Found problems: 85335
2018 Bosnia And Herzegovina - Regional Olympiad, 4
Let $ABCD$ be a cyclic quadrilateral and let $k_1$ and $k_2$ be circles inscribed in triangles $ABC$ and $ABD$. Prove that external common tangent of those circles (different from $AB$) is parallel with $CD$
2014 IMS, 3
Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.
1969 IMO Longlists, 66
$(USS 3)$ $(a)$ Prove that if $0 \le a_0 \le a_1 \le a_2,$ then $(a_0 + a_1x - a_2x^2)^2 \le (a_0 + a_1 + a_2)^2\left(1 +\frac{1}{2}x+\frac{1}{3}x^2+\frac{1}{2}x^3+x^4\right)$
$(b)$ Formulate and prove the analogous result for polynomials of third degree.
1996 All-Russian Olympiad, 4
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),
$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.
[i]O. Musin[/i]
2007 AMC 10, 11
The numbers from $ 1$ to $ 8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum?
$ \textbf{(A)}\ 14 \qquad \textbf{(B)}\ 16 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 20 \qquad \textbf{(E)}\ 24$
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.
KoMaL A Problems 2022/2023, A. 856
In a rock-paper-scissors round robin tournament any two contestants play against each other ten times in a row. Each contestant has a favourite strategy, which is a fixed sequence of ten hands (for example, RRSPPRSPPS), which they play against all other contestants. At the end of the tournament it turned out that every player won at least one hand (out of the ten) against any other player.
Prove that at most $1024$ contestants participated in the tournament.
[i]Submitted by Dávid Matolcsi, Budapest[/i]
2018 Stanford Mathematics Tournament, 1
Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.
1965 IMO Shortlist, 4
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.
May Olympiad L1 - geometry, 2012.3
From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half.
[img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]
1949-56 Chisinau City MO, 30
Through the point of intersection of the diagonals of the trapezoid, a straight line is drawn parallel to its bases. Determine the length of the segment of this straight line, enclosed between the lateral sides of the trapezoid, if the lengths of the bases of the trapezoid are equal to $a$ and $b$.
2015 Purple Comet Problems, 4
Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.
2022 Sharygin Geometry Olympiad, 9
The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$. Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$, and $RM$ meet $DI$ at point $S$.
Prove that $P,N$ and $S$ are collinear.
2015 Thailand TSTST, 2
Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.
2023 Ecuador NMO (OMEC), 1
Find all reals $(a, b, c)$ such that
$$\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}$$
PEN A Problems, 18
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.