Found problems: 85335
2019 Israel National Olympiad, 1
In kindergarden, there are 32 children and three classes: Judo, Agriculture, and Math. Every child is in exactly one class and every class has at least one participant.
One day, the teacher gathered 6 children to clean up the classroom. The teacher counted and found that exactly 1/2 of the Judo students, 1/4 of the Agriculture students and 1/8 of the Math students are cleaning.
How many children are in each class?
1987 Austrian-Polish Competition, 1
Three pairwise orthogonal chords of a sphere $S$ are drawn through a given point $P$ inside $S$. Prove that the sum of the squares of their lengths does not depend on their directions.
2024 Assara - South Russian Girl's MO, 8
Given a set $S$ of $2024$ natural numbers satisfying the following condition: if you select any $10$ (different) numbers from $S$, then you can select another number from $S$ so that the sum of all $11$ selected numbers is divisible by $10$. Prove that one of the numbers can be thrown out of $S$ so that the resulting set $S'$ of $2023$ numbers satisfies the condition: if you choose any $9$ (different) numbers from $S'$, then you can choose another number from $S'$ so that the sum of all $10$ selected numbers is divisible by $10$.
[i]K.A.Sukhov[/i]
1990 Baltic Way, 18
Numbers $1, 2,\dots , 101$ are written in the cells of a $101\times 101$ square board so that each number is repeated $101$ times. Prove that there exists either a column or a row containing at least $11$ different numbers.
2018 PUMaC Team Round, 15
Aaron the Ant is somewhere on the exterior of a hollow cube of side length $2$ inches, and Fred the Flea is on the inside, at one of the vertices. At some instant, Fred flies in a straight line towards the opposite vertex, and simultaneously Aaron begins crawling on the exterior of the cube towards that same vertex. Fred moves at $\sqrt{3}$ inches per second and Aaron moves at $\sqrt{2}$ inches per second. If Aaron arrives before Fred, the area of the surface on the cube from which Aaron could have started can be written as $a\pi+\sqrt{b}+c$ where $a$, $b$, and $c$ are integers. Find $a+b+c.$
2011 Akdeniz University MO, 1
Let $m,n$ positive integers and $p$ prime number with $p=3k+2$. If $p \mid {(m+n)^2-mn}$ , prove that
$$p \mid m,n$$
2008 China Team Selection Test, 3
Let $ S$ be a set that contains $ n$ elements. Let $ A_{1},A_{2},\cdots,A_{k}$ be $ k$ distinct subsets of $ S$, where $ k\geq 2, |A_{i}| \equal{} a_{i}\geq 1 ( 1\leq i\leq k)$. Prove that the number of subsets of $ S$ that don't contain any $ A_{i} (1\leq i\leq k)$ is greater than or equal to $ 2^n\prod_{i \equal{} 1}^k(1 \minus{} \frac {1}{2^{a_{i}}}).$
2002 Singapore Team Selection Test, 3
For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).
1999 Harvard-MIT Mathematics Tournament, 10
$A, B, C, D,$ and $E$ are relatively prime integers (i.e., have no single common factor) such that the polynomials $5Ax^4 +4Bx^3 +3Cx^2 +2Dx+E$ and $10Ax^3 +6Bx^2 +3Cx+D$ together have $7$ distinct integer roots. What are all possible values of $A$?
[i]Your team has been given a sealed envelope that contains a hint for this problem. If you open the envelope, the value of this problem decreases by 20 points. To get full credit, give the sealed envelope to the judge before presenting your solution.[/i]
2009 ELMO Problems, 2
Let $ABC$ be a triangle such that $AB < AC$. Let $P$ lie on a line through $A$ parallel to line $BC$ such that $C$ and $P$ are on the same side of line $AB$. Let $M$ be the midpoint of segment $BC$. Define $D$ on segment $BC$ such that $\angle BAD = \angle CAM$, and define $T$ on the extension of ray $CB$ beyond $B$ so that $\angle BAT = \angle CAP$. Given that lines $PC$ and $AD$ intersect at $Q$, that lines $PD$ and $AB$ intersect at $R$, and that $S$ is the midpoint of segment $DT$, prove that if $A$,$P$,$Q$, and $R$ lie on a circle, then $Q$, $R$, and $S$ are collinear.
[i]David Rush[/i]
2020 USMCA, 19
Call a right triangle [i]peri-prime[/i] if it has relatively prime integer side lengths, perimeter a multiple of $65$, and at least one leg with length less than $100$. Compute the sum of all possible lengths for the smallest leg of a peri-prime triangle.
2014 Indonesia MO Shortlist, G3
Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.
2015 Benelux, 4
Let $n$ be a positive integer. For each partition of the set $\{1,2,\dots,3n\}$ into arithmetic progressions, we consider the sum $S$ of the respective common differences of these arithmetic progressions. What is the maximal value that $S$ can attain?
(An [i]arithmetic progression[/i] is a set of the form $\{a,a+d,\dots,a+kd\}$, where $a,d,k$ are positive integers, and $k\geqslant 2$; thus an arithmetic progression has at least three elements, and successive elements have difference $d$, called the [i]common difference[/i] of the arithmetic progression.)
2010 IMO, 2
Given a triangle $ABC$, with $I$ as its incenter and $\Gamma$ as its circumcircle, $AI$ intersects $\Gamma$ again at $D$. Let $E$ be a point on the arc $BDC$, and $F$ a point on the segment $BC$, such that $\angle BAF=\angle CAE < \dfrac12\angle BAC$. If $G$ is the midpoint of $IF$, prove that the meeting point of the lines $EI$ and $DG$ lies on $\Gamma$.
[i]Proposed by Tai Wai Ming and Wang Chongli, Hong Kong[/i]
2020 Iranian Geometry Olympiad, 2
Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$.
[i]Proposed by Ali Zamani[/i]
2014 NIMO Problems, 2
How many $2 \times 2 \times 2$ cubes must be added to a $8 \times 8 \times 8$ cube to form a $12 \times 12 \times 12$ cube?
[i]Proposed by Evan Chen[/i]
1950 Miklós Schweitzer, 2
Consider three different planes and consider also one point on each of them. Give necessary and sufficient conditions for the existence of a quadratic which passes through the given points and whose tangent-plane at each of these points is the respective given plane.
2021 Durer Math Competition (First Round), 4
Find all pairs of polynomials $(p, q)$ with integer coefficients that satisfy the equation $$p(x^2) + q(x^2) = p(x)q(x)$$ such that $p$ is of degree $n$ and has $n$ nonnegative real roots (with multiplicity).
1975 IMO Shortlist, 9
Let $f(x)$ be a continuous function defined on the closed interval $0 \leq x \leq 1$. Let $G(f)$ denote the graph of $f(x): G(f) = \{(x, y) \in \mathbb R^2 | 0 \leq$$ x \leq 1, y = f(x) \}$. Let $G_a(f)$ denote the graph of the translated function $f(x - a)$ (translated over a distance $a$), defined by $G_a(f) = \{(x, y) \in \mathbb R^2 | a \leq x \leq a + 1, y = f(x - a) \}$. Is it possible to find for every $a, \ 0 < a < 1$, a continuous function $f(x)$, defined on $0 \leq x \leq 1$, such that $f(0) = f(1) = 0$ and $G(f)$ and $G_a(f)$ are disjoint point sets ?
2021 Grand Duchy of Lithuania, 4
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers.
a) Find at least one primary triplet.
b) Show that there are infinitely many primary triplets.
2008 IberoAmerican Olympiad For University Students, 7
Let $A$ be an abelian additive group such that all nonzero elements have infinite order and for each prime number $p$ we have the inequality $|A/pA|\leq p$, where $pA = \{pa |a \in A\}$, $pa = a+a+\cdots+a$ (where the sum has $p$ summands) and $|A/pA|$ is the order of the quotient group $A/pA$ (the index of the subgroup $pA$).
Prove that each subgroup of $A$ of finite index is isomorphic to $A$.
MIPT student olimpiad spring 2022, 3
Prove that for any two linear subspaces $V, W \subset R^n$ the same
dimension there is an orthogonal transformation $A:R^n\to R^n$, such that $A(V )=W$ and $A(W) = V$
2009 Today's Calculation Of Integral, 448
Evaluate $ \int_0^{\ln 2} \frac {2e^x \plus{} 1}{e^{3x} \plus{} 2e^{2x} \plus{} e^{x} \minus{} e^{ \minus{} x}}\ dx.$
2016 Estonia Team Selection Test, 6
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
1951 AMC 12/AHSME, 47
If $ r$ and $ s$ are the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$, the value of $ \frac {1}{r^2} \plus{} \frac {1}{s^2}$ is:
$ \textbf{(A)}\ b^2 \minus{} 4ac \qquad\textbf{(B)}\ \frac {b^2 \minus{} 4ac}{2a} \qquad\textbf{(C)}\ \frac {b^2 \minus{} 4ac}{c^2} \qquad\textbf{(D)}\ \frac {b^2 \minus{} 2ac}{c^2}$
$ \textbf{(E)}\ \text{none of these}$