Found problems: 85335
2017 Harvard-MIT Mathematics Tournament, 5
[b]E[/b]ach of the integers $1,2,...,729$ is written in its base-$3$ representation without leading zeroes. The numbers are then joined together in that order to form a continuous string of digits: $12101112202122...$ How many times in this string does the substring $012$ appear?
2017 Romania Team Selection Test, P3
Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.
2023 European Mathematical Cup, 2
Let $ABC$ be a triangle such that $\angle BAC = 90^{\circ}$. The incircle of triangle $ABC$ is tangent to the sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D,E,F$ respectively. Let $M$ be the midpoint of $\overline{EF}$. Let $P$ be the projection of $A$ onto $BC$ and let $K$ be the intersection of $MP$ and $AD$. Prove that the circumcircles of triangles $AFE$ and $PDK$ have equal radius.
[i]Kyprianos-Iason Prodromidis[/i]
2019 Slovenia Team Selection Test, 4
Let $P$ be the set of all prime numbers. Let $A$ be some subset of $P$ that has at least two elements. Let's say that for every positive integer $n$ the following statement holds: If we take $n$ different elements $p_1,p_2...p_n \in A$, every prime number that divides $p_1 p_2 \cdots p_n-1$ is also an element of $A$. Prove, that $A$ contains all prime numbers.
2007 Princeton University Math Competition, 5
Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $.0001$ of the correct answer will be accepted.
2007 Balkan MO Shortlist, A1
Find the minimum and maximum value of the function
\begin{align*} f(x,y)=ax^2+cy^2 \end{align*}
Under the condition $ax^2-bxy+cy^2=d$, where $a,b,c,d$ are positive real numbers such that $b^2 -4ac <0$
2000 Czech and Slovak Match, 3
Let $n$ be a positive integer. Prove that $n$ is a power of two if and only if there exists an integer $m$ such that $2^n-1$ is a divisor of $m^2 +9$.
2016 Vietnam National Olympiad, 1
Find all $a\in\mathbb{R}$ such that there is function $f:\mathbb{R}\to\mathbb{R}$
i) $f(1)=2016$
ii) $f(x+y+f(y))=f(x)+ay\quad\forall x,y\in\mathbb{R}$
2005 iTest, 25
Consider the set $\{1!, 2!, 3!, 4!, …, 2004!, 2005!\}$. How many elements of this set are divisible by $2005$?
2002 Stanford Mathematics Tournament, 3
A clockmaker wants to design a clock such that the area swept by each hand (second, minute, and hour) in one minute is the same (all hands move continuously). What is the length of the hour hand divided by the length of the second hand?
1994 Baltic Way, 15
Does there exist a triangle such that the lengths of all its sides and altitudes are integers and its perimeter is equal to $1995$?
2008 Hanoi Open Mathematics Competitions, 7
Find all triples $(a, b,c)$ of consecutive odd positive integers such that $a < b < c$ and $a^2 + b^2 + c^2$ is a four digit
number with all digits equal.
1993 Korea - Final Round, 6
Consider a triangle $ABC$ with $BC = a, CA = b, AB = c.$ Let $D$ be the midpoint of $BC$ and $E$ be the intersection of the bisector of $A$ with $BC$ . The circle through $A, D, E$ meets $AC, AB$ again at $F, G$ respectively. Let $H\not = B$ be a point on $AB$ with $BG = GH$ . Prove that triangles $EBH$ and $ABC$ are similar and find the ratio of their areas.
2014 ASDAN Math Tournament, 22
Apples cost $2$ dollars. Bananas cost $3$ dollars. Oranges cost $5$ dollars. Compute the number of distinct baskets of fruit such that there are $100$ pieces of fruit and the basket costs $300$ dollars. Two baskets are distinct if and only if, for some type of fruit, the two baskets have differing amounts of that fruit.
1992 All Soviet Union Mathematical Olympiad, 568
A cinema has its seats arranged in $n$ rows $\times m$ columns. It sold mn tickets but sold some seats more than once. The usher managed to allocate seats so that every ticket holder was in the correct row or column. Show that he could have allocated seats so that every ticket holder was in the correct row or column and at least one person was in the correct seat. What is the maximum $k$ such that he could have always put every ticket holder in the correct row or column and at least $k$ people in the correct seat?
2002 IMO Shortlist, 4
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.
2001 Tuymaada Olympiad, 6
On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.
2022 IFYM, Sozopol, 3
The set of quadruples $(a,b,c,d)$ where each of $a,b,c,d$ is either $0$ or $1$ is [i]called vertices of the four dimensional unit cube[/i] or [i]4-cube[/i] for short. Two vertices are called [i]adjacent[/i], if their respective quadruples differ by one variable only. Each two adjacent vertices are connected by an edge. A robot is moving through the edges of the 4-cube starting from $(0,0,0,0)$ and each turn consists of passing an edge and moving to adjacent vertex. In how many ways can the robot go back to $(0,0,0,0)$ after $4042$ turns? Note that it is [u]NOT[/u] forbidden for the robot to pass through $(0,0,0,0)$ before the $4042$-nd turn.
Gheorghe Țițeica 2024, P4
Let $f:\mathbb{R}\rightarrow (0,\infty)$ be continuous function of period $1$. Prove that for any $a\in\mathbb{R}$ $$\int_0^1\frac{f(x)}{f(x+a)}dx\geq 1.$$
2021 China Second Round A2, 3
Given $n\geq 2$, $a_1$, $a_2$, $\cdots$, $a_n\in\mathbb {R}$ satisfy
$$a_1\geqslant a_2\geqslant \cdots \geqslant a_n\geqslant 0,a_1+a_2+\cdots +a_n=n.$$
Find the minimum value of $a_1+a_1a_2+\cdots +a_1a_2\cdots a_n$.
1998 All-Russian Olympiad Regional Round, 8.2
Given a parallelogram ABCD, let M and N be the midpoints of the sides BC and CD.
Can the lines AM, AN divide the angle BAD into three equal angles?
2019 Online Math Open Problems, 10
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m+n$ is divided by $10^6$.
[i]Proposed by Yannick Yao[/i]
2023 Malaysian IMO Training Camp, 5
Let $n\ge 3$, $d$ be positive integers. For an integer $x$, denote $r(x)$ be the remainder of $x$ when divided by $n$ such that $0\le r(x)\le n-1$. Let $c$ be a positive integer with $1<c<n$ and $\gcd(c,n)=1$, and suppose $a_1, \cdots, a_d$ are positive integers with $a_1+\cdots+a_d\le n-1$. \\
(a) Prove that if $n<2d$, then $\displaystyle\sum_{i=1}^d r(ca_i)\ge n.$ \\
(b) For each $n$, find the smallest $d$ such that $\displaystyle\sum_{i=1}^d r(ca_i)\ge n$ always holds.
[i]Proposed by Yeoh Zi Song & Anzo Teh Zhao Yang[/i]
2006 Purple Comet Problems, 21
In triangle $ABC$, $AB = 52$, $BC = 56$, $CA = 60$. Let $D$ be the foot of the altitude from $A$ and $E$ be the intersection of the internal angle bisector of $\angle BAC$ with $BC$. Find $DE$.
1998 All-Russian Olympiad Regional Round, 8.4
A set of $n\ge 9$ points is given on the plane. For any 9 it points can be selected from all circles so that all these points end up on selected circles. Prove that all n points lie on two circles