Found problems: 85335
2003 Flanders Math Olympiad, 3
A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.
2018 Azerbaijan IZhO TST, 1
Problem 3. Suppose that the equation x^3-ax^2+bx-a=0 has three positive real roots (b>0). Find the minimum value of the expression:
(b-a)(b^3+3a^3)
2014 Cuba MO, 8
Let $a$ and $b$ be real numbers. It is known that the graph of the parabola $y =ax^2 +b$ cuts the graph of the curve $y = x+1/x$ in exactly three points. Prove that $3ab < 1$.
2015 Cuba MO, 4
Let $A = \overline{abcd}$ be a $4$-digit positive integer, such that $a\ge 7$ and $a > b >c > d > 0$. Let us consider a positive integer $B = \overline{dcba}$. If all digits of $A+B$ are odd, determine all possible values of $A$.
2015 Saudi Arabia IMO TST, 3
Let $a_1, a_2, ...,a_n$ be positive real numbers such that $$a_1 + a_2 + ... + a_n = a_1^2 + a_2^2 + ... + a_n^2$$ Prove that $$\sum_{1\le i<j\le n} a_ia_j(1 - a_ia_j) \ge 0$$
Võ Quốc Bá Cẩn.
2009 Federal Competition For Advanced Students, P2, 5
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.
How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime?
2000 Slovenia National Olympiad, Problem 4
Three boxes with at least one marble in each are given. In each step we double the number of marbles in one of the boxes, taking the required number of boxes from one of the other two boxes. Is it always possible to have one of the boxes empty after several steps?
2010 Macedonia National Olympiad, 2
Let $a,b,c$ be positive real numbers for which $a+b+c=3$. Prove the inequality
\[\frac{a^3+2}{b+2}+\frac{b^3+2}{c+2}+\frac{c^3+2}{a+2}\ge3\]
2006 Costa Rica - Final Round, 2
If $ a$, $ b$, $ c$ are the sidelengths of a triangle, then prove that
$ \frac {3\left(a^4 \plus{} b^4 \plus{} c^4\right)}{\left(a^2 \plus{} b^2 \plus{} c^2\right)^2} \plus{} \frac {bc \plus{} ca \plus{} ab}{a^2 \plus{} b^2 \plus{} c^2}\geq 2$.
2022 Sharygin Geometry Olympiad, 7
A square with center $F$ was constructed on the side $AC$ of triangle $ABC$ outside it. After this, everything was erased except $F$ and the midpoints $N,K$ of sides $BC,AB$.
Restore the triangle.
2007 Singapore MO Open, 2
Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial
$f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients
and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.
1979 Putnam, B5
In the plane, let $C$ be a closed convex set that contains $(0,0)$ but no other point with integer coordinates. Suppose that $A(C)$, the area of $C$, is equally distributed among the four quadrants. Prove that $A(C) \leq 4.$
2011 National Olympiad First Round, 32
Two players are playing a game with $n$ pieces. At each turn, the player takes $2^i$ pieces where $i \geq 0$. The player who takes the last piece will win the game. If the game is played for each $n=1000, 2000, 2011, 3000, 4000$ once, in how many of them the first player can guarantee to win?
$\textbf{(A)}\ 4 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$
2021 Philippine MO, 1
In convex quadrilateral $ABCD$, $\angle CAB = \angle BCD$. $P$ lies on line $BC$ such that $AP = PC$, $Q$ lies on line $AP$ such that $AC$ and $DQ$ are parallel, $R$ is the point of intersection of lines $AB$ and $CD$, and $S$ is the point of intersection of lines $AC$ and $QR$. Line $AD$ meets the circumcircle of $AQS$ again at $T$. Prove that $AB$ and $QT$ are parallel.
2011 Canadian Open Math Challenge, 9
ABC is a triangle with coordinates A =(2, 6), B =(0, 0), and C =(14, 0).
(a) Let P be the midpoint of AB. Determine the equation of the line perpendicular to AB passing through P.
(b) Let Q be the point on line BC for which PQ is perpendicular to AB. Determine the length of AQ.
(c) There is a (unique) circle passing through the points A, B, and C. Determine the radius of this circle.
2019 Balkan MO Shortlist, A3
Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$
Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases.
(Edit: Proposed by sir Leonard Giugiuc, Romania)
2008 HMNT, 5
A triangle has altitudes of length $15$, $21$, and $35$. Find its area.
2021 New Zealand MO, 7
Let $a, b, c, d$ be integers such that $a > b > c > d \ge -2021$ and $$\frac{a + b}{b + c}=\frac{c + d}{d + a}$$ (and $b + c \ne 0 \ne d + a$). What is the maximum possible value of $ac$?
Russian TST 2014, P1
The inscribed circle of the triangle $ABC{}$ touches the sides $BC,CA$ and $AB{}$ at $A',B'$ and $C'{}$ respectively. Let $I_a$ be the $A$-excenter of $ABC{}.$ Prove that $I_aA'$ is perpendicular to the line determined by the circumcenters of $I_aBC'$ and $I_aCB'.$
2015 Germany Team Selection Test, 2
Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.
Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.
[i](Notation: $[\cdot]$ denotes the line segment.)[/i]
2021 JBMO Shortlist, A3
Let $n$ be a positive integer. A finite set of integers is called $n$-divided if there are exactly $n$ ways to partition this set into two subsets with equal sums. For example, the set $\{1, 3, 4, 5, 6, 7\}$ is $2$-divided because the only ways to partition it into two subsets with equal sums is by dividing it into $\{1, 3, 4, 5\}$ and $\{6, 7\}$, or $\{1, 5, 7\}$
and $\{3, 4, 6\}$. Find all the integers $n > 0$ for which there exists a $n$-divided set.
Proposed by [i]Martin Rakovsky, France[/i]
2015 Indonesia MO Shortlist, C6
Let $k$ be a fixed natural number. In the infinite number of real line, each integer is colored with color ..., red, green, blue, red, green, blue, ... and so on. A number of flea settles at first at integer points. On each turn, a flea will jump over the other tick so that the distance $k$ is the original distance. Formally, we may choose $2$ tails $A, B$ that are spaced $n$ and move $A$ to the different side of $B$ so the current distance is $kn$. Some fleas may occupy the same point because we consider the size of fleas very small. Determine all the values of $k$ so that, whatever the initial position of the ticks, we always get a position where all ticks land on the same color.
1970 All Soviet Union Mathematical Olympiad, 143
The vertices of the regular $n$-gon are marked with some colours (each vertex -- with one colour) in such a way, that the vertices of one colour represent the right polygon. Prove that there are two equal ones among the smaller polygons.
2022 MOAA, 2
While doing her homework for a Momentum Learning class, Valencia draws two intersecting segments $AB = 10$ and $CD = 7$ on a plane. Across all possible configurations of those two segments, determine the maximum possible area of quadrilateral $ACBD$.
2011 Bosnia Herzegovina Team Selection Test, 1
Find maximum value of number $a$ such that for any arrangement of numbers $1,2,\ldots ,10$ on a circle, we can find three consecutive numbers such their sum bigger or equal than $a$.