Found problems: 85335
1991 Turkey Team Selection Test, 3
Let $f$ be a function on defined on $|x|<1$ such that $f\left (\tfrac1{10}\right )$ is rational and $f(x)= \sum_{i=1}^{\infty} a_i x^i $ where $a_i\in{\{0,1,2,3,4,5,6,7,8,9\}}$. Prove that $f$ can be written as $f(x)= \frac{p(x)}{q(x)}$ where $p(x)$ and $q(x)$ are polynomials with integer coefficients.
2015 British Mathematical Olympiad Round 1, 2
Let $ABCD$ be a cyclic quadrilateral and let the lines $CD$ and $BA$ meet at $E$. The line through $D$ which is tangent to the circle $ADE$ meets the line $CB$ at $F$. Prove that triangle $CDF$ is isosceles.
2016 Junior Regional Olympiad - FBH, 3
Prove that when dividing a prime number with $30$, remainder is always not a composite number
2016 Romania Team Selection Tests, 4
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
CVM 2020, Problem 2
Find all $(x,y,z)\in\mathbb R^3$ such that
$$x+y+z=xy+yz+zx=3$$
[i]Proposed by Ezra Guerrero, Francisco Morazan[/i]
1990 Romania Team Selection Test, 10
Let $p,q$ be positive prime numbers and suppose $q>5$. Prove that if $q \mid 2^{p}+3^{p}$, then $q>p$.
[i]Laurentiu Panaitopol[/i]
2019 Macedonia National Olympiad, 1
In an acute-angled triangle $ABC$, point $M$ is the midpoint of side $BC$ and the centers of the $M$- excircles of triangles $AMB$ and $AMC$ are $D$ and $E$, respectively. The circumcircle of triangle $ABD$ intersects line $BC$ at points $B$ and $F$. The circumcircle of triangle $ACE$ intersects line $BC$ at points $C$ and $G$. Prove that $BF\hspace{0.25mm} = \hspace{0.25mm} CG$ .
1959 Miklós Schweitzer, 1
[b]1.[/b] Let $p_n$ be the $n$th prime number. Prove that
$\sum_{n=2}^{\infty} \frac{1}{np_n-(n-1)p_{n-1}}= \infty$
[b](N.17)[/b]
2015 JBMO TST - Turkey, 1
Let $p,q$ be prime numbers such that their sum isn't divisible by $3$. Find the all $(p,q,r,n)$ positive integer quadruples satisfy:
$$p+q=r(p-q)^n$$
[i]Proposed by Şahin Emrah[/i]
2016 SDMO (High School), 4
Let triangle $ABC$ be an isosceles triangle with $AB = AC$. Suppose that the angle bisector of its angle $\angle B$ meets the side $AC$ at a point $D$ and that $BC = BD+AD$.
Determine $\angle A$.
1997 All-Russian Olympiad Regional Round, 8.7
Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.
IV Soros Olympiad 1997 - 98 (Russia), 11.11
An arbitrary point $M$ is taken on the basis of a regular triangular pyramid. Let $K$, $L$, $N$ be the projections of $M$ onto the lateral faces of this pyramid, and $P$ be the intersection point of the medians of the triangle $KLN$. Prove that the straight line passing through the points $M$ and$ P$ intersects the height of the pyramid (or its extension). Let us denote this intersection point by $E$. Find $MP: PE$ if the dihedral angles at the base of the pyramid are equal to $a$.
2019 Sharygin Geometry Olympiad, 6
Two quadrilaterals $ABCD$ and $A_1B_1C_1D_1$ are mutually symmetric with respect to the point $P$. It is known that $A_1BCD$, $AB_1CD$ and $ABC_1D$ are cyclic quadrilaterals. Prove that the quadrilateral $ABCD_1$ is also cyclic
2009 Stanford Mathematics Tournament, 4
How many ways are there to write $657$ as a sum of powers of two where each power of two is used at
most twice in the sum? For example, $256+256+128+16+1$ is a valid sum.
2021 Estonia Team Selection Test, 1
Let $n$ be a positive integer. Find the number of permutations $a_1$, $a_2$, $\dots a_n$ of the
sequence $1$, $2$, $\dots$ , $n$ satisfying
$$a_1 \le 2a_2\le 3a_3 \le \dots \le na_n$$.
Proposed by United Kingdom
2018 IFYM, Sozopol, 8
Find all positive integers $n$ for which a square[b][i] n x n[/i][/b] can be covered with rectangles [b][i]k x 1[/i][/b] and one square [b][i]1 x 1[/i][/b] when:
a) $k = 4$ b) $k = 8$
2020 Peru IMO TST, 6
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
2006 Cezar Ivănescu, 3
[b]a)[/b] Let be a sequence $ \left( x_n \right)_{n\ge 1} $ defined by the recursion $ x_{n+1}=\frac{1+x_n}{1-x_n} , $ with $ x_1=2006. $ Calculate $ \lim_{n\to\infty } \frac{x_1+x_2+\cdots +x_n}{n} . $
[b]b)[/b] Prove that if a convergent sequence $ \left( s_n \right)_{n\ge 1} $ verifies $ a_{2^n} =na_n , $ for any natural numbers $ n, $ then $ a_n=0, $ for any natural numbers $ n. $
[i]Cornel Stoicescu[/i]
1994 IMO, 1
Let $ m$ and $ n$ be two positive integers. Let $ a_1$, $ a_2$, $ \ldots$, $ a_m$ be $ m$ different numbers from the set $ \{1, 2,\ldots, n\}$ such that for any two indices $ i$ and $ j$ with $ 1\leq i \leq j \leq m$ and $ a_i \plus{} a_j \leq n$, there exists an index $ k$ such that $ a_i \plus{} a_j \equal{} a_k$. Show that
\[ \frac {a_1 \plus{} a_2 \plus{} ... \plus{} a_m}{m} \geq \frac {n \plus{} 1}{2}.
\]
1978 IMO Longlists, 39
$A$ is a $2m$-digit positive integer each of whose digits is $1$. $B$ is an $m$-digit positive integer each of whose digits is $4$. Prove that $A+B +1$ is a perfect square.
2024 IFYM, Sozopol, 7
Consider a finite undirected graph in which each edge belongs to at most three cycles. Prove that its vertices can be colored with three colors so that any two vertices connected by an edge have different colors.
[i](A cycle in a graph is a sequence of distinct vertices \( v_1, v_2, \ldots, v_k \), \( k \geq 3 \), such that \( v_i v_{i+1} \) is an edge for each \( i = 1, 2, \ldots, k \); we consider \( v_{k+1} = v_1 \). The edges \( v_i v_{i+1} \) belong to the cycle.)[/i]
2023 Romania National Olympiad, 3
We consider triangle $ABC$ and variables points $M$ on the half-line $BC$, $N$ on the half-line $CA$, and $P$ on the half-line $AB$, each start simultaneously from $B,C$ and respectively $A$, moving with constant speeds $ v_1, v_2, v_3 > 0 $, where $v_1$, $v_2$, and $v_3$ are expressed in the same unit of measure.
a) Given that there exist three distinct moments in which triangle $MNP$ is equilateral, prove that triangle $ABC$ is equilateral and that $v_1 = v_2 = v_3$.
b) Prove that if $v_1 = v_2 = v_3$ and there exists a moment in which triangle $MNP$ is equilateral, then triangle $ABC$ is also equilateral.
2018 German National Olympiad, 3
Given a positive integer $n$, Susann fills a square of $n \times n$ boxes. In each box she inscribes an integer, taking care that each row and each column contains distinct numbers. After this an imp appears and destroys some of the boxes.
Show that Susann can choose some of the remaining boxes and colour them red, satisfying the following two conditions:
1) There are no two red boxes in the same column or in the same row.
2) For each box which is neither destroyed nor coloured, there is a red box with a larger number in the same row or a red box with a smaller number in the same column.
[i]Proposed by Christian Reiher[/i]
2003 IMC, 1
Let $A,B \in \mathbb{R}^{n\times n}$ such that $AB+B+A=0$. Prove that $AB=BA$.
2010 Math Prize For Girls Problems, 1
If $a$ and $b$ are nonzero real numbers such that $\left| a \right| \ne \left| b \right|$, compute the value of the expression
\[
\left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
\left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
\left(
\frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
- \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
\right).
\]