Found problems: 85335
2004 District Olympiad, 4
Divide a $ 2\times 4 $ rectangle into $ 8 $ unit squares to obtain a set of $ 15 $ vertices denoted by $ \mathcal{M} . $ Find the points $ A\in\mathcal{M} $ that have the property that the set $ \mathcal{M}\setminus \{ A\} $ can form $ 7 $ pairs $ \left( A_1,B_1\right) ,\left( A_2,B_2\right) ,\ldots ,\left( A_7,B_7\right)\in\mathcal{M}\times\mathcal{M} $ such that
$$ \overrightarrow{A_1B_1} +\overrightarrow{A_2B_2} +\cdots +\overrightarrow{A_7B_7} =\overrightarrow{O} . $$
PEN M Problems, 25
Let $\{a_{n}\}_{n \ge 1}$ be a sequence of positive integers such that \[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\] Show that there are infinitely many pairs $(p, q)$ of positive integers such that $p>q$ and $a_{q}\; \vert \; a_{p}$.
2021 India National Olympiad, 3
Betal marks $2021$ points on the plane such that no three are collinear, and draws all possible segments joining these. He then chooses any $1011$ of these segments, and marks their midpoints. Finally, he chooses a segment whose midpoint is not marked yet, and challenges Vikram to construct its midpoint using [b]only[/b] a straightedge. Can Vikram always complete this challenge?
[i]Note.[/i] A straightedge is an infinitely long ruler without markings, which can only be used to draw the line joining any two given distinct points.
[i]Proposed by Prithwijit De and Sutanay Bhattacharya[/i]
2020 Turkey Team Selection Test, 1
Find all pairs of $(a,b)$ positive integers satisfying the equation:
$$\frac {a^3+b^3}{ab+4}=2020$$
2017 ASDAN Math Tournament, 10
Let $\zeta=e^{2\pi i/36}$. Compute
$$\prod_{\stackrel{a=1}{\gcd(a,36)=1}}^{35}(\zeta^a-2).$$
2024 CCA Math Bonanza, I10
Let $ABC$ be a triangle with side lengths $AB = 7$, $BC = 8$, and $CA = 9$. Let $O$ be the circumcenter of $\triangle ABC$, and let $AO$, $BO$, $CO$ intersect the circumcircle of $\triangle ABC$ again at $D$, $E$, and $F$, respectively. The area of convex hexagon $AFBDCE$ can be expressed as $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is square-free. Find $m + n$.
[i]Individual #10[/i]
2003 Chile National Olympiad, 5
Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)
2005 China Western Mathematical Olympiad, 6
In isosceles right-angled triangle $ABC$, $CA = CB = 1$. $P$ is an arbitrary point on the sides of $ABC$. Find the maximum of $PA \cdot PB \cdot PC$.
2005 Alexandru Myller, 4
Let $K$ be a finite field and $f:K\to K^*$. Prove that there is a reducible polynomial $P\in K[X]$ s.t. $P(x)=f(x),\forall x\in K$.
[i]Marian Andronache[/i]
2024 Princeton University Math Competition, 11
Austen has a regular icosahedron ($20$-sided polyhedron with all triangular faces). He randomly chooses $3$ distinct points among the vertices and constructs the circle through these three points. The expected value of the total number of the icosahedron’s vertices that lie on this circle can be written as $m/n$ for relatively prime positive integers $m$ and $n.$ Find $m + n.$
2017 Korea Winter Program Practice Test, 3
Let $\triangle ABC$ be a triangle with $\angle A \neq 60^\circ$. Let $I_B, I_C$ be the $B, C$-excenters of triangle $ABC$, let $B^\prime$ be the reflection of $B$ with respect to $AC$, and let $C^\prime$ be the reflection of $C$ with respect to $AB$. Let $P$ be the intersection of $I_C B^\prime$ and $I_B C^\prime$. Denote by $P_A, P_B, P_C$ the reflections of the point $P$ with respect to $BC, CA, AB$. Show that the three lines $A P_A, B P_B, C P_C$ meet at a single point.
2013 Danube Mathematical Competition, 3
Show that, for every integer $r \ge 2$, there exists an $r$-chromatic simple graph (no loops, nor multiple edges) which has no cycle of less than $6$ edges
1998 Dutch Mathematical Olympiad, 2
Let $TABCD$ be a pyramid with top vertex $T$, such that its base $ABCD$ is a square of side length 4. It is given that, among the triangles $TAB$, $TBC$, $TCD$ and $TDA$, one can find an isosceles triangle and a right-angled triangle. Find all possible values for the volume of the pyramid.
2016 China National Olympiad, 2
In $\triangle AEF$, let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$. Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively.
Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.
2018 Caucasus Mathematical Olympiad, 8
Let $a, b, c$ be the lengths of sides of a triangle. Prove the inequality
$$(a+b)\sqrt{ab}+(a+c)\sqrt{ac}+(b+c)\sqrt{bc} \geq (a+b+c)^2/2.$$
2018 USAMTS Problems, 2:
Let $n>1$ be an integer. There are $n$ orangutoads, conveniently numbered $1,2,\dots{},n$, each sitting at an integer position on the number line. They take turns moving in the order $1,2,\dots{},n$, and then going back to $1$ to start the process over; they stop if any orangutoad is ever unable to move. To move, an orangutoad chooses another orangutoad who is at least $2$ units away from her towards them by a a distance of $1$ unit. (Multiple orangutoads can be at the same position.) Show that eventually some orangutoad will be unable to move.
2022 JBMO TST - Turkey, 6
Let $c$ be a real number. If the inequality
$$f(c)\cdot f(-c)\ge f(a)$$
holds for all $f(x)=x^2-2ax+b$ where $a$ and $b$ are arbitrary real numbers, find all possible values of $c$.
2020 SJMO, 3
Let $O$ and $\Omega$ denote the circumcenter and circumcircle, respectively, of scalene triangle $\triangle ABC$. Furthermore, let $M$ be the midpoint of side $BC$. The tangent to $\Omega$ at $A$ intersects $BC$ and $OM$ at points $X$ and $Y$, respectively. If the circumcircle of triangle $\triangle OXY$ intersects $\Omega$ at two distinct points $P$ and $Q$, prove that $PQ$ bisects $\overline{AM}$.
[i]Proposed by Andrew Wen[/i]
PEN A Problems, 28
Prove that the expression \[\frac{\gcd(m, n)}{n}{n \choose m}\] is an integer for all pairs of positive integers $(m, n)$ with $n \ge m \ge 1$.
1994 Turkey Team Selection Test, 3
Find all integer pairs $(a,b)$ such that $a\cdot b$ divides $a^2+b^2+3$.
2011 India Regional Mathematical Olympiad, 2
Let $n$ be a positive integer such that $2n+1$ and $3n+1$ are both perfect squares. Show that $5n+3$ is a composite number.
2003 Finnish National High School Mathematics Competition, 4
Find pairs of positive integers $(n, k)$ satisfying \[(n + 1)^k - 1 = n!\]
2013 Saudi Arabia BMO TST, 4
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ be a function which satisfies for all integer $n \ge 0$:
(a) $f(2n + 1)^2 - f(2n)^2 = 6f(n) + 1$, (b) $f(2n) \ge f(n)$
where $Z_{\ge 0}$ is the set of nonnegative integers. Solve the equation $f(n) = 1000$
1990 Tournament Of Towns, (275) 3
There are two identical clocks on the wall, one showing the current Moscow time and the other showing current local time. The minimum distance between the ends of their hour hands equals $m$ and the maximum distance equals $M$. Find the distance between the centres of the clocks.
(S Fomin, Leningrad)
2019 Saudi Arabia JBMO TST, 3
Let $d$ be a positive divisor of the number $A = 1024^{1024}+5$ and suppose that $d$ can be expressed as $d = 2x^2+2xy+3y^2$ for some integers $x,y$. Which remainder we can have when divide $d$ by $20$ ?