Found problems: 85335
2021 Turkey Junior National Olympiad, 2
We are numbering the rows and columns of a $29 \text{x} 29$ chess table with numbers $1, 2, ..., 29$ in order (Top row is numbered with $1$ and first columns is numbered with $1$ as well). We choose some of the squares in this chess table and for every selected square, we know that there exist at most one square having a row number greater than or equal to this selected square's row number and a column number greater than or equal to this selected square's column number. How many squares can we choose at most?
2007 iTest Tournament of Champions, 5
Let $s=a+b+c$, where $a$, $b$, and $c$ are integers that are lengths of the sides of a box. The volume of the box is numerically equal to the sum of the lengths of the twelve edges of the box plus its surface area. Find the sum of the possible values of $s$.
2013 India IMO Training Camp, 1
Let $n \ge 2$ be an integer. There are $n$ beads numbered $1, 2, \ldots, n$. Two necklaces made out of some of these beads are considered the same if we can get one by rotating the other (with no flipping allowed). For example, with $n \ge 5$, the necklace with four beads $1, 5, 3, 2$ in the clockwise order is same as the one with $5, 3, 2, 1$ in the clockwise order, but is different from the one with $1, 2, 3, 5$ in the clockwise order.
We denote by $D_0(n)$ (respectively $D_1(n)$) the number of ways in which we can use all the beads to make an even number (resp. an odd number) of necklaces each of length at least $3$. Prove that $n - 1$ divides $D_1(n) - D_0(n)$.
2009 Vietnam Team Selection Test, 2
Let a circle $ (O)$ with diameter $ AB$. A point $ M$ move inside $ (O)$. Internal bisector of $ \widehat{AMB}$ cut $ (O)$ at $ N$, external bisector of $ \widehat{AMB}$ cut $ NA,NB$ at $ P,Q$. $ AM,BM$ cut circle with diameter $ NQ,NP$ at $ R,S$.
Prove that: median from $ N$ of triangle $ NRS$ pass over a fix point.
2007 Bulgarian Autumn Math Competition, Problem 10.1
Find all integers $b$ and $c$ for which the equation $x^2-bx+c=0$ has two real roots $x_{1}$ and $x_{2}$ satisfying $x_{1}^2+x_{2}^2=5$.
2015 Romania National Olympiad, 2
Let $a, b, c $ be distinct positive integers.
a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$.
b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that
$$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$
1990 Polish MO Finals, 3
In a tournament, every two of the $n$ players played exactly one match with each other (no
draws). Prove that it is possible either
(i) to partition the league in two groups $A$ and $B$ such that everybody in $A$ defeated everybody in $B$; or
(ii) to arrange all the players in a chain $x_1, x_2, . . . , x_n, x_1$ in such a way that each player defeated his successor.
2015 Puerto Rico Team Selection Test, 8
Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.
2024 Princeton University Math Competition, A7
Let $F_1=1, F_2=1,$ and $F_{n+2}=F_{n+1}+F_n.$ Then, $$S = \sum_{n=1}^{\infty} \arctan\left(\frac{1}{F_n}\right)\arctan\left(\frac{1}{F_{n+1}}\right)$$ Find $\lfloor 80S \rfloor.$
(Hint: it may be useful to note that $\arctan(\tfrac{1}{1}) = \arctan(\tfrac{1}{2})+\arctan(\tfrac{1}{3}).$)
2008 Junior Balkan Team Selection Tests - Moldova, 11
Let $ABCD$ be a convex quadrilateral with $AD = BC, CD \nparallel AB, AD \nparallel BC$. Points $M$ and $N$ are the midpoints of the sides $CD$ and $AB$, respectively.
a) If $E$ and $F$ are points, such that $MCBF$ and $ADME$ are parallelograms, prove that $\vartriangle BF N \equiv \vartriangle AEN$.
b) Let $P = MN \cap BC$, $Q = AD \cap MN$, $R = AD \cap BC$. Prove that the triangle $PQR$ is iscosceles.
1991 Romania Team Selection Test, 3
Let $C$ be a coloring of all edges and diagonals of a convex $n$−gon in red and blue (in Romanian, rosu and albastru). Denote by $q_r(C)$ (resp. $q_a(C)$) the number of quadrilaterals having all its edges and diagonals red (resp. blue).
Prove: $ \underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}$
1989 IMO Longlists, 83
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
2018 Hanoi Open Mathematics Competitions, 13
For a positive integer $n$, let $S(n), P(n)$ denote the sum and the product of all the digits of $n$ respectively.
1) Find all values of n such that $n = P(n)$:
2) Determine all values of n such that $n = S(n) + P(n)$.
2016 AMC 10, 24
How many four-digit integers $abcd$, with $a \neq 0$, have the property that the three two-digit integers $ab<bc<cd$ form an increasing arithmetic sequence? One such number is $4692$, where $a=4$, $b=6$, $c=9$, and $d=2$.
$\textbf{(A)}\ 9\qquad\textbf{(B)}\ 15\qquad\textbf{(C)}\ 16\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 20$
1997 Akdeniz University MO, 1
Let $m \in {\mathbb R}$ and
$$x^2+(m-4)x+(m^2-3m+3)=0$$
equations roots are $x_1$ and $x_2$ and $x_1^2+x_2^2=6$. Find all $m$ values.
2002 Junior Balkan Team Selection Tests - Moldova, 5
For any natural number $m \ge 1$ and any real number $x \ge 0$ we define expression
$$E (x, m) = \frac{(1^4 + x) (3^4 + x) (5^4 + x) ... [(2m -1)^ 4 + x]}{(2^4 + x) (4^4 + x) (6^4 + x) ... [(2m )^ 4 + x]}.$$
It is known that $E\left(\frac{1}{4},m\right)=\frac{1}{1013}.$ . Determine the value of $m$
2018 China Team Selection Test, 5
Given a positive integer $k$, call $n$ [i]good[/i] if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.
2007 Purple Comet Problems, 5
The repeating decimal $0.328181818181...$ can equivalently be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2008 iTest Tournament of Champions, 3
For how many integers $1\leq n\leq 9999$ is there a solution to the congruence \[\phi(n)\equiv 2\,\,\,\pmod{12},\] where $\phi(n)$ is the Euler phi-function?
1995 USAMO, 3
Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.
2000 All-Russian Olympiad Regional Round, 10.8
There are $2000$ cities in the country, some pairs of cities are connected by roads. It is known that no more than $N$ different non-self-intersecting cyclic routes of odd length. Prove that the country can be divided into $2N +2$ republics so that no two cities from the same republic are connected by a road.
2019 Polish Junior MO First Round, 5
A parallelogram $ABCD$ is given. On the diagonal BD, a point $P$ is selected such that $AP = BD$ is satisfied. Point $Q$ is the midpoint of segment $CP$. Prove that $\angle BQD = 90^o$.
[img]https://cdn.artofproblemsolving.com/attachments/2/0/4bc69ec0330e2afa6b560c56da5dd783b16efb.png[/img]
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2007 China Girls Math Olympiad, 6
For $ a,b,c\geq 0$ with $ a\plus{}b\plus{}c\equal{}1$, prove that
$ \sqrt{a\plus{}\frac{(b\minus{}c)^2}{4}}\plus{}\sqrt{b}\plus{}\sqrt{c}\leq \sqrt{3}$
PEN A Problems, 78
Determine all ordered pairs $(m, n)$ of positive integers such that \[\frac{n^{3}+1}{mn-1}\] is an integer.
2014 Online Math Open Problems, 12
Let $a$, $b$, $c$ be positive real numbers for which \[
\frac{5}{a} = b+c, \quad
\frac{10}{b} = c+a, \quad \text{and} \quad
\frac{13}{c} = a+b. \] If $a+b+c = \frac mn$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Evan Chen[/i]