Found problems: 85335
1984 Polish MO Finals, 6
Cities $P_1,...,P_{1025}$ are connected to each other by airlines $A_1,...,A_{10}$ so that for any two distinct cities $P_k$ and $P_m$ there is an airline offering a direct flight between them. Prove that one of the airlines can offer a round trip with an odd number of flights.
1953 Putnam, B5
Show that the roots of $x^4 +ax^3 +bx^2 +cx +d$, if suitably numbered, satisfy the relation $\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},$ provided $a^2 d=c^2 \ne 0.$
2025 239 Open Mathematical Olympiad, 8
The incircle of a right triangle $ABC$ touches its hypotenuse $BC$ at point $D$. The line $AD$ intersects the circumscribed circle at point $X$. Prove that $ |BX-CX| \geqslant |AD - DX|$.
2013 India IMO Training Camp, 3
Let $h \ge 3$ be an integer and $X$ the set of all positive integers that are greater than or equal to $2h$. Let $S$ be a nonempty subset of $X$ such that the following two conditions hold:
[list]
[*]if $a + b \in S$ with $a \ge h, b \ge h$, then $ab \in S$;
[*]if $ab \in S$ with $a \ge h, b \ge h$, then $a + b \in S$.[/list]
Prove that $S = X$.
1997 Vietnam Team Selection Test, 1
Let $ ABCD$ be a given tetrahedron, with $ BC \equal{} a$, $ CA \equal{} b$, $ AB \equal{} c$, $ DA \equal{} a_1$, $ DB \equal{} b_1$, $ DC \equal{} c_1$. Prove that there is a unique point $ P$ satisfying
\[ PA^2 \plus{} a_1^2 \plus{} b^2 \plus{} c^2 \equal{} PB^2 \plus{} b_1^2 \plus{} c^2 \plus{} a^2 \equal{} PC^2 \plus{} c_1^2 \plus{} a^2 \plus{} b^2 \equal{} PD^2 \plus{} a_1^2 \plus{} b_1^2 \plus{} c_1^2
\]
and for this point $ P$ we have $ PA^2 \plus{} PB^2 \plus{} PC^2 \plus{} PD^2 \ge 4R^2$, where $ R$ is the circumradius of the tetrahedron $ ABCD$. Find the necessary and sufficient condition so that this inequality is an equality.
2004 Nicolae Păun, 2
The following geometry is embedded in the Cartesian plane.
[b]a)[/b] Prove that any line that passes through at least two lattice points, passes through at least three lattice points.
[b]b)[/b] Find a point on the plane which doesn't lie on any line that passes through at least two lattice points.
[b]c)[/b] Show that any point with rational coordinates lie on a line that passes through two lattice points.
[i]Lavinia Savu[/i]
2021 Junior Macedonian Mathematical Olympiad, Problem 5
Let $ABC$ be an acute triangle and let $X$ and $Y$ be points on the segments $AB$ and $AC$ such that $BX = CY$. If $I_{B}$ and $I_{C}$ are centers of inscribed circles in triangles $ABY$ and $ACX$, and $T$ is the second intersection point of the circumcircles of $ABY$ and $ACX$, show that:
$$\frac{TI_{B}}{TI_{C}} = \frac{BY}{CX}.$$
[i]Proposed by Nikola Velov[/i]
2002 All-Russian Olympiad Regional Round, 10.4
(10.4) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $0 \le a_{k+1}- a_k \le 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \le \left(\sum_{k=1}^n a_k \right)^2$$
(11.3) A set of numbers $a_0, a_1,..., a_n$ satisfies the conditions: $a_0 = 0$, $a_{k+1} \ge a_k + 1$ for $k = 0, 1, .. , n -1$. Prove the inequality $$\sum_{k=1}^n a^3_k \ge \left(\sum_{k=1}^n a_k \right)^2$$
1991 Canada National Olympiad, 3
Let $C$ be a circle and $P$ a given point in the plane. Each line through $P$ which intersects $C$ determines a chord of $C$. Show that the midpoints of these chords lie on a circle.
2010 Today's Calculation Of Integral, 525
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.
2018 Brazil Undergrad MO, 9
How many functions $f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}$ satisfy $f(f(x))=f(f(f(x)))$ for every $ x $?
1977 IMO Shortlist, 3
Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$
2008 Junior Balkan Team Selection Tests - Romania, 1
From numbers $ 1,2,3,...,37$ we randomly choose 10 numbers. Prove that among these exist four distinct numbers, such that sum of two of them equals to the sum of other two.
Kyiv City MO 1984-93 - geometry, 1986.9.5
Prove that inside any convex hexagon with pairs of parallel sides of area $1$, you can draw a triangle of area $1/2$.
2005 MOP Homework, 7
Find all positive integers $n$ for which there are distinct integers $a_1$, ..., $a_n$ such that
$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1+a_2+...+a_n}{2}$.
2022 Israel National Olympiad, P6
Let $x,y,z$ be non-negative real numbers. Prove that:
\[\sqrt{(2x+y)(2x+z)}+\sqrt{(2y+x)(2y+z)}+\sqrt{(2z+x)(2z+y)}\geq \]
\[\geq \sqrt{(x+2y)(x+2z)}+\sqrt{(y+2x)(y+2z)}+\sqrt{(z+2x)(z+2y)}.\]
2004 Harvard-MIT Mathematics Tournament, 1
There are 1000 rooms in a row along a long corridor. Initially the first room contains 1000 people and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different
rooms will have people in them?
2013 ELMO Shortlist, 1
Let $n\ge2$ be a positive integer. The numbers $1,2,..., n^2$ are consecutively placed into squares of an $n\times n$, so the first row contains $1,2,...,n$ from left to right, the second row contains $n+1,n+2,...,2n$ from left to right, and so on. The [i]magic square value[/i] of a grid is defined to be the number of rows, columns, and main diagonals whose elements have an average value of $\frac{n^2 + 1}{2}$. Show that the magic-square value of the grid stays constant under the following two operations: (1) a permutation of the rows; and (2) a permutation of the columns. (The operations can be used multiple times, and in any order.)
[i]Proposed by Ray Li[/i]
1994 Poland - Second Round, 4
Each vertex of a cube is assigned $1$ or $-1$. Each face is assigned the product of the four numbers at its vertices. Determine all possible values that can be obtained as the sum of all the $14$ assigned numbers.
2004 May Olympiad, 2
Inside an $11\times 11$ square, Pablo drew a rectangle and extending its sides divided the square into $5$ rectangles, as shown in the figure.
[img]https://cdn.artofproblemsolving.com/attachments/5/a/7774da7085f283b3aae74fb5ff472572571827.gif[/img]
Sofía did the same, but she also managed to make the lengths of the sides of the $5$ rectangles be whole numbers between $1$ and $10$, all different. Show a figure like the one Sofia made.
MOAA Team Rounds, 2022.4
Angeline flips three fair coins, and if there are any tails, she then flips all coins that landed tails each one more time. The probability that all coins are now heads can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2016 CMIMC, 10
Let $\triangle ABC$ be a triangle with circumcircle $\Omega$ and let $N$ be the midpoint of the major arc $\widehat{BC}$. The incircle $\omega$ of $\triangle ABC$ is tangent to $AC$ and $AB$ at points $E$ and $F$ respectively. Suppose point $X$ is placed on the same side of $EF$ as $A$ such that $\triangle XEF\sim\triangle ABC$. Let $NX$ intersect $BC$ at a point $P$. Given that $AB=15$, $BC=16$, and $CA=17$, compute $\tfrac{PX}{XN}$.
2017 BMO TST, 3
Find all functions $f : \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that : $f(x)f(y)f(z)=9f(z+xyf(z))$, where $x$, $y$, $z$, are three positive real numbers.
1988 IMO Shortlist, 4
An $ n \times n, n \geq 2$ chessboard is numbered by the numbers $ 1, 2, \ldots, n^2$ (and every number occurs). Prove that there exist two neighbouring (with common edge) squares such that their numbers differ by at least $ n.$
2004 VJIMC, Problem 3
Let $\sum_{n=1}^\infty a_n$ be a divergent series with positive nonincreasing terms. Prove that the series
$$\sum_{n=1}^\infty\frac{a_n}{1+na_n}$$diverges.