Found problems: 85335
2016 Belarus Team Selection Test, 2
Let $ABC$ be a triangle with $\angle{C} = 90^{\circ}$, and let $H$ be the foot of the altitude from $C$. A point $D$ is chosen inside the triangle $CBH$ so that $CH$ bisects $AD$. Let $P$ be the intersection point of the lines $BD$ and $CH$. Let $\omega$ be the semicircle with diameter $BD$ that meets the segment $CB$ at an interior point. A line through $P$ is tangent to $\omega$ at $Q$. Prove that the lines $CQ$ and $AD$ meet on $\omega$.
2019 JBMO Shortlist, N1
Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number
$x^p + y^p + z^p - x - y - z$
is a product of exactly three distinct prime numbers.
1987 Putnam, B6
Let $F$ be the field of $p^2$ elements, where $p$ is an odd prime. Suppose $S$ is a set of $(p^2-1)/2$ distinct nonzero elements of $F$ with the property that for each $a\neq 0$ in $F$, exactly one of $a$ and $-a$ is in $S$. Let $N$ be the number of elements in the intersection $S \cap \{2a: a \in S\}$. Prove that $N$ is even.
2010 Portugal MO, 3
On each day, more than half of the inhabitants of Évora eats [i]sericaia[/i] as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate [i]sericaia[/i] as dessert.
2004 Swedish Mathematical Competition, 2
In one country there are coins of value $1,2,3,4$ or $5$. Nisse wants to buy a pair of shoes. While paying, he tells the seller that he has $100$ coins in the bag, but that he does not know the exact number of coins of each value. ”Fine, then you will have the exact amount”, the seller responds. What is the price of the shoes, and how did the seller conclude that Nisse would have the exact amount?
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.