Found problems: 85335
2022 Princeton University Math Competition, A7
Let $\vartriangle ABC$ be a triangle with $BC = 7$, $CA = 6$, and, $AB = 5$. Let $I$ be the incenter of $\vartriangle ABC$. Let the incircle of $\vartriangle ABC$ touch sides $BC$, $CA$, and $AB$ at points $D,E$, and $F$. Let the circumcircle of $\vartriangle AEF$ meet the circumcircle of $\vartriangle ABC$ for a second time at point $X\ne A$. Let $P$ denote the intersection of $XI$ and $EF$. If the product $XP \cdot IP$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.
1947 Putnam, A2
A real valued continuous function $f$ satisfies for all real $x$ and $y$ the functional equation
$$ f(\sqrt{x^2 +y^2 })= f(x)f(y).$$
Prove that
$$f(x) =f(1)^{x^{2}}.$$
2024 CCA Math Bonanza, TB1
Let $S$ be the set of all positive integers $n$ such that $3n$ and $n+225$ share a divisor that is not $1$. Find the $100$th smallest element in $S$.
[i]Tiebreaker #1[/i]
2003 AMC 10, 7
The symbolism $ \lfloor x\rfloor$ denotes the largest integer not exceeding $ x$. For example. $ \lfloor3\rfloor\equal{}3$, and $ \lfloor 9/2\rfloor\equal{}4$. Compute
\[ \lfloor\sqrt1\rfloor\plus{}\lfloor\sqrt2\rfloor\plus{}\lfloor\sqrt3\rfloor\plus{}\cdots\plus{}\lfloor\sqrt{16}\rfloor.
\]$ \textbf{(A)}\ 35 \qquad
\textbf{(B)}\ 38 \qquad
\textbf{(C)}\ 40 \qquad
\textbf{(D)}\ 42 \qquad
\textbf{(E)}\ 136$
1997 Vietnam Team Selection Test, 2
There are $ 25$ towns in a country. Find the smallest $ k$ for which one can set up two-direction flight routes connecting these towns so that the following conditions are satisfied:
1) from each town there are exactly $ k$ direct routes to $ k$ other towns;
2) if two towns are not connected by a direct route, then there is a town which has direct routes to these two towns.
2019 CMI B.Sc. Entrance Exam, 5
Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$
2023 JBMO Shortlist, G2
Let $ABC$ be a triangle with $AB<AC$ and $\omega$ be its circumcircle. The tangent line to $\omega$ at $A$ intersects line $BC$ at $D$ and let $E$ be a point on $\omega$ such that $BE$ is parallel to $AD$. $DE$ intersects segment $AB$ and $\omega$ at $F$ and $G$, respectively. The circumcircle of $BGF$ intersects $BE$ at $N$. The line $NF$ intersects lines $AD$ and $EA$ at $S$ and $T$, respectively. Prove that $DGST$ is cyclic.
2019 India IMO Training Camp, P3
Let $k$ be a positive integer. The organising commitee of a tennis tournament is to schedule the matches for $2k$ players so that every two players play once, each day exactly one match is played, and each player arrives to the tournament site the day of his first match, and departs the day of his last match. For every day a player is present on the tournament, the committee has to pay $1$ coin to the hotel. The organisers want to design the schedule so as to minimise the total cost of all players' stays. Determine this minimum cost.
Kyiv City MO 1984-93 - geometry, 1993.8.4
The diameter of a circle of radius $R$ is divided into $4$ equal parts. The point $M$ is taken on the circle. Prove that the sum of the squares of the distances from the point $M$ to the points of division (together with the ends of the diameter) does not depend on the choice of the point $M$. Calculate this sum.
2010 Saudi Arabia Pre-TST, 2.4
Let $AMNB$ be a quadrilateral inscribed in a semicircle of diameter $AB = x$. Denote $AM = a$, $MN = b$, $NB = c$. Prove that $x^3- (a^2 + b^2 + c^2)x -2abc = 0$.
2024 IMC, 3
For which positive integers $n$ does there exist an $n \times n$ matrix $A$ whose entries are all in $\{0,1\}$, such that $A^2$ is the matrix of all ones?
2018 South East Mathematical Olympiad, 4
Does there exist a set $A\subseteq\mathbb{N}^*$ such that for any positive integer $n$, $A\cap\{n,2n,3n,...,15n\}$ contains exactly one element and there exists infinitely many positive integer $m$ such that $\{m,m+2018\}\subset A$? Please prove your conclusion.
1988 IberoAmerican, 5
Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that
\[(x+yt+z^2)(u+vt+wt^2)=1\]
2015 Irish Math Olympiad, 1
In the triangle $ABC$, the length of the altitude from $A$ to $BC$ is equal to $1$. $D$ is the midpoint of $AC$. What are the possible lengths of $BD$?
2002 Tuymaada Olympiad, 5
Prove that for all $ x, y \in \[0, 1\] $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.
1951 Moscow Mathematical Olympiad, 203
A sphere is inscribed in an $n$-angled pyramid. Prove that if we align all side faces of the pyramid with the base plane, flipping them around the corresponding edges of the base, then
(1) all tangent points of these faces to the sphere would coincide with one point, $H$, and
(2) the vertices of the faces would lie on a circle centered at $H$.
2018 Math Prize for Girls Olympiad, 4
For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that
\[
\sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} )
\le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, .
\]
1983 Swedish Mathematical Competition, 3
The systems of equations
\[\left\{ \begin{array}{l}
2x_1 - x_2 = 1 \\
-x_1 + 2x_2 - x_3 = 1 \\
-x_2 + 2x_3 - x_4 = 1 \\
-x_3 + 3x_4 - x_5 =1 \\
\cdots\cdots\cdots\cdots\\
-x_{n-2} + 2x_{n-1} - x_n = 1 \\
-x_{n-1} + 2x_n = 1 \\
\end{array} \right.
\]
has a solution in positive integers $x_i$. Show that $n$ must be even.
2005 Iran Team Selection Test, 2
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:
\[PX || AC \ , \ PY ||AB \]
Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
2012 Grand Duchy of Lithuania, 3
How many ways are there to line up $19$ girls (all of different heights) in a row so that no girl has a shorter girl both in front of and behind her?
2022 VN Math Olympiad For High School Students, Problem 7
Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $s$ is a positive integer. Prove that:
a) ${F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})$ and ${F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).$
b) $k({2^s}) = {3.2^{s - 1}}.$
2022 Saint Petersburg Mathematical Olympiad, 7
Given are $n$ distinct natural numbers. For any two of them, the one is obtained from the other by permuting its digits (zero cannot be put in the first place). Find the largest $n$ such that it is possible all these numbers to be divisible by the smallest of them?
2002 Germany Team Selection Test, 2
Let $A_1$ be the center of the square inscribed in acute triangle $ABC$ with two vertices of the square on side $BC$. Thus one of the two remaining vertices of the square is on side $AB$ and the other is on $AC$. Points $B_1,\ C_1$ are defined in a similar way for inscribed squares with two vertices on sides $AC$ and $AB$, respectively. Prove that lines $AA_1,\ BB_1,\ CC_1$ are concurrent.
2012 NIMO Summer Contest, 5
In the diagram below, three squares are inscribed in right triangles. Their areas are $A$, $M$, and $N$, as indicated in the diagram. If $M = 5$ and $N = 12$, then $A$ can be expressed as $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers and $c$ is not divisible by the square of any prime. Compute $a + b + c$.
[asy]
size(250);
defaultpen (linewidth (0.7) + fontsize (10));
pair O = origin, A = (1, 1), B = (4/3, 1/3), C = (2/3, 5/3), P = (3/2, 0), Q = (0,3);
draw (P--O--Q--cycle^^(0, 5/3)--C--(2/3,1)^^(0,1)--A--(1,0)^^(1,1/3)--B--(4/3,0));
label("$A$", (.5,.5));
label("$M$", (7/6, 1/6));
label("$N$", (1/3, 4/3));[/asy]
[i]Proposed by Aaron Lin[/i]
1992 Taiwan National Olympiad, 4
For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.