Found problems: 85335
V Soros Olympiad 1998 - 99 (Russia), grade7
[b]p1.[/b] There are eight different dominoes in the box (fig.), but the boundaries between them are not visible. Draw the boundaries.
[img]https://cdn.artofproblemsolving.com/attachments/6/f/6352b18c25478d68a23820e32a7f237c9f2ba9.png[/img]
[b]p2.[/b] The teacher drew a quadrilateral $ABCD$ on the board. Vanya and Vitya marked points $X$ and $Y$ inside it, from which all sides of the quadrilateral are visible at equal angles. What is the distance between points $X$ and $Y$? (From point $X$, side $AB$ is visible at angle $AXB$.)
[b]pЗ.[/b] Several identical black squares, perhaps partially overlapping, were placed on a white plane. The result was a black polygonal figure, possibly with holes or from several pieces. Could it be that this figure does not have a single right angle?
[b]p4.[/b] The bus ticket number consists of six digits (the first digits may be zeros). A ticket is called [i]lucky [/i] if the sum of the first three digits is equal to the sum of the last three. Prove that the sum of the numbers of all lucky tickets is divisible by $13$.
[b]p5.[/b] The Meandrovka River, which has many bends, crosses a straight highway under thirteen bridges. Prove that there are two neighboring bridges along both the highway and the river. (Bridges are called river neighbors if there are no other bridges between them on the river section; bridges are called highway neighbors if there are no other bridges between them on the highway section.)
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]
2018 Belarusian National Olympiad, 11.8
The vertices of the regular $n$-gon are marked. Two players play the following game: they, in turn, select a vertex and connect it by a segment to either the adjacent vertex or the center of the $n$-gon. The winner is a player if after his move it is possible to get any vertex from any other vertex moving along segments.
For each integer $n\geqslant 3$ determine who has a winning strategy.
1973 Czech and Slovak Olympiad III A, 4
For any integer $n\ge2$ evaluate the sum \[\sum_{k=1}^{n^2-1}\bigl\lfloor\sqrt k\bigr\rfloor.\]
1997 ITAMO, 4
Let $ABCD$ be a tetrahedron. Let $a$ be the length of $AB$ and let $S$ be the area of the projection of the tetrahedron onto a plane perpendicular to $AB$. Determine the volume of the tetrahedron in terms of $a$ and $S$.
BIMO 2022, 3
Let $\omega$ be the circumcircle of an actue triangle $ABC$ and let $H$ be the feet of aliitude from $A$ to $BC$. Let $M$ and $N$ be the midpoints of the sides $AC$ and $AB$. The lines $BM$ and $CN$ intersect each other at $G$ and intersect $\omega$ at $P$ and $Q$ respectively. The circles $(HMG)$ and $(HNG)$ intersect the segments $HP$ and $HQ$ again at $R$ and $S$ respectively. Prove that $PQ\parallel RS$.
2020 Brazil Undergrad MO, Problem 1
Let $R > 0$, be an integer, and let $n(R)$ be the number um triples $(x, y, z) \in \mathbb{Z}^3$ such that $2x^2+3y^2+5z^2 = R$. What is the value of
$\lim_{ R \to \infty}\frac{n(1) + n(2) + \cdots + n(R)}{R^{3/2}}$?
2019 China Team Selection Test, 2
Fix a positive integer $n\geq 3$. Does there exist infinitely many sets $S$ of positive integers $\lbrace a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n\rbrace$, such that $\gcd (a_1,a_2,\ldots, a_n$, $b_1,b_2,\ldots,b_n)=1$, $\lbrace a_i\rbrace _{i=1}^n$, $\lbrace b_i\rbrace _{i=1}^n$ are arithmetic progressions, and $\prod_{i=1}^n a_i = \prod_{i=1}^n b_i$?
1991 Arnold's Trivium, 55
Investigate topologically the Riemann surface of the function
\[w=\arctan z\]
1996 China Team Selection Test, 2
$S$ is the set of functions $f:\mathbb{N} \to \mathbb{R}$ that satisfy the following conditions:
[b]I.[/b] $f(1) = 2$
[b]II.[/b] $f(n+1) \geq f(n) \geq \frac{n}{n + 1} f(2n)$ for $n = 1, 2, \ldots$
Find the smallest $M \in \mathbb{N}$ such that for any $f \in S$ and any $n \in \mathbb{N}, f(n) < M$.
1978 Canada National Olympiad, 1
Let $n$ be an integer. If the tens digit of $n^2$ is 7, what is the units digit of $n^2$?
2013 Iran MO (3rd Round), 7
An equation $P(x)=Q(y)$ is called [b]Interesting[/b] if $P$ and $Q$ are polynomials with degree at least one and integer coefficients and the equations has an infinite number of answers in $\mathbb{N}$.
An interesting equation $P(x)=Q(y)$ [b]yields in[/b] interesting equation $F(x)=G(y)$ if there exists polynomial $R(x) \in \mathbb{Q} [x]$ such that $F(x) \equiv R(P(x))$ and $G(x) \equiv R(Q(x))$.
(a) Suppose that $S$ is an infinite subset of $\mathbb{N} \times \mathbb{N}$.$S$ [i]is an answer[/i] of interesting equation $P(x)=Q(y)$ if each element of $S$ is an answer of this equation. Prove that for each $S$ there's an interesting equation $P_0(x)=Q_0(y)$ such that if there exists any interesting equation that $S$ is an answer of it, $P_0(x)=Q_0(y)$ yields in that equation.
(b) Define the degree of an interesting equation $P(x)=Q(y)$ by $max\{deg(P),deg(Q)\}$. An interesting equation is called [b]primary[/b] if there's no other interesting equation with lower degree that yields in it.
Prove that if $P(x)=Q(y)$ is a primary interesting equation and $P$ and $Q$ are monic then $(deg(P),deg(Q))=1$.
Time allowed for this question was 2 hours.
2002 Polish MO Finals, 2
There is given a triangle $ABC$ in a space. A sphere does not intersect the plane of $ABC$. There are $4$ points $K, L, M, P$ on the sphere such that $AK, BL, CM$ are tangent to the sphere and $\frac{AK}{AP} = \frac{BL}{BP} = \frac{CM}{CP}$. Show that the sphere touches the circumsphere of $ABCP$.
2013 Princeton University Math Competition, 3
Consider all planes through the center of a $2\times2\times2$ cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in the form $a\sqrt b+c$, where $b$ is a square-free positive integer. Find $a+b+c$.
2004 Oral Moscow Geometry Olympiad, 2
Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?
2013 Princeton University Math Competition, 16
Is $\cos 1^\circ$ rational? Prove.
2011 Postal Coaching, 4
Let $n > 1$ be a positive integer. Find all $n$-tuples $(a_1 , a_2 ,\ldots, a_n )$ of positive integers which are pairwise distinct, pairwise coprime, and such that for each $i$ in the range $1 \le i \le n$,
\[(a_1 + a_2 + \ldots + a_n )|(a_1^i + a_2^i + \ldots + a_n^i )\].
1993 Mexico National Olympiad, 4
$f(n,k)$ is defined by
(1) $f(n,0) = f(n,n) = 1$ and
(2) $f(n,k) = f(n-1,k-1) + f(n-1,k)$ for $0 < k < n$.
How many times do we need to use (2) to find $f(3991,1993)$?
2007 AMC 8, 1
Theresa's parents have agreed to buy her tickets to see her favorite band if she spends an average of $10$ hours per week helping around the house for $6$ weeks. For the first $5$ weeks, she helps around the house for $8$, $11$, $7$, $12$ and $10$ hours. How many hours must she work during the final week to earn the tickets?
$\textbf{(A)}\ 9 \qquad
\textbf{(B)}\ 10 \qquad
\textbf{(C)}\ 11 \qquad
\textbf{(D)}\ 12 \qquad
\textbf{(E)}\ 13$
2023 Harvard-MIT Mathematics Tournament, 28
Suppose $ABCD$ is a convex quadrilateral with $\angle{ABD}=105^\circ, \angle{ADB}=15^\circ, AC=7,$ and $BC=CD=5.$ Compute the sum of all possible values of $BD.$
2020 Nigerian Senior MO Round 2, 3
$N$ straight lines are drawn on a plane. The $N$ lines can be partitioned into set of lines such that if a line $l$ belongs to a partition set then all lines parallel to $l$ make up the rest of that set. For each $n>=1$,let $a_n$ denote the number of partition sets of size $n$. Now that $N$ lines intersect at certain points on the plane. For each $n>=2$ let $b_n$ denote the number of points that are intersection of exactly $n$ lines. Show that
$\sum_{n>= 2}(a_n+b_n)$$\binom{n}{2}$ $=$ $\binom{N}{2}$
2023 Serbia Team Selection Test, P5
For positive integers $a$ and $b$, define \[a!_b=\prod_{1\le i\le a\atop i \equiv a \mod b} i\]
Let $p$ be a prime and $n>3$ a positive integer. Show that there exist at least 2 different positive integers $t$ such that $1<t<p^n$ and $t!_p\equiv 1\pmod {p^n}$.
2023 Assam Mathematics Olympiad, 8
If $n$ is a positive even number, find the last two digits of $(2^{6n}+26)-(6^{2n}-62)$.
JBMO Geometry Collection, 2013
Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.
1989 USAMO, 3
Let $P(z)= z^n + c_1 z^{n-1} + c_2 z^{n-2} + \cdots + c_n$ be a polynomial in the complex variable $z$, with real coefficients $c_k$. Suppose that $|P(i)| < 1$. Prove that there exist real numbers $a$ and $b$ such that $P(a + bi) = 0$ and $(a^2 + b^2 + 1)^2 < 4 b^2 + 1$.
Indonesia MO Shortlist - geometry, g12
In triangle $ABC$, the incircle is tangent to $BC$ at $D$, to $AC$ at $E$, and to $AB$ at $F$. Prove that:
$$\frac{CE-EA}{\sqrt{AB}}+\frac{AF-FB}{\sqrt{BC}} +\frac{BD-DC}{\sqrt{CA}} \ge \frac{BD-DC}{\sqrt{AB}}
+\frac{CE-EA}{\sqrt{BC}} +\frac{AF-FB}{\sqrt{CA}}$$