Found problems: 85335
MathLinks Contest 7th, 7.1
Find all pairs of positive integers $ a,b$ such that \begin{align*} b^2 + b+ 1 & \equiv 0 \pmod a \\ a^2+a+1 &\equiv 0 \pmod b . \end{align*}
1999 Chile National Olympiad, 2
In an acute triangle $ABC$, let $ \overline {AK}, \overline {BL}, \overline {CM} $ be the altitudes of the triangle concurrent at the point $ H $ and let $ P $ the midpoint of $ \overline {AH} $. Let's define $ S = \overline {BH} \cap \overline {MK} $ and $ T = \overline {LP} \cap \overline {AB} $. Show that $ \overline {TS} \perp \overline {BC} $
2014 IMO, 5
For each positive integer $n$, the Bank of Cape Town issues coins of denomination $\frac1n$. Given a finite collection of such coins (of not necessarily different denominations) with total value at most most $99+\frac12$, prove that it is possible to split this collection into $100$ or fewer groups, such that each group has total value at most $1$.
Russian TST 2019, P2
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
2011 Saudi Arabia Pre-TST, 3
Find all integers $n \ge 2$ for which $\sqrt[n]{3^n+ 4^n+5^n+8^n+10^n}$ is an integer.
2017 Czech And Slovak Olympiad III A, 2
Find all pairs of real numbers $k, l$ such that inequality $ka^2 + lb^2> c^2$ applies to the lengths of sides $a, b, c$ of any triangle.
2021 Durer Math Competition Finals, 6
Bertalan thought about a $4$-digit positive number. Then he draw a simple graph on $4$ vertices and wrote the digits of the number to the vertices of the graph in such a way that every vertex received exactly the degree of the vertex. In how many ways could he think about? In a simple graph every edge connects two different vertices, and between two vertices at most one edge can go.
1999 Harvard-MIT Mathematics Tournament, 11
Circles $C_1$, $C_2$, $C_3$ have radius $ 1$ and centers $O, P, Q$ respectively. $C_1$ and $C_2$ intersect at $A$, $C_2$ and $C_3$ intersect at $B$, $C_3$ and $C_1$ intersect at $C$, in such a way that $\angle APB = 60^o$ , $\angle BQC = 36^o$ , and $\angle COA = 72^o$ . Find angle $\angle ABC$ (degrees).
2023 CIIM, 5
Given a positive integer $k > 1$, find all positive integers $n$ such that the polynomial $$P(z) = z^n + \sum_{j=0}^{2^k-2} z^j = 1 +z +z^2 + \cdots +z^{2^k-2} + z^n$$ has a complex root $w$ such that $|w| = 1$.
II Soros Olympiad 1995 - 96 (Russia), 11.9
In triangle $ABC$, the side $BC = a$ and the radius $r$ of the circle tangent to the side BC and the extensions of $AB$ and $AC$ ($A$-excircle) are known. It is also known that inside the triangle there is a point $M$ such that $$BC - AM = CA - BM = AB - CM$$ Find the radius of the circle inscribed in the triangle $BMC$.
2016 Dutch IMO TST, 4
Determine the number of sets $A = \{a_1,a_2,...,a_{1000}\}$ of positive integers satisfying $a_1 < a_2 <...< a_{1000} \le 2014$, for which we have that the set
$S = \{a_i + a_j | 1 \le i, j \le 1000$ with $i + j \in A\}$ is a subset of $A$.
2021 Polish Junior MO Second Round, 1
The numbers $a, b$ satisfy the condition $2a + a^2= 2b + b^2$. Prove that if $a$ is an integer, $b$ is also an integer.
2023 CMIMC Algebra/NT, 3
Compute
$$
2022^{\left(2022^{\cdot ^ {\cdot ^{\cdot ^{\left(2022^{2022}\right)}}}}\right)} \pmod{111}
$$
where there are $2022$ $2022$s. (Give the answer as an integer from $0$ to $110$).
[i]Proposed by David Tang[/i]
1999 Romania Team Selection Test, 8
Let $a$ be a positive real number and $\{x_n\}_{n\geq 1}$ a sequence of real numbers such that $x_1=a$ and
\[ x_{n+1} \geq (n+2)x_n - \sum^{n-1}_{k=1}kx_k, \ \forall \ n\geq 1. \]
Prove that there exists a positive integer $n$ such that $x_n > 1999!$.
[i]Ciprian Manolescu[/i]
2011 Uzbekistan National Olympiad, 1
Find the minimum value of
$|x-y|+\sqrt{(x+2)^2+(y-4)^4}$
2019 Portugal MO, 1
In a square of side $10$ cm , the vertices are joined to the midpoints on the opposite sides, as shown in the figure. How much does the area of the colored region measure?
[img]https://1.bp.blogspot.com/-bHrc1Nu0PQI/X4KaJysLAcI/AAAAAAAAMk0/LLGv1fotQO0Tk1AXqQymG_nNdpyWcbjyACLcBGAsYHQ/s109/2019%2BPortugal%2Bp1.png[/img]
2017 Iran Team Selection Test, 3
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$.
Prove that $X,Y,Z$ are collinear.
[i]Proposed by Hooman Fattahi[/i]
2009 Poland - Second Round, 1
$ABCD$ is a cyclic quadrilateral inscribed in the circle $\Gamma$ with $AB$ as diameter. Let $E$ be the intersection of the diagonals $AC$ and $BD$. The tangents to $\Gamma$ at the points $C,D$ meet at $P$. Prove that $PC=PE$.
PEN O Problems, 53
Suppose that the set $M=\{1,2,\cdots,n\}$ is split into $t$ disjoint subsets $M_{1}$, $\cdots$, $M_{t}$ where the cardinality of $M_i$ is $m_{i}$, and $m_{i} \ge m_{i+1}$, for $i=1,\cdots,t-1$. Show that if $n>t!\cdot e$ then at least one class $M_z$ contains three elements $x_{i}$, $x_{j}$, $x_{k}$ with the property that $x_{i}-x_{j}=x_{k}$.
2002 India IMO Training Camp, 6
Determine the number of $n$-tuples of integers $(x_1,x_2,\cdots ,x_n)$ such that $|x_i| \le 10$ for each $1\le i \le n$ and $|x_i-x_j| \le 10$ for $1 \le i,j \le n$.
1997 Balkan MO, 1
Suppose that $O$ is a point inside a convex quadrilateral $ABCD$ such that \[ OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , \] where by $\mathcal A[ABCD]$ we have denoted the area of $ABCD$. Prove that $ABCD$ is a square and $O$ is its center.
[i]Yugoslavia[/i]
1998 Singapore Team Selection Test, 3
Suppose $f(x)$ is a polynomial with integer coefficients satisfying the condition $0 \le f(c) \le 1997$ for each $c \in \{0, 1, ..., 1998\}$. Is is true that $f(0) = f(1) = ... = f(1998)$?
(variation of [url=https://artofproblemsolving.com/community/c6h49788p315649]1997 IMO Shortlist p12[/url])
1992 IMTS, 4
An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm?
2005 Moldova Team Selection Test, 3
\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\]
where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.
2012-2013 SDML (Middle School), 4
In the puzzle below, dots are placed in some of the empty squares (one dot per square) so that each number gives the combined number of dots in its row and column. How many dots must be placed to complete the puzzle?
[asy]
size(5cm,0);
draw((0,0)--(5,0));
draw((0,1)--(5,1));
draw((0,2)--(5,2));
draw((0,3)--(5,3));
draw((0,4)--(5,4));
draw((0,5)--(5,5));
draw((0,0)--(0,5));
draw((1,0)--(1,5));
draw((2,0)--(2,5));
draw((3,0)--(3,5));
draw((4,0)--(4,5));
draw((5,0)--(5,5));
label("$1$",(2.5,4.5));
label("$2$",(4.5,4.5));
label("$3$",(2.5,1.5));
label("$4$",(2.5,3.5));
label("$5$",(4.5,3.5));
label("$6$",(0.5,2.5));
label("$7$",(3.5,0.5));
[/asy]