Found problems: 85335
2024 Greece Junior Math Olympiad, 3
Examine if we can put the sixteen positive divisors of $2024$ on the cells of the table shown such that the sum of the four numbers of any line or row to be a multiple of $3$.
$ \begin{tabular}{ | l | c | c | r| }
\hline
& & & \\ \hline
& & & \\ \hline
& & & \\ \hline
& & & \\
\hline
\end{tabular}
$
2002 Estonia National Olympiad, 3
John takes seven positive integers $a_1,a_2,...,a_7$ and writes the numbers $a_i a_j$, $a_i+a_j$ and $|a_i -a_j |$ for all $i \ne j$ on the blackboard. Find the greatest possible number of distinct odd integers on the blackboard.
1958 AMC 12/AHSME, 8
Which of these four numbers $ \sqrt{\pi^2},\,\sqrt[3]{.8},\,\sqrt[4]{.00016},\,\sqrt[3]{\minus{}1}\cdot \sqrt{(.09)^{\minus{}1}}$, is (are) rational:
$ \textbf{(A)}\ \text{none}\qquad
\textbf{(B)}\ \text{all}\qquad
\textbf{(C)}\ \text{the first and fourth}\qquad
\textbf{(D)}\ \text{only the fourth}\qquad
\textbf{(E)}\ \text{only the first}$
2015 India PRMO, 16
$16.$ In an acute angle triangle $ABC,$ let $D$ be the foot of the altitude from $A,$ and $E$ be the midpoint of $BC.$ Let $F$ be the midpoint of $AC.$ Suppose $\angle{BAE}=40^o. $ If $\angle{DAE}=\angle{DFE},$ What is the magnitude of $\angle{ADF}$ in degrees $?$
2004 Romania National Olympiad, 1
On the sides $AB,AD$ of the rhombus $ABCD$ are the points $E,F$ such that $AE=DF$. The lines $BC,DE$ intersect at $P$ and $CD,BF$ intersect at $Q$. Prove that:
(a) $\frac{PE}{PD} + \frac{QF}{QB} = 1$;
(b) $P,A,Q$ are collinear.
[i]Virginia Tica, Vasile Tica[/i]
2021 Romania National Olympiad, 1
In the cuboid $ABCDA'B'C'D'$ with $AB=a$, $AD=b$ and $AA'=c$ such that $a>b>c>0$, the points $E$ and $F$ are the orthogonal projections of $A$ on the lines $A'D$ and $A'B$, respectively, and the points $M$ and $N$ are the orthogonal projections of $C$ on the lines $C'D$ and $C'B$, respectively. Let $DF\cap BE=\{G\}$ and $DN\cap BM=\{P\}$.
[list=a]
[*] Show that $(A'AG)\parallel (C'CP)$ and determine the distance between these two planes;
[*] Show that $GP\parallel (ABC)$ and determine the distance between the line $GP$ and the plane $(ABC)$.
[/list]
[i]Petre Simion, Nicolae Victor Ioan[/i]
1978 Canada National Olympiad, 2
Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.
2023 Thailand TST, 3
For a positive integer $n$ we denote by $s(n)$ the sum of the digits of $n$. Let $P(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a polynomial, where $n \geqslant 2$ and $a_i$ is a positive integer for all $0 \leqslant i \leqslant n-1$. Could it be the case that, for all positive integers $k$, $s(k)$ and $s(P(k))$ have the same parity?
1998 North Macedonia National Olympiad, 3
A triangle $ABC$ is given. For every positive numbers $p,q,r$, let $A',B',C'$ be the points such that $\overrightarrow{BA'} = p\overrightarrow{AB}, \overrightarrow{CB'}=q\overrightarrow{BC} $, and $\overrightarrow{AC'}=r\overrightarrow{CA}$. Define $f(p,q,r)$ as the ratio of the area of $\vartriangle A'B'C'$ to that of $\vartriangle ABC$. Prove that for all positive numbers $x,y,z$ and every positive integer $n$, $\sum_{k=0}^{n-1}f(x+k,y+k,z+k) = n^3f\left(\frac{x}{n},\frac{y}{n},\frac{z}{n}\right)$.
2021 Saudi Arabia IMO TST, 9
For a positive integer $n$, let $d(n)$ be the number of positive divisors of $n$, and let $\varphi(n)$ be the number of positive integers not exceeding $n$ which are coprime to $n$. Does there exist a constant $C$ such that
$$ \frac {\varphi ( d(n))}{d(\varphi(n))}\le C$$
for all $n\ge 1$
[i]Cyprus[/i]
1984 Bundeswettbewerb Mathematik, 1
Let $n$ be a positive integer and $M = \{1, 2, 3, 4, 5, 6\}$. Two persons $A$ and $B$ play in the following Way: $A$ writes down a digit from $M$, $B$ appends a digit from $M$, and so it becomes alternately one digit from $M$ is appended until the $2n$-digit decimal representation of a number has been created. If this number is divisible by $9$, $B$ wins, otherwise $A$ wins.
For which $n$ can $A$ and for which $n$ can $B$ force the win?
1953 Moscow Mathematical Olympiad, 252
Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.
2008 JBMO Shortlist, 4
Find all triples $(x,y,z)$ of real numbers that satisfy the system
$\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}$
Estonia Open Junior - geometry, 2001.2.2
In a triangle $ABC$, the lengths of the sides are consecutive integers and median drawn from $A$ is perpendicular to the bisector drawn from $B$. Find the lengths of the sides of triangle $ABC$.
2006 Iran MO (3rd Round), 8
We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$
a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$.
b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$.
c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$.
d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$.
You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)
2021 CCA Math Bonanza, L3.4
Compute the sum of $x^2+y^2$ over all four ordered pairs $(x,y)$ of real numbers satisfying $x=y^2-20$ and $y=x^2+x-21$.
[i]2021 CCA Math Bonanza Lightning Round #3.4[/i]
1979 IMO Longlists, 71
Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$
KoMaL A Problems 2021/2022, A. 811
Let $A$ be a given set with $n$ elements. Let $k<n$ be a given positive integer. Find the maximum value of $m$ for which it is possible to choose sets $B_i$ and $C_i$ for $i=1,2,\ldots,m$ satisfying the following conditions:
[list=1]
[*]$B_i\subset A,$ $|B_i|=k,$
[*]$C_i\subset B_i$ (there is no additional condition for the number of elements in $C_i$), and
[*]$B_i\cap C_j\neq B_j\cap C_i$ for all $i\neq j.$
[/list]
Denmark (Mohr) - geometry, 2007.1
Triangle $ABC$ lies in a regular decagon as shown in the figure.
What is the ratio of the area of the triangle to the area of the entire decagon?
Write the answer as a fraction of integers.
[img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]
2014 Contests, 2
Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that \[ \angle BCQ = \dfrac{1}{2} \angle AQP .\]
2020/2021 Tournament of Towns, P4
It is well-known that a quadratic equation has no more than 2 roots. Is it possible for the equation $\lfloor x^2\rfloor+px+q=0$ with $p\neq 0$ to have more than 100 roots?
[i]Alexey Tolpygo[/i]
2014 Spain Mathematical Olympiad, 2
Let $M$ be the set of all integers in the form of $a^2+13b^2$, where $a$ and $b$ are distinct itnegers.
i) Prove that the product of any two elements of $M$ is also an element of $M$.
ii) Determine, reasonably, if there exist infinite pairs of integers $(x,y)$ so that $x+y\not\in M$ but $x^{13}+y^{13}\in M$.
Ukrainian TYM Qualifying - geometry, III.11
A circle centered at point $O$ is separated by points $A_1,A_2,...,A_n$ on $n$ equal parts (points are listed sequentially clockwise) and the rays $OA_1,OA_2,...,OA_n$ are constructed. The angle $A_2OA_3$ is divided by rays into two equal angles at vertex $O$, the angle $A_3OA_4$ is divided into three equal angles, and so on, finally, the angle $A_nOA_1$ divided into $n$ equal angles at vertex $O$. A point belonging to the ray other than $OA_1$, is connected by a segment with its orthogonal projection $B_0$ on the neighboring (clockwise) arrow) with ray $OA_1$, point$ B_1$ is connected by a segment with its orthogonal projection on the next (clockwise) ray, etc. As a result of such process it turns out the broken line $B_0B_1B_2B_3...$ infinitely "twists". Consider the question of giving the thus obtained broken numerical value of "length" $L (n)$ and explore the value of $L(n)$ depending on $n$.
Denmark (Mohr) - geometry, 2010.1
Four right triangles, each with the sides $1$ and $2$, are assembled to a figure as shown.
How large a fraction does the area of the small circle make up of that of the big one?
[img]https://1.bp.blogspot.com/-XODK1XKCS0Q/XzXDtcA-xAI/AAAAAAAAMWA/zSLPpf3IcX0rgaRtOxm_F2begnVdUargACLcBGAsYHQ/s0/2010%2BMohr%2Bp1.png[/img]
2021 CCA Math Bonanza, T4
Let $ABCD$ be a unit square. Points $M$ and $N$ are the midpoints of sides $AB$ and $BC$ respectively. Let $P$ and $Q$ be the midpoints of line segments $AM$ and $BN$ respectively. Find the reciprocal of the area of the triangle enclosed by the three line segments $PQ$, $MN$, and $DB$.
[asy]
size(5 cm);
pair A=(0,0); pair B=(1,0); pair C=(1,1); pair D=(0,1); pair M=(0.5,0); pair N=(1,0.5); pair P=(0.25,0); pair Q=(1,0.25);
draw(A--B--C--D--cycle);
draw(M--N); draw(P--Q); draw(B--D);
label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$M$",M,S); label("$N$",N,E); label("$P$",P,S); label("$Q$",Q,E);
fill((0.8125,0.1875)--(0.75,0.25)--(0.625,0.125)--cycle, gray);[/asy]
[i]2021 CCA Math Bonanza Team Round #4[/i]