Found problems: 85335
1968 IMO Shortlist, 4
Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations:
\[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.
2017 USAMO, 1
Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$.
1955 Miklós Schweitzer, 6
[b]6.[/b] For a prime factorisation of a positive integer $N$ let us call the exponent of a prime $p$ the integer $k$ for which $p^{k} \mid N$ but $p^{k+1} \nmid N$; let, further, the power $p^{k}$ be called the "contribution" of $p$ to $N$. Show that for any positive integer $n$ and for any primes $p$ and $q$ the contibution of $p$ to $n!$ is greater than the contribution of $q$ if and only if the exponent of $p$ is greater than that of $q$.
1995 USAMO, 5
Suppose that in a certain society, each pair of persons can be classified as either [i]amicable [/i]or [i]hostile[/i]. We shall say that each member of an amicable pair is a [i]friend[/i] of the other, and each member of a hostile pair is a [i]foe[/i] of the other. Suppose that the society has $\, n \,$ persons and $\, q \,$ amicable pairs, and that for every set of three persons, at least one pair is hostile. Prove that there is at least one member of the society whose foes include $\, q(1 - 4q/n^2) \,$ or fewer amicable pairs.
2012 NIMO Summer Contest, 14
A set of lattice points is called [i]good[/i] if it does not contain two points that form a line with slope $-1$ or slope $1$. Let $S = \{(x, y)\ |\ x, y \in \mathbb{Z}, 1 \le x, y \le 4\}$. Compute the number of non-empty good subsets of $S$.
[i]Proposed by Lewis Chen[/i]
1998 Akdeniz University MO, 1
Prove that, for $k \in {\mathbb Z^+}$
$$k(k+1)(k+2)(k+3)$$
is not a perfect square.
2019 Oral Moscow Geometry Olympiad, 4
Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.
2022 CCA Math Bonanza, I12
Find the number of 8-tuples of binary inputs $\{A, B, C, D, E, F, G, H\}$ such that \[ \{ (A \text{ AND } B)\text{ OR } (C \text{ AND } D)\} \text{ AND } \{ (E \text{ AND } F)\text{ OR } (G \text{ AND } H)\}\]\[ = \{ (A \text{ OR } B)\text{ AND } (C \text{ OR } D)\} \text{ OR } \{ (E \text{ OR } F)\text{ AND } (G \text{ OR } H)\}\]
The AND gates produce an output that is ON only if both the inputs are ON, and the OR gates produce an output that is OFF only if both inputs are OFF.
[i]2022 CCA Math Bonanza Individual Round #12[/i]
2008 Baltic Way, 17
Assume that $ a$, $ b$, $ c$ and $ d$ are the sides of a quadrilateral inscribed in a given circle. Prove that the product $ (ab \plus{} cd)(ac \plus{} bd)(ad \plus{} bc)$ acquires its maximum when the quadrilateral is a square.
2018 PUMaC Live Round, Estimation 3
Andrew starts with the $2018$-tuple of binary digits $(0,0,\dots,0)$. On each turn, he randomly chooses one index (between $1$ and $2018$) and flips the digit at that index (makes it $1$ if it was a $0$ and vice versa). What is the smallest $k$ such that, after $k$ steps, the expected number of ones in the sequence is greater than $1008?$
You must give your answer as a nonnegative integer. If your answer is $A$ and the correct answer is $C$, then your score will be $\max\{\lfloor18.5-\tfrac{|A-C|^{1.8}}{40}\rfloor,0\}.$
2022 New Zealand MO, 4
Triangle $ABC$ is right-angled at $B$ and has incentre $I$. Points $D$, $E$ and $F$ are the points where the incircle of the triangle touches the sides $BC$, $AC$ and AB respectively. Lines $CI$ and $EF$ intersect at point $P$. Lines $DP$ and $AB$ intersect at point $Q$. Prove that $AQ = BF$.
1985 IMO Longlists, 27
Let $O$ be a point on the oriented Euclidean plane and $(\mathbf i, \mathbf j)$ a directly oriented orthonormal basis. Let $C$ be the circle of radius $1$, centered at $O$. For every real number $t$ and non-negative integer$ n$ let $M_n$ be the point on $C$ for which $\langle \mathbf i , \overrightarrow{OM_n} \rangle = \cos 2^n t.$ (or $\overrightarrow{OM_n} =\cos 2^n t \mathbf i +\sin 2^n t \mathbf j$).
Let $k \geq 2$ be an integer. Find all real numbers $t \in [0, 2\pi)$ that satisfy
[b](i)[/b] $M_0 = M_k$, and
[b](ii)[/b] if one starts from $M0$ and goes once around $C$ in the positive direction, one meets successively the points $M_0,M_1, \dots,M_{k-2},M_{k-1}$, in this order.
2011 Olympic Revenge, 3
Let $E$ to be an infinite set of congruent ellipses in the plane, and $r$ a fixed line. It is known that each line parallel to $r$ intersects at least one ellipse belonging to $E$. Prove that there exist infinitely many triples of ellipses belonging to $E$, such that there exists a line that intersect the triple of ellipses.
1973 All Soviet Union Mathematical Olympiad, 184
The king have revised the chess-board $8\times 8$ having visited all the fields once only and returned to the starting point. When his trajectory was drawn (the centres of the squares were connected with the straight lines), a closed broken line without self-intersections appeared.
a) Give an example that the king could make $28$ steps parallel the sides of the board only.
b) Prove that he could not make less than $28$ such a steps.
c) What is the maximal and minimal length of the broken line if the side of a field is $1$?
1986 IMO Longlists, 73
Let $(a_i)_{i\in \mathbb N}$ be a strictly increasing sequence of positive real numbers such that $\lim_{i \to \infty} a_i = +\infty$ and $a_{i+1}/a_i \leq 10$ for each $i$. Prove that for every positive integer $k$ there are infinitely many pairs $(i, j)$ with $10^k \leq a_i/a_j \leq 10^{k+1}.$
2012 May Olympiad, 1
A four digit number is called [i]stutterer[/i] if its first two digits are the same and its last two digits are also the same, e.g. $3311$ and $2222$ are stutterer numbers. Find all stutterer numbers that are square numbers.
2014 Singapore Senior Math Olympiad, 35
Two circles intersect at the points $C$ and $D$. The straight lines $CD$ and $BYXA$ intersect at the point $Z$. Moreever, the straight line $WB$ is tangent to both of the circles. Suppose $ZX=ZY$ and $AB\cdot AX=100$. Find the value of $BW$.
Russian TST 2017, P1
Let $\mathbb{N}$ denote the set of positive integers. Find all functions $f:\mathbb{N}\longrightarrow\mathbb{N}$ such that
\[n+f(m)\mid f(n)+nf(m)\]
for all $m,n\in \mathbb{N}$
[i]Proposed by Dorlir Ahmeti, Albania[/i]
1996 AMC 12/AHSME, 28
On a $4 \times 4 \times 3$ rectangular parallelepiped, vertices $A$, $B$, and $C$ are adjacent to vertex $D$. The perpendicular distance from $D$ to the plane containing
$A$, $B$, and $C$ is closest to
$\text{(A)}\ 1.6 \qquad \text{(B)}\ 1.9 \qquad \text{(C)}\ 2.1 \qquad \text{(D)}\ 2.7 \qquad \text{(E)}\ 2.9$
2016 Israel National Olympiad, 5
The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$.
Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.
2004 Argentina National Olympiad, 4
Determine all positive integers $a$ and $b$ such that each square on the $a\times b$ board can be colored red, blue, or green such that each red square has exactly one blue neighbor and one green neighbor, each blue square has exactly one red and one green neighbor and each green square has exactly one red and one blue neighbor.
Clarification: Two squares are neighbors if they have a common side.
2008 Princeton University Math Competition, A9/B10
If $p(x)$ is a polynomial with integer coeffcients, let $q(x) = \frac{p(x)}{x(1-x)}$ . If $q(x) = q\left(\frac{1}{1-x}\right)$ for every $x \ne 0$, and $p(2) = -7, p(3) = -11$, find $p(10)$.
2017 Yasinsky Geometry Olympiad, 5
Find the area of the section of a unit cube $ABCDA_1B_1C_1D_1$, when a plane passes through the midpoints of the edges $AB, AD$ and $CC_1$.
2023 Puerto Rico Team Selection Test, 5
Six fruit baskets contain peaches, apples and pears. The number of peaches in each basket is equal to the total number of apples in the other baskets. The number of apples in each basket is equal to the total number of pears in the other baskets.
(a) Find a way to place $31$ fruits in the baskets, satisfying the conditions of the statement.
(b) Explain why the total number of fruits must always be multiple of $31$.
2002 Iran MO (3rd Round), 13
$f,g$ are two permutations of set $X=\{1,\dots,n\}$. We say $f,g$ have common points iff there is a $k\in X$ that $f(k)=g(k)$.
a) If $m>\frac{n}{2}$, prove that there are $m$ permutations $f_{1},f_{2},\dots,f_{m}$ from $X$ that for each permutation $f\in X$, there is an index $i$ that $f,f_{i}$ have common points.
b) Prove that if $m\leq\frac{n}{2}$, we can not find permutations $f_{1},f_{2},\dots,f_{m}$ satisfying the above condition.