Found problems: 85335
2024 Macedonian TST, Problem 5
Let \(P\) be a convex polyhedron with the following properties:
[b]1)[/b] \(P\) has exactly \(666\) edges.
[b]2)[/b] The degrees of all vertices of \(P\) differ by at most \(1\).
[b]3)[/b] There is an edge‐coloring of \(P\) with \(k\) colors such that for each color \(c\) and any two distinct vertices \(V_1,V_2\), there exists a path from \(V_1\) to \(V_2\) all of whose edges have color \(c\).
Determine the largest positive integer \(k\) for which such a polyhedron \(P\) exists.
2019 ELMO Shortlist, G5
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
1997 Romania Team Selection Test, 3
Let $n\ge 4$ be a positive integer and let $M$ be a set of $n$ points in the plane, where no three points are collinear and not all of the $n$ points being concyclic. Find all real functions $f:M\to\mathbb{R}$ such that for any circle $\mathcal{C}$ containing at least three points from $M$, the following equality holds:
\[\sum_{P\in\mathcal{C}\cap M} f(P)=0\]
[i]Dorel Mihet[/i]
2018 Rio de Janeiro Mathematical Olympiad, 4
Let $ABC$ be an acute triangle inscribed on the circumference $\Gamma$. Let $D$ and $E$ be points on $\Gamma$ such that $AD$ is perpendicular to $BC$ and $AE$ is diameter. Let $F$ be the intersection between $AE$ and $BC$.
Prove that, if $\angle DAC = 2 \angle DAB$, then $DE = CF$.
2010 Pan African, 1
Seven distinct points are marked on a circle of circumference $c$. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal $\frac{c}{24}$.
1994 China Team Selection Test, 1
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.
2025 Belarusian National Olympiad, 8.1
In a rectangle $ABCD$ two not intersecting circles $\omega_1$ and $\omega_2$ are drawn such that $\omega_1$ is tangent to $AB$ and $AD$ at points $P$ and $S$ respectively, and $\omega_2$ is tangent to $CB$ and $CD$ at $T$ and $Q$ respectively. It is known that $PQ=11, ST=10, BD=14$.
Find the distance between centers of circles $\omega_1$ and $\omega_2$.
[i]I. Voronovich[/i]
Estonia Open Junior - geometry, 2007.1.4
Call a scalene triangle K [i]disguisable[/i] if there exists a triangle K′ similar to K with two shorter sides precisely as long as the two longer sides of K, respectively. Call a disguisable triangle [i]integral[/i] if the lengths of all its sides are integers.
(a) Find the side lengths of the integral disguisable triangle with the smallest possible perimeter.
(b) Let K be an arbitrary integral disguisable triangle for which no smaller integral
disguisable triangle similar to it exists. Prove that at least two side lengths of K are
perfect squares.
2013 National Chemistry Olympiad, 58
Which statement does not describe benzene, $\ce{C6H6}$?
$ \textbf{(A)}\ \text{It is an aromatic hydrocarbon.} \qquad$
$\textbf{(B)}\ \text{It exists in two isomeric forms.} \qquad$
$\textbf{(C)}\ \text{It undergoes substitution reactions.} \qquad$
$\textbf{(D)}\ \text{It can react to form three different products with the formula C}_6\text{H}_4\text{Cl}_2\qquad$
2005 Romania National Olympiad, 1
Let $ABCD$ be a convex quadrilateral with $AD\not\parallel BC$. Define the points $E=AD \cap BC$ and $I = AC\cap BD$. Prove that the triangles $EDC$ and $IAB$ have the same centroid if and only if $AB \parallel CD$ and $IC^{2}= IA \cdot AC$.
[i]Virgil Nicula[/i]
2023 Putnam, B3
A sequence $y_1, y_2, \ldots, y_k$ of real numbers is called $\textit{zigzag}$ if $k=1$, or if $y_2-y_1, y_3-y_2, \ldots, y_k-y_{k-1}$ are nonzero and alternate in sign. Let $X_1, X_2, \ldots, X_n$ be chosen independently from the uniform distribution on $[0,1]$. Let $a\left(X_1, X_2, \ldots, X_n\right)$ be the largest value of $k$ for which there exists an increasing sequence of integers $i_1, i_2, \ldots, i_k$ such that $X_{i_1}, X_{i_2}, \ldots X_{i_k}$ is zigzag. Find the expected value of $a\left(X_1, X_2, \ldots, X_n\right)$ for $n \geq 2$.
2020 Taiwan APMO Preliminary, P5
Let $S$ is the set of permutation of {1,2,3,4,5,6,7,8}
(1)For all $\sigma=\sigma_1\sigma_2...\sigma_8\in S$
Evaluate the sum of S=$\sigma_1\sigma_2+\sigma_3\sigma_4+\sigma_5\sigma_6+\sigma_7\sigma_8$. Then for all elements in $S$,what is the arithmetic mean of S?
(Notice $S$ and S are different.)
(2)In $S$, how many permutations are there which satisfies "For all $k=1,2,...,7$,the digit after k is [b]not[/b] (k+1)"?
2004 Nicolae Păun, 4
[b]a)[/b] Show that the solution of the equation $ |z-i|=1 $ in $ \mathbb{C} $ is the set $ \{ 2e^{i\alpha} \sin\alpha |\alpha\in [0,\pi ) \} . $
[b]b)[/b] Let be $ n\ge 1 $ complex numbers $ z_1,z_2,\ldots ,z_n $ that verify the inequalities
$$ \left| z_k-i \right|\le 1,\quad\forall k\in\{ 1,2,\ldots ,n \} . $$
Prove that there exists a complex number $ w $ such that $ |w-i|\le 1 $ and $ w^n=z_1z_2\cdots z_n. $
[i]Dan-Ștefan Marinescu[/i]
2024 Belarusian National Olympiad, 8.7
On the diagonal $AC$ of the convex quadrilateral $ABCD$ points $P$,$Q$ are chosen such that triangles $ABD$,$PCD$ and $QBD$ are similar to each other in this order.
Prove that $AQ=PC$
[i]M. Zorka[/i]
1958 AMC 12/AHSME, 11
The number of roots satisfying the equation $ \sqrt{5 \minus{} x} \equal{} x\sqrt{5 \minus{} x}$ is:
$ \textbf{(A)}\ \text{unlimited}\qquad
\textbf{(B)}\ 3\qquad
\textbf{(C)}\ 2\qquad
\textbf{(D)}\ 1\qquad
\textbf{(E)}\ 0$
2019 USAMTS Problems, 4
Let $FIG$ be a triangle and let $D$ be a point on $\overline{FG}$. The line perpendicular to $\overline{FI}$ passing through the midpoint of $\overline{FD}$ and the line perpendicular to $\overline{IG}$ passing through the midpoint of $\overline{DG}$ intersect at $T$. Prove that $FT = GT$ if and only if $\overline{ID}$ is perpendicular to $\overline{FG}$.
2000 Switzerland Team Selection Test, 3
An equilateral triangle of side $1$ is covered by five congruent equilateral triangles of side $s < 1$ with sides parallel to those of the larger triangle. Show that some four of these smaller triangles also cover the large triangle.
1985 Tournament Of Towns, (095) 4
The convex set $F$ does not cover a semi-circle of radius $R$.
Is it possible that two sets, congruent to $F$, cover the circle of radius $R$ ?
What if $F$ is not convex?
( N . B . Vasiliev , A. G . Samosvat)
Russian TST 2019, P3
Inside the acute-angled triangle $ABC$ we take $P$ and $Q$ two isogonal conjugate points. The perpendicular lines on the interior angle-bisector of $\angle BAC$ passing through $P$ and $Q$ intersect the segments $AC$ and $AB$ at the points $B_p\in AC$, $B_q\in AC$, $C_p\in AB$ and $C_q\in AB$, respectively. Let $W$ be the midpoint of the arc $BAC$ of the circle $(ABC)$. The line $WP$ intersects the circle $(ABC)$ again at $P_1$ and the line $WQ$ intersects the circle $(ABC)$ again at $Q_1$. Prove that the points $P_1$, $Q_1$, $B_p$, $B_q$, $C_p$ and $C_q$ lie on a circle.
[i]Proposed by P. Bibikov[/i]
1982 Tournament Of Towns, (023) 1
There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits.
(A Merkov, Moscow)
2002 National Olympiad First Round, 27
The keys of a safe with five locks are cloned and distributed among eight people such that any of five of eight people can open the safe. What is the least total number of keys?
$
\textbf{a)}\ 18
\qquad\textbf{b)}\ 20
\qquad\textbf{c)}\ 22
\qquad\textbf{d)}\ 24
\qquad\textbf{e)}\ 25
$
2010 Iran MO (3rd Round), 1
Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)
2025 Romania National Olympiad, 3
Define the functions $g_k \colon \mathbb{Z} \to \mathbb{Z}$, $g_k(x) = x^k$, where $k$ is a positive integer.
Find the set $M_k$ of positive integers $n$ for which there exist injective functions $f_1,f_2, \dots ,f_n \colon \mathbb{Z} \to \mathbb{Z}$ such that $g_k=f_1\cdot f_2 \cdot \ldots \cdot f_n$.
(Here, $\cdot$ denotes component-wise function multiplication)
2022 Saint Petersburg Mathematical Olympiad, 4
There are two piles of stones: $1703$ stones in one pile and $2022$ in the other. Sasha and Olya
play the game, making moves in turn, Sasha starts. Let before the player's move the heaps contain $a$ and $b$ stones, with $a \geq b$. Then, on his own move, the player is allowed take from the pile with $a$ stones any number of stones from $1$ to $b$. A player loses if he can't make a move. Who wins?
Remark: For 10.4, the initial numbers are $(444,999)$
1966 Miklós Schweitzer, 10
For a real number $ x$ in the interval $ (0,1)$ with decimal representation
\[ 0.a_1(x)a_2(x)...a_n(x)...,\]
denote by $ n(x)$ the smallest nonnegative integer such that
\[ \overline{a_{n(x)\plus{}1}a_{n(x)\plus{}2}a_{n(x)\plus{}3}a_{n(x)\plus{}4}}\equal{}1966 .\]
Determine $ \int_0^1n(x)dx$. ($ \overline{abcd}$ denotes the decimal number with digits $ a,b,c,d .$)
[i]A. Renyi[/i]