Found problems: 85335
1994 AMC 12/AHSME, 23
In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0), (0,3), (3,3), (3,1), (5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is
[asy]
size(200);
Label l;
l.p=fontsize(6);
xaxis("$x$",0,6,Ticks(l,1.0,0.5),EndArrow);
yaxis("$y$",0,4,Ticks(l,1.0,0.5),EndArrow);
draw((0,3)--(3,3)--(3,1)--(5,1)--(5,0)--(0,0)--cycle,black+linewidth(2));[/asy]
$ \textbf{(A)}\ \frac{2}{7} \qquad\textbf{(B)}\ \frac{1}{3} \qquad\textbf{(C)}\ \frac{2}{3} \qquad\textbf{(D)}\ \frac{3}{4} \qquad\textbf{(E)}\ \frac{7}{9} $
2004 Tuymaada Olympiad, 3
Zeroes and ones are arranged in all the squares of $n\times n$ table.
All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form
[asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy]
(consisting of a square and its neighbours from left and from below)
is even.
Prove that no two rows of the table are identical.
[i]Proposed by O. Vanyushina[/i]
2023 Princeton University Math Competition, A2 / B4
If $\theta$ is the unique solution in $(0,\pi)$ to the equation $2\sin(x)+3\sin(\tfrac{3x}{2})+\sin(2x)+3\sin(\tfrac{5x}{2})=0,$ then $\cos(\theta)=\tfrac{a-\sqrt{b}}{c}$ for positive integers $a,b,c$ such that $a$ and $c$ are relatively prime. Find $a+b+c.$
2016 LMT, 24
Let $S$ be a set consisting of all positive integers less than or equal to $100$. Let $P$ be a subset of $S$ such that there do not exist two elements $x,y\in P$ such that $x=2y$. Find the maximum possible number of elements of $P$.
[i]Proposed by Nathan Ramesh
2014 France Team Selection Test, 3
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
1988 AMC 12/AHSME, 19
Simplify \[\frac{bx(a^2x^2 + 2a^2y^2 + b^2y^2) + ay(a^2x^2 + 2b^2x^2 + b^2y^2)}{bx + ay}.\]
$ \textbf{(A)}\ a^2x^2 + b^2y^2\qquad\textbf{(B)}\ (ax + by)^2\qquad\textbf{(C)}\ (ax + by)(bx + ay)\qquad\textbf{(D)}\ 2(a^2x^2 + b^2y^2)\qquad\textbf{(E)}\ (bx + ay)^2 $
2002 Pan African, 6
If $a_1 \geq a_2 \geq \cdots \geq a_n \geq 0$ and $a_1+a_2+\cdots+a_n=1$, then prove:
\[a_1^2+3a_2^2+5a_3^2+ \cdots +(2n-1)a_n^2 \leq 1\]
2024 Belarusian National Olympiad, 10.3
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that for every $x,y \in \mathbb{R}$ the following equation holds:$$1+f(xy)=f(x+f(y))+(y-1)f(x-1)$$
[i]M. Zorka[/i]
2017 Mathematical Talent Reward Programme, MCQ: P 7
Let $ABCD$ be a quadrilateral with sides $AB=2$, $BC=CD=4$ and $DA=5$. The opposite angles $A$ and $C$ are equal. The length of diagonal $BD$ equals
[list=1]
[*] $2\sqrt{6}$
[*] $3\sqrt{3}$
[*] $3\sqrt{6}$
[*] $2\sqrt{3}$
[/list]
2010 IFYM, Sozopol, 3
Through vertex $C$ of $\Delta ABC$ are constructed lines $l_1$ and $l_2$ which are symmetrical about the angle bisector $CL_c$. Prove that the projections of $A$ and $B$ on lines $l_1$ and $l_2$ lie on one circle.