Found problems: 85335
2023 MMATHS, 6
Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$
2012 Belarus Team Selection Test, 2
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy \[g(f(x+y)) = f(x) + (2x + y)g(y)\] for all real numbers $x$ and $y$.
[i]Proposed by Japan[/i]
2023 Junior Balkan Team Selection Tests - Romania, P1
Determine the real numbers $x$, $y$, $z > 0$ for which
$xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$
2018 IMO Shortlist, C6
Let $a$ and $b$ be distinct positive integers. The following infinite process takes place on an initially empty board.
[list=i]
[*] If there is at least a pair of equal numbers on the board, we choose such a pair and increase one of its components by $a$ and the other by $b$.
[*] If no such pair exists, we write two times the number $0$.
[/list]
Prove that, no matter how we make the choices in $(i)$, operation $(ii)$ will be performed only finitely many times.
Proposed by [I]Serbia[/I].
2010 Belarus Team Selection Test, 1.2
Points $H$ and $T$ are marked respectively on the sides $BC$ abd $AC$ of triangle $ABC$ so that $AH$ is the altitude and $BT$ is the bisectrix $ABC$. It is known that the gravity center of $ABC$ lies on the line $HT$.
a) Find $AC$ if $BC$=a nad $AB$=c.
b) Determine all possible values of $\frac{c}{b}$ for all triangles $ABC$ satisfying the given condition.
2019 India PRMO, 18
Find the number of ordered triples $(a, b)$ of positive integers with $a < b$ and $100 \leq a, b \leq 1000$ satisfy $\gcd(a, b) : \mathrm{lcm}(a, b) = 1 : 495$?
2024 Taiwan TST Round 2, N
For any positive integer $n$, consider its binary representation. Denote by $f(n)$ the number we get after removing all the $0$'s in its binary representation, and $g(n)$ the number of $1$'s in the binary representation. For example, $f(19) = 7$ and $g(19) = 3.$
Find all positive integers $n$ that satisfy
$$n = f(n)^{g(n)}.$$
[i]
Proposed by usjl[/i]
2017 BMT Spring, 9
Let $\vartriangle ABC$ be a triangle. Let $D$ be the point on $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ be the point on the circumcircle of $ABC$ such that $DE$ is tangent to the circumcircle of $ABC$, but $E \ne A$. Let $F$ be the intersection of $AE$ and $BC$. Given that $BF/F C = 4/5$, find the maximum possible value for $\sin \angle ACB$/
1994 National High School Mathematics League, 5
In regular $n$-regular pyramid, the range value of dihedral angle of two adjacent sides is
$\text{(A)}\left(\frac{n-2}{n}\pi,\pi\right)\qquad\text{(B)}\left(\frac{n-1}{n}\pi,\pi\right)\qquad\text{(C)}\left(0,\frac{\pi}{2}\right)\qquad\text{(D)}\left(\frac{n-2}{n}\pi,\frac{n-1}{n}\pi\right)$
2008 Purple Comet Problems, 13
Let $A_1A_2A_3...A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set ${A_1A_2A_3...A_{12}}$
2008 South africa National Olympiad, 6
Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$,
\[f(a+b)=f(a)g(b)+g(a)f(b)\\
g(a+b)=g(a)g(b)-f(a)f(b).\]
2009 Balkan MO Shortlist, G4
Let $ MN$ be a line parallel to the side $ BC$ of a triangle $ ABC$, with $ M$ on the side $ AB$ and $ N$ on the side $ AC$. The lines $ BN$ and $ CM$ meet at point $ P$. The circumcircles of triangles $ BMP$ and $ CNP$ meet at two distinct points $ P$ and $ Q$. Prove that $ \angle BAQ = \angle CAP$.
[i]Liubomir Chiriac, Moldova[/i]
2013 AMC 8, 13
When Clara totaled her scores, she inadvertently reversed the units digit and the tens digit of one score. By which of the following might her incorrect sum have differed from the correct one?
$\textbf{(A)}\ 45 \qquad \textbf{(B)}\ 46 \qquad \textbf{(C)}\ 47 \qquad \textbf{(D)}\ 48 \qquad \textbf{(E)}\ 49$
2024/2025 TOURNAMENT OF TOWNS, P3
It is known that each rectangular parallelepiped has the following property: the square of its volume is equal to the product of areas of its three faces sharing a common vertex. Does there exist a parallelepiped which has the same property but is not rectangular?
Alexandr Bufetov
2011 Postal Coaching, 3
Construct a triangle, by straight edge and compass, if the three points where the extensions of the medians intersect the circumcircle of the triangle are given.
2009 Tournament Of Towns, 2
(a) Find a polygon which can be cut by a straight line into two congruent parts so that one side of the polygon is divided in half while another side at a ratio of $1 : 2$.
(b) Does there exist a convex polygon with this property?
2014 PUMaC Geometry B, 2
Consider the pyramid $OABC$. Let the equilateral triangle $ABC$ with side length $6$ be the base. Also $9=OA=OB=OC$. Let $M$ be the midpoint of $AB$. Find the square of the distance from $M$ to $OC$.
2018 Tuymaada Olympiad, 1
Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$, the roots of the equation $g(x)=h(x)$ are $2$ and $5$, and the roots of the equation $h(x)=f(x)$ are $3$ and $6$?
[i]Proposed by A. Golovanov[/i]
2009 Abels Math Contest (Norwegian MO) Final, 3a
In the triangle $ABC$ the edge $BC$ has length $a$, the edge $AC$ length $b$, and the edge $AB$ length $c$. Extend all the edges at both ends – by the length $a$ from the vertex $A, b$ from $B$, and $c$ from $C$. Show that the six endpoints of the extended edges all lie on a common circle.
[img]https://cdn.artofproblemsolving.com/attachments/8/7/14c8c6a4090d4fade28893729a510d263e7abb.png[/img]
2005 AMC 8, 23
Isosceles right triangle $ ABC$ encloses a semicircle of area $ 2\pi$. The circle has its center $ O$ on hypotenuse $ \overline{AB}$ and is tangent to sides $ \overline{AC}$ and $ \overline{BC}$. What is the area of triangle $ ABC$?
[asy]defaultpen(linewidth(0.8));pair a=(4,4), b=(0,0), c=(0,4), d=(4,0), o=(2,2);
draw(circle(o, 2));
clip(a--b--c--cycle);
draw(a--b--c--cycle);
dot(o);
label("$C$", c, NW);
label("$A$", a, NE);
label("$O$", o, SE);
label("$B$", b, SW);[/asy]
$ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 3\pi\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 4\pi $
2024 Abelkonkurransen Finale, 3b
A $2024$-[i]table [/i]is a table with two rows and $2024$ columns containg all the numbers $1,2,\dots,4048$. Such a table is [i]evenly coloured[/i] if exactly half of the numbers in each row, and one number in each column, is coloured red. The [i]red sum[/i] in an evenly coloured $2024$-table is the sum of all the red numbers in the table.
Let $N$ be the largest number such that every $2024$-table has an even colouring with red sum $\ge N$. Determine $N$, and find the number of $2024$-tables such that every even colouring of the table has red sum $\le N$.
1999 AMC 12/AHSME, 20
The sequence $ a_1$, $ a_2$, $ a_3$, $ \dots$ satisfies $ a_1 \equal{} 19$, $ a_9 \equal{} 99$, and, for all $ n \ge 3$, $ a_n$ is the arithmetic mean of the first $ n \minus{} 1$ terms. Find $ a_2$.
$ \textbf{(A)}\ 29\qquad
\textbf{(B)}\ 59\qquad
\textbf{(C)}\ 79\qquad
\textbf{(D)}\ 99\qquad
\textbf{(E)}\ 179$
2005 Rioplatense Mathematical Olympiad, Level 3, 1
Find all numbers $n$ that can be expressed in the form $n=k+2\lfloor\sqrt{k}\rfloor+2$ for some nonnegative integer $k$.
2011 Tournament of Towns, 2
Peter buys a lottery ticket on which he enters an $n$-digit number, none of the digits being $0$. On the draw date, the lottery administrators will reveal an $n\times n$ table, each cell containing one of the digits from $1$ to $9$. A ticket wins a prize if it does not match any row or column of this table, read in either direction. Peter wants to bribe the administrators to reveal the digits on some cells chosen by Peter, so that Peter can guarantee to have a winning ticket. What is the minimum number of digits Peter has to know?
1985 Traian Lălescu, 1.3
Let $ a,b,c $ denote the lengths of a right triangle ($ a $ being the hypothenuse) that satisfy the equality $ a=2\sqrt{bc} . $
Find the angles of this triangle.