Found problems: 85335
2015 Spain Mathematical Olympiad, 3
Let $ABC$ be a triangle. $M$, and $N$ points on $BC$, such that $BM=CN$, with $M$ in the interior of $BN$. Let $P$ and $Q$ be points in $AN$ and $AM$ respectively such that $\angle PMC= \angle MAB$, and $\angle QNB= \angle NAC$. Prove that $ \angle QBC= \angle PCB$.
2020 Balkan MO Shortlist, A3
Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$.
[i]Demetres Christofides, Cyprus[/i]
2005 iTest, 4
How many multiples of $2005$ are factors of $(2005)^2$?
2020 Thailand Mathematical Olympiad, 9
Let $n,k$ be positive integers such that $n>k$. There is a square-shaped plot of land, which is divided into $n\times n$ grid so that each cell has the same size. The land needs to be plowed by $k$ tractors; each tractor will begin on the lower-left corner cell and keep moving to the cell sharing a common side until it reaches the upper-right corner cell. In addition, each tractor can only move in two directions: up and right. Determine the minimum possible number of unplowed cells.
Kyiv City MO Juniors 2003+ geometry, 2021.8.4
Let $BM$ be the median of the triangle $ABC$, in which $AB> BC$. Point $P$ is chosen so that $AB \parallel PC$ and $PM \perp BM$. Prove that $\angle ABM = \angle MBP$.
(Mikhail Standenko)
2020 USEMO, 6
Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^
2 + b$, then the following conditions hold:
$\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$;
$\bullet$ the number $Q(0)$ is divisible by $n$; and
$\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.
1972 Bundeswettbewerb Mathematik, 2
Prove: out of $ 79$ consecutive positive integers, one can find at least one whose sum of digits is divisible by $ 13$. Show that this isn't true for $ 78$ consecutive integers.
2016 CCA Math Bonanza, I13
Let $P(x)$ be a polynomial with integer coefficients, leading coefficient 1, and $P(0) = 3$. If the polynomial $P(x)^2 + 1$ can be factored as a product of two non-constant polynomials with integer coefficients, and the degree of $P$ is as small as possible, compute the largest possible value of $P(10)$.
[i]2016 CCA Math Bonanza Individual #13[/i]
2010 Kurschak Competition, 3
For what positive integers $n$ and $k$ do there exits integers $a_1,a_2,\dots,a_n$ and $b_1,b_2,\dots,b_k$ such that the products $a_ib_j$ ($1\le i\le n,1\le j\le k$) give pairwise different residues modulo $nk$?
2006 Turkey Team Selection Test, 1
For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.
2014 Contests, 1
Let $ABC$ be a triangle, let ${A}'$, ${B}'$, ${C}'$ be the orthogonal projections of the vertices $A$ ,$B$ ,$C$ on the lines $BC$, $CA$ and $AB$, respectively, and let $X$ be a point on the line $A{A}'$.Let $\gamma_{B}$ be the circle through $B$ and $X$, centred on the line $BC$, and let $\gamma_{C}$ be the circle through $C$ and $X$, centred on the line $BC$.The circle $\gamma_{B}$ meets the lines $AB$ and $B{B}'$ again at $M$ and ${M}'$, respectively, and the circle $\gamma_{C}$ meets the lines $AC$ and $C{C}'$ again at $N$ and ${N}'$, respectively.Show that the points $M$, ${M}'$, $N$ and ${N}'$ are collinear.
2013 AIME Problems, 3
Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC}$, respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two non square rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD$. Find $\frac{AE}{EB} + \frac{EB}{AE}$.
1962 Swedish Mathematical Competition, 5
Find the largest cube which can be placed inside a regular tetrahedron with side $1$ so that one of its faces lies on the base of the tetrahedron.
2025 AIME, 4
The product \[\prod^{63}_{k=4} \frac{\log_k (5^{k^2 - 1})}{\log_{k + 1} (5^{k^2 - 4})} = \frac{\log_4 (5^{15})}{\log_5 (5^{12})} \cdot \frac{\log_5 (5^{24})}{\log_6 (5^{21})}\cdot \frac{\log_6 (5^{35})}{\log_7 (5^{32})} \cdots \frac{\log_{63} (5^{3968})}{\log_{64} (5^{3965})}\] is equal to $\tfrac mn,$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$
2023 Romania EGMO TST, P3
Let $D{}$ be a point inside the triangle $ABC$. Let $E{}$ and $F{}$ be the projections of $D{}$ onto $AB$ and $AC$, respectively. The lines $BD$ and $CD$ intersect the circumcircle of $ABC$ the second time at $M{}$ and $N{}$, respectively. Prove that \[\frac{EF}{MN}\geqslant \frac{r}{R},\]where $r{}$ and $R{}$ are the inradius and circumradius of $ABC$, respectively.
2016 Silk Road, 3
Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ .
(Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)
2019 Stars of Mathematics, 4
Given a positive integer $n$. A triangular array $(a_{i,j})$ of zeros and ones, where $i$ and $j$ run through the positive integers such that $i+j\leqslant n+1$ is called a [i]binary anti-Pascal $n$-triangle[/i] if $a_{i,j}+a_{i,j+1}+a_{i+1,j}\equiv 1\pmod{2}$ for all possible values $i$ and $j$ may take on. Determine the minimum number of ones a binary anti-Pascal $n$-triangle may contain.
2016 Indonesia TST, 3
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2016 China Team Selection Test, 6
Let $m,n$ be naturals satisfying $n \geq m \geq 2$ and let $S$ be a set consisting of $n$ naturals. Prove that $S$ has at least $2^{n-m+1}$ distinct subsets, each whose sum is divisible by $m$. (The zero set counts as a subset).
2003 Alexandru Myller, 4
A professor organized five exams for a class consisting of at least two students. Before starting the first test, he deduced that there will be at least two students from that class that will have the same amount of passed exams.
What is the minimum numer of students that class could have had such that the conclusion of the professor's reasoning was correct.
2018 Hanoi Open Mathematics Competitions, 5
Find all $3$-digit numbers $\overline{abc}$ ($a,b \ne 0$) such that $\overline{bcd} \times a = \overline{1a4d}$ for some integer $d$ from $1$ to $9$
2011 Ukraine Team Selection Test, 4
Suppose an ordered set of $ ({{a} _{1}}, \ {{a} _{2}},\ \ldots,\ {{a} _{n}}) $ real numbers, $n \ge 3 $. It is possible to replace the number $ {{a} _ {i}} $, $ i = \overline {2, \ n-1} $ by the number $ a_ {i} ^ {*} $ that $ {{a} _ {i}} + a_ {i} ^ {*} = {{a} _ {i-1}} + {{a} _ {i + 1}} $. Let $ ({{b} _ {1}},\ {{b} _ {2}}, \ \ldots, \ {{b} _ {n}}) $ be the set with the largest sum of numbers that can be obtained from this, and $ ({{c} _ {1}},\ {{c} _ {2}}, \ \ldots, \ {{c} _ {n}}) $ is a similar set with the least amount.
For the odd $n \ge 3 $ and set $ (1,\ 3, \ \ldots, \ n, \ 2, \ 4, \ \ldots,\ n-1) $ find the values of the expressions $ {{b} _ {1}} + {{b} _ {2}} + \ldots + {{b} _ {n}} $ and $ {{c} _ {1}} + {{c} _ {2}} + \ldots + {{c} _ {n}} $.
2016 Putnam, A2
Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that
\[\binom{m}{n-1}>\binom{m-1}{n}.\]
Evaluate
\[\lim_{n\to\infty}\frac{M(n)}{n}.\]
2018 AMC 12/AHSME, 17
Let $p$ and $q$ be positive integers such that \[\frac{5}{9} < \frac{p}{q} < \frac{4}{7}\] and $q$ is as small as possible. What is $q-p$?
$\textbf{(A) } 7 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 17 \qquad \textbf{(E) } 19 $
1999 Bosnia and Herzegovina Team Selection Test, 5
For any nonempty set $S$, we define $\sigma(S)$ and $\pi(S)$ as sum and product of all elements from set $S$, respectively. Prove that
$a)$ $\sum \limits_{} \frac{1}{\pi(S)} =n$
$b)$ $\sum \limits_{} \frac{\sigma(S)}{\pi(S)} =(n^2+2n)-\left(1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}\right)(n+1)$
where $\sum$ denotes sum by all nonempty subsets $S$ of set $\{1,2,...,n\}$