Found problems: 85335
2015 Regional Olympiad of Mexico Southeast, 2
In a acutangle triangle $ABC, \angle B>\angle C$. Let $D$ the foot of the altitude from $A$ to $BC$ and $E$ the foot of the perpendicular from $D$ to $AC$. Let $F$ a point in $DE$. Prove that $AF$ and $BF$ are perpendiculars if and only if $EF\cdot DC=BD\cdot DE$.
1998 Baltic Way, 18
Determine all positive integers $n$ for which there exists a set $S$ with the following properties:
(i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$;
(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.
2024 Harvard-MIT Mathematics Tournament, 9
Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*}
& f(g(h(x))) = h(g(f(x))) = g(x) \\
& g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\
& h(f(g(x))) = g(f(h(x))) = f(x), \\
\end{align*} for all $x\in \{1,2,3,4,5\}$.
2017 Indonesia MO, 6
Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$.
2023 Malaysia IMONST 2, 6
Ivan has a parallelogram whose interior angles are $60^{\circ}, 120^{\circ}, 60^{\circ}, 120^{\circ}$ respectively, and all side lengths are integers. Is it possible that one of the diagonals has length $\sqrt{2024}$?
Brazil L2 Finals (OBM) - geometry, 2017.5
Let $ABC$ be a triangle with, $AB$ ≠ $AC$, and let $K$ is your incenter. The points $P$ and $Q$ are the points of the intersections of the circumcicle($BCK$) with the line(s) $AB$ and $AC$, respectively. Let $D$ be intersection of $AK$ and $BC$.
Show that $P, Q, D$ are collinears.
2017 Rioplatense Mathematical Olympiad, Level 3, 5
Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.
2023 USA IMO Team Selection Test, 4
Let $\lfloor \bullet \rfloor$ denote the floor function. For nonnegative integers $a$ and $b$, their [i]bitwise xor[/i], denoted $a \oplus b$, is the unique nonnegative integer such that $$ \left \lfloor \frac{a}{2^k} \right \rfloor+ \left\lfloor\frac{b}{2^k} \right\rfloor - \left\lfloor \frac{a\oplus b}{2^k}\right\rfloor$$ is even for every $k \ge 0$. Find all positive integers $a$ such that for any integers $x>y\ge 0$, we have \[ x\oplus ax \neq y \oplus ay. \]
[i]Carl Schildkraut[/i]
2009 Ukraine National Mathematical Olympiad, 2
In acute-angled triangle $ABC,$ let $M$ be the midpoint of $BC$ and let $K$ be a point on side $AB.$ We know that $AM$ meet $CK$ at $F.$ Prove that if $AK = KF$ then $AB = CF .$
2012 Today's Calculation Of Integral, 835
Evaluate the following definite integrals.
(a) $\int_1^2 \frac{x-1}{x^2-2x+2}\ dx$
(b) $\int_0^1 \frac{e^{4x}}{e^{2x}+2}\ dx$
(c) $\int_1^e x\ln \sqrt{x}\ dx$
(d) $\int_0^{\frac{\pi}{3}} \left(\cos ^ 2 x\sin 3x-\frac 14\sin 5x\right)\ dx$
2011 CentroAmerican, 5
If $x$, $y$, $z$ are positive numbers satisfying
\[x+\frac{y}{z}=y+\frac{z}{x}=z+\frac{x}{y}=2.\]
Find all the possible values of $x+y+z$.
2024 Miklos Schweitzer, 3
Do there exist continuous functions $f, g: \mathbb{R} \to \mathbb{R}$, both nowhere differentiable, such that $f \circ g$ is differentiable?
1986 Tournament Of Towns, (123) 5
Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .
(A. Andjans, Riga)
1993 AMC 8, 18
The rectangle shown has length $AC=32$, width $AE=20$, and $B$ and $F$ are midpoints of $\overline{AC}$ and $\overline{AE}$, respectively. The area of quadrilateral $ABDF$ is
[asy]
pair A,B,C,D,EE,F;
A = (0,20); B = (16,20); C = (32,20); D = (32,0); EE = (0,0); F = (0,10);
draw(A--C--D--EE--cycle);
draw(B--D--F);
dot(A); dot(B); dot(C); dot(D); dot(EE); dot(F);
label("$A$",A,NW);
label("$B$",B,N);
label("$C$",C,NE);
label("$D$",D,SE);
label("$E$",EE,SW);
label("$F$",F,W);
[/asy]
$\text{(A)}\ 320 \qquad \text{(B)}\ 325 \qquad \text{(C)}\ 330 \qquad \text{(D)}\ 335 \qquad \text{(E)}\ 340$
2016 India IMO Training Camp, 2
Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.
2016 Online Math Open Problems, 23
$S$ be the set of all $2017^2$ lattice points $(x,y)$ with $x,y\in \{0\}\cup\{2^{0},2^{1},\cdots,2^{2015}\}$. A subset $X\subseteq S$ is called BQ if it has the following properties:
(a) $X$ contains at least three points, no three of which are collinear.
(b) One of the points in $X$ is $(0,0)$.
(c) For any three distinct points $A,B,C \in X$, the orthocenter of $\triangle ABC$ is in $X$.
(d) The convex hull of $X$ contains at least one horizontal line segment.
Determine the number of BQ subsets of $S$.
[i] Proposed by Vincent Huang [/i]
2022 IFYM, Sozopol, 5
Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.
2014 PUMaC Algebra B, 5
Given that $a_na_{n-2}-a_{n-1}^2+a_n-na_{n-2}=-n^2+3n-1$ and $a_0=1$, $a_1=3$, find $a_{20}$.
2016 HMNT, 8
Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?
KoMaL A Problems 2024/2025, A. 891
Let $ABC$ be an acute triangle. Points $B'$ and $C'$ are located on the interior of sides $AB$ and $AC$, respectively. Let $M$ denote the second intersection of the circumcircles of triangles $ABC$ and $AB'C'$, while let $N$ denote the second intersection of the circumcircles of triangles $ABC'$ and $AB'C$. Reflect $M$ across lines $AB$ and $AC$, and let $l$ denote the line through the reflections.
a) Prove that the line through $M$ perpendicular to $AM$, the line $AK$, and $l$ are either concurrent or all parallel.
b) Show that if the three lines are concurrent at $S$, then triangles $SBC'$ and $SCB'$ have equal areas.
[i]Proposed by Áron Bán-Szabó, Budapest[/i]
MathLinks Contest 4th, 5.2
Let $ABCD$ be a convex quadrilateral, and let $K$ be a point on side$ AB$ such that $\angle KDA = \angle BCD$. Let $L$ be a point on the diagonal $AC$ such that $KL \parallel BC$. Prove that $\angle KDB = \angle LDC$.
2016 Sharygin Geometry Olympiad, P21
The areas of rectangles $P$ and $Q$ are equal, but the diagonal of $P$ is greater. Rectangle $Q$ can be covered by two copies of $P$. Prove that $P$ can be covered by two copies of $Q$.
1991 Tournament Of Towns, (281) 1
$N$ integers are given. Prove that the sum of their squares is divisible by $N$ if it is known that the difference between the product of any $N - 1$ of them and the last one is divisible by $N$.
(D. Fomin, Leningrad)
2017 Canadian Mathematical Olympiad Qualification, 2
For any positive integer n, let $\varphi(n)$ be the number of integers in the set $\{1, 2, \ldots , n\}$ whose greatest common divisor with $n$ is 1. Determine the maximum value of $\frac{n}{\varphi(n)}$ for $n$ in the set $\{2, \ldots, 1000\}$ and all values of $n$ for which this maximum is attained.
2019 Saudi Arabia JBMO TST, 4
Let $n$ be positive integer and let $a_1, a_2,...,a_n$ be real numbers. Prove that there exist positive integers $m, k$ $<=n$ , $|$ $(a_1+a_2+...+a_m)$ $-$ $(a_{m+1}+a_{m+2}+...+a_n)$ $|$ $<=$ $|$ $a_k$ $|$