Found problems: 85335
2008 Purple Comet Problems, 24
Each of the distinct letters in the following addition problem represents a different digit. Find the number represented by the word MEET.
$ \begin{array}{cccccc}P&U&R&P&L&E\\&C&O&M&E&T\\&&M&E&E&T\\ \hline Z&Z&Z&Z&Z&Z\end{array} $
2021 Moldova EGMO TST, 12
Find all real numbers $y$, for which there exists at least one real number $x$ such that $y=\frac{\sqrt{x^2+4}}{\sqrt{x^2+1}+\sqrt{x^2+9}}.$
2008 Gheorghe Vranceanu, 1
At what index the harmonic series has a fractional part of $ 1/12? $
2019 Singapore Senior Math Olympiad, 4
Positive integers $m,n,k$ satisfy $1+2+3++...+n=mk$ and $m \ge n$.
Show that we can partite $\{1,2,3,...,n \}$ into $k$ subsets (Every element belongs to exact one of these $k$ subsets), such that the sum of elements in each subset is equal to $m$.
2018 Estonia Team Selection Test, 11
Let $k$ be a positive integer. Find all positive integers $n$, such that it is possible to mark $n$ points on the sides of a triangle (different from its vertices) and connect some of them with a line in such a way that the following conditions are satisfied:
1) there is at least $1$ marked point on each side,
2) for each pair of points $X$ and $Y$ marked on different sides, on the third side there exist exactly $k$ marked points which are connected to both $X$ and $Y$ and exactly k points which are connected to neither $X$ nor $Y$
1989 AMC 12/AHSME, 18
The set of all numbers x for which \[x+\sqrt{x^{2}+1}-\frac{1}{x+\sqrt{x^{2}+1}}\] is a rational number is the set of all:
$\textbf{(A)}\ \text{ integers } x \qquad
\textbf{(B)}\ \text{ rational } x \qquad
\textbf{(C)}\ \text{ real } x\qquad
\textbf{(D)}\ x \text{ for which } \sqrt{x^2+1} \text{ is rational} \qquad
\textbf{(E)}\ x \text{ for which } x+\sqrt{x^2+1} \text{ is rational }$
2010 Germany Team Selection Test, 2
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
1995 AMC 12/AHSME, 25
A list of five positive integers has mean $12$ and range $18$. The mode and median are both $8$. How many different values are possible for the second largest element of the list?
$\textbf{(A)}\ 4\qquad
\textbf{(B)}\ 6 \qquad
\textbf{(C)}\ 8\qquad
\textbf{(D)}\ 10\qquad
\textbf{(E)}\ 12$
1952 Poland - Second Round, 5
The vertical mast located on the tower can be seen at the greatest angle from a point on the ground whose distance from the mast axis is $ a $; this angle equals the given angle $ \alpha $. Calculate the height of the tower and the height of the mast.
2022 MIG, 21
A herder has forgotten the number of cows she has, and does not want to count them all of them. She remembers these four facts about the number of cows:
[list]
[*]It has $3$ digits.
[*]It is a palindrome.
[*]The middle digit is a multiple of $4$.
[*]It is divisible by $11$.
[/list]
What is the sum of all possible numbers of cows that the herder has?
$\textbf{(A) }343\qquad\textbf{(B) }494\qquad\textbf{(C) }615\qquad\textbf{(D) }635\qquad\textbf{(E) }726$
2017 Dutch IMO TST, 2
The incircle of a non-isosceles triangle $ABC$ has centre $I$ and is tangent to $BC$ and $CA$ in $D$ and $E$, respectively. Let $H$ be the orthocentre of $ABI$, let $K$ be the intersection of $AI$ and $BH$ and let $L$ be the intersection of $BI$ and $AH$. Show that the circumcircles of $DKH$ and $ELH$ intersect on the incircle of $ABC$.
1987 Bulgaria National Olympiad, Problem 5
Let $E$ be a point on the median $AD$ of a triangle $ABC$, and $F$ be the projection of $E$ onto $BC$. From a point $M$ on $EF$ the perpendiculars $MN$ to $AC$ and $MP$ to $AB$ are drawn. Prove that if the points $N,E,P$ lie on a line, then $M$ lies on the bisector of $\angle BAC$.
2012 Online Math Open Problems, 34
$p,q,r$ are real numbers satisfying \[\frac{(p+q)(q+r)(r+p)}{pqr} = 24\] \[\frac{(p-2q)(q-2r)(r-2p)}{pqr} = 10.\] Given that $\frac{p}{q} + \frac{q}{r} + \frac{r}{p}$ can be expressed in the form $\frac{m}{n}$, where $m,n$ are relatively prime positive integers, compute $m+n$.
[i]Author: Alex Zhu[/i]
2019 Iran Team Selection Test, 1
A table consisting of $5$ columns and $32$ rows, which are filled with zero and one numbers, are "varied", if no two lines are filled in the same way.\\
On the exterior of a cylinder, a table with $32$ rows and $16$ columns is constructed. Is it possible to fill the numbers cells of the table with numbers zero and one, such that any five consecutive columns, table $32\times5$ created by these columns, is a varied one?
[i]Proposed by Morteza Saghafian[/i]
2012 Baltic Way, 16
Let $n$, $m$, and $k$ be positive integers satisfying $(n - 1)n(n + 1) = m^k$. Prove that $k = 1$.
2003 Junior Balkan Team Selection Tests - Moldova, 7
The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.
2011 Germany Team Selection Test, 3
Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that:
$i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$.
$ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$.
a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist.
b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.
2012 Indonesia TST, 1
Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that
\[f(x+t) - f(x) = P(x)\]
for all $x \in \mathbb{R}$.
2018 Harvard-MIT Mathematics Tournament, 2
Compute the positive real number $x$ satisfying $$x^{(2x^6)}=3.$$
2023 Indonesia TST, 1
Let $k\ge2$ be an integer. Find the smallest integer $n \ge k+1$ with the property that there exists a set of $n$ distinct real numbers such that each of its elements can be written as a sum of $k$ other distinct elements of the set.
2011 AMC 8, 24
In how many ways can 10001 be written as the sum of two primes?
$ \textbf{(A)}0\qquad\textbf{(B)}1\qquad\textbf{(C)}2\qquad\textbf{(D)}3\qquad\textbf{(E)}4 $
2022 Iran MO (2nd round), 4
There is an $n*n$ table with some unit cells colored black and the others are white.
In each step , Amin takes a $row$ with exactly one black cell in it , and color all cells in that black cell's $column$ red.
While Ali , takes a $column$ with exactly one black cell in it , and color all cells in that black cell's $row$ red.
Prove that Amin can color all the cells red , iff Ali can do so.
VII Soros Olympiad 2000 - 01, 9.7
Sides $AB$ and $CD$ of quadrilateral $ABCD$ intersect at point $E$. On the diagonals$ AC$ and $BD$ points $M$ and $N$ are taken, respectively, so that $AM / AC = BN / BD = k$. Find the area of a triangle $EMN$ if the area of $ABCD$ is $S$.
1979 IMO Longlists, 31
Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions:
(i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ;
(ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$
(iii) $\bigcup_{X \in F} X = R$
2018 AMC 12/AHSME, 18
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
$
\textbf{(A) }60 \qquad
\textbf{(B) }65 \qquad
\textbf{(C) }70 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }80 \qquad
$