Found problems: 85335
2005 QEDMO 1st, 2 (G2)
Let $ABC$ be a triangle. Let $C^{\prime}$ and $A^{\prime}$ be the reflections of its vertices $C$ and $A$, respectively, in the altitude of triangle $ABC$ issuing from $B$. The perpendicular to the line $BA^{\prime}$ through the point $C^{\prime}$ intersects the line $BC$ at $U$; the perpendicular to the line $BC^{\prime}$ through the point $A^{\prime}$ intersects the line $BA$ at $V$. Prove that $UV \parallel CA$.
Darij
2018 Mexico National Olympiad, 6
Let $ABC$ be an acute-angled triangle with circumference $\Omega$. Let the angle bisectors of $\angle B$ and $\angle C$ intersect $\Omega$ again at $M$ and $N$. Let $I$ be the intersection point of these angle bisectors. Let $M'$ and $N'$ be the respective reflections of $M$ and $N$ in $AC$ and $AB$. Prove that the center of the circle passing through $I$, $M'$, $N'$ lies on the altitude of triangle $ABC$ from $A$.
[i]Proposed by Victor DomÃnguez and Ariel GarcÃa[/i]
2024 Korea - Final Round, P1
Let $a, b, c, d$ be odd positive integers and pairwise coprime. For a positive integer $n$, let $$f(n) = \left[\frac{n}{a}
\right]+\left[\frac{n}{b}\right]+\left[\frac{n}{c}\right]+\left[\frac{n}{d}\right]$$ Prove that $$\sum_{n=1}^{abcd}(-1)^{f(n)}=1$$
DMM Individual Rounds, 2022 Tie
[b]p1.[/b] The sequence $\{x_n\}$ is defined by $$x_{n+1} = \begin{cases} 2x_n - 1, \,\, if \,\, \frac12 \le x_n < 1 \\ 2x_n, \,\, if \,\, 0 \le x_n < \frac12 \end{cases}$$ where $0 \le x_0 < 1$ and $x_7 = x_0$. Find the number of sequences satisfying these conditions.
[b]p2.[/b] Let $M = \{1, . . . , 2022\}$. For any nonempty set $X \subseteq M$, let $a_X$ be the sum of the maximum and the minimum number of $X$. Find the average value of $a_X$ across all nonempty subsets $X$ of $M$.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1966 Spain Mathematical Olympiad, 5
The length of the hypotenuse $BC$ of a right triangle $ABC$ is $a$, and on it the points $M$ and $N$ are taken such that $BM = NC = k$, with $k < a/2$. Assuming that (only) the data $a$ and $k$ are known, calculate:
a) The value of the sum of the squares of the lengths $AM$ and $AN$.
b) The ratio of the areas of triangles $ABC$ and $AMN$.
c) The area enclosed by the circle that passes through the points $A, M' , N'$ , where $M'$ is the orthogonal projection of $M$ onto $AC$ and $N'$ that of $N$ onto $AB$.
2003 AMC 10, 1
What is the difference between the sum of the first $ 2003$ even counting numbers and the sum of the first $ 2003$ odd counting numbers?
$ \textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 2003 \qquad
\textbf{(E)}\ 4006$
Kvant 2024, M2810
The positive integer $n \geqslant 2$ is given. How many ways can the cells of the $n\times n$ square be colored in four colors so that any two cells with a common side or vertex are colored in different colors?
[i] I. Efremov [/i]
2022 CMIMC, 2.3
We say that a set $S$ of $3$ unit squares is \textit{commutable} if $S = \{s_1,s_2,s_3\}$ for some $s_1,s_2,s_3$ where $s_2$ shares a side with each of $s_1,s_3$. How many ways are there to partition a $3\times 3$ grid of unit squares into $3$ pairwise disjoint commutable sets?
[i]Proposed by Srinivasan Sathiamurthy[/i]
2003 District Olympiad, 1
Find the disjoint sets $B$ and $C$ such that $B \cup C = \{1,2,..., 10\}$ and the product of the elements of $C$ equals the sum of elements of $B$.
2013 Romania Team Selection Test, 2
Let $n$ be an integer larger than $1$ and let $S$ be the set of $n$-element subsets of the set $\{1,2,\ldots,2n\}$. Determine
\[\max_{A\in S}\left (\min_{x,y\in A, x \neq y} [x,y]\right )\] where $[x,y]$ is the least common multiple of the integers $x$, $y$.
2012 District Olympiad, 3
Let be a sequence of natural numbers $ \left( a_n \right)_{n\ge 1} $ such that $ a_n\le n $ for all natural numbers $ n, $ and
$$ \sum_{j=1}^{k-1} \cos \frac{\pi a_j}{k} =0, $$
for all natural $ k\ge 2. $
[b]a)[/b] Find $ a_2. $
[b]b)[/b] Determine this sequence.
2015 Kosovo Team Selection Test, 3
It's given system of equations
$a_{11}x_1+a_{12}x_2+a_{1n}x_n=b_1$
$a_{21}x_1+a_{22}x_2+a_{2n}x_n=b_2$
..........
$a_{n1}x_1+a_{n2}x_2+a_{nn}x_n=b_n$
such that $a_{11},a_{12},...,a_{1n},b_1,a_{21},a_{22},...,a_{2n},b_2,...,a_{n1},a_{n2},...,a_{nn},b_n,$ form an arithmetic sequence.If system has one solution find it
2003 India National Olympiad, 1
Let $P$ be an interior point of an acute-angled triangle $ABC$. The line $BP$ meets the line $AC$ at $E$, and the line $CP$ meets the line $AB$ at $F$. The lines $AP$ and $EF$ intersect each other at $D$. Let $K$ be the foot of the perpendicular from the point $D$ to the line $BC$. Show that the line $KD$ bisects the angle $\angle EKF$.
2014 ASDAN Math Tournament, 10
Let $p(x)=c_1+c_2\cdot2^x+c_3\cdot3^x+c_4\cdot5^x+c_5\cdot8^x$. Given that $p(k)=k$ for $k=1,2,3,4,5$, compute $p(6)$.
1998 German National Olympiad, 6b
Prove that the following statement holds for all odd integers $n \ge 3$:
If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.
2002 All-Russian Olympiad, 1
For positive real numbers $a, b, c$ such that $a+b+c=3$, show that:
\[\sqrt{a} + \sqrt{b} + \sqrt{c} \ge ab+bc+ca.\]
2019 ELMO Shortlist, G6
Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$.
a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$.
b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$.
[i]Proposed by Vincent Huang[/i]
2021 Bolivia Ibero TST, 1
Let $n$ be a posititve integer. On a $n \times n$ grid there are $n^2$ unit squares and on these we color the sides with blue such that every unit square has exactly one side with blue.
[b]a)[/b] Find the maximun number of blue unit sides we can have on the $n \times n$ grid.
[b]b)[/b] Find the minimun number of blue unit sides we can have on the $n \times n$ grid.
2014 Iran Team Selection Test, 3
we named a $n*n$ table $selfish$ if we number the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down)
for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$.
for example we have such table for $n=5$
1 0 3 3 4
1 3 2 1 1
0 1 0 1 0
2 1 0 0 0
1 0 0 0 0
prove that for $n>5$ there is no $selfish$ table
2022 VN Math Olympiad For High School Students, Problem 4
Assume that $\triangle ABC$ is acute. Let $a=BC, b=CA, c=AB$.
a) Denote $H$ by the orthocenter of $\triangle ABC$. Prove that:$$a.\frac{{\overrightarrow {HA} }}{{HA}} + b.\frac{{\overrightarrow {HB} }}{{HB}} + c.\frac{{\overrightarrow {HC} }}{{HC}} = \overrightarrow 0 .$$
b) Consider a point $P$ lying on the plane. Prove that the sum:$$aPa+bPB+cPC$$ get its minimum value iff $P\equiv H$.
2017 Azerbaijan Team Selection Test, 3
Find all functions $f : \mathbb R\to\mathbb R $ such that \[f(x+yf(x^2))=f(x)+xf(xy)\] for all real numbers $x$ and $y$.
2010 Contests, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
Russian TST 2016, P3
Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]
ICMC 6, 5
A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$?
[i]Proposed by Dylan Toh[/i]
2022 Sharygin Geometry Olympiad, 6
The incircle and the excircle of triangle $ABC$ touch the side $AC$ at points $P$ and $Q$ respectively. The lines $BP$ and $BQ$ meet the circumcircle of triangle $ABC$ for the second time at points $P'$ and $Q'$ respectively.
Prove that
$$PP' > QQ'$$