Found problems: 85335
2023 Czech-Polish-Slovak Match, 1
Given an integer $n\geq 3$, determine the smallest positive number $k$ such that any two points in any $n$-gon (or at its boundary) in the plane can be connected by a polygonal path consisting of $k$ line segments contained in the $n$-gon (including its boundary).
2019 Purple Comet Problems, 4
The diagram below shows a sequence of equally spaced parallel lines with a triangle whose vertices lie on these lines. The segment $\overline{CD}$ is $6$ units longer than the segment $\overline{AB}$. Find the length of segment $\overline{EF}$.
[img]https://cdn.artofproblemsolving.com/attachments/8/0/abac87d63d366bf4c4e913fdb1022798379a73.png[/img]
2017 Balkan MO Shortlist, C2
Let $n,a,b,c$ be natural numbers. Every point on the coordinate plane with integer coordinates is colored in one of $n$ colors. Prove there exists $c$ triangles whose vertices are colored in the same color, which are pairwise congruent, and which have a side whose lenght is divisible by $a$ and a side whose lenght is divisible by $b$.
2021 Taiwan TST Round 1, 5
For each prime $p$, construct a graph $G_p$ on $\{1,2,\ldots p\}$, where $m\neq n$ are adjacent if and only if $p$ divides $(m^{2} + 1-n)(n^{2} + 1-m)$. Prove that $G_p$ is disconnected for infinitely many $p$
1986 AMC 12/AHSME, 13
A parabola $y = ax^{2} + bx + c$ has vertex $(4,2)$. If $(2,0)$ is on the parabola, then $abc$ equals
$ \textbf{(A)}\ -12\qquad\textbf{(B)}\ -6\qquad\textbf{(C)}\ 0\qquad\textbf{(D)}\ 6\qquad\textbf{(E)}\ 12$
1996 All-Russian Olympiad, 6
Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders:
(a) Each day, at least one task must be issued to a soldier.
(b) No soldier may have more than two task or receive more than one tasks in a single day.
(c) The lists of soldiers receiving tasks for two different days must not be the same.
(d) The first sergeant violating any of these orders will be jailed.
Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed?
[i]M. Kulikov[/i]
2008 Princeton University Math Competition, A8/B9
Find the polynomial $f$ with the following properties:
$\bullet$ its leading coefficient is $1$,
$\bullet$ its coefficients are nonnegative integers,
$\bullet$ $72|f(x)$ if $x$ is an integer,
$\bullet$ if $g$ is another polynomial with the same properties, then $g - f$ has a nonnegative leading coecient.
2011 HMNT, 9
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. The inscribed circles of triangles $ABD$ and $ACD$ are tangent to $AD$ at $P$ and $Q$, respectively, and are tangent to $BC$ at $X$ and $Y$ , respectively. Let $PX$ and $QY$ meet at $Z$. Determine the area of triangle $XY Z$.
2008 Dutch Mathematical Olympiad, 1
Suppose we have a square $ABCD$ and a point $S$ in the interior of this square.
Under homothety with centre $S$ and ratio of magnification $k > 1$, this square becomes another square $A'B'C'D'$.
Prove that the sum of the areas of the two quadrilaterals $A'ABB'$ and $C'CDD'$ are equal to the sum of the areas of the two quadrilaterals $B'BCC'$ and $D'DAA'$.
[asy]
unitsize(3 cm);
pair[] A, B, C, D;
pair S;
A[1] = (0,1);
B[1] = (0,0);
C[1] = (1,0);
D[1] = (1,1);
S = (0.3,0.6);
A[0] = interp(S,A[1],2/3);
B[0] = interp(S,B[1],2/3);
C[0] = interp(S,C[1],2/3);
D[0] = interp(S,D[1],2/3);
draw(A[0]--B[0]--C[0]--D[0]--cycle);
draw(A[1]--B[1]--C[1]--D[1]--cycle);
draw(A[1]--S, dashed);
draw(B[1]--S, dashed);
draw(C[1]--S, dashed);
draw(D[1]--S, dashed);
dot("$A$", A[0], N);
dot("$B$", B[0], SE);
dot("$C$", C[0], SW);
dot("$D$", D[0], SE);
dot("$A'$", A[1], NW);
dot("$B'$", B[1], SW);
dot("$C'$", C[1], SE);
dot("$D'$", D[1], NE);
dot("$S$", S, dir(270));
[/asy]
2015 ASDAN Math Tournament, 3
Place points $A$, $B$, $C$, $D$, $E$, and $F$ evenly spaced on a unit circle. Compute the area of the shaded $12$-sided region, where the region is bounded by line segments $AD$, $DF$, $FB$, $BE$, $EC$, and $CA$.
[center]<see attached>[/center]
2007 Cuba MO, 2
A prism is called [i]binary [/i] if it can be assigned to each of its vertices a number from the set $\{-1, 1\}$, such that the product of the numbers assigned to the vertices of each face is equal to $-1$.
a) Prove that the number of vertices of the binary prisms is divisible for $8$.
b) Prove that a prism with $2000$ vertices is binary.
2023 BMT, Tie 3
Compute the real solution for$ x$ to the equation $$(4^x + 8)^4 - (8^x - 4)^4 = (4 + 8^x + 4^x)^4.$$
2025 Harvard-MIT Mathematics Tournament, 8
Define $\text{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0.$ Compute $$\sum_{n=1}^{\infty} \frac{\text{sgn}(\sin(2^n))}{2^n}.$$ (The arguments to $\sin$ are in radians.)
Oliforum Contest I 2008, 2
Let $ \{a_n\}_{n \in \mathbb{N}_0}$ be a sequence defined as follows: $ a_1=0$, $ a_n=a_{[\frac{n}{2}]}+(-1)^{n(n+1)/2}$, where $ [x]$ denotes the floor function. For every $ k \ge 0$, find the number $ n(k)$ of positive integers $ n$ such that $ 2^k \le n < 2^{k+1}$ and $ a_n=0$.
2002 Turkey Team Selection Test, 3
Consider $2n+1$ points in space, no four of which are coplanar where $n>1$. Each line segment connecting any two of these points is either colored red, white or blue. A subset $M$ of these points is called a [i]connected monochromatic[/i] subset, if for each $a,b \in M$, there are points $a=x_0,x_1, \dots, x_l = b$ that belong to $M$ such that the line segments $x_0x_1, x_1x_2, \dots, x_{l-1}x_1$ are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic $k-$subset, find the largest value of $k$. ($l > 1$)
1999 National Olympiad First Round, 3
Four boxes with ball capacity $3, 5, 7,$ and $8$ are given. In how many ways can $19$ same balls be put into these boxes?
$\textbf{(A)}\ 34 \qquad\textbf{(B)}\ 35 \qquad\textbf{(C)}\ 36 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ \text{None}$
2012 Switzerland - Final Round, 9
Let $a, b, c > 0$ be real numbers with $abc = 1$. Show
$$1 + ab + bc + ca \ge \min \left\{ \frac{(a + b)^2}{ab} , \frac{(b+c)^2}{bc} , \frac{(c + a)^2}{ca}\right\}.$$
When does equality holds?
2006 Portugal MO, 5
Determine all the natural numbers $n$ such that exactly one fifth of the natural numbers $1,2,...,n$ are divisors of $n$.
2004 AMC 10, 6
Bertha has $ 6$ daughters and no sons. Some of her daughters have $ 6$ daughters and the rest have none. Bertha has a total of $ 30$ daughters and granddaughters, and no great-grand daughters. How many of Bertha's daughters and granddaughters have no daughters?
$ \textbf{(A)}\ 22\qquad
\textbf{(B)}\ 23\qquad
\textbf{(C)}\ 24\qquad
\textbf{(D)}\ 25\qquad
\textbf{(E)}\ 26$
1975 IMO Shortlist, 6
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)
2015 HMNT, 4
Chords $AB$ and $CD$ of a circle are perpendicular and intersect at a point $P$. If $AP = 6, BP = 12$, and $CD = 22$, find the area of the circle.
2010 All-Russian Olympiad Regional Round, 11.4
We call a triple of natural numbers $(a, b, c)$ [i]square [/i] if they form an arithmetic progression (in exactly this order), the number $b$ is coprime to each of the numbers $a$ and $c$, and the number $abc$ is a perfect square. Prove that for any given a square triple, there is another square triple that has at least one common number with it.
2007 Stanford Mathematics Tournament, 14
Let there be 50 natural numbers $ a_i$ such that $ 0 < a_1 < a_2 < ... < a_{50} < 150$. What is the greatest possible sum of the differences $ d_j$ where each $ d_j \equal{} a_{j \plus{} 1} \minus{} a_j$?
2025 239 Open Mathematical Olympiad, 1
There are $100$ points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
2021 IMO Shortlist, G2
Let $\Gamma$ be a circle with centre $I$, and $A B C D$ a convex quadrilateral such that each of the segments $A B, B C, C D$ and $D A$ is tangent to $\Gamma$. Let $\Omega$ be the circumcircle of the triangle $A I C$. The extension of $B A$ beyond $A$ meets $\Omega$ at $X$, and the extension of $B C$ beyond $C$ meets $\Omega$ at $Z$. The extensions of $A D$ and $C D$ beyond $D$ meet $\Omega$ at $Y$ and $T$, respectively. Prove that \[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]
[i]Proposed by Dominik Burek, Poland and Tomasz Ciesla, Poland[/i]