Found problems: 85335
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]
2017 Balkan MO Shortlist, C1
A grasshopper is sitting at an integer point in the Euclidean plane. Each second it jumps to another integer point in such a way that the jump vector is constant. A hunter that knows neither the starting point of the grasshopper nor the jump vector (but knows that the jump vector for each second is constant) wants to catch the grasshopper. Each second the hunter can choose one integer point in the plane and, if the grasshopper is there, he catches it. Can the hunter always catch the grasshopper in a finite amount of time?
2011 VJIMC, Problem 3
Prove that
$$\sum_{k=0}^\infty x^k\frac{1+x^{2k+2}}{(1-x^{2k+2})^2}=\sum_{k=0}^\infty(-1)^k\frac{x^k}{(1-x^{k+1})^2}$$for all $x\in(-1,1)$.