Found problems: 85335
2019 JBMO Shortlist, A1
Real numbers $a$ and $b$ satisfy $a^3+b^3-6ab=-11$. Prove that $-\frac{7}{3}<a+b<-2$.
[i]Proposed by Serbia[/i]
2014 Thailand TSTST, 2
Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.
2016 PUMaC Combinatorics A, 4
A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an chess $\text{L}$-shape ($2$ units parallel to one axis and $1$ unit parallel to the other) to one of eight possible location, chosen at random. After $2016$ such turns, what is the expected value of the square of the distance of the knight from the origin?
2013 All-Russian Olympiad, 4
A square with horizontal and vertical sides is drawn on the plane. It held several segments parallel to the sides, and there are no two segments which lie on one line or intersect at an interior point for both segments. It turned out that the segments cuts square into rectangles, and any vertical line intersecting the square and not containing segments of the partition intersects exactly $ k $ rectangles of the partition, and any horizontal line intersecting the square and not containing segments of the partition intersects exactly $\ell$ rectangles. How much the number of rectangles can be?
[i]I. Bogdanov, D. Fon-Der-Flaass[/i]
2012 AIME Problems, 6
Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$.
Find $c+d$.
2004 Switzerland Team Selection Test, 6
Find all finite sequences $(x_0, x_1, \ldots,x_n)$ such that for every $j$, $0 \leq j \leq n$, $x_j$ equals the number of times $j$ appears in the sequence.
Swiss NMO - geometry, 2006.2
Let $ABC$ be an equilateral triangle and let $D$ be an inner point of the side $BC$. A circle is tangent to $BC$ at $D$ and intersects the sides $AB$ and $AC$ in the inner points $M, N$ and $P, Q$ respectively. Prove that $|BD| + |AM| + |AN| = |CD| + |AP| + |AQ|$.
Kvant 2025, M2826
In the square $ABCD$, points $E$ and $F$ were chosen on the sides $AB$ and $BC$ respectively, such that $BE=BF$. Let $L$ be midpoint of $EF$, $N$ be midpoint of $DF$, $O$ be the center of the square and $K=AL \cap DF$ (look at picture). Prove that points $C, K, L, O, N$ are lies on one circle.
[i]A. Paleev[/i]
2012 NIMO Problems, 5
The hour and minute hands on a certain 12-hour analog clock are indistinguishable. If the hands of the clock move continuously, compute the number of times strictly between noon and midnight for which the information on the clock is not sufficient to determine the time.
[i]Proposed by Lewis Chen[/i]
1992 Austrian-Polish Competition, 2
Each point on the boundary of a square has to be colored in one color. Consider all right triangles with the vertices on the boundary of the square. Determine the least number of colors for which there is a coloring such that no such triangle has all its vertices of the same color.
2015 Tuymaada Olympiad, 8
There are $\frac{k(k+1)}{2}+1$ points on the planes, some are connected by disjoint segments ( also point can not lies on segment, that connects two other points). It is true, that plane is divided to some parallelograms and one infinite region. What maximum number of segments can be drawn ?
[i] A.Kupavski, A. Polyanski[/i]
MIPT Undergraduate Contest 2019, 1.3
Given a natural number $n$, for what maximal value $k$ it is possible to construct a matrix of size $k \times n$ consisting only of elements $\pm 1$ in such a way that for any interchange of a $+1$ with a $-1$ or vice versa, its rank is equal to $k$?
2019 Auckland Mathematical Olympiad, 1
Given a convex quadrilateral $ABCD$ in which $\angle BAC = 20^o$, $\angle CAD = 60^o$, $\angle ADB = 50^o$ , and $\angle BDC = 10^o$. Find $\angle ACB$.
2019 ABMC, Accuracy
[b]p1.[/b] Compute $45\times 45 - 6$.
[b]p2.[/b] Consecutive integers have nice properties. For example, $3$, $4$, $5$ are three consecutive integers, and $8$, $9$, $10$ are three consecutive integers also. If the sum of three consecutive integers is $24$, what is the smallest of the three numbers?
[b]p3.[/b] How many positive integers less than $25$ are either multiples of $2$ or multiples of $3$?
[b]p4.[/b] Charlotte has $5$ positive integers. Charlotte tells you that the mean, median, and unique mode of his five numbers are all equal to $10$. What is the largest possible value of the one of Charlotte's numbers?
[b]p5.[/b] Mr. Meeseeks starts with a single coin. Every day, Mr. Meeseeks goes to a magical coin converter where he can either exchange $1$ coin for $5$ coins or exchange $5$ coins for $3$ coins. What is the least number of days Mr. Meeseeks needs to end with $15$ coins?
[b]p6.[/b] Twelve years ago, Violet's age was twice her sister Holo's age. In $7$ years, Holo's age will be $13$ more than a third of Violet's age. $3$ years ago, Violet and Holo's cousin Rindo's age was the sum of their ages. How old is Rindo?
[b]p7.[/b] In a $2 \times 3$ rectangle composed of $6$ unit squares, let $S$ be the set of all points $P$ in the rectangle such that a unit circle centered at $P$ covers some point in exactly $3$ of the unit squares. Find the area of the region $S$. For example, the diagram below shows a valid unit circle in a $2 \times 3$ rectangle.
[img]https://cdn.artofproblemsolving.com/attachments/d/9/b6e00306886249898c2bdb13f5206ced37d345.png[/img]
[b]p8.[/b] What are the last four digits of $2^{1000}$?
[b]p9.[/b] There is a point $X$ in the center of a $2 \times 2 \times 2$ box. Find the volume of the region of points that are closer to $X$ than to any of the vertices of the box.
[b]p10.[/b] Evaluate $\sqrt{37 \cdot 41 \cdot 113 \cdot 290 - 4319^2}$
[b]p11.[/b] (Estimation) A number is abundant if the sum of all its divisors is greater than twice the number. One such number is $12$, because $1+2+3+4+6+12 = 28 > 24$: How many abundant positive integers less than $20190$ are there?
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1988 Bundeswettbewerb Mathematik, 2
Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle.
Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.
2016 PUMaC Algebra Individual B, B3
Bob draws the graph of $y = x^3 - 13x^2 + 40x + 25$ and is dismayed to find out that it only has one root. Alice comes to the rescue, translating (without rotating or dilating) the axes so that the origin is at the point that used to be $(-20, 16)$. This new graph has three $x$-intercepts; compute their sum.
2005 Putnam, B4
For positive integers $ m$ and $ n$, let $ f\left(m,n\right)$ denote the number of $ n$-tuples $ \left(x_1,x_2,\dots,x_n\right)$ of integers such that $ \left|x_1\right| \plus{} \left|x_2\right| \plus{} \cdots \plus{} \left|x_n\right|\le m$. Show that $ f\left(m,n\right) \equal{} f\left(n,m\right)$.
1958 November Putnam, B7
Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.
Russian TST 2016, P2
$ABCDEF$ is a cyclic hexagon with $AB=BC=CD=DE$. $K$ is a point on segment $AE$ satisfying $\angle BKC=\angle KFE, \angle CKD = \angle KFA$. Prove that $KC=KF$.
2019 Romania National Olympiad, 4
A piece of rectangular paper $20 \times 19$, divided into four units, is cut into several square pieces, the cuts being along the sides of the unit squares. Such a square piece is called odd square if the length of its side is an odd number.
a) What is the minimum possible number of odd squares?
b) What is the smallest value that the sum of the perimeters of the odd squares can take?
2015 South East Mathematical Olympiad, 6
In $\triangle ABC$, we have three edges with lengths $BC=a, \, CA=b \, AB=c$, and $c<b<a<2c$. $P$ and $Q$ are two points of the edges of $\triangle ABC$, and the straight line $PQ$ divides $\triangle ABC$ into two parts with the same area. Find the minimum value of the length of the line segment $PQ$.
2002 AIME Problems, 8
Find the smallest integer $k$ for which the conditions
$(1)$ $a_1, a_2, a_3, \ldots$ is a nondecreasing sequence of positive integers
$(2)$ $a_n=a_{n-1}+a_{n-2}$ for all $n>2$
$(3)$ $a_9=k$
are satisfied by more than one sequence.
2007 District Olympiad, 2
Let $A\in \mathcal{M}_n(\mathbb{R}^*)$. If $A\cdot\ ^t A=I_n$, prove that:
a)$|\text{Tr}(A)|\le n$;
b)If $n$ is odd, then $\det(A^2-I_n)=0$.
1976 Canada National Olympiad, 5
Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.
1967 Putnam, B4
a) A certain locker room contains $n$ lockers numbered $1,2,\ldots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1, T_2 ,\ldots, T_n$, whereby with the operation $T_k$ the state of those lockers whose number is divisible by $k$ is swapped. After all $n$ operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this.
b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $2 m^2 $, or the set of numbers of the form $m^2 +1$, or some nontrivial similar set of your own selection.