Found problems: 85335
2014 Rioplatense Mathematical Olympiad, Level 3, 2
El ChapulÃn observed that the number $2014$ has an unusual property. By placing its eight positive divisors in increasing order, the fifth divisor is equal to three times the third minus $4$. A number of eight divisors with this unusual property is called the [i]red[/i] number . How many [i]red[/i] numbers smaller than $2014$ exist?
2008 Romania Team Selection Test, 2
Let $ ABC$ be an acute triangle with orthocenter $ H$ and let $ X$ be an arbitrary point in its plane. The circle with diameter $ HX$ intersects the lines $ AH$ and $ AX$ at $ A_{1}$ and $ A_{2}$, respectively. Similarly, define $ B_{1}$, $ B_{2}$, $ C_{1}$, $ C_{2}$. Prove that the lines $ A_{1}A_{2}$, $ B_{1}B_{2}$, $ C_{1}C_{2}$ are concurrent.
[hide][i]Remark[/i]. The triangle obviously doesn't need to be acute.[/hide]
2010 Dutch Mathematical Olympiad, 5
Amber and Brian are playing a game using $2010$ coins. Throughout the game, the coins are divided into a number of piles of at least 1 coin each. A move consists of choosing one or more piles and dividing each of them into two smaller piles. (So piles consisting of only $1$ coin cannot be chosen.)
Initially, there is only one pile containing all $2010$ coins. Amber and Brian alternatingly take turns to make a move, starting with Amber. The winner is the one achieving the situation where all piles have only one coin.
Show that Amber can win the game, no matter which moves Brian makes.
1999 AMC 12/AHSME, 17
Let $ P(x)$ be a polynomial such that when $ P(x)$ is divided by $ x \minus{} 19$, the remainder is $ 99$, and when $ P(x)$ is divided by $ x \minus{} 99$, the remainder is $ 19$. What is the remainder when $ P(x)$ is divided by $ (x \minus{} 19)(x \minus{} 99)$?
$ \textbf{(A)}\ \minus{}x \plus{} 80 \qquad
\textbf{(B)}\ x \plus{} 80 \qquad
\textbf{(C)}\ \minus{}x \plus{} 118 \qquad
\textbf{(D)}\ x \plus{} 118 \qquad
\textbf{(E)}\ 0$
1982 Spain Mathematical Olympiad, 2
By composing a symmetry of axis $r$ with a right angle rotation around from a point $P$ that does not belong to the line, another movement $M$ results. Is $M$ an axis symmetry? Is there any line invariant through $M$?
2014 Saudi Arabia BMO TST, 3
Let $n \ge 2$ be a positive integer, and write in a digit form \[\frac{1}{n}=0.a_1a_2\dots.\] Suppose that $n = a_1 + a_2 + \cdots$. Determine all possible values of $n$.
2003 China National Olympiad, 3
Suppose $a,b,c,d$ are positive reals such that $ab+cd=1$ and $x_i,y_i$ are real numbers such that $x_i^2+y_i^2=1$ for $i=1,2,3,4$. Prove that
\[(ax_1+bx_2+cx_3+dx_4)^2+(ay_4+by_3+cy_2+dy_1)^2\le 2\left(\frac{a^2+b^2}{ab}+\frac{c^2+d^2}{cd}\right).\]
[i]Li Shenghong[/i]
2014 Contests, 2 juniors
Let $ABCD$ be a parallelogram with an acute angle at $A$. Let $G$ be a point on the line $AB$, distinct from $B$, such that $|CG| = |CB|$. Let $H$ be a point on the line $BC$, distinct from $B$, such that $|AB| =|AH|$. Prove that triangle $DGH$ is isosceles.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, G, H;
A = (0,0);
B = (2,0);
D = (0.5,1.5);
C = B + D - A;
G = reflect(A,B)*(C) + C - B;
H = reflect(B,C)*(H) + A - B;
draw(H--A--D--C--G);
draw(interp(A,G,-0.1)--interp(A,G,1.1));
draw(interp(C,H,-0.1)--interp(C,H,1.1));
draw(D--G--H--cycle, dashed);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, E);
dot("$D$", D, NW);
dot("$G$", G, NE);
dot("$H$", H, SE);
[/asy]
2010 South africa National Olympiad, 4
Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that
\[\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.\]
2010 Balkan MO, 2
Let $ABC$ be an acute triangle with orthocentre $H$, and let $M$ be the midpoint of $AC$. The point $C_1$ on $AB$ is such that $CC_1$ is an altitude of the triangle $ABC$. Let $H_1$ be the reflection of $H$ in $AB$. The orthogonal projections of $C_1$ onto the lines $AH_1$, $AC$ and $BC$ are $P$, $Q$ and $R$, respectively. Let $M_1$ be the point such that the circumcentre of triangle $PQR$ is the midpoint of the segment $MM_1$.
Prove that $M_1$ lies on the segment $BH_1$.
2006 Peru MO (ONEM), 1
Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$.
2022 Kazakhstan National Olympiad, 2
Given a prime number $p$. It is known that for each integer $a$ such that $1<a<p/2$ there exist integer $b$ such that $p/2<b<p$ and $p|ab-1$. Find all such $p$.
2023 Princeton University Math Competition, A8
Let $S_0 = 0, S_1 = 1,$ and for $n \ge 2,$ let $S_n = S_{n-1}+5S_{n-2}.$ What is the sum of the five smallest primes $p$ such that $p \mid S_{p-1}$?
2005 Morocco TST, 2
Consider the set $A=\{1,2,...,49\}$. We partitionate $A$ into three subsets. Prove that there exist a set from these subsets containing three distincts elements $a,b,c$ such that $a+b=c$
1985 Austrian-Polish Competition, 4
Solve the system of equations:
$\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$
1992 All Soviet Union Mathematical Olympiad, 577
Find all integers $k > 1$ such that for some distinct positive integers $a, b$, the number $k^a + 1$ can be obtained from $k^b + 1$ by reversing the order of its (decimal) digits.
Ukraine Correspondence MO - geometry, 2014.12
Let $\omega$ be the circumscribed circle of triangle $ABC$, and let $\omega'$ 'be the circle tangent to the side $BC$ and the extensions of the sides $AB$ and $AC$. The common tangents to the circles $\omega$ and $\omega'$ intersect the line $BC$ at points $D$ and $E$. Prove that $\angle BAD = \angle CAE$.
1994 National High School Mathematics League, 10
If $0<\theta<\pi$, then the maximum value of $\sin\frac{\theta}{2}(1+\cos\theta)$ is________.
2004 Mexico National Olympiad, 6
What is the maximum number of possible change of directions in a path traveling on the edges of a rectangular array of $2004 \times 2004$, if the path does not cross the same place twice?.
2017 Princeton University Math Competition, A6/B8
Jackson begins at $1$ on the number line. At each step, he remains in place with probability $85\%$ and increases his position on the number line by $1$ with probability $15\%$. Let $d_n$ be his position on the number line after $n$ steps, and let $E_n$ be the expected value of $\tfrac{1}{d_n}$. Find the least $n$ such that $\tfrac{1}{E_n}
> 2017$.
2019 Thailand TST, 2
Define the sequence $a_0,a_1,a_2,\hdots$ by $a_n=2^n+2^{\lfloor n/2\rfloor}$. Prove that there are infinitely many terms of the sequence which can be expressed as a sum of (two or more) distinct terms of the sequence, as well as infinitely many of those which cannot be expressed in such a way.
2011 Germany Team Selection Test, 2
Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC + AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
1996 Vietnam Team Selection Test, 1
In the plane we are given $3 \cdot n$ points ($n>$1) no three collinear, and the distance between any two of them is $\leq 1$. Prove that we can construct $n$ pairwise disjoint triangles such that: The vertex set of these triangles are exactly the given 3n points and the sum of the area of these triangles $< 1/2$.
1974 All Soviet Union Mathematical Olympiad, 202
Given a convex polygon. You can put no triangle with area $1$ inside it. Prove that you can put the polygon inside a triangle with the area $4$.
2011 IberoAmerican, 1
The number $2$ is written on the board. Ana and Bruno play alternately. Ana begins. Each one, in their turn, replaces the number written by the one obtained by applying exactly one of these operations: multiply the number by $2$, multiply the number by $3$ or add $1$ to the number. The first player to get a number greater than or equal to $2011$ wins. Find which of the two players has a winning strategy and describe it.