Found problems: 85335
2022 Thailand Mathematical Olympiad, 10
For each positive integers $u$ and $n$, say that $u$ is a [i]friend[/i] of $n$ if and only if there exists a positive integer $N$ that is a multiple of $n$ and the sum of digits of $N$ (in base 10) is equal to $u$. Determine all positive integers $n$ that only finitely many positive integers are not a friend of $n$.
2020 MMATHS, I12
Let $p(x)$ be the monic cubic polynomial with roots $\sin^2(1^{\circ})$, $\sin^2(3^{\circ})$, and $\sin^2(9^{\circ})$. Suppose that $p\left(\frac{1}{4}\right)=\frac{\sin(a^{\circ})}{n\sin(b^{\circ})}$, where $0 <a,b \le 90$ and $a,b,n$ are positive integers. What is $a+b+n$?
[i]Proposed by Andrew Yuan[/i]
1989 IMO Shortlist, 31
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation
\[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\]
where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]
2024 CMIMC Geometry, 6
Andrew Mellon found a piece of melon that is shaped like a octagonal prism where the bases are regular. Upon slicing it in half once, he found that he created a cross-section that is an equilateral hexagon. What is the minimum possible ratio of the height of the melon piece to the side length of the base?
[i]Proposed by Lohith Tummala[/i]
2021 Saint Petersburg Mathematical Olympiad, 1
Solve the following system of equations $$\sin^2{x} + \cos^2{y} = y^4. $$ $$\sin^2{y} + \cos^2{x} = x^2. $$
[i]A. Khrabov[/i]
2006 Cezar Ivănescu, 1
Solve the equation
[b]a)[/b] $ \log_2^2 +(x-1)\log_2 x =6-2x $ in $ \mathbb{R} . $
[b]b)[/b] $ 2^{x+1}+3^{x+1} +2^{1/x^2}+3^{1/x^2}=18 $ in $ (0,\infty ) . $
[i]Cristinel Mortici[/i]
2001 AMC 8, 11
Points $A, B, C$ and $D$ have these coordinates: $A(3,2), B(3,-2), C(-3,-2)$ and $D(-3, 0)$. The area of quadrilateral $ABCD$ is
[asy]
for (int i = -4; i <= 4; ++i)
{
for (int j = -4; j <= 4; ++j)
{
dot((i,j));
}
}
draw((0,-4)--(0,4),linewidth(1));
draw((-4,0)--(4,0),linewidth(1));
for (int i = -4; i <= 4; ++i)
{
draw((i,-1/3)--(i,1/3),linewidth(0.5));
draw((-1/3,i)--(1/3,i),linewidth(0.5));
}[/asy]
$ \text{(A)}\ 12\qquad\text{(B)}\ 15\qquad\text{(C)}\ 18\qquad\text{(D)}\ 21\qquad\text{(E)}\ 24 $
1998 Belarus Team Selection Test, 1
Let $S(n)$ be the sum of all different natural divisors of odd natural number $n> 1$ (including $n$ and $1$).
Prove that $(S(n))^3 <n^4$.
2004 Thailand Mathematical Olympiad, 12
Let $n$ be a positive integer and define $A_n = \{1, 2, ..., n\}$. How many functions $f : A_n \to A_n$ are there such that for all $x, y \in A_n$, if $x < y$ then $f(x) \ge f(y)$?
2023 AMC 10, 7
Square $ABCD$ is rotated $20^\circ$ clockwise about its center to obtain square $EFGH$, as shown below. What is the degree measure of $\angle EAB$?
[asy]
size(170);
defaultpen(linewidth(0.6));
real r = 25;
draw(dir(135)--dir(45)--dir(315)--dir(225)--cycle);
draw(dir(135-r)--dir(45-r)--dir(315-r)--dir(225-r)--cycle);
label("$A$",dir(135),NW);
label("$B$",dir(45),NE);
label("$C$",dir(315),SE);
label("$D$",dir(225),SW);
label("$E$",dir(135-r),N);
label("$F$",dir(45-r),E);
label("$G$",dir(315-r),S);
label("$H$",dir(225-r),W);
[/asy]
$\textbf{(A) }20^\circ\qquad\textbf{(B) }30^\circ\qquad\textbf{(C) }32^\circ\qquad\textbf{(D) }35^\circ\qquad\textbf{(E) }45^\circ$
2005 Sharygin Geometry Olympiad, 13
A triangle $ABC$ and two lines $\ell_1, \ell_2$ are given. Through an arbitrary point $D$ on the side $AB$, a line parallel to $\ell_1$ intersects the $AC$ at point $E$ and a line parallel to $\ell_2$ intersects the $BC$ at point $F$. Construct a point $D$ for which the segment $EF$ has the smallest length.
2024 Indonesia TST, N
Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that for every prime number $p$ and natural number $x$,
$$\{ x,f(x),\cdots f^{p-1}(x) \} $$
is a complete residue system modulo $p$. With $f^{k+1}(x)=f(f^k(x))$ for every natural number $k$ and $f^1(x)=f(x)$.
[i]Proposed by IndoMathXdZ[/i]
OIFMAT III 2013, 7
Define $ a \circledast b = a + b-2ab $. Calculate the value of
$$A=\left( ...\left(\left(\frac{1}{2014}\circledast \frac{2}{2014}\right)\circledast\frac{3}{2014}\right)...\right)\circledast\frac{2013}{2014}$$
2008 Singapore MO Open, 5
consider a $2008 \times 2008$ chess board. let $M$ be the smallest no of rectangles that can be drawn on the chess board so that sides of every cell of the board is contained in the sides of one of the rectangles. find the value of $M$. (eg for $2\times 3$ chessboard, the value of $M$ is 3.)
Indonesia MO Shortlist - geometry, g6
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$
2015 CCA Math Bonanza, T4
Evaluate the continued fraction $$1+\frac{2}{2+\frac{2}{2+\ldots}}$$
[i]2015 CCA Math Bonanza Team Round #4[/i]
2021 CMIMC, 1.5
There are exactly 7 possible tetrominos (groups of 4 connected squares in a grid):
[img]https://cdn.discordapp.com/attachments/813077401265242143/816189385859006474/tetris.png[/img]
Daniel has a $2 \times 20210$ rectangle and wants to tile the interior with tetrominos without overlaps, pieces sticking out, or extra pieces left over. Note that you are allowed to rotate tetrominos but not reflect them.
For how many multisets of tetrominos (ie. an ordered tuple of how many of each tile he has) is it possible to exactly tile his $2\times20210$ rectangle?
[i]Proposed by Dilhan Salgado[/i]
2019 AIME Problems, 13
Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \neq C$ is a point of intersection of the circumcircles of $\triangle ACD$ and $\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expressed as $\tfrac{a+b\sqrt{c}}{d}$, where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$.
2000 Moldova National Olympiad, Problem 6
Find all real values of the parameter $a$ for which the system
\begin{align*}
&1+\left(4x^2-12x+9\right)^2+2^{y+2}=a\\
&\log_3\left(x^2-3x+\frac{117}4\right)+32=a+\log_3(2y+3)
\end{align*}has a unique real solution. Solve the system for these values of $a$.
1999 Junior Balkan MO, 1
Let $ a,b,c,x,y$ be five real numbers such that $ a^3 \plus{} ax \plus{} y \equal{} 0$, $ b^3 \plus{} bx \plus{} y \equal{} 0$ and $ c^3 \plus{} cx \plus{} y \equal{} 0$. If $ a,b,c$ are all distinct numbers prove that their sum is zero.
[i]Ciprus[/i]
1969 IMO Shortlist, 47
$C$ is a point on the semicircle diameter $AB$, between $A$ and $B$. $D$ is the foot of the perpendicular from $C$ to $AB$. The circle $K_1$ is the incircle of $ABC$, the circle $K_2$ touches $CD,DA$ and the semicircle, the circle $K_3$ touches $CD,DB$ and the semicircle. Prove that $K_1,K_2$ and $K_3$ have another common tangent apart from $AB$.
1980 Polish MO Finals, 4
Show that for every polynomial $W$ in three variables there exist polynomials $U$ and $V$ such that:
$$W(x,y,z) = U(x,y,z)+V(x,y,z),$$
$$U(x,y,z) = U(y,x,z),$$
$$V(x,y,z) = -V(x,z,y).$$
2016 Math Prize for Girls Problems, 3
Compute the least possible value of $ABCD - AB \times CD$, where $ABCD$ is a 4-digit positive integer, and $AB$ and $CD$ are 2-digit positive integers. (Here $A$, $B$, $C$, and $D$ are digits, possibly equal. Neither $A$ nor $C$ can be zero.)
2023 Iran Team Selection Test, 2
Suppose $\frac{1}{2} < s < 1$ . An insect flying on $[0,1]$ . If it is on point $a$ , it jump into point $ a\times s$ or $(a-1) \times s +1$ . For every real number $0 \le c \le 1$, Prove that insect can jump that after some jumps , it has a distance less than $\frac {1}{1402}$ from point $c$.
[i]Proposed by Navid Safaei [/i]
1995 Irish Math Olympiad, 4
Consider the following one-person game played on the real line. During the game disks are piled at some of the integer points on the line. To perform a move in the game, the player chooses a point $ j$ at which at least two disks are piled and then takes two disks from the point $ j$ and places one of them at $ j\minus{}1$ and one at $ j\plus{}1$. Initially, $ 2n\plus{}1$ disks are placed at point $ 0$. The player proceeds to perform moves as long as possible. Prove that after $ \frac{1}{6} n(n\plus{}1)(2n\plus{}1)$ moves no further moves will be possible and that at this stage, one disks remains at each of the positions $ \minus{}n,\minus{}n\plus{}1,...,0,...n$.