Found problems: 85335
1997 May Olympiad, 4
Joaquín and his brother Andrés go to class every day on the $62$ bus. Joaquín always pays for the tickets. Each ticket has a $5$-digit number printed on it. One day, Joaquín observes that the numbers on his tickets - his and his brother's - as well as being consecutive, are such that the sum of the ten digits is precisely $62$. Andrés asks him if the sum of the digits of any of the tickets is $35$ and, knowing the answer, he can directly say the number of each ticket. What were those numbers?
2021 HMNT, 7
De ne the function $f : R \to R$ by $$f(x) =\begin{cases}
\dfrac{1}{x^2+\sqrt{x^4+2x}}\,\,\,
\text{if} \,\,\,x \notin (- \sqrt[3]{2}, 0] \\
\,\,\, 0 \,\,\,, \,\,\, \text{otherwise}
\end{cases}$$
The sum of all real numbers $x$ for which $f^{10}(x) = 1$ can be written as $\frac{a+b\sqrt{c}}{d}$ , where $a, b,c, d$ are integers, $d$ is positive, $c$ is square-free, and gcd$(a,b, d) = 1$. Find $1000a + 100b + 10c + d.$
(Here, $f^n(x)$ is the function $f(x)$ iterated $n$ times. For example, $f^3(x) = f(f(f(x)))$.)
2008 Sharygin Geometry Olympiad, 1
(B.Frenkin, 8) Does a regular polygon exist such that just half of its diagonals are parallel to its sides?
1983 National High School Mathematics League, 6
Let $a,b,c,d,m,n$ be positive real numbers. $P=\sqrt{ab}+\sqrt{cd},Q=\sqrt{ma+nc}\cdot\sqrt{\frac{b}{m}+\frac{d}{n}}$. Then
$\text{(A)}P\geq Q\qquad\text{(B)}P\leq Q\qquad\text{(C)}P<Q\qquad\text{(D)}$Not sure
1986 USAMO, 2
During a certain lecture, each of five mathematicians fell asleep exactly twice. For each pair of mathematicians, there was some moment when both were asleep simultaneously. Prove that, at some moment, three of them were sleeping simultaneously.
2021 Saudi Arabia JBMO TST, 1
Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that
$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$.
2001 Argentina National Olympiad, 3
Let $a$ and $b$ be positive integers, $a < b$, such that in the decimal expansion of the fraction $\dfrac{a}{b} $ the five digits $1,4,2,8,6$ appear somewhere, in that order and consecutively. Determine the lowest possible value $b$ can take .
May Olympiad L2 - geometry, 2012.4
Six points are given so that there are not three on the same line and that the lengths of the segments determined by these points are all different. We consider all the triangles that they have their vertices at these points. Show that there is a segment that is both the shortest side of one of those triangles and the longest side of another.
2021 Dutch IMO TST, 4
Let $p > 10$ be prime. Prove that there are positive integers $m$ and $n$ with $m + n < p$ exist for which $p$ is a divisor of $5^m7^n-1$.
2024 Sharygin Geometry Olympiad, 10.2
For which greatest $n$ there exists a convex polyhedron with $n$ faces having the following property: for each face there exists a point outside the polyhedron such that the remaining $n - 1$ faces are seen from this point?
2017 Costa Rica - Final Round, 5
Consider two circles $\Pi_1$ and $\Pi_1$ tangent externally at point $S$, such that the radius of $\Pi_2$ is triple the radius of $\Pi_1$. Let $\ell$ be a line that is tangent to $\Pi_1$ at point $ P$ and tangent to $\Pi_2$ at point $Q$, with $P$ and $Q$ different from $S$. Let $T$ be a point at $\Pi_2$, such that the segment $TQ$ is diameter of $\Pi_2$ and let point $R$ be the intersection of the bisector of $\angle SQT$ with $ST$. Prove that $QR = RT$.
1995 South africa National Olympiad, 4
Three circles, with radii $p$, $q$ and $r$ and centres $A$, $B$ and $C$ respectively, touch one another externally at points $D$, $E$ and $F$. Prove that the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ equals
\[\frac{2pqr}{(p+q)(q+r)(r+p)}.\]
OMMC POTM, 2023 3
Three natural numbers are such that the product of any two of them is divisible by the sum of those two numbers. Prove that these numbers have a common divisor greater than $1$.
[i]Proposed by Evan Chang (squareman), USA[/i]
2000 Switzerland Team Selection Test, 5
Consider all words of length $n$ consisting of the letters $I,O,M$.
How many such words are there, which contain no two consecutive $M$’s?
Croatia MO (HMO) - geometry, 2018.3
Let $k$ be a circle centered at $O$. Let $\overline{AB}$ be a chord of that circle and $M$ its midpoint. Tangent on $k$ at points $A$ and $B$ intersect at $T$. The line $\ell$ goes through $T$, intersect the shorter arc $AB$ at the point $C$ and the longer arc $AB$ at the point $D$, so that $|BC| = |BM|$. Prove that the circumcenter of the triangle $ADM$ is the reflection of $O$ across the line $AD$
2025 Spain Mathematical Olympiad, 5
Let $S$ be a finite set of cells in a square grid. On each cell of $S$ we place a grasshopper. Each grasshopper can face up, down, left or right. A grasshopper arrangement is Asturian if, when each grasshopper moves one cell forward in the direction in which it faces, each cell of $S$ still contains one grasshopper.
[list]
[*] Prove that, for every set $S$, the number of Asturian arrangements is a perfect square.
[*] Compute the number of Asturian arrangements if $S$ is the following set:
1948 Moscow Mathematical Olympiad, 142
Find all possible arrangements of $4$ points on a plane, so that the distance between each pair of points is equal to either $a$ or $b$. For what ratios of $a : b$ are such arrangements possible?
2008 Silk Road, 3
Let $ G$ be a graph with $ 2n$ vertexes and $ 2n(n\minus{}1)$ edges.If we color some edge to red,then vertexes,which are connected by this edge,must be colored to red too. But not necessary that all edges from the red vertex are red.
Prove that it is possible to color some vertexes and edges in $ G$,such that all red vertexes has exactly $ n$ red edges.
2001 Korea - Final Round, 2
In a triangle $ABC$ with $\angle B < 45^{\circ}$, $D$ is a point on $BC$ such that the incenter of $\triangle ABD$ coincides with the circumcenter $O$ of $\triangle ABC$. Let $P$ be the intersection point of the tangent lines to the circumcircle $\omega$ of $\triangle AOC$ at points $A$ and $C$. The lines $AD$ and $CO$ meet at $Q$. The tangent to $\omega$ at $O$ meets $PQ$ at $X$. Prove that $XO=XD$.
2002 District Olympiad, 3
Let be two real numbers $ a,b, $ that satisfy $ 3^a+13^b=17^a $ and $ 5^a+7^b=11^b. $
Show that $ a<b. $
2019 Bulgaria National Olympiad, 2
Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$
2019 ELMO Shortlist, N3
Let $S$ be a nonempty set of positive integers such that, for any (not necessarily distinct) integers $a$ and $b$ in $S$, the number $ab+1$ is also in $S$. Show that the set of primes that do not divide any element of $S$ is finite.
[i]Proposed by Carl Schildkraut[/i]
2023 BMT, Tie 1
Wen finds $17$ consecutive positive integers that sum to $2023$. Compute the smallest of these integers.
2022-2023 OMMC, 1
John has cut out these two polygons made out of unit squares. He joins them to each other to form a larger polygon (but they can't overlap). Find the smallest possible perimeter this larger polygon can have. He can rotate and reflect the cut out polygons.
1988 Federal Competition For Advanced Students, P2, 4
Let $ a_{ij}$ be nonnegative integers such that $ a_{ij}\equal{}0$ if and only if $ i>j$ and that $ \displaystyle\sum_{j\equal{}1}^{1988}a_{ij}\equal{}1988$ holds for all $ i\equal{}1,...,1988$. Find all real solutions of the system of equations:
$ \displaystyle\sum_{j\equal{}1}^{1988} (1\plus{}a_{ij})x_j\equal{}i\plus{}1, 1 \le i \le 1988$.