Found problems: 85335
2002 Korea - Final Round, 1
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\]
For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.
2022 Saint Petersburg Mathematical Olympiad, 1
The positive integers $a$ and $b$ are such that $a+k$ is divisible by $b+k$ for all positive integers numbers $k<b$. Prove that $a-k$ is divisible by $b-k$ for all positive integers $k<b$.
2017 Irish Math Olympiad, 1
Does there exist an even positive integer $n$ for which $n+1$ is divisible by $5$ and the two numbers $2^n + n$ and $2^n -1$ are co-prime?
1995 Tournament Of Towns, (469) 3
Let $AK$, $BL$ and $CM$ be the angle bisectors of a triangle $ABC$, with $K$ on $BC$. Let $P$ and $Q$ be the points on the lines $BL$ and $CM$ respectively such that $AP = PK$ and $AQ = QK$. Prove that $\angle PAQ = 90^o -\frac12 \angle B AC.$
(I Sharygin)
Novosibirsk Oral Geo Oly VIII, 2020.1
Three squares of area $4, 9$ and $36$ are inscribed in the triangle as shown in the figure. Find the area of the big triangle
[img]https://cdn.artofproblemsolving.com/attachments/9/7/3e904a9c78307e1be169ec0b95b1d3d24c1aa2.png[/img]
2018 Romania National Olympiad, 2
In the square $ABCD$ the point $E$ is located on the side $[AB]$, and $F$ is the foot of the perpendicular from $B$ on the line $DE$. The point $L$ belongs to the line $DE$, such that $F$ is between $E$ and $L$, and $FL = BF$. $N$ and $P$ are symmetric of the points $A , F$ with respect to the lines $DE, BL$, respectively. Prove that:
a) The quadrilateral $BFLP$ is square and the quadrilateral $ALND$ is rhombus.
b) The area of the rhombus $ALND$ is equal to the difference between the areas of the squares $ABCD$ and $BFLP$.
2015 CHMMC (Fall), 1
$3$ players take turns drawing lines that connect vertices of a regular $n$-gon. No player may draw a line that intersects another line at a point other than a vertex of the $n-$gon. The last player able to draw a line wins. For how many $n$ in the range $4\le n \le 100$ does the first player have a winning strategy?
2005 Postal Coaching, 9
In how many ways can $n$ identical balls be distributed to nine persons $A,B,C,D,E,F,G,H,I$ so that the number of balls recieved by $A$ is the same as the total number of balls recieved by $B,C,D,E$ together,.
2022 Kosovo & Albania Mathematical Olympiad, 3
Let $ABCD$ be a square and let $M$ be the midpoint of $BC$. Let $X$ and $Y$ be points on the segments $AB$ and $CD$, respectively. Prove that $\angle XMY = 90^\circ$ if and only if $BX + CY = XY$.
[i]Note: In the competition, students were only asked to prove the 'only if' direction.[/i]
2006 ISI B.Stat Entrance Exam, 5
Let $A,B$ and $C$ be three points on a circle of radius $1$.
(a) Show that the area of the triangle $ABC$ equals
\[\frac12(\sin(2\angle ABC)+\sin(2\angle BCA)+\sin(2\angle CAB))\]
(b) Suppose that the magnitude of $\angle ABC$ is fixed. Then show that the area of the triangle $ABC$ is maximized when $\angle BCA=\angle CAB$
(c) Hence or otherwise, show that the area of the triangle $ABC$ is maximum when the triangle is equilateral.
2005 National Olympiad First Round, 19
What is the greatest real root of the equation $x^3-x^2-x-\frac 13 = 0$?
$
\textbf{(A)}\ \dfrac{\sqrt {3} - \sqrt{2}}{2}
\qquad\textbf{(B)}\ \dfrac{\sqrt [3]{3} - \sqrt[3]{2}}{2}
\qquad\textbf{(C)}\ \dfrac 1{\sqrt[3] {3} - 1}
\qquad\textbf{(D)}\ \dfrac 1{\sqrt[3] {4} - 1}
\qquad\textbf{(E)}\ \text{None of above}
$
Ukrainian TYM Qualifying - geometry, 2020.10
In triangle $ABC$, point $I$ is the center, point $I_a$ is the center of the excircle tangent to the side $BC$. From the vertex $A$ inside the angle $BAC$ draw rays $AX$ and $AY$. The ray $AX$ intersects the lines $BI$, $CI$, $BI_a$, $CI_a$ at points $X_1$, $...$, $X_4$, respectively, and the ray $AY$ intersects the same lines at points $Y_1$, $...$, $Y_4$ respectively. It turned out that the points $X_1,X_2,Y_1,Y_2$ lie on the same circle. Prove the equality $$\frac{X_1X_2}{X_3X_4}= \frac{Y_1Y_2}{Y_3Y_4}.$$
1977 AMC 12/AHSME, 4
[asy]
size(130);
pair A = (2, 2.4), C = (0, 0), B = (4.3, 0),
E = 0.7*A, F = 0.57*A + 0.43*B, D = (2.4, 0);
draw(A--B--C--cycle);
draw(E--D--F);
label("$A$", A, N);
label("$B$", B, E);
label("$C$", C, W);
label("$D$", D, S);
label("$E$", E, NW);
label("$F$", F, NE);
//Credit to MSTang for the diagram[/asy]
In triangle $ABC$, $AB=AC$ and $\measuredangle A=80^\circ$. If points $D$, $E$, and $F$ lie on sides $BC$, $AC$ and $AB$, respectively, and $CE=CD$ and $BF=BD$, then $\measuredangle EDF$ equals
$\textbf{(A) }30^\circ\qquad\textbf{(B) }40^\circ\qquad\textbf{(C) }50^\circ\qquad\textbf{(D) }65^\circ\qquad \textbf{(E) }\text{none of these}$
1992 IMO Longlists, 66
A circle of radius $\rho$ is tangent to the sides $AB$ and $AC$ of the triangle $ABC$, and its center $K$ is at a distance $p$ from $BC$.
[i](a)[/i] Prove that $a(p - \rho) = 2s(r - \rho)$, where $r$ is the inradius and $2s$ the perimeter of $ABC$.
[i](b)[/i] Prove that if the circle intersect $BC$ at $D$ and $E$, then
\[DE=\frac{4\sqrt{rr_1(\rho-r)(r_1-\rho)}}{r_1-r}\]
where $r_1$ is the exradius corresponding to the vertex $A.$
2011 Tuymaada Olympiad, 3
An excircle of triangle $ABC$ touches the side $AB$ at $P$ and the extensions of sides $AC$ and $BC$ at $Q$ and $R$, respectively. Prove that if the midpoint of $PQ$ lies on the circumcircle of $ABC$, then the midpoint of $PR$ also lies on that circumcircle.
1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 6
The number of 3-digit numbers not containing the digit 0 and such that one of the digits is the sum of the two others is
$ \text{(A)}\ 96 \qquad \text{(B)}\ 100 \qquad \text{(C)}\ 104 \qquad \text{(D)}\ 106 \qquad \text{(E)}\ 108$
2022 China Second Round, 3
Let $a_1,a_2,\cdots ,a_{100}$ be non-negative integers such that
$(1)$ There are positive integers$ k\leq 100$ such that $a_1\leq a_2\leq \cdots\leq a_{k}$
and $a_i=0$ $(i>k);$
$(2)$ $ a_1+a_2+a_3+\cdots +a_{100}=100;$
$(3)$ $ a_1+2a_2+3a_3+\cdots +100a_{100}=2022.$
Find the minimum of $ a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.$
2013 Chile TST Ibero, 1
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.
2021 Ukraine National Mathematical Olympiad, 2
Denote by $P^{(n)}$ the set of all polynomials of degree $n$ the coefficients of which is a permutation of the set of numbers $\{2^0, 2^1,..., 2^n\}$. Find all pairs of natural numbers $(k,d)$ for which there exists a $n$ such that for any polynomial $p \in P^{(n)}$, number $P(k)$ is divisible by the number $d$.
(Oleksii Masalitin)
2020 LMT Fall, B2
The area of a square is $144$. An equilateral triangle has the same perimeter as the square. The area of a regular hexagon is $6$ times the area of the equilateral triangle. What is the perimeter of the hexagon?
2019 Junior Balkan Team Selection Tests - Romania, 3
Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.
1965 Putnam, A5
In how many ways can the integers from $1$ to $n$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by $1$ from some integer to the left of it?
2019 Moldova Team Selection Test, 2
Prove that $E_n=\frac{\arccos {\frac{n-1}{n}} } {\text{arccot} {\sqrt{2n-1} }}$ is a natural number for any natural number $n$.
(A natural number is a positive integer)
2024 All-Russian Olympiad Regional Round, 9.4
The positive integers $1, 2, \ldots, 1000$ are written in some order on one line. Show that we can find a block of consecutive numbers, whose sum is in the interval $(100000; 100500]$.
2022 Girls in Math at Yale, 2
How many ways are there to fill in a $2\times 2$ square grid with the numbers $1,2,3,$ and $4$ such that the numbers in any two grid squares that share an edge have an absolute difference of at most $2$?
[i]Proposed by Andrew Wu[/i]