Found problems: 85335
1987 Iran MO (2nd round), 3
Let $L_1, L_2, L_3, L_4$ be four lines in the space such that no three of them are in the same plane. Let $L_1, L_2$ intersect in $A$, $L_2,L_3$ intersect in $B$ and $L_3, L_4$ intersect in $C.$ Find minimum and maximum number of lines in the space that intersect $L_1, L_2, L_3$ and $L_4.$ Justify your answer.
2008 Greece National Olympiad, 2
Find all integers $x$ and prime numbers $p$ satisfying $x^8 + 2^{2^x+2} = p$.
2008 Kurschak Competition, 1
Denote by $d(n)$ the number of positive divisors of a positive integer $n$. Find the smallest constant $c$ for which $d(n)\le c\sqrt n$ holds for all positive integers $n$.
2023 Sharygin Geometry Olympiad, 8.7
The bisector of angle $A$ of triangle $ABC$ meet its circumcircle $\omega$ at point $W$. The circle $s$ with diameter $AH$ ($H$ is the orthocenter of $ABC$) meets $\omega$ for the second time at point $P$. Restore the triangle $ABC$ if the points $A$, $P$, $W$ are given.
2018 Thailand TSTST, 7
Evaluate $\sum_{n=2017}^{2030}\sum_{k=1}^{n}\left\{\frac{\binom{n}{k}}{2017}\right\}$.
[i]Note: $\{x\}=x-\lfloor x\rfloor$ for every real numbers $x$.[/i]
2025 Korea - Final Round, P5
$S={1,2,...,1000}$ and $T'=\left\{ 1001-t|t \in T\right\}$.
A set $P$ satisfies the following three conditions:
$1.$ All elements of $P$ are a subset of $S$.
$2. A,B \in P \Rightarrow A \cap B \neq \O$
$3. A \in P \Rightarrow A' \in P$
Find the maximum of $|P|$.
1981 IMO, 3
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
2015 Stars Of Mathematics, 4
Let $n\ge 5$ be a positive integer and let $\{a_1,a_2,...,a_n\}=\{1,2,...,n\}$.Prove that at least $\lfloor \sqrt{n}\rfloor +1$ numbers from $a_1,a_1+a_2,...,a_1+a_2+...+a_n$ leave different residues when divided by $n$.
2011 NIMO Problems, 10
The edges and diagonals of convex pentagon $ABCDE$ are all colored either red or blue. How many ways are there to color the segments such that there is exactly one monochromatic triangle with vertices among $A$, $B$, $C$, $D$, $E$; that is, triangles, whose edges are all the same color?
[i]Proposed by Eugene Chen
[/i]
1984 Kurschak Competition, 3
Given are $n$ integers, not necessarily distinct, and two positive integers $p$ and $q$. If the $n$ numbers are not all distinct, choose two equal ones. Add $p$ to one of them and subtract $q$ from the other. If there are still equal ones among the $n$ numbers, repeat this procedure. Prove that after a finite number of steps, all $n$ numbers are distinct.
2017 Balkan MO Shortlist, A2
Consider the sequence of rational numbers defined by $x_1=\frac{4}{3}$ and $x_{n+1}=\frac{x_n^2}{x_n^2-x_n+1}$ , $n\geq 1$.
Show that the numerator of the lowest term expression of each sum $\sum_{k=1}^{n}x_k$ is a perfect square.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
2019 CCA Math Bonanza, I11
Let $G$ be the centroid of triangle $ABC$ with $AB=13,BC=14,CA=15$. Calculate the sum of the distances from $G$ to the three sides of the triangle.
Note: The [i]centroid[/i] of a triangle is the point that lies on each of the three line segments between a vertex and the midpoint of its opposite side.
[i]2019 CCA Math Bonanza Individual Round #11[/i]
2005 Sharygin Geometry Olympiad, 10.2
A triangle can be cut into three similar triangles.
Prove that it can be cut into any number of triangles similar to each other.
2016 Israel National Olympiad, 7
Find all functions $f:\mathbb{Z}\rightarrow\mathbb{C}$ such that $f(x(2y+1))=f(x(y+1))+f(x)f(y)$ holds for any two integers $x,y$.
2013 Rioplatense Mathematical Olympiad, Level 3, 3
A division of a group of people into various groups is called $k$-regular if the number of groups is less or equal to $k$ and two people that know each other are in different groups.
Let $A$, $B$, and $C$ groups of people such that there are is no person in $A$ and no person in $B$ that know each other. Suppose that the group $A \cup C$ has an $a$-regular division and the group $B \cup C$ has a $b$-regular division.
For each $a$ and $b$, determine the least possible value of $k$ for which it is guaranteed that the group $A \cup B \cup C$ has a $k$-regular division.
2016 Ukraine Team Selection Test, 9
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
1994 China National Olympiad, 4
Let $f(z)=c_0z^n+c_1z^{n-1}+ c_2z^{n-2}+\cdots +c_{n-1}z+c_n$ be a polynomial with complex coefficients. Prove that there exists a complex number $z_0$ such that $|f(z_0)|\ge |c_0|+|c_n|$, where $|z_0|\le 1$.
2015 Estonia Team Selection Test, 10
Let $n$ be an integer and $a, b$ real numbers such that $n > 1$ and $a > b > 0$. Prove that $$(a^n - b^n) \left ( \frac{1}{b^{n- 1}} - \frac{1}{a^{n -1}}\right) > 4n(n -1)(\sqrt{a} - \sqrt{b})^2$$
2019 India PRMO, 22
What is the greatest integer not exceeding the sum $\sum^{1599}_{n=1} \dfrac{1}{\sqrt{n}}$?
2012 National Olympiad First Round, 28
At the beginning, three boxes contain $m$, $n$, and $k$ pieces, respectively. Ayşe and Burak are playing a turn-based game with these pieces. At each turn, the player takes at least one piece from one of the boxes. The player who takes the last piece will win the game. Ayşe will be the first player. They are playing the game once for each $(m,n,k)=(1,2012,2014)$, $(2011,2011,2012)$, $(2011,2012,2013)$, $(2011,2012,2014)$, $(2011,2013,2013)$. In how many of them can Ayşe guarantee to win the game?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2009 Peru Iberoamerican Team Selection Test, P1
A set $P$ has the following property: “For any positive integer $k$, if $p$ is a prime factor of $k^3+6$, then $p$ belongs to $P$ ”. Prove that $P$ is infinite.
2010 AMC 12/AHSME, 3
Rectangle $ ABCD$, pictured below, shares $50\%$ of its area with square $ EFGH$. Square $ EFGH$ shares $20\%$ of its area with rectangle $ ABCD$. What is $ \frac{AB}{AD}$?
[asy]unitsize(5mm);
defaultpen(linewidth(0.8pt)+fontsize(10pt));
pair A=(0,3), B=(8,3), C=(8,2), D=(0,2), Ep=(0,4), F=(4,4), G=(4,0), H=(0,0);
fill(shift(0,2)*xscale(4)*unitsquare,grey);
draw(Ep--F--G--H--cycle);
draw(A--B--C--D);
label("$A$",A,W);
label("$B$",B,E);
label("$C$",C,E);
label("$D$",D,W);
label("$E$",Ep,NW);
label("$F$",F,NE);
label("$G$",G,SE);
label("$H$",H,SW);[/asy]$ \textbf{(A)}\ 4\qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 10$
2003 Purple Comet Problems, 14
Let $a$, $b$, $c$ be real numbers such that $a^2 - 2 = 3b - c$, $b^2 + 4 = 3 + a$, and $c^2 + 4 = 3a - b$. Find $a^4 + b^4 + c^4$.
1998 Greece JBMO TST, 6
Prove that if the number $A = 111 \cdots 1$ ($n$ digits) is prime, then $n$ is prime. Is the converse true?
2018 Taiwan TST Round 2, 1
Given positive integers $a_1,a_2,\ldots, a_n$ with $a_1<a_2<\cdots<a_n)$, and a positive real $k$ with $k\geq 1$. Prove that
\[\sum_{i=1}^{n}a_i^{2k+1}\geq \left(\sum_{i=1}^{n}a_i^k\right)^2.\]