Found problems: 85335
2021 JHMT HS, 2
Call a positive integer [i]almost square[/i] if it is not a perfect square, but all of its digits are perfect squares. For example, both $149$ and $904$ are almost square, but $144$ and $936$ are not. Find the number of positive integers less than $1000$ that are not almost square.
2012 AMC 12/AHSME, 22
Distinct planes $p_1,p_2,....,p_k$ intersect the interior of a cube $Q$. Let $S$ be the union of the faces of $Q$ and let $ P =\bigcup_{j=1}^{k}p_{j} $. The intersection of $P$ and $S$ consists of the union of all segments joining the midpoints of every pair of edges belonging to the same face of $Q$. What is the difference between the maximum and minimum possible values of $k$?
$ \textbf{(A)}\ 8\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 23\qquad\textbf{(E)}\ 24 $
2003 Bulgaria Team Selection Test, 3
Some of the vertices of a convex $n$-gon are connected by segments, such that any two of them have no common interior point. Prove that, for any $n$ points in general position, there exists a one-to-one correspondence between the points and the vertices of the $n$ gon, such that any two segments between the points, corresponding to the respective segments from the $n$ gon, have no common interior point.
2011 Junior Balkan Team Selection Tests - Romania, 3
Let n be a positive integer and let $x_1, x_2,...,x_n$ and $y_1, y_2,...,y_n$ be real numbers. Prove that there exists a number $i, i = 1, 2,...,n$, such that $$\sum_{j=1}^n |x_i - x_j | \le \sum_{j=1}^n |x_i - y_j | $$
2017 Harvard-MIT Mathematics Tournament, 8
Consider all ordered pairs of integers $(a,b)$ such that $1\le a\le b\le 100$ and $$\frac{(a+b)(a+b+1)}{ab}$$ is an integer.
Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose $(3,85)$ over $(2,85)$ over $(4,84)$. Note that your answer should be an ordered pair.
2025 Ukraine National Mathematical Olympiad, 10.2
Given $12$ segments, it is known that they can be divided into $4$ groups of $3$ segments each in such a way that a triangle can be formed from the segments of each triplet. Is it always possible to divide these $12$ segments into $3$ groups of $4$ segments each in such a way that a quadrilateral can be formed from the segments of each quartet?
[i]Proposed by Mykhailo Shtandenko[/i]
1983 IMO Longlists, 47
In a plane, three pairwise intersecting circles $C_1, C_2, C_3$ with centers $M_1,M_2,M_3$ are given. For $i = 1, 2, 3$, let $A_i$ be one of the points of intersection of $C_j$ and $C_k \ (\{i, j, k \} = \{1, 2, 3 \})$. Prove that if $ \angle M_3A_1M_2 = \angle M_1A_2M_3 = \angle M_2A_3M_1 = \frac{\pi}{3}$(directed angles), then $M_1A_1, M_2A_2$, and $M_3A_3$ are concurrent.
2001 Vietnam Team Selection Test, 1
Let a sequence of integers $\{a_n\}$, $n \in \mathbb{N}$ be given, defined by
\[a_0 = 1, a_n= a_{n-1} + a_{[n/3]}\]
for all $n \in \mathbb{N}^{*}$.
Show that for all primes $p \leq 13$, there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$.
(Here $[x]$ denotes the integral part of real number $x$).
2009 Mediterranean Mathematics Olympiad, 3
Decide whether the integers $1,2,\ldots,100$ can be arranged in the cells $C(i, j)$ of a $10\times10$ matrix (where $1\le i,j\le 10$), such that the following conditions are fullfiled:
i) In every row, the entries add up to the same sum $S$.
ii) In every column, the entries also add up to this sum $S$.
iii) For every $k = 1, 2, \ldots, 10$ the ten entries $C(i, j)$ with $i-j\equiv k\bmod{10}$ add up to $S$.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
2024 Turkey Team Selection Test, 8
For an integer $n$, $\sigma(n)$ denotes the sum of postitive divisors of $n$. A sequence of positive integers $(a_i)_{i=0}^{\infty}$ with $a_0 =1$ is defined as follows: For each $n>1$, $a_n$ is the smallest integer greater than $1$ that satisfies
$$\sigma{(a_0a_1\dots a_{n-1})} \vert \sigma{(a_0a_1\dots a_{n})}.$$
Determine the number of divisors of $2024^{2024}$ amongst the sequence.
1997 IMO, 4
An $ n \times n$ matrix whose entries come from the set $ S \equal{} \{1, 2, \ldots , 2n \minus{} 1\}$ is called a [i]silver matrix[/i] if, for each $ i \equal{} 1, 2, \ldots , n$, the $ i$-th row and the $ i$-th column together contain all elements of $ S$. Show that:
(a) there is no silver matrix for $ n \equal{} 1997$;
(b) silver matrices exist for infinitely many values of $ n$.
2024 Canadian Mathematical Olympiad Qualification, 3
Let $\vartriangle ABC$ be an acute triangle with $AB < AC$. Let $H$ be its orthocentre and $M$ be the midpoint of arc $BAC$ on the circumcircle. It is given that $B$, $H$, $M$ are collinear, the length of the altitude from $M$ to $AB$ is $1$, and the length of the altitude from $M$ to $BC$ is $6$. Determine all possible areas for $\vartriangle ABC$ .
1952 AMC 12/AHSME, 44
If an integer of two digits is $ k$ times the sum of its digits, the number formed by interchanging the digits is the sum of the digits multiplied by:
$ \textbf{(A)}\ (9 \minus{} k) \qquad\textbf{(B)}\ (10 \minus{} k) \qquad\textbf{(C)}\ (11 \minus{} k) \qquad\textbf{(D)}\ (k \minus{} 1) \qquad\textbf{(E)}\ (k \plus{} 1)$
2017 Czech-Polish-Slovak Junior Match, 2
Given is the triangle $ABC$, with $| AB | + | AC | = 3 \cdot | BC | $. Let's denote $D, E$ also points that $BCDA$ and $CBEA$ are parallelograms. On the sides $AC$ and $AB$ sides, $F$ and $G$ are selected respectively so that $| AF | = | AG | = | BC |$. Prove that the lines $DF$ and $EG$ intersect at the line segment $BC$
2003 Swedish Mathematical Competition, 2
In a lecture hall some chairs are placed in rows and columns, forming a rectangle. In each row there are $6$ boys sitting and in each column there are $8$ girls sitting, whereas $15$ places are not taken. What can be said about the number of rows and that of columns?
Kvant 2024, M2823
A parabola $p$ is drawn on the coordinate plane — the graph of the equation $y =-x^2$, and a point $A$ is marked that does not lie on the parabola $p$. All possible parabolas $q$ of the form $y = x^2+ax+b$ are drawn through point $A$, intersecting $p$ at two points $X$ and $Y$ . Prove that all possible $XY$ lines pass through a fixed point in the plane.
[i]P.A.Kozhevnikov[/i]
2017 Hanoi Open Mathematics Competitions, 14
Put $P = m^{2003}n^{2017} - m^{2017}n^{2003}$ , where $m, n \in N$.
a) Is $P$ divisible by $24$?
b) Do there exist $m, n \in N$ such that $P$ is not divisible by $7$?
2023 Belarusian National Olympiad, 11.4
Denote by $R_{>0}$ the set of all positive real numbers. Find all functions $f: R_{>0} \to R_{>0}$ such that for all $x,y \in R_{>0}$ the following equation holds $$f(y)f(x+f(y))=f(1+xy)$$
2015 Irish Math Olympiad, 8
In triangle $\triangle ABC$, the angle $\angle BAC$ is less than $90^o$. The perpendiculars from $C$ on $AB$ and from $B$ on $AC$ intersect the circumcircle of $\triangle ABC$ again at $D$ and $E$ respectively. If $|DE| =|BC|$, find the measure of the angle $\angle BAC$.
Estonia Open Senior - geometry, 2019.1.1
Juri and Mari play the following game. Juri starts by drawing a random triangle on a piece of paper. Mari then draws a line on the same paper that goes through the midpoint of one of the midsegments of the triangle. Then Juri adds another line that also goes through the midpoint of the same midsegment. These two lines divide the triangle into four pieces. Juri gets the piece with maximum area (or one of those with maximum area) and the piece with minimum area (or one of those with minimum area), while Mari gets the other two pieces. The player whose total area is bigger wins. Does either of the players have a winning strategy, and if so, who has it?
2001 Moldova Team Selection Test, 8
A group of $n{}$ $(n>1)$ people each visited $k{}$ $(k>1)$ citites. Each person makes a list of these $k$ cities in the order they want to visit them. A permutation $(a_1,a_2,\ldots,a_k)$ is called $m-prefered$ $(m\in\mathbb{N})$, if for every $i=1,2,\ldots,k$ there are at least $m$ people that would prefer to visit the city $a_i$ before the city $a_{i+1}$, $(a_{k+1}=a_1)$. Prove that there exists an m-prefered permutation if and only if $km\leq n(k-1)$.
1998 All-Russian Olympiad, 4
Let $k$ be a positive integer. Some of the $2k$-element subsets of a given set are marked. Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element. Show that all the marked subsets have a common element.
2019 HMNT, 4
In $\vartriangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\vartriangle ABC$ so that each $n$-gon shares a side with a distinct side of $\vartriangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$?
1989 IMO Shortlist, 30
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2006 AIME Problems, 9
Circles $\mathcal{C}_1$, $\mathcal{C}_2$, and $\mathcal{C}_3$ have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line $t_1$ is a common internal tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ and has a positive slope, and line $t_2$ is a common internal tangent to $\mathcal{C}_2$ and $\mathcal{C}_3$ and has a negative slope. Given that lines $t_1$ and $t_2$ intersect at $(x,y)$, and that $x=p-q\sqrt{r}$, where $p$, $q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r$.