Found problems: 85335
2020 Princeton University Math Competition, B1
Find all pairs of natural numbers $(n, k)$ with the following property:
Given a $k\times k$ array of cells, such that every cell contains one integer, there always exists a path from the left to the right edges such that the sum of the numbers on the path is a multiple of $n$.
Note: A path from the left to the right edge is a sequence of cells of the array $a_1, a_2, ... , a_m$ so that $a_1$ is a cell of the leftmost column, $a_m$ is the cell of the rightmost column, and $a_{i}$, $a_{i+1}$ share an edge for all $i = 1, 2, ... , m -1$.
1997 Pre-Preparation Course Examination, 6
A polygon can be dissected into $100$ rectangles, but it cannot be dissected into $99$ rectangles. Prove that this polygon cannot be dissected into $100$ triangles.
2001 Canada National Olympiad, 4
Let $n$ be a positive integer. Nancy is given a rectangular table in which each entry is a positive integer. She is permitted to make either of the following two moves:
(1) select a row and multiply each entry in this row by $n$;
(2) select a column and subtract $n$ from each entry in this column.
Find all possible values of $n$ for which the following statement is true:
Given any rectangular table, it is possible for Nancy to perform a finite sequence of moves to create a table in which each entry is $0$.
2005 Taiwan TST Round 3, 2
Find all primes $p$ such that the number of distinct positive factors of $p^2+2543$ is less than 16.
1980 Miklós Schweitzer, 1
For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{}
m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\]
[i]V. T. Sos[/i]
2020 New Zealand MO, 5
Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$
$f(x+f(y))=2x+2f(y+1)$
2022 Francophone Mathematical Olympiad, 2
We consider an $n \times n$ table, with $n\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable?
2020 Malaysia IMONST 1, 5
Determine the last digit of $5^5+6^6+7^7+8^8+9^9$.
1991 Federal Competition For Advanced Students, 1
Suppose that $ a,b,$ and $ \sqrt[3]{a}\plus{}\sqrt[3]{b}$ are rational numbers. Prove that $ \sqrt[3]{a}$ and $ \sqrt[3]{b}$ are also rational.
1994 Tournament Of Towns, (433) 3
Let $a, b, c$ and $d$ be real numbers such that
$$a^3+b^3+c^3+d^3=a+b+c+d=0$$
Prove that the sum of a pair of these numbers is equal to $0$.
(LD Kurliandchik)
1990 IMO Longlists, 8
Let $ n \geq 3$ and consider a set $ E$ of $ 2n \minus{} 1$ distinct points on a circle. Suppose that exactly $ k$ of these points are to be colored black. Such a coloring is [b]good[/b] if there is at least one pair of black points such that the interior of one of the arcs between them contains exactly $ n$ points from $ E$. Find the smallest value of $ k$ so that every such coloring of $ k$ points of $ E$ is good.
1975 Kurschak Competition, 1
Transform the equation $$ab^2 \left(\frac{1}{(a + c)^2} +\frac{1}{(a- c)^2} \right) = (a -b)$$ into a simpler form, given that $a > c \ge 0$, $b > 0$.
1987 Greece National Olympiad, 4
Consider a convex $100$-gon $A_1A_2...A_{100}$. Draw the diagonal $A_{43}A_{81}$ which divides it into two convex polygons $P_1,P_2$. How many vertices and how diagonals, has each of the polygons $P_1,P_2$?
2014 Purple Comet Problems, 17
In the figure below $\triangle ABC$, $\triangle DEF$, and $\triangle GHI$ are overlapping equilateral triangles, $C$ and $F$ lie on $\overline{BD}$, $F$ and $I$ lie on $\overline{EG}$, and $C$ and $I$ lie on $\overline{AH}$. Length $AB = 2FC$, $DE = 3FC$, and $GH = 4FC$. Given that the area of $\triangle FCI$ is $3$, find the area of the hexagon $ABGHDE$.
[asy]
size(5cm);
pen dps = fontsize(10);
defaultpen(dps);
pair A,B,C,D,E,F,G,H,I;
G=origin;
H=(4,0);
I=(2,2*sqrt(3));
F=(1.5,3*sqrt(3)/2);
C=F+(1,0);
B=F-(1,0);
D=C+(2,0);
A=F+(0,sqrt(3));
E=C+(0.5,3*sqrt(3)/2);
draw(A--H--G--E--D--B--cycle);
label("$A$",A,N*.5);
label("$B$",B,S*.5);
label("$C$",C,SW*.5);
label("$D$",D,S*.5);
label("$E$",E,N*.5);
label("$F$",F,SE*.5);
label("$G$",G,S*.5);
label("$H$",H,S*.5);
label("$I$",I,N*2);
[/asy]
2010 Canada National Olympiad, 2
Let $A,B,P$ be three points on a circle. Prove that if $a,b$ are the distances from $P$ to the tangents at $A,B$ respectively, and $c$ is the distance from $P$ to the chord $AB$, then $c^2 =ab$.
1998 Croatia National Olympiad, Problem 3
Ivan and Krešo started to travel from Crikvenica to Kraljevica, whose distance is $15$ km, and at the same time Marko started from Kraljevica to Crikvenica. Each of them can go either walking at a speed of $5$ km/h, or by bicycle with the speed of $15$ km/h. Ivan started walking, and Krešo was driving a bicycle until meeting Marko. Then Krešo gave the bicycle to Marko and continued walking to Kraljevica, while Marko continued to Crikvenica by bicycle. When Marko met Ivan, he gave him the bicycle and continued on foot, so Ivan arrived at Kraljevica by bicycle. Find, for each of them, the time he spent in travel as well as the time spent in walking.
2002 Iran MO (3rd Round), 20
$a_{0}=2,a_{1}=1$ and for $n\geq 1$ we know that : $a_{n+1}=a_{n}+a_{n-1}$
$m$ is an even number and $p$ is prime number such that $p$ divides $a_{m}-2$. Prove that $p$ divides $a_{m+1}-1$.
2019 Dutch Mathematical Olympiad, 3
Points $A, B$, and $C$ lie on a circle with centre $M$. The reflection of point $M$ in the line $AB$ lies inside triangle $ABC$ and is the intersection of the angle bisectors of angles $A$ and $B$. Line $AM$ intersects the circle again in point $D$.
Show that $|CA| \cdot |CD| = |AB| \cdot |AM|$.
2017-2018 SDML (Middle School), 1
Let $N = \frac{1}{3} + \frac{3}{5} + \frac{5}{7} + \frac{7}{9} + \frac{9}{11}$. What is the greatest integer which is less than $N$?
2013 Taiwan TST Round 1, 6
Let $ABCD$ be a convex quadrilateral with non-parallel sides $BC$ and $AD$. Assume that there is a point $E$ on the side $BC$ such that the quadrilaterals $ABED$ and $AECD$ are circumscribed. Prove that there is a point $F$ on the side $AD$ such that the quadrilaterals $ABCF$ and $BCDF$ are circumscribed if and only if $AB$ is parallel to $CD$.
2005 USAMTS Problems, 3
We play a game. The pot starts at $\$0$. On every turn, you flip a fair coin. If you flip heads, I add $\$100$ to the pot. If you flip tails, I take all of the money out of the pot, and you are assessed a "strike". You can stop the game before any flip and collect the contents of the pot, but if you get 3 strikes, the game is over and you win nothing. Find, with proof, the expected value of your winnings if you follow an optimal strategy.
2018 CMIMC Individual Finals, 1
The [i]distance[/i] between two vertices in a connected graph is defined to be the length of the shortest path between them. How many graphs with the vertex set $\{0,1,2,\dots,6\}$ satisfy the following property: there are $3$ vertices of distance $1$ away from vertex $0$, $2$ of distance $2$ away, and $1$ of distance $3$ away?
2014 Albania Round 2, 3
In a right $\Delta ABC$ ($\angle C = 90^{\circ} $), $CD$ is the height. Let $r_1$ and $r_2$ be the radii of inscribed circles of $\Delta ACD$ and $\Delta DCB$. Find the radius of inscribed circle of $\Delta ABC$
2001 India National Olympiad, 1
Let $ABC$ be a triangle in which no angle is $90^{\circ}$. For any point $P$ in the plane of the triangle, let $A_1, B_1, C_1$ denote the reflections of $P$ in the sides $BC,CA,AB$ respectively. Prove that
(i) If $P$ is the incenter or an excentre of $ABC$, then $P$ is the circumenter of $A_1B_1C_1$;
(ii) If $P$ is the circumcentre of $ABC$, then $P$ is the orthocentre of $A_1B_1C_1$;
(iii) If $P$ is the orthocentre of $ABC$, then $P$ is either the incentre or an excentre of $A_1B_1C_1$.
2006 Switzerland Team Selection Test, 1
Let $n$ be natural number and $1=d_1<d_2<\ldots <d_k=n$ be the positive divisors of $n$.
Find all $n$ such that $2n = d_5^2+ d_6^2 -1$.