Found problems: 85335
1949-56 Chisinau City MO, 4
Prove that the product of four consecutive integers plus $1$ is a perfect square.
2023 LMT Spring, 7
Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?
2020 South East Mathematical Olympiad, 1
Let $f(x)=a(3a+2c)x^2-2b(2a+c)x+b^2+(c+a)^2$ $(a,b,c\in R, a(3a+2c)\neq 0).$ If $$f(x)\leq 1$$for any real $x$, find the maximum of $|ab|.$
2011 Princeton University Math Competition, B1
What is the largest prime factor of $7999488$?
2012 Sharygin Geometry Olympiad, 4
Given triangle $ABC$. Point $M$ is the midpoint of side $BC$, and point $P$ is the projection of $B$ to the perpendicular bisector of segment $AC$. Line $PM$ meets $AB$ in point $Q$. Prove that triangle $QPB$ is isosceles.
1970 Canada National Olympiad, 4
a) Find all positive integers with initial digit 6 such that the integer formed by deleting 6 is $1/25$ of the original integer.
b) Show that there is no integer such that the deletion of the first digit produces a result that is $1/35$ of the original integer.
2016 CCA Math Bonanza, L1.2
What is the largest prime factor of $729-64$?
[i]2016 CCA Math Bonanza Lightning #1.2[/i]
1992 IMO Shortlist, 14
For any positive integer $ x$ define $ g(x)$ as greatest odd divisor of $ x,$ and
\[ f(x) \equal{} \begin{cases} \frac {x}{2} \plus{} \frac {x}{g(x)} & \text{if \ \(x\) is even}, \\
2^{\frac {x \plus{} 1}{2}} & \text{if \ \(x\) is odd}. \end{cases}
\]
Construct the sequence $ x_1 \equal{} 1, x_{n \plus{} 1} \equal{} f(x_n).$ Show that the number 1992 appears in this sequence, determine the least $ n$ such that $ x_n \equal{} 1992,$ and determine whether $ n$ is unique.
1982 Czech and Slovak Olympiad III A, 4
In a circle with a radius of $1$, $64$ mutually different points are selected. Prove that $10$ mutually different points can be selected from them, which lie in a circle with a radius $\frac12$.
2011 IMO Shortlist, 6
Let $ABC$ be a triangle with $AB=AC$ and let $D$ be the midpoint of $AC$. The angle bisector of $\angle BAC$ intersects the circle through $D,B$ and $C$ at the point $E$ inside the triangle $ABC$. The line $BD$ intersects the circle through $A,E$ and $B$ in two points $B$ and $F$. The lines $AF$ and $BE$ meet at a point $I$, and the lines $CI$ and $BD$ meet at a point $K$. Show that $I$ is the incentre of triangle $KAB$.
[i]Proposed by Jan Vonk, Belgium and Hojoo Lee, South Korea[/i]
2004 APMO, 5
Prove that the inequality \[\left(a^{2}+2\right)\left(b^{2}+2\right)\left(c^{2}+2\right) \geq 9\left(ab+bc+ca\right)\] holds for all positive reals $a$, $b$, $c$.
2023 MOAA, 6
Let $b$ be a positive integer such that 2032 has 3 digits when expressed in base $b$. Define the function $S_k(n)$ as the sum of the digits of the base $k$ representation of $n$. Given that $S_b(2032)+S_{b^2}(2032) = 14$, find $b$.
[i]Proposed by Anthony Yang[/i]
2016 Bosnia and Herzegovina Team Selection Test, 5
Let $k$ be a circumcircle of triangle $ABC$ $(AC<BC)$. Also, let $CL$ be an angle bisector of angle $ACB$ $(L \in AB)$, $M$ be a midpoint of arc $AB$ of circle $k$ containing the point $C$, and let $I$ be an incenter of a triangle $ABC$. Circle $k$ cuts line $MI$ at point $K$ and circle with diameter $CI$ at $H$. If the circumcircle of triangle $CLK$ intersects $AB$ again at $T$, prove that $T$, $H$ and $C$ are collinear.
.
2010 N.N. Mihăileanu Individual, 2
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the property that there exists a continuous and bounded function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that verifies the equality
$$ f(x)=\int_0^x f(\xi )g(\xi )d\xi , $$
for any real number $ x. $ Prove that $ f=0. $
[i]Nelu Chichirim[/i]
1968 Vietnam National Olympiad, 2
$L$ and $M$ are two parallel lines a distance $d$ apart. Given $r$ and $x$, construct a triangle $ABC$, with $A$ on $L$, and $B$ and $C$ on $M$, such that the inradius is $r$, and angle $A = x$. Calculate angles $B$ and $C$ in terms of $d$, $r$ and $x$. If the incircle touches the side $BC$ at $D$, find a relation between $BD$ and $DC$
1971 IMO, 2
Let $P_1$ be a convex polyhedron with vertices $A_1,A_2,\ldots,A_9$. Let $P_i$ be the polyhedron obtained from $P_1$ by a translation that moves $A_1$ to $A_i$. Prove that at least two of the polyhedra $P_1,P_2,\ldots,P_9$ have an interior point in common.
2025 Harvard-MIT Mathematics Tournament, 17
Let $f$ be a quadratic polynomial with real coefficients, and let $g_1, g_2, g_3, \ldots$ be a geometric progression of real numbers. Define $a_n=f(n)+g_n.$ Given that $a_1, a_2, a_3, a_4,$ and $a_5$ are equal to $1, 2, 3, 14,$ and $16,$ respectively, compute $\tfrac{g_2}{g_1}.$
2017 JBMO Shortlist, A4
Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$
When does the equality hold?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1994 Tournament Of Towns, (432) 2
Prove that one can construct two triangles from six edges of an arbitrary tetrahedron.
(VV Proizvolov)
2016 SEEMOUS, Problem 1
SEEMOUS 2016 COMPETITION PROBLEMS
2021 Vietnam National Olympiad, 3
Let $\bigtriangleup ABC$ is not an isosceles triangle and is an acute triangle, $AD,BE,CF$ be the altitudes and $H$ is the orthocenter .Let $I$ is the circumcenter of $\bigtriangleup HEF$ and let $K,J$ is the midpoint of $BC,EF$ respectively.Let $HJ$ intersects $(I)$ again at $G$ and $GK$ intersects $(I)$ at $L\neq G$.
a) Prove that $AL$ is perpendicular to $EF$.
b) Let $AL$ intersects $EF$ at $M$, the line $IM$ intersects the circumcircle $\bigtriangleup IEF$ again at $N$, $DN$ intersects $AB,AC$ at $P$ and $Q$ respectively then prove that $PE,QF,AK$ are concurrent.
2006 South East Mathematical Olympiad, 2
In $\triangle ABC$, $\angle ABC=90^{\circ}$. Points $D,G$ lie on side $AC$. Points $E, F$ lie on segment $BD$, such that $AE \perp BD $ and $GF \perp BD$. Show that if $BE=EF$, then $\angle ABG=\angle DFC$.
2008 BAMO, 3
A triangle is constructed with the lengths of the sides chosen from the set $\{2, 3, 5, 8, 13, 21, 34, 55, 89, 144\}$. Show that this triangle must be isosceles.
(A triangle is isosceles if it has at least two sides the same length.)
2002 AMC 10, 8
Suppose July of year $ N$ has five Mondays. Which of the following must occur five times in August of year $ N$? (Note: Both months have $ 31$ days.)
$ \textbf{(A)}\ \text{Monday} \qquad
\textbf{(B)}\ \text{Tuesday} \qquad
\textbf{(C)}\ \text{Wednesday} \qquad
\textbf{(D)}\ \text{Thursday} \qquad
\textbf{(E)}\ \text{Friday}$
2014 Sharygin Geometry Olympiad, 22
Does there exist a convex polyhedron such that it has diagonals and each of them is shorter than each of its edges?