Found problems: 85335
2006 Singapore Junior Math Olympiad, 4
In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle A = 60^o$.
2006 Germany Team Selection Test, 1
We denote by $\mathbb{R}^\plus{}$ the set of all positive real numbers.
Find all functions $f: \mathbb R^ \plus{} \rightarrow\mathbb R^ \plus{}$ which have the property:
\[f(x)f(y)\equal{}2f(x\plus{}yf(x))\]
for all positive real numbers $x$ and $y$.
[i]Proposed by Nikolai Nikolov, Bulgaria[/i]
2019 Harvard-MIT Mathematics Tournament, 10
Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_1, a_2, a_3, a_4)$ to some position $(x_1, x_2, x_3, x_4)$ with integer coordinates satisfying
\[(x_1-a_1)^2 + (x_2-a_2)^2 + (x_3-a_3)^2 + (x_4-a_4)^2 = 4
\quad \text{and} \quad
|(x_1 + x_2 + x_3 + x_4) - (a_1 + a_2 + a_3 + a_4)| = 2.\]
In how many ways can Fred reach $(10, 10, 10, 10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?
2010 May Olympiad, 2
Let $ABCD$ be a rectangle and the circle of center $D$ and radius $DA$, which cuts the extension of the side $AD$ at point $P$. Line $PC$ cuts the circle at point $Q$ and the extension of the side $AB$ at point $R$. Show that $QB = BR$.
2022 Purple Comet Problems, 2
Call a date mm/dd/yy $\textit{multiplicative}$ if its month number times its day number is a two-digit integer equal to its year expressed as a two-digit year. For example, $01/21/21$, $03/07/21$, and $07/03/21$ are multiplicative. Find the number of dates between January 1, 2022 and December 31, 2030 that are multiplicative.
2024 AMC 12/AHSME, 3
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
2000 Stanford Mathematics Tournament, 25
How many points does one have to place on a unit square to guarantee that two of them are strictly less than 1/2 unit apart?
2007 All-Russian Olympiad, 5
Two numbers are written on each vertex of a convex $100$-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different.
[i]F. Petrov [/i]
2006 AMC 8, 3
Elisa swims laps in the pool. When she first started, she completed 10 laps in 25 minutes. Now she can finish 12 laps in 24 minutes. By how many minutes has she improved her lap time?
$ \textbf{(A)}\ \dfrac{1}{2} \qquad \textbf{(B)}\ \dfrac{3}{4} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 3$
2004 Mediterranean Mathematics Olympiad, 1
Find all natural numbers $m$ such that
\[1! \cdot 3! \cdot 5! \cdots (2m-1)! = \biggl( \frac{m(m+1)}{2}\biggr) !.\]
2010 Contests, 1
In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$
2016 Dutch IMO TST, 4
Let $\Gamma_1$ be a circle with centre $A$ and $\Gamma_2$ be a circle with centre $B$, with $A$ lying on $\Gamma_2$. On $\Gamma_2$ there is a (variable) point $P$ not lying on $AB$. A line through $P$ is a tangent of $\Gamma_1$ at $S$, and it intersects $\Gamma_2$ again in $Q$, with $P$ and $Q$ lying on the same side of $AB$. A different line through $Q$ is tangent to $\Gamma_1$ at $T$. Moreover, let $M$ be the foot of the perpendicular to $AB$ through $P$. Let $N$ be the intersection of $AQ$ and $MT$.
Show that $N$ lies on a line independent of the position of $P$ on $\Gamma_2$.
2010 ELMO Shortlist, 3
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
2023 Harvard-MIT Mathematics Tournament, 7
Svitlana writes the number $147$ on a blackboard. Then, at any point, if the number on the blackboard is $n$, she can perform one of the following three operations:
$\bullet$ if $n$ is even, she can replace $n$ with $\frac{n}{2}$
$\bullet$ if $n$ is odd, she can replace $n$ with $\frac{n+255}{2}$ and
$\bullet$ if $n \ge 64$, she can replace $n$ with $n - 64$.
Compute the number of possible values that Svitlana can obtain by doing zero or more operations.
2012 239 Open Mathematical Olympiad, 4
For positive real numbers $a$, $b$, and $c$ with $a+b+c=1$, prove that:
$$ (a-b)^2 + (b-c)^2 + (c-a)^2 \geq \frac{1-27abc}{2}. $$
2014 Taiwan TST Round 2, 2
Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying
\[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \]
for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)
2015 District Olympiad, 1
[b]a)[/b] Solve the equation $ x^2-x+2\equiv 0\pmod 7. $
[b]b)[/b] Determine the natural numbers $ n\ge 2 $ for which the equation $ x^2-x+2\equiv 0\pmod n $ has an unique solution modulo $ n. $
2022 OMpD, 2
We say that a sextuple of positive real numbers $(a_1, a_2, a_3, b_1, b_2, b_3)$ is $\textit{phika}$ if $a_1 + a_2 + a_3 = b_1 + b_2 + b_3 = 1$.
(a) Prove that there exists a $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$ such that:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) > 1 - \frac{1}{2022^{2022}}$$
(b) Prove that for every $\textit{phika}$ sextuple $(a_1, a_2, a_3, b_1, b_2, b_3)$, we have:
$$a_1(\sqrt{b_1} + a_2) + a_2(\sqrt{b_2} + a_3) + a_3(\sqrt{b_3} + a_1) < 1$$
1999 USAMTS Problems, 2
Let $a$ be a positive real number, $n$ a positive integer, and define the [i]power tower[/i] $a\uparrow n$ recursively with $a\uparrow 1=a$, and $a\uparrow(i+1)=a^{a\uparrow i}$ for $i=1,2,3,\ldots$. For example, we have $4\uparrow 3=4^{(4^4)}=4^{256}$, a number which has $155$ digits. For each positive integer $k$, let $x_k$ denote the unique positive real number solution of the equation $x\uparrow k=10\uparrow (k+1)$. Which is larger: $x_{42}$ or $x_{43}$?
2017 India IMO Training Camp, 2
Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.
2025 239 Open Mathematical Olympiad, 8
Positive integer numbers $n$ and $k > 1$ are given. Losyash likes some of the cells of the $n \times n$ checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of $2n + 2$, the upper-left corner of which coincides with the upper-left corner of the board (there are $n$ such rectangles in total). Given $n$ and $k$, determine whether Losyash can color each cell he likes in one of $k$ colors so that in any rectangle of interest to him the number of cells of any two colors differ by no more than $1$.
2008 Ukraine Team Selection Test, 5
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.
[i]Proposed by Paisan Nakmahachalasint, Thailand[/i]
1988 Romania Team Selection Test, 2
Let $OABC$ be a trihedral angle such that \[ \angle BOC = \alpha, \quad \angle COA = \beta, \quad \angle AOB = \gamma , \quad \alpha + \beta + \gamma = \pi . \] For any interior point $P$ of the trihedral angle let $P_1$, $P_2$ and $P_3$ be the projections of $P$ on the three faces. Prove that $OP \geq PP_1+PP_2+PP_3$.
[i]Constantin Cocea[/i]
1972 All Soviet Union Mathematical Olympiad, 161
Find the maximal $x$ such that the expression $4^{27} + 4^{1000} + 4^x$ is the exact square.
1980 Bundeswettbewerb Mathematik, 3
In a triangle $ABC$, points $P, Q$ and $ R$ distinct from the vertices of the triangle are chosen on sides $AB, BC$ and $CA$, respectively. The circumcircles of the triangles $APR$, $BPQ$, and $CQR$ are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle $ABC$.