Found problems: 85335
1983 National High School Mathematics League, 6
Let $a,b,c,d,m,n$ be positive real numbers. $P=\sqrt{ab}+\sqrt{cd},Q=\sqrt{ma+nc}\cdot\sqrt{\frac{b}{m}+\frac{d}{n}}$. Then
$\text{(A)}P\geq Q\qquad\text{(B)}P\leq Q\qquad\text{(C)}P<Q\qquad\text{(D)}$Not sure
2016 IFYM, Sozopol, 4
Let $ABCD$ be a convex quadrilateral. The circle $\omega_1$ is tangent to $AB$ in $S$ and the continuations after $A$ and $B$ of sides $DA$ and $CB$, circle $\omega_2$ with center $I$ is tangent to $BC$ and the continuations after $B$ and $C$ of sides $AB$ and $DC$, circle $\omega_3$ is tangent to $CD$ in $T$ and the continuations after $C$ and $D$ of sides $BC$ and $AD$, and circle $\omega_4$ with center $J$ is tangent to $DA$ and the continuations after $D$ and $A$ of sides $CD$ and $BA$. Prove that points $S$ and $T$ are on equal distance from the middle point of segment $IJ$.
1994 Vietnam National Olympiad, 2
$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.
2010 Belarus Team Selection Test, 3.3
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
KoMaL A Problems 2022/2023, A. 848
Let $G$ be a planar graph, which is also bipartite. Is it always possible to assign a vertex to each face of the graph such that no two faces have the same vertex assigned to them?
[i]Submitted by Dávid Matolcsi, Budapest[/i]
2020 LIMIT Category 2, 9
Three points are chosen randomly and independently on a circle. The probability that all three pairwise distance between the points are less than the radius of the circle is $\frac{1}{K}$, $K\in\mathbb{N}$. Find $K$.
1994 AMC 8, 19
Around the outside of a $4$ by $4$ square, construct four semicircles (as shown in the figure) with the four sides of the square as their diameters. Another square, $ABCD$, has its sides parallel to the corresponding sides of the original square, and each side of $ABCD$ is tangent to one of the semicircles. The area of the square $ABCD$ is
[asy]
pair A,B,C,D;
A = origin; B = (4,0); C = (4,4); D = (0,4);
draw(A--B--C--D--cycle);
draw(arc((2,1),(1,1),(3,1),CCW)--arc((3,2),(3,1),(3,3),CCW)--arc((2,3),(3,3),(1,3),CCW)--arc((1,2),(1,3),(1,1),CCW));
draw((1,1)--(3,1)--(3,3)--(1,3)--cycle);
dot(A); dot(B); dot(C); dot(D); dot((1,1)); dot((3,1)); dot((1,3)); dot((3,3));
label("$A$",A,SW);
label("$B$",B,SE);
label("$C$",C,NE);
label("$D$",D,NW);
[/asy]
$\text{(A)}\ 16 \qquad \text{(B)}\ 32 \qquad \text{(C)}\ 36 \qquad \text{(D)}\ 48 \qquad \text{(E)}\ 64$
2003 Tournament Of Towns, 5
What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?
IV Soros Olympiad 1997 - 98 (Russia), 10.8
In triangle $ABC$, angle $B$ is different from a right angle, $AB : BC = k$. Let $M$ be the midpoint of $AC$. Lines symmetric to $BM$ wrt $AB$ and $BC$ intersect line $AC$ at points $D$ and $E$. Find $BD : BE$.
2009 Moldova Team Selection Test, 2
$ f(x)$ and $ g(x)$ are two polynomials with nonzero degrees and integer coefficients, such that $ g(x)$ is a divisor of $ f(x)$ and the polynomial $ f(x)\plus{}2009$ has $ 50$ integer roots. Prove that the degree of $ g(x)$ is at least $ 5$.
2007 AMC 10, 23
A pyramid with a square base is cut by a plane that is parallel to its base and is $ 2$ units from the base. The surface area of the smaller pyramid that is cut from the top is half the surface area of the original pyramid. What is the altitude of the original pyramid?
$ \textbf{(A)}\ 2\qquad
\textbf{(B)}\ 2 \plus{} \sqrt{2}\qquad
\textbf{(C)}\ 1 \plus{} 2\sqrt{2}\qquad
\textbf{(D)}\ 4\qquad
\textbf{(E)}\ 4 \plus{} 2\sqrt{2}$
2024 Philippine Math Olympiad, P4
Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?
2010 Indonesia MO, 7
Given 2 positive reals $a$ and $b$. There exists 2 polynomials $F(x)=x^2+ax+b$ and $G(x)=x^2+bx+a$ such that all roots of polynomials $F(G(x))$ and $G(F(x))$ are real. Show that $a$ and $b$ are more than $6$.
[i]Raja Oktovin, Pekanbaru[/i]
2022 BMT, 15
Let $f(x)$ be a function acting on a string of $0$s and $1$s, defined to be the number of substrings of $x$ that have at least one $1$, where a substring is a contiguous sequence of characters in $x$. Let $S$ be the set of binary strings with $24$ ones and $100$ total digits. Compute the maximum possible value of $f(s)$ over all $s\in S$.
For example, $f(110) = 5$ as $\underline{1}10$, $1\underline{1}0$, $\underline{11}0$, $1\underline{10}$, and $\underline{110}$ are all substrings including a $1$. Note that $11\underline{0}$ is not such a substring.
2012 South East Mathematical Olympiad, 1
A nonnegative integer $m$ is called a “six-composited number” if $m$ and the sum of its digits are both multiples of $6$. How many “six-composited numbers” that are less than $2012$ are there?
2014 Math Prize For Girls Problems, 2
Let $x_1$, $x_2$, …, $x_{10}$ be 10 numbers. Suppose that $x_i + 2 x_{i + 1} = 1$ for each $i$ from 1 through 9. What is the value of $x_1 + 512 x_{10}$?
2009 IMO Shortlist, 2
For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such that the following two conditions are satisfied:
[list][*] $a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$,
[*] If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$[/list]
Determine $N(n)$ for all $n\geq 2$.
[i]Proposed by Dan Schwarz, Romania[/i]
1939 Moscow Mathematical Olympiad, 043
Solve the system $\begin{cases} 3xyz -x^3 - y^3-z^3 = b^3 \\
x + y+ z = 2b \\
x^2 + y^2-z^2 = b^2
\end{cases}$ in $C$
2014 Postal Coaching, 5
Let $(x_j,y_j)$, $1\le j\le 2n$, be $2n$ points on the half-circle in the upper half-plane. Suppose $\sum_{j=1}^{2n}x_j$ is an odd integer. Prove that $\displaystyle{\sum_{j=1}^{2n}y_j \ge 1}$.
2025 Caucasus Mathematical Olympiad, 7
It is known that from segments of lengths $a$, $b$ and $c$, a triangle can be formed. Could it happen that from segments of lengths $$\sqrt{a^2 + \frac{2}{3} bc},\quad \sqrt{b^2 + \frac{2}{3} ca}\quad \text{and} \quad \sqrt{c^2 + \frac{2}{3} ab},$$ a right-angled triangle can be formed?
2012-2013 SDML (High School), 9
Sammy and Tammy run laps around a circular track that has a radius of $1$ kilometer. They begin and end at the same point and at the same time. Sammy runs $3$ laps clockwise while Tammy runs $4$ laps counterclockwise. How many times during their run is the straight-line distance between Sammy and Tammy exactly $1$ kilometer?
$\text{(A) }7\qquad\text{(B) }8\qquad\text{(C) }13\qquad\text{(D) }14\qquad\text{(E) }21$
2018 Math Prize for Girls Problems, 6
Martha writes down a random mathematical expression consisting of 3 single-digit positive integers with an addition sign "$+$" or a multiplication sign "$\times$" between each pair of adjacent digits. (For example, her expression could be $4 + 3\times 3$, with value 13.) Each positive digit is equally likely, each arithmetic sign ("$+$" or "$\times$") is equally likely, and all choices are independent. What is the expected value (average value) of her expression?
2006 AIME Problems, 12
Equilateral $\triangle ABC$ is inscribed in a circle of radius 2. Extend $\overline{AB}$ through $B$ to point $D$ so that $AD=13$, and extend $\overline{AC}$ through $C$ to point $E$ so that $AE=11$. Through $D$, draw a line $l_1$ parallel to $\overline{AE}$, and through $E$, draw a line ${l}_2$ parallel to $\overline{AD}$. Let $F$ be the intersection of ${l}_1$ and ${l}_2$. Let $G$ be the point on the circle that is collinear with $A$ and $F$ and distinct from $A$. Given that the area of $\triangle CBG$ can be expressed in the form $\frac{p\sqrt{q}}{r}$, where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is not divisible by the square of any prime, find $p+q+r$.
2015 Saudi Arabia GMO TST, 3
Let $ABC$ be a triangle, with $AB < AC$, $D$ the foot of the altitude from $A, M$ the midpoint of $BC$, and $B'$ the symmetric of $B$ with respect to $D$. The perpendicular line to $BC$ at $B'$ intersects $AC$ at point $P$ . Prove that if $BP$ and $AM$ are perpendicular then triangle $ABC$ is right-angled.
Liana Topan
2025 India STEMS Category A, 4
Alice and Bob play a game on a connected graph with $2n$ vertices, where $n\in \mathbb{N}$ and $n>1$.. Alice and Bob have tokens named A and B respectively. They alternate their turns with Alice going first. Alice gets to decide the starting positions of A and B. Every move, the player with the turn moves their token to an adjacent vertex. Bob's goal is to catch Alice, and Alice's goal is to prevent this. Note that positions of A, B are visible to both Alice and Bob at every moment.
Provided they both play optimally, what is the maximum possible number of edges in the graph if Alice is able to evade Bob indefinitely?
[i]Proposed by Shashank Ingalagavi and Vighnesh Sangle[/i]