Found problems: 85335
2019 Ukraine Team Selection Test, 3
Let $a_0,a_1,a_2,\dots $ be a sequence of real numbers such that $a_0=0, a_1=1,$ and for every $n\geq 2$ there exists $1 \leq k \leq n$ satisfying \[ a_n=\frac{a_{n-1}+\dots + a_{n-k}}{k}. \]Find the maximum possible value of $a_{2018}-a_{2017}$.
2006 Princeton University Math Competition, 1
What is the greatest possible number of edges in a planar graph with $12$ vertices? A planar graph is one that can be drawn in a plane with none of the edges crossing (they intersect only at vertices).
2014 AMC 12/AHSME, 8
In the addition shown below $A$, $B$, $C$, and $D$ are distinct digits. How many different values are possible for $D$?
\[\begin{array}{lr}
&ABBCB \\
+& BCADA \\
\hline
& DBDDD
\end{array}\]
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
1978 Miklós Schweitzer, 9
Suppose that all subspaces of cardinality at most $ \aleph_1$ of a topological space are second-countable. Prove that the whole space is second-countable.
[i]A. Hajnal, I. Juhasz[/i]
2016 Postal Coaching, 6
Let $K$ and $L$ be the centers of the excircles of a non-isosceles triangle $ABC$ opposite $B$ and $C$ respectively. Let $M$ and $N$ be points in the plane of the triangle such that $BM$ bisects $AC$ and $CN$ bisects $AB$. Prove that the lines $KM$ and $NK$ meet on $BC$.
[hide=Note]The problem in its current formulation is trivially wrong. No possible rectification is known to OP / was sent to the participants.[/hide]
2014 ASDAN Math Tournament, 6
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$.
Ukraine Correspondence MO - geometry, 2008.7
On the sides $AC$ and $AB$ of the triangle $ABC$, the points $D$ and $E$ were chosen such that $\angle ABD =\angle CBD$ and $3 \angle ACE = 2\angle BCE$. Let $H$ be the point of intersection of $BD$ and $CE$, and $CD = DE = CH$. Find the angles of triangle $ABC$.
2013 CHMMC (Fall), 3
Let $p_n$ be the product of the $n$th roots of $1$. For integral $x > 4$, let $f(x) = p_1 - p_2 + p_3 - p_4 + ... + (-1)^{x+1}p_x$. What is $f(2010)$?
2010 IFYM, Sozopol, 1
The inscribed circle of $\Delta ABC$ is tangent to $AC$ and $BC$ in points $M$ and $N$ respectively. Line $MN$ intersects line $AB$ in point $P$, so that $B$ is between $A$ and $P$. Determine $\angle ABC$, if $BP=CM$.
1996 Argentina National Olympiad, 4
Let $ABCD$ be a parallelogram with center $O$ such that $\angle BAD <90^o$ and $\angle AOB> 90^o$. Consider points $A_1$ and $B_1$ on the rays $OA$ and $OB$ respectively, such that $A_1B_1$ is parallel to $AB$ and $\angle A_1B_1C = \frac12 \angle ABC$. Prove that $A_1D$ is perpendicular to $B_1C$.
2012 Grigore Moisil Intercounty, 4
A real continuous function has the property that its evaluation at any point is nilpotent under composition with itself.
Prove that this function is $ 0. $
[i]Vasile Pop[/i]
2006 USAMO, 6
Let $ABCD$ be a quadrilateral, and let $E$ and $F$ be points on sides $AD$ and $BC$, respectively, such that $\frac{AE}{ED} = \frac{BF}{FC}$. Ray $FE$ meets rays $BA$ and $CD$ at $S$ and $T$, respectively. Prove that the circumcircles of triangles $SAE$, $SBF$, $TCF$, and $TDE$ pass through a common point.
2014 Grand Duchy of Lithuania, 2
An isosceles triangle $ABC$ with $AC = BC$ is given. Let $M$ be the midpoint of the side $AB$ and let $P$ be a point inside the triangle such that $\angle PAB = \angle PBC$. Prove that $\angle APM + \angle BPC = 180 \textdegree $
2011 Bosnia And Herzegovina - Regional Olympiad, 3
Triangle $AOB$ is rotated in plane around point $O$ for $90^{\circ}$ and it maps in triangle $A_1OB_1$ ($A$ maps to $A_1$, $B$ maps to $B_1$). Prove that median of triangle $OAB_1$ of side $AB_1$ is orthogonal to $A_1B$
1991 IMTS, 5
Show that it is impossible to dissect an arbitary tetrahedron into six parts by planes or portions thereof so that each of the parts has a plane of symmetry.
2011 Hanoi Open Mathematics Competitions, 2
What is the smallest number ?
(A) $3$ (B) $2^{\sqrt2}$ (C) $2^{1+\frac{1}{\sqrt2}}$ (D) $2^{\frac12} + 2^{\frac23}$ (E) $2^{\frac53}$
2016 Chile National Olympiad, 3
The [i]giraffe[/i] is a chess piece that moves $4$ squares in one direction and then a box in a perpendicular direction. What is the smallest value of $n$ such that the giraffe that starts from a corner on an $n \times n$ board can visit all the squares of said board?
2018 USAMTS Problems, 4:
Right triangle $\triangle{}ABC$ has $\angle{}C=90^{\circ{}}$. A fly is trapped inside $\triangle{}ABC$. It starts at point $D$, the foot of the altitude from $C$ to $\overline{AB}$, and then makes a (finite) sequence of moves. In each move, it flies in a direction parallel to either $\overline{AC}$ or $\overline{BC}$; upon reaching a leg of the triangle, it then flies to a point on $\overline{AB}$ in a direction parallel to $\overline{CD}$. For example, on its first move, the fly can move to either of the points $Y_1$ or $Y_2$, as shown.
[asy]
pair C = (0,0);
pair A = (0,4);
pair B = (5,0);
draw(C--A);
draw(C--B);
draw(B--A);
dot(A);
dot(B);
dot(C);
label("$A$",A,NW);
label("$C$",C,SW);
label("$B$",B,SE);
pair D = foot(C,A,B);
draw(C--D,dotted);
label("$D$",D,NE);
dot(D);
draw(rightanglemark(A,C,B));
pair B1 = foot(D,C,B);
draw(D--B1,dotted);
pair A1 = foot(D,A,C);
draw(D--A1,dotted);
pair Y1 = foot(A1,A,D);
draw(A1--Y1,dotted);
dot(Y1);
label("$Y_1$",Y1,NE);
pair Y2 = foot(B1,D,B);
draw(B1--Y2,dotted);
dot(Y2);
label("$Y_2$",Y2,NE);
draw(rightanglemark(C,A1,D));
draw(rightanglemark(C,B1,D));
draw(rightanglemark(B1,Y2,D));
draw(rightanglemark(A1,Y1,D));
draw(rightanglemark(C,D,A));
[/asy]
Let $P$ and $Q$ be distinct points on $\overline{AB}$. Show that the fly can reach some point on $\overline{PQ}$.
2008 Gheorghe Vranceanu, 1
Find the complex numbers $ a,b $ having the properties that $ |a|=|b|=1=\bar{a} +\bar{b} -ab. $
1955 Miklós Schweitzer, 4
[b]4.[/b] Find all positive integers $\alpha , \beta (\alpha >1)$ and all prime numbers $p, q, r$ which satisfy the equation $p^{\alpha}= q^{\beta}+r^{\alpha}$ ($\alpha , \beta , p, q, r$ need not necessarily be different). [b](N. 12)[/b]
2008 Tuymaada Olympiad, 8
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23.
[i]Author: K. Kokhas[/i]
Estonia Open Junior - geometry, 2016.2.5
On the plane three different points $P, Q$, and $R$ are chosen. It is known that however one chooses another point $X$ on the plane, the point $P$ is always either closer to $X$ than the point $Q$ or closer to $X$ than the point $R$. Prove that the point $P$ lies on the line segment $QR$.
1975 Polish MO Finals, 4
All decimal digits of some natural number are $1,3,7$, and $9$. Prove that one can rearrange its digits so as to obtain a number divisible by $7$.
2024 CCA Math Bonanza, L1.3
Find the number of $10$ digit palindromes that are not divisible by $11$.
[i]Lightning 1.3[/i]
1991 IMTS, 4
Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that
\[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]