Found problems: 85335
2007 Bulgaria Team Selection Test, 1
In isosceles triangle $ABC(AC=BC)$ the point $M$ is in the segment $AB$ such that $AM=2MB,$ $F$ is the midpoint of $BC$ and $H$ is the orthogonal projection of $M$ in $AF.$ Prove that $\angle BHF=\angle ABC.$
1995 AMC 12/AHSME, 30
A large cube is formed by stacking $27$ unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. The number of unit cubes that the plane intersects is
[asy]
size(120); defaultpen(linewidth(0.7)); pair slant = (2,1);
for(int i = 0; i < 4; ++i)
draw((0,i)--(3,i)^^(i,0)--(i,3)^^(3,i)--(3,i)+slant^^(i,3)--(i,3)+slant);
for(int i = 1; i < 4; ++i)
draw((0,3)+slant*i/3--(3,3)+slant*i/3^^(3,0)+slant*i/3--(3,3)+slant*i/3);[/asy]
$\textbf{(A)}\ 16\qquad
\textbf{(B)}\ 17 \qquad
\textbf{(C)}\ 18 \qquad
\textbf{(D)}\ 19 \qquad
\textbf{(E)}\ 20$
2015 Geolympiad Summer, 2.
Let $ABC$ be a triangle. Let line $\ell$ be the line through the tangency points that are formed when the tangents from $A$ to the circle with diameter $BC$ are drawn. Let line $m$ be the line through the tangency points that are formed when the tangents from $B$ to the circle with diameter $AC$ are drawn. Show that the $\ell$, $m$, and the $C$-altitude concur.
2016 Portugal MO, 6
The natural numbers are colored green or blue so that:
$\bullet$ The sum of a green and a blue is blue;
$\bullet$ The product of a green and a blue is green.
How many ways are there to color the natural numbers with these rules, so that $462$ are blue and $2016$ are green?
2004 Iran MO (2nd round), 4
$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that:
\[f(m)+f(n) \ | \ m+n .\]
2024 Brazil Cono Sur TST, 3
Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy:
\[
\sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil
\]
2000 USAMO, 4
Find the smallest positive integer $n$ such that if $n$ squares of a $1000 \times 1000$ chessboard are colored, then there will exist three colored squares whose centers form a right triangle with sides parallel to the edges of the board.
1983 IMO Longlists, 58
In a test, $3n$ students participate, who are located in three rows of $n$ students in each. The students leave the test room one by one. If $N_1(t), N_2(t), N_3(t)$ denote the numbers of students in the first, second, and third row respectively at time $t$, find the probability that for each t during the test,
\[|N_i(t) - N_j(t)| < 2, i \neq j, i, j = 1, 2, \dots .\]
2019 HMIC, 5
Let $p = 2017$ be a prime and $\mathbb{F}_p$ be the integers modulo $p$. A function $f: \mathbb{Z}\rightarrow\mathbb{F}_p$ is called [i]good[/i] if there is $\alpha\in\mathbb{F}_p$ with $\alpha\not\equiv 0\pmod{p}$ such that
\[f(x)f(y) = f(x + y) + \alpha^y f(x - y)\pmod{p}\]
for all $x, y\in\mathbb{Z}$. How many good functions are there that are periodic with minimal period $2016$?
[i]Ashwin Sah[/i]
2020 Iran Team Selection Test, 2
Alice and Bob take turns alternatively on a $2020\times2020$ board with Alice starting the game. In every move each person colours a cell that have not been coloured yet and will be rewarded with as many points as the coloured cells in the same row and column. When the table is coloured completely, the points determine the winner. Who has the wining strategy and what is the maximum difference he/she can grantees?
[i]Proposed by Seyed Reza Hosseini[/i]
2020 Bulgaria Team Selection Test, 5
Given is a function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $|f(x+y)-f(x)-f(y)|\leq 1$.
Prove the existence of an additive function $g:\mathbb{R}\rightarrow \mathbb{R}$ (that is $g(x+y)=g(x)+g(y)$) such that $|f(x)-g(x)|\leq 1$ for any $x \in \mathbb{R}$
2013 Online Math Open Problems, 47
Let $f(x,y)$ be a function from ordered pairs of positive integers to real numbers
such that
\[ f(1,x) = f(x,1) = \frac{1}{x} \quad\text{and}\quad f(x+1,y+1)f(x,y)-f(x,y+1)f(x+1,y) = 1 \]
for all ordered pairs of positive integers $(x,y)$. If $f(100,100) = \frac{m}{n}$ for two relatively prime positive integers $m,n$, compute $m+n$.
[i]David Yang[/i]
2015 İberoAmerican, 3
Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$:
$s_n = \alpha^n + \beta^n$
$t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$
Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.
2001 All-Russian Olympiad Regional Round, 8.6
We call a natural number $n$ good if each of the numbers $n$, $ n+1$, $n+2$ and $n+3$ are divided by the sum of their digits. (For example, $n = 60398$ is good.) Does the penultimate digit of a good number ending in eight have to be nine?
1996 Italy TST, 1
1-Let $A$ and $B$ be two diametrically opposite points on a circle with radius $1$. Points $P_1,P_2,...,P_n$ are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of $P_1,P_2,...,P_n$ from $A$ and $B$, respectively. Show that at least one of the numbers $a$ and $b$ does not exceed $\sqrt{2}$
2019 Greece Team Selection Test, 2
Let a triangle $ABC$ inscribed in a circle $\Gamma$ with center $O$. Let $I$ the incenter of triangle $ABC$ and $D, E, F$ the contact points of the incircle with sides $BC, AC, AB$ of triangle $ABC$ respectively . Let also $S$ the foot of the perpendicular line from $D$ to the line $EF$.Prove that line $SI$ passes from the antidiametric point $N$ of $A$ in the circle $\Gamma$.( $AN$ is a diametre of the circle $\Gamma$).
2022 Czech-Polish-Slovak Junior Match, 1
Determine the largest possible value of the expression $ab+bc+ 2ac$ for non-negative real numbers $a, b, c$ whose sum is $1$.
2002 Miklós Schweitzer, 2
Let $G$ be a simple $k$ edge-connected graph on $n$ vertices and let $u$ and $v$ be different vertices of $G$. Prove that there are $k$ edge-disjoint paths from $u$ to $v$ each having at most $\frac{20n}{k}$ edges.
2017 China Northern MO, 8
Let \(n>1\) be an integer, and let \(x_1, x_2, ..., x_n\) be real numbers satisfying \(x_1, x_2, ..., x_n \in [0,n]\) with \(x_1x_2...x_n = (n-x_1)(n-x_2)...(n-x_n)\). Find the maximum value of \(y = x_1 + x_2 + ... + x_n\).
2010 National Olympiad First Round, 32
At least two of any three students of a school with $1001$ students are friends. How many of the numbers $334,412,450,499$ can be the number of friends of the one with the highest number of friends?
$ \textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ \text{None}
$
2013 Putnam, 1
Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.
2014 Iran Team Selection Test, 4
Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set at last in one of the permutation
$b$ comes exactly after $a$
2018 Belarusian National Olympiad, 10.4
Some cells of a checkered plane are marked so that figure $A$ formed by marked cells satisfies the following condition:$1)$ any cell of the figure $A$ has exactly two adjacent cells of $A$; and $2)$ the figure $A$ can be divided into isosceles trapezoids of area $2$ with vertices at the grid nodes (and acute angles of trapezoids are equal to $45$) . Prove that the number of marked cells is divisible by $8$.
2013 Saudi Arabia GMO TST, 1
An acute triangle $ABC$ is inscribed in circle $\omega$ centered at $O$. Line $BO$ and side $AC$ meet at $B_1$. Line $CO$ and side $AB$ meet at $C_1$. Line $B_1C_1$ meets circle $\omega$ at $P$ and $Q$. If $AP = AQ$, prove that $AB = AC$.
2017 Baltic Way, 19
For an integer $n\geq 1$ let $a(n)$ denote the total number of carries which arise when adding $2017$ and $n\cdot 2017$. The first few values are given by $a(1)=1$, $a(2)=1$, $a(3)=0$, which can be seen from the following:
\begin{align*}
001 &&001 && 000 \\
2017 &&4034 &&6051 \\
+2017 &&+2017 &&+2017\\
=4034 &&=6051 &&=8068\\
\end{align*}
Prove that
$$a(1)+a(2)+...+a(10^{2017}-1)=10\cdot\frac{10^{2017}-1}{9}.$$