Found problems: 85335
2003 Manhattan Mathematical Olympiad, 3
One hundred pins are arranged to form a square grid as shown. Jimmy wants to mark these pins using four letters $a,b,c,d,$ so that:
(I) every horizontal line and every vertical line contains all four letters;
(ii) each small square (such as the one shown) has its vertices marked by four different letters.
[asy]
unitsize(.5cm);
for(int a=1; a<11; ++a)
{
for(int b=1; b<11; ++b)
{
draw(Circle((a,b),.1));
}
}
draw((5.1, 5)--(5.9,5));
draw((5.1, 6)--(5.9,6));
draw((5, 5.1)--(5, 5.9));
draw((6, 5.1)--(6, 5.9));
[/asy]
Can he do this?
2018 Brazil Undergrad MO, 17
In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?
2020 JHMT, 1
In a country named Fillip, there are three major cities called Alenda, Breda, Chenida. This country uses the unit of "FP". The distance between Alenda and Chenida is $100$ FP. Breda is $70$ FP from Alenda and $30$ FP from Chenida. Let us say that we take a road trip from Alenda to Chenida. After $2$ hours of driving, we are currently at $50$ FP away from Alenda and $50$ FP away from Chenida. How many FP are we away from Breda?
2016 Hanoi Open Mathematics Competitions, 9
Let rational numbers $a, b, c$ satisfy the conditions $a + b + c = a^2 + b^2 + c^2 \in Z$.
Prove that there exist two relative prime numbers $m, n$ such that $abc =\frac{m^2}{n^3}$ .
2005 Georgia Team Selection Test, 11
On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.
2023 India Regional Mathematical Olympiad, 5
Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, \ldots, a_n$ which satisfy
$$
\sum_{i=1}^n \sqrt{\frac{k a_i^k}{(k-1) a_i^k+1}}=\sum_{i=1}^n a_i=n .
$$
1996 ITAMO, 5
Given a circle $C$ and an exterior point $A$. For every point $P$ on the circle construct the square $APQR$ (in counterclock order). Determine the locus of the point $Q$ when $P$ moves on the circle $C$.
2008 Bosnia and Herzegovina Junior BMO TST, 2
Let $ x,y,z$ be positive integers. If $ 7$ divides $ (x\plus{}6y)(2x\plus{}5y)(3x\plus{}4y)$ than prove that $ 343$ also divides it.
2013 Cuba MO, 2
An equilateral triangle with side $3$ is divided into $9$ small equal equilateral triangles with sides of length $1$. Each vertex of a triangle small (bold dots) is numbered with a different number than the $1$ to $10$. Inside each small triangle, write the sum of the numbers corresponding to its three vertices. Prove that there are three small triangles for which it is verified that the sum of the numbers written inside is at least $48$.
[img]https://cdn.artofproblemsolving.com/attachments/2/1/b2f58b6d59cb26e2fe29d0df59c1a42639a496.png[/img]
2008 Harvard-MIT Mathematics Tournament, 2
Let $ f(n)$ be the number of times you have to hit the $ \sqrt {\ }$ key on a calculator to get a number less than $ 2$ starting from $ n$. For instance, $ f(2) \equal{} 1$, $ f(5) \equal{} 2$. For how many $ 1 < m < 2008$ is $ f(m)$ odd?
2012 239 Open Mathematical Olympiad, 6
Let $G$ be a planar graph all of whose vertices are of degree $4$. Vasya and Petya walk along its edges. The first time each of them goes as he pleases, and then each of them goes straight (from the three roads they have to choose the middle one). As the result, each vertex was visited by exactly one of them and exactly once. Prove that this graph has an even number of vertices.
2011 IFYM, Sozopol, 5
A circle is inscribed in a quadrilateral $ABCD$, which is tangent to its sides $AB$, $BC$, $CD$, and $DC$ in points $M$, $N$, $P$, and $Q$ respectively. Prove that the lines $MP$, $NQ$, $AC$, and $BD$ intersect in one point.
2022 Stanford Mathematics Tournament, 7
Let $n_0$ be the product of the first $25$ primes. Now, choose a random divisor $n_1$ of $n_0$, where a choice $n_1$ is taken with probability proportional to $\phi(n_1)$. ($\phi(m)$ is the number of integers less than $m$ which are relatively prime to $m$.) Given this $n_1$, we let $n_2$ be a random divisor of $n_1$, again chosen with probability proportional to $\phi(n_2)$. Compute the probability that $n_2\equiv0\pmod{2310}$.
1999 AMC 12/AHSME, 22
The graphs of $ y \equal{} \minus{}|x \minus{} a| \plus{} b$ and $ y \equal{} |x \minus{} c| \plus{} d$ intersect at points $ (2,5)$ and $ (8,3)$. Find $ a \plus{} c$.
$ \textbf{(A)}\ 7\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 13\qquad
\textbf{(E)}\ 18$
2007 IMO, 1
Real numbers $ a_{1}$, $ a_{2}$, $ \ldots$, $ a_{n}$ are given. For each $ i$, $ (1 \leq i \leq n )$, define
\[ d_{i} \equal{} \max \{ a_{j}\mid 1 \leq j \leq i \} \minus{} \min \{ a_{j}\mid i \leq j \leq n \}
\]
and let $ d \equal{} \max \{d_{i}\mid 1 \leq i \leq n \}$.
(a) Prove that, for any real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$,
\[ \max \{ |x_{i} \minus{} a_{i}| \mid 1 \leq i \leq n \}\geq \frac {d}{2}. \quad \quad (*)
\]
(b) Show that there are real numbers $ x_{1}\leq x_{2}\leq \cdots \leq x_{n}$ such that the equality holds in (*).
[i]Author: Michael Albert, New Zealand[/i]
1989 China Team Selection Test, 2
Let $v_0 = 0, v_1 = 1$ and $v_{n+1} = 8 \cdot v_n - v_{n-1},$ $n = 1,2, ...$. Prove that in the sequence $\{v_n\}$ there aren't terms of the form $3^{\alpha} \cdot 5^{\beta}$ with $\alpha, \beta \in \mathbb{N}.$
1982 Tournament Of Towns, (031) 5
The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.
2013 239 Open Mathematical Olympiad, 6
Convex polyhedron $M$ with triangular faces is cut into tetrahedrons; all the vertices of the tetrahedrons are the vertices of the polyhedron, and any two tetrahedrons either do not intersect, or they intersect along a common vertex, common edge, or common face. Prove that it it's not possible that each tetrahedron has exactly one face on the surface of $M$.
1994 Portugal MO, 4
To date, in each Mathematics Olympiad Final, no participant has been able to solve all the problems, but every problem has been solved by at least one participant. Prove that in each Final, there was a participant $A$ who solved a problem $P_A$ and another participant $B$ who solved a problem $P_B$ such that $A$ did not solve $P_B$ and $B$ did not solve $P_A$.
2004 Poland - Second Round, 3
Determine all sequences $ a_1,a_2,a_3,...$ of $ 1$ and $ \minus{}1$ such that $ a_{mn}\equal{}a_ma_n$ for all $ m,n$ and among any three successive terms $ a_n,a_{n\plus{}1},a_{n\plus{}2}$ both $ 1$ and $ \minus{}1$ occur.
1996 Canadian Open Math Challenge, 4
Determine the smallest positive integer, $n$, which satisfies the equation $n^3+2n^2 = b$, where $b$ is the square of an odd integer.
2017 Czech-Polish-Slovak Junior Match, 4
Given is a right triangle $ABC$ with perimeter $2$, with $\angle B=90^o$ . Point $S$ is the center of the excircle to the side $AB$ of the triangle and $H$ is the intersection of the heights of the triangle $ABS$ . Determine the smallest possible length of the segment $HS $.
2020 Estonia Team Selection Test, 3
The prime numbers $p$ and $q$ and the integer $a$ are chosen such that $p> 2$ and $a \not\equiv 1$ (mod $q$), but $a^p \equiv 1$ (mod $q$). Prove that $(1 + a^1)(1 + a^2)...(1 + a^{p - 1})\equiv 1$ (mod $q$) .
Estonia Open Junior - geometry, 2012.1.3
A rectangle $ABEF$ is drawn on the leg $AB$ of a right triangle $ABC$, whose apex $F$ is on the leg $AC$. Let $X$ be the intersection of the diagonal of the rectangle $AE$ and the hypotenuse $BC$ of the triangle. In what ratio does point $X$ divide the hypotenuse $BC$ if it is known that $| AC | = 3 | AB |$ and $| AF | = 2 | AB |$?
2018 Dutch IMO TST, 1
Suppose a grid with $2m$ rows and $2n$ columns is given, where $m$ and $n$ are positive integers. You may place one pawn on any square of this grid, except the bottom left one or the top right one. After placing the pawn, a snail wants to undertake a journey on the grid. Starting from the bottom left square, it wants to visit every square exactly once, except the one with the pawn on it, which the snail wants to avoid. Moreover, it wants to fi nish in the top right square. It can only move horizontally or vertically on the grid.
On which squares can you put the pawn for the snail to be able to finish its journey?