009 Romanian Masters In Mathematics, 4
For a finite set $ X$ of positive integers, let $ \Sigma(X) \equal{} \sum_{x \in X} \arctan \frac{1}{x}.$ Given a finite set $ S$ of positive integers for which $ \Sigma(S) < \frac{\pi}{2},$ show that there exists at least one finite set $ T$ of positive integers for which $ S \subset T$ and $ \Sigma(S) \equal{} \frac{\pi}{2}.$
[i]Kevin Buzzard, United Kingdom[/i]
1985 IMO Longlists, 77
Two equilateral triangles are inscribed in a circle with radius $r$. Let $A$ be the area of the set consisting of all points interior to both triangles. Prove that $2A \geq r^2 \sqrt 3.$
2016 USAMTS Problems, 5:
Consider the set $S = \{ q + \frac{1}{q}, \text{ where } q \text{ ranges over all positive rational numbers} \}$.
(a) Let $N$ be a positive integer. Show that $N$ is the sum of two elements of $S$ if and only if $N$ is the product of two elements of $S$.
(b) Show that there exist infinitely many positive integers $N$ that cannot be written as the sum of two elements of $S$.
(c)Show that there exist infinitely many positive integers $N$ that can be written as the sum of two elements of $S$.
1992 Mexico National Olympiad, 1
The tetrahedron $OPQR$ has the $\angle POQ = \angle POR = \angle QOR = 90^o$. $X, Y, Z$ are the midpoints of $PQ, QR$ and $RP.$ Show that the four faces of the tetrahedron $OXYZ$ have equal area.
2016 Online Math Open Problems, 1
Let $A_n$ denote the answer to the $n$th problem on this contest ($n=1,\dots,30$); in particular, the answer to this problem is $A_1$. Compute $2A_1(A_1+A_2+\dots+A_{30})$.
[i]Proposed by Yang Liu[/i]
2010 Princeton University Math Competition, 2
On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.
1990 Rioplatense Mathematical Olympiad, Level 3, 3
Let $ABCD$ be a trapezium with bases $AB$ and $CD$ such that $AB = 2 CD$. From $A$ the line $r$ is drawn perpendicular to $BC$ and from $B$ the line $t$ is drawn perpendicular to $AD$. Let $P$ be the intersection point of $r$ and $t$. From $C$ the line $s$ is drawn perpendicular to $BC$ and from $D$ the line $u$ perpendicular to $AD$. Let $Q$ be the intersection point of $s$ and $u$. If $R$ is the intersection point of the diagonals of the trapezium, prove that points $P, Q$ and $R$ are collinear.
2000 Putnam, 2
Prove that the expression \[ \dfrac {\text {gcd}(m, n)}{n} \dbinom {n}{m} \] is an integer for all pairs of integers $ n \ge m \ge 1 $.
2014 Peru IMO TST, 13
Let $r$ be a positive integer and let $N$ be the smallest positive integer such that the numbers $\frac{N}{n+r}\binom{2n}{n}$,
$n=0,1,2,\ldots $, are all integer. Show that $N=\frac{r}{2}\binom{2r}{r}$.
2012 Czech-Polish-Slovak Match, 2
City of Mar del Plata is a square shaped $WSEN$ land with $2(n + 1)$ streets that divides it into $n \times n$ blocks, where $n$ is an even number (the leading streets form the perimeter of the square). Each block has a dimension of $100 \times 100$ meters. All streets in Mar del Plata are one-way. The streets which are parallel and adjacent to each other are directed in opposite direction. Street $WS$ is driven in the direction from $W$ to $S$ and the street $WN$ travels from $W$ to $N$. A street cleaning car starts from point $W$. The driver wants to go to the point $E$ and in doing so, he must cross as much as possible roads. What is the length of the longest route he can go, if any $100$-meter stretch cannot be crossed more than once? (The figure shows a plan of the city for $n=6$ and one of the possible - but not the longest - routes of the street cleaning car. See http://goo.gl/maps/JAzD too.)
[img]http://s14.postimg.org/avfg7ygb5/CPS_2012_P5.jpg[/img]
2005 JBMO Shortlist, 2
Let $ABCD$ be an isosceles trapezoid with $AB=AD=BC, AB//CD, AB>CD$. Let $E= AC \cap BD$ and $N$ symmetric to $B$ wrt $AC$. Prove that the quadrilateral $ANDE$ is cyclic.
2024 Iran Team Selection Test, 3
For any real numbers $x , y ,z$ prove that :
$$(x+y+z)^2 + \sum_{cyc}{\frac{(x+y)(y+z)}{1+|x-z|}} \ge xy+yz+zx$$
[i]Proposed by Navid Safaei[/i]
2005 Taiwan National Olympiad, 2
Ten test papers are to be prepared for the National Olympiad. Each paper has 4 problems, and no two papers have more than 1 problem in common. At least how many problems are needed?
2016 China Team Selection Test, 3
In cyclic quadrilateral $ABCD$, $AB>BC$, $AD>DC$, $I,J$ are the incenters of $\triangle ABC$,$\triangle ADC$ respectively. The circle with diameter $AC$ meets segment $IB$ at $X$, and the extension of $JD$ at $Y$. Prove that if the four points $B,I,J,D$ are concyclic, then $X,Y$ are the reflections of each other across $AC$.
1990 Brazil National Olympiad, 3
Each face of a tetrahedron is a triangle with sides $a, b,$c and the tetrahedon has circumradius 1. Find $a^2 + b^2 + c^2$.
2018 Putnam, A5
Let $f: \mathbb{R} \to \mathbb{R}$ be an infinitely differentiable function satisfying $f(0) = 0$, $f(1) = 1$, and $f(x) \ge 0$ for all $x \in \mathbb{R}$. Show that there exist a positive integer $n$ and a real number $x$ such that $f^{(n)}(x) < 0$.
2010 Contests, 4
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.