Found problems: 85335
2005 Purple Comet Problems, 24
$\triangle ABC$ has area $240$. Points $X, Y, Z$ lie on sides $AB$, $BC$, and $CA$, respectively. Given that $\frac{AX}{BX} = 3$, $\frac{BY}{CY} = 4$, and $\frac{CZ}{AZ} = 5$, find the area of $\triangle XYZ$.
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(0,15),B=(0,-5),C=(25,0.5),X=origin,Y=(4C+B)/5,Z=(5A+C)/6;
draw(A--B--C--cycle^^X--Y--Z--cycle);
label("$A$",A,N);
label("$B$",B,S);
label("$C$",C,E);
label("$X$",X,W);
label("$Y$",Y,S);
label("$Z$",Z,NE);[/asy]
2022 Auckland Mathematical Olympiad, 9
Does there exist a function $f(n)$, which maps the set of natural numbers into itself and such that for each natural number $n > 1$ the following equation is satisfi ed $$f(n) = f(f(n - 1)) + f(f(n + 1))?$$
2014 Saudi Arabia BMO TST, 4
Let $n$ be an integer greater than $2$. Consider a set of $n$ different points, with no three collinear, in the plane. Prove that we can label the points $P_1,~ P_2, \dots , P_n$ such that $P_1P_2 \dots P_n$ is not a self-intersecting polygon. ([i]A polygon is self-intersecting if one of its side intersects the interior of another side. The polygon is not necessarily convex[/i] )
Kvant 2023, M2750
Let $D, E$ and $F{}$ be the midpoints of the sides $BC, CA$ and $AB{}$ of the acute-angled triangle $ABC$ and let $H_a, H_b$ and $H_c{}$ be the orthocenters of the triangles $ABD, BCE$ and $CAF{}$ respectively. Prove that the triangles $H_aH_bH_c$ and $DEF$ have equal areas.
[i]Proposed by Tran Quang Hung[/i]
Taiwan TST 2015 Round 1, 2
Given any triangle $ABC.$ Let $O_1$ be it's circumcircle, $O_2$ be it's nine point circle, $O_3$ is a circle with orthocenter of $ABC$, $H$, and centroid $G$, be it's diameter. Prove that: $O_1,O_2,O_3$ share axis. (i.e. chose any two of them, their axis will be the same one, if $ABC$ is an obtuse triangle, the three circle share two points.)
2019 Stanford Mathematics Tournament, 9
Let $ABCD$ be a quadrilateral with $\angle ABC = \angle CDA = 45^o$ , $AB = 7$, and $BD = 25$. If $AC$ is perpendicular to $CD$, compute the length of $BC$.
1994 National High School Mathematics League, 5
In regular $n$-regular pyramid, the range value of dihedral angle of two adjacent sides is
$\text{(A)}\left(\frac{n-2}{n}\pi,\pi\right)\qquad\text{(B)}\left(\frac{n-1}{n}\pi,\pi\right)\qquad\text{(C)}\left(0,\frac{\pi}{2}\right)\qquad\text{(D)}\left(\frac{n-2}{n}\pi,\frac{n-1}{n}\pi\right)$
2013 Kosovo National Mathematical Olympiad, 2
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
1992 IMTS, 2
Prove that if $a,b,c$ are positive integers such that $c^2 = a^2+b^2$, then both $c^2+ab$ and $c^2-ab$ are also expressible as the sums of squares of two positive integers.
1996 All-Russian Olympiad, 4
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),
$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.
[i]O. Musin[/i]
1994 Bulgaria National Olympiad, 2
Find all functions $f : R \to R$ such that $x f(x)-y f(y) = (x-y)f(x+y)$ for all $x,y \in R$.
2021 Indonesia TST, G
The circles $k_1$ and $k_2$ intersect at points $A$ and $B$, and $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ at the points $K ,O$ and $k_2$ at the points $L ,M$ so that $L$ lies between $K$ and $O$. The point $P$ is the projection of $L$ on the line $AB$. Prove that $KP$ is parallel to the median of triangle $ABM$ drawn from the vertex $M$.
1982 Miklós Schweitzer, 9
Suppose that $ K$ is a compact Hausdorff space and $ K\equal{} \cup_{n\equal{}0}^{\infty}A_n$, where $ A_n$ is metrizable and $ A_n \subset A_m$ for $ n<m$. Prove that $ K$ is metrizable.
[i]Z. Balogh[/i]
2018 China Second Round Olympiad, 3
Let set $A=\{1,2,\ldots,n\} ,$ and $X,Y$ be two subsets (not necessarily distinct) of $A.$ Define that $\textup{max} X$ and $\textup{min} Y$ represent the greatest element of $X$ and the least element of $Y,$ respectively. Determine the number of two-tuples $(X,Y)$ which satisfies $\textup{max} X>\textup{min} Y.$
2020 CIIM, 2
Find all triples of positive integers $(a,b,c)$ such that the following equations are both true:
I- $a^2+b^2=c^2$
II- $a^3+b^3+1=(c-1)^3$
1985 AMC 8, 14
The difference between a $ 6.5 \%$ sales tax and a $ 6 \%$ sales tax on an item priced at $ \$20$ before tax is
\[ \textbf{(A)}\ \$.01 \qquad
\textbf{(B)}\ \$.10 \qquad
\textbf{(C)}\ \$.50 \qquad
\textbf{(D)}\ \$1 \qquad
\textbf{(E)}\ \$10
\]
2023 BMT, 1
Lakshay chooses two numbers, $m$ and $n$, and draws two lines, $y = mx + 3$ and $y = nx + 23$. Given that the two lines intersect at $(20, 23)$, compute $m + n$.
1998 Poland - First Round, 5
Find all pairs of positive integers $ x,y$ satisfying the equation
\[ y^x \equal{} x^{50}\]
2019 AMC 8, 20
How many different real numbers $x$ satisfy the equation $$(x^2-5)^2=16?$$
$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$
1982 IMO Longlists, 22
Let $M$ be the set of real numbers of the form $\frac{m+n}{\sqrt{m^2+n^2}}$, where $m$ and $n$ are positive integers. Prove that for every pair $x \in M, y \in M$ with $x < y$, there exists an element $z \in M$ such that $x < z < y.$
2012 Benelux, 3
In triangle $ABC$ the midpoint of $BC$ is called $M$. Let $P$ be a variable interior point of the triangle such that $\angle CPM=\angle PAB$. Let $\Gamma$ be the circumcircle of triangle $ABP$. The line $MP$ intersects $\Gamma$ a second time at $Q$. Define $R$ as the reflection of $P$ in the tangent to $\Gamma$ at $B$. Prove that the length $|QR|$ is independent of the position of $P$ inside the triangle.
2002 Vietnam Team Selection Test, 1
Find all triangles $ABC$ for which $\angle ACB$ is acute and the interior angle bisector of $BC$ intersects the trisectors $(AX, (AY$ of the angle $\angle BAC$ in the points $N,P$ respectively, such that $AB=NP=2DM$, where $D$ is the foot of the altitude from $A$ on $BC$ and $M$ is the midpoint of the side $BC$.
2006 Baltic Way, 8
The director has found out that six conspiracies have been set up in his department, each of them involving exactly $3$ persons. Prove that the director can split the department in two laboratories so that none of the conspirative groups is entirely in the same laboratory.
2013 District Olympiad, 3
Take the function $f:\mathbb{R}\to \mathbb{R}$, $f\left( x \right)=ax,x\in \mathbb{Q},f\left( x \right)=bx,x\in \mathbb{R}\backslash \mathbb{Q}$, where $a$ and $b$ are two real numbers different from 0.
Prove that $f$ is injective if and only if $f$ is surjective.
2016 Online Math Open Problems, 4
Given that $x$ is a real number, find the minimum value of $f(x)=|x+1|+3|x+3|+6|x+6|+10|x+10|.$
[i]Proposed by Yannick Yao[/i]