Found problems: 85335
1990 Bundeswettbewerb Mathematik, 4
Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.
1999 North Macedonia National Olympiad, 4
Do there exist $100$ straight lines on a plane such that they intersect each other in exactly $1999$ points?
2009 India IMO Training Camp, 3
Let $ a,b$ be two distinct odd natural numbers.Define a Sequence $ { < a_n > }_{n\ge 0}$ like following:
$ a_1 \equal{} a \\
a_2 \equal{} b \\
a_n \equal{} \text{largest odd divisor of }(a_{n \minus{} 1} \plus{} a_{n \minus{} 2})$.
Prove that there exists a natural number $ N$ such that $ a_n \equal{} gcd(a,b) \forall n\ge N$.
1955 Czech and Slovak Olympiad III A, 1
Consider a trapezoid $ABCD,AB\parallel CD,AB>CD.$ Let us denote intersections of lines as follows: $E=AC\cap BD, F=AD\cap BC.$ Let $GH$ be a line such that $G\in AD,H\in BC, E\in GH,GH\parallel AB.$ Moreover, denote $K,L$ midpoints of the bases $AB,CD$ respectively. Show that
(a) the points $K,L$ lie on the line $EF,$
(b) lines $AC,KH$ and $BD,KG$ are not parallel (denote $M=AC\cap KH,N=BD\cap KG$),
(c) the points $F,M,N$ are collinear.
2012 Turkmenistan National Math Olympiad, 5
Let $O$ be the center of $\bigtriangleup ABC$'s circumcircle. $CO$ line intersect $AB$ at $D$ and $BO$ line intersect $AC$ at $E$. If $\angle A=\angle CDE=50$° then find $\angle ADE$
2006 Pre-Preparation Course Examination, 3
There is a right angle whose vertex moves on a fixed circle and one of it's sides passes a fixed point. What is the curve that the other side of the angle is always tangent to it.
2019 Purple Comet Problems, 23
Find the number of ordered pairs of integers $(x, y)$ such that $$\frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right)$$
2020 Princeton University Math Competition, 1
Consider a $2021$-by-$2021$ board of unit squares. For some integer $k$, we say the board is tiled by $k$-by-$k$ squares if it is completely covered by (possibly overlapping) $k$-by-$k$ squares with their corners on the corners of the unit squares. What is the largest integer k such that the minimum number of $k$-by-$k$ squares needed to tile the $2021$-by-$2021$ board is exactly equal to $100$?
2013 Stanford Mathematics Tournament, 9
Big candles cost 16 cents and burn for exactly 16 minutes. Small candles cost 7 cents and burn for exactly 7 minutes. The candles burn at possibly varying and unknown rates, so it is impossible to predictably modify the amount of time for which a candle will burn except by burning it down for a known amount of time. Candles may be arbitrarily and instantly put out and relit. Compute the cost in cents of the cheapest set of big and small candles you need to measure exactly 1 minute.
2017 China Northern MO, 2
Prove that there exist infinitely many integers \(n\) which satisfy \(2017^2 | 1^n + 2^n + ... + 2017^n\).
1977 Poland - Second Round, 4
A pyramid with a quadrangular base is given such that each pair of circles inscribed in adjacent faces has a common point. Prove that the touchpoints of these circles with the base of the pyramid lie on one circle.
Kvant 2024, M2802
The positive numbers $a_1, a_2, \ldots , a_{2024}$ are placed on a circle clockwise in this order. Let $A_i$ be the arithmetic mean of the number $a_i$ and one or several following it clockwise. Prove that the largest of the numbers $A_1, A_2, \ldots , A_{2024}$ is not less than the arithmetic mean of all numbers $a_1, a_2, \ldots , a_{2024}$.
2019 Jozsef Wildt International Math Competition, W. 11
Let $(s_n)_{n\geq 1}$ be a sequence given by $s_n=-2\sqrt{n}+\sum \limits_{k=1}^n\frac{1}{\sqrt{k}}$ with $\lim \limits_{n \to \infty}s_n=s=$Ioachimescu constant and $(a_n)_{n\geq 1}$ , $(b_n)_{n\geq 1}$ be a positive real sequences such that $$\lim \limits_{n\to \infty}\frac{a_{n+1}}{na_n}=a\in \mathbb{R}^*_+, \lim \limits_{n\to \infty}\frac{b_{n+1}}{b_n\sqrt{n}}=b\in \mathbb{R}^*_+$$Compute$$\lim \limits_{n\to \infty}\left(1+e^{s_n}-e^{s_{n+1}}\right)^{\sqrt[n]{a_nb_n}}$$
2002 Moldova National Olympiad, 4
Prove that there are infinitely many triplets $ (a,b,c)$ that satisfy the following equalities:
$ \dfrac{2a\minus{}b\plus{}6}{4a\plus{}c\plus{}2}\equal{}\dfrac{b\minus{}2c}{a\minus{}c}\equal{}\dfrac{2a\plus{}b\plus{}2c\minus{}2}{6a\plus{}2c\minus{}2}$
2019 CMIMC, 11
Let $S$ be a subset of the natural numbers such that $0\in S$, and for all $n\in\mathbb N$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
2016 Brazil Team Selection Test, 1
Let $ABC$ be an acute triangle with orthocenter $H$. Let $G$ be the point such that the quadrilateral $ABGH$ is a parallelogram. Let $I$ be the point on the line $GH$ such that $AC$ bisects $HI$. Suppose that the line $AC$ intersects the circumcircle of the triangle $GCI$ at $C$ and $J$. Prove that $IJ = AH$.
2020 Sharygin Geometry Olympiad, 21
The diagonals of bicentric quadrilateral $ABCD$ meet at point $L$. Given are three segments equal to $AL$, $BL$, $CL$. Restore the quadrilateral using a compass and a ruler.
2006 India National Olympiad, 4
Some 46 squares are randomly chosen from a $9 \times 9$ chess board and colored in [color=red]red[/color]. Show that there exists a $2\times 2$ block of 4 squares of which at least three are colored in [color=red]red[/color].
1996 Tournament Of Towns, (516) 3
The parabola $y = x^2$ is drawn in the coordinate plane and then the axes are erased so that the whole parabola stays on the picture but the origin is not shown on it. Reconstruct the axes with compass and ruler alone.
(A Egorov)
2005 Federal Math Competition of S&M, Problem 2
Suppose that in a convex hexagon, each of the three lines connecting the midpoints of two opposite sides divides the hexagon into two parts of equal area. Prove that these three lines intersect in a point.
2020 Harvard-MIT Mathematics Tournament, 1
Let $n$ be a positive integer. Define a sequence by $a_0 = 1$, $a_{2i+1} = a_i$, and $a_{2i+2} = a_i + a_{i+1}$ for each $i \ge 0$. Determine, with proof, the value of $a_0 + a_1 + a_2 + \dots + a_{2^n-1}$.
[i]Proposed by Kevin Ren.[/i]
1997 Israel Grosman Mathematical Olympiad, 3
Find all real solutions of $\sqrt[4]{13+x}+ \sqrt[4]{14-x} = 3$.
2013 AMC 10, 24
Central High School is competing against Northern High School in a backgammon match. Each school has three players, and the contest rules require that each player play two games against each of the other's school's players. The match takes place in six rounds, with three games played simultaneously in each round. In how many different ways can the match be scheduled?
$\textbf{(A)} \ 540 \qquad \textbf{(B)} \ 600 \qquad \textbf{(C)} \ 720 \qquad \textbf{(D)} \ 810 \qquad \textbf{(E)} \ 900$
2021 IMC, 6
For a prime number $p$, let $GL_2(\mathbb{Z}/p\mathbb{Z})$ be the group of invertible $2 \times 2$ matrices of residues modulo $p$, and let $S_p$ be the symmetric group (the group of all permutations) on $p$ elements. Show that there is no injective group homomorphism $\phi : GL_2(\mathbb{Z}/p\mathbb{Z}) \rightarrow S_p$.
2018 Junior Balkan MO, 2
Find max number $n$ of numbers of three digits such that :
1. Each has digit sum $9$
2. No one contains digit $0$
3. Each $2$ have different unit digits
4. Each $2$ have different decimal digits
5. Each $2$ have different hundreds digits