Found problems: 85335
Ukrainian TYM Qualifying - geometry, 2011.2
Eight circles of radius $r$ located in a right triangle $ABC$ (angle $C$ is right) as shown in figure (each of the circles touches the respactive sides of the triangle and the other circles). Find the radius of the inscribed circle of triangle $ABC$.
[img]https://cdn.artofproblemsolving.com/attachments/4/7/1b1cd7d6bc7f5004b8e94468d723ed16e9a039.png[/img]
Kvant 2025, M2826
In the square $ABCD$, points $E$ and $F$ were chosen on the sides $AB$ and $BC$ respectively, such that $BE=BF$. Let $L$ be midpoint of $EF$, $N$ be midpoint of $DF$, $O$ be the center of the square and $K=AL \cap DF$ (look at picture). Prove that points $C, K, L, O, N$ are lies on one circle.
[i]A. Paleev[/i]
2018 Taiwan APMO Preliminary, 1
Let trapezoid $ABCD$ inscribed in a circle $O$, $AB||CD$. Tangent at $D$ wrt $O$ intersects line $AC$ at $F$, $DF||BC$. If $CA=5, BC=4$, then find $AF$.
2020-IMOC, A1
$\definecolor{A}{RGB}{190,0,60}\color{A}\fbox{A1.}$ Find all $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$\definecolor{A}{RGB}{80,0,200}\color{A} x^4+y^4+z^4\ge f(xy)+f(yz)+f(zx)\ge xyz(x+y+z)$$holds for all $a,b,c\in\mathbb{R}$.
[i]Proposed by [/i][b][color=#FFFF00]usjl[/color][/b].
[color=#B6D7A8]#1733[/color]
2017 IFYM, Sozopol, 5
We are given a convex quadrilateral $ABCD$ with $AD=CD$ and $\angle BAD=\angle ABC.$
Points $K$ and $L$ are middle points of $AB$ and $BC$, respectively. The rays $\overrightarrow{DL}$ and $\overrightarrow{AB}$ intersect in $M$ and the rays $\overrightarrow{DK}$ and $\overrightarrow{BC}$ – in $N$. On segment $AN$ a point $X$ is chosen, such that $AX=CM$, and on segment $AC$ – point $Y$, such that $AY=MN$. Prove that the line $AB$ bisects segment $XY$.
JOM 2015 Shortlist, A5
Let $ a, b, c $ be the side length of a triangle, with $ ab + bc + ca = 18 $ and $ a, b, c > 1 $. Prove that $$ \sum_{cyc}\frac{1}{(a - 1)^3} > \frac{1}{a + b + c - 3} $$
2011 ELMO Problems, 3
Determine whether there exist two reals $x,y$ and a sequence $\{a_n\}_{n=0}^{\infty}$ of nonzero reals such that $a_{n+2}=xa_{n+1}+ya_n$ for all $n\ge0$ and for every positive real number $r$, there exist positive integers $i,j$ such that $|a_i|<r<|a_j|$.
[i]Alex Zhu.[/i]
2001 National High School Mathematics League, 8
Complex numbers $z_1,z_2$ satisfy that $|z_1|=2,|z_2|=3,3z_1-2z_2=\frac{3}{2}-\text{i}$, then $z_1\cdot z_2=$________.
2018 Adygea Teachers' Geometry Olympiad, 3
Two circles intersect at points $A$ and $B$. Through point $B$, a straight line intersects the circles at points $C$ and $D$, and then tangents to the circles are drawn through points $C$ and $D$. Prove that the points $A, D, C$ and $P$ - the intersection point of the tangents - lie on the same circle.
2001 239 Open Mathematical Olympiad, 4
Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of
the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.
1982 Polish MO Finals, 2
In a cyclic quadrilateral $ABCD$ the line passing through the midpoint of $AB$ and the intersection point of the diagonals is perpendicular to $CD$. Prove that either the sides $AB$ and $CD$ are parallel or the diagonals are perpendicular.
1996 Vietnam Team Selection Test, 3
Find all reals $a$ such that the sequence $\{x(n)\}$, $n=0,1,2, \ldots$ that satisfy: $x(0)=1996$ and $x_{n+1} = \frac{a}{1+x(n)^2}$ for any natural number $n$ has a limit as n goes to infinity.
2021 CCA Math Bonanza, T7
Find the sum of all positive integers $n$ with the following properties:
[list]
[*] $n$ is not divisible by any primes larger than $10$.
[*] For some positive integer $k$, the positive divisors of $n$ are
\[1=d_1<d_2<d_3\cdots<d_{2k}=n.\]
[*] The divisors of $n$ have the property that
\[d_1+d_2+\cdots+d_k=3k.\]
[/list]
[i]2021 CCA Math Bonanza Team Round #7[/i]
2013 Brazil Team Selection Test, 2
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
2017 Turkey Team Selection Test, 2
There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.
2012 Princeton University Math Competition, A2 / B5
How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?
2022 May Olympiad, 3
Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.
2014 Korea Junior Math Olympiad, 7
In a parallelogram $\Box ABCD$ $(AB < BC)$
The incircle of $\triangle ABC$ meets $\overline {BC}$ and $\overline {CA}$ at $P, Q$.
The incircle of $\triangle ACD$ and $\overline {CD}$ meets at $R$.
Let $S$ = $PQ$ $\cap$ $AD$
$U$ = $AR$ $\cap$ $CS$
$T$, a point on $\overline {BC}$ such that $\overline {AB} = \overline {BT}$
Prove that $AT, BU, PQ$ are concurrent
2005 Today's Calculation Of Integral, 85
Evaluate
\[\lim_{n\to\infty} \int_0^{\frac{\pi}{2}} \frac{[n\sin x]}{n}\ dx\]
where $ [x] $ is the integer equal to $ x $ or less than $ x $.
2014 Contests, 1
For every $3$-digit natural number $n$ (leading digit of $n$ is nonzero), we consider the number $n_0$ obtained from $n$ eliminating all possible digits that are zero. For example, if $n = 207$, then $n_0 = 27$. Determine the number of three-digit positive integers $n$, for which $n_0$ is a divisor of $n$ different from $n$.
2001 India IMO Training Camp, 2
A strictly increasing sequence $(a_n)$ has the property that $\gcd(a_m,a_n) = a_{\gcd(m,n)}$ for all $m,n\in \mathbb{N}$. Suppose $k$ is the least positive integer for which there exist positive integers $r < k < s$ such that $a_k^2 = a_ra_s$. Prove that $r | k$ and $k | s$.
Kvant 2023, M2752
A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$?
2021 Romania Team Selection Test, 1
Find all pairs $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.
the 13th XMO, P2
Given $n\in\mathbb N_+,n\ge 3,a_1,a_2,\cdots ,a_n\in\mathbb R_+.$ Let $b_1,b_2,\cdots ,b_n\in\mathbb R_+$ satisfy that for $\forall k\in\{1,2,\cdots ,n\},$
$$\sum_{\substack{i,j\in\{1,2,\cdots ,n\}\backslash \{k\}\\i\neq j}}a_ib_j=0.$$
Prove that $b_1=b_2=\cdots =b_n=0.$
1989 IMO Longlists, 56
Let $ P_1(x), P_2(x), \ldots, P_n(x)$ be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials $ A_r(x),B_r(x) \quad (r \equal{} 1, 2, 3)$ such that
\[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2\]
\[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2\]
\[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2\]