Found problems: 85335
2012 AMC 10, 15
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$?
[asy]
size(200);
defaultpen(linewidth(.6pt)+fontsize(12pt));
dotfactor=4;
draw((0,0)--(0,2));
draw((0,0)--(1,0));
draw((1,0)--(1,2));
draw((0,1)--(2,1));
draw((0,0)--(1,2));
draw((0,2)--(2,1));
draw((0,2)--(2,2));
draw((2,1)--(2,2));
label("$A$",(0,2),NW);
label("$B$",(1,2),N);
label("$C$",(4/5,1.55),W);
dot((0,2));
dot((1,2));
dot((4/5,1.6));
dot((2,1));
dot((0,0));
[/asy]
$ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{2}{9}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{\sqrt2}{4} $
2019 PUMaC Team Round, 11
The game Prongle is played with a special deck of cards: on each card is a nonempty set of distinct colors. No two cards in the deck contain the exact same set of colors. In this game, a “Prongle” is a set of at least $2$ cards such that each color is on an even number of cards in the set. Let k be the maximum possible number of prongles in a set of $2019$ cards. Find $\lfloor \log 2 (k) \rfloor$.
2013 JBMO TST - Turkey, 3
Two players $A$ and $B$ play a game with a ball and $n$ boxes placed onto the vertices of a regular $n$-gon where $n$ is a positive integer. Initially, the ball is hidden in a box by player $A$. At each step, $B$ chooses a box, then player $A$ says the distance of the ball to the selected box to player $B$ and moves the ball to an adjacent box. If $B$ finds the ball, then $B$ wins. Find the least number of steps for which $B$ can guarantee to win.
1949 Miklós Schweitzer, 4
Let $ A$ and $ B$ be two disjoint sets in the interval $ (0,1)$ . Denoting by $ \mu$ the Lebesgue measure on the real line, let $ \mu(A)>0$ and $ \mu(B)>0$ . Let further $ n$ be a positive integer and $ \lambda \equal{}\frac1n$ . Show that there exists a subinterval $ (c,d)$ of $ (0,1)$ for which $ \mu(A\cap (c,d))\equal{}\lambda \mu(A)$ and $ \mu(B\cap (c,d))\equal{}\lambda \mu(B)$ . Show further that this is not true if $ \lambda$ is not of the form $ \frac1n$.
2010 Belarus Team Selection Test, 7.1
Find the smallest value of the expression $|3 \cdot 5^m - 11 \cdot 13^n|$ for all $m,n \in N$.
(Folklore)
1985 Traian Lălescu, 1.4
Two planes, $ \alpha $ and $ \beta, $ form a dihedral angle of $ 30^{\circ} , $ and their intersection is the line $ d. $ A point $ A $ situated at the exterior of this angle projects itself in $ P\not\in d $ on $ \alpha , $ and in $ Q\not\in d $ on $ \beta $ such that $ AQ<AP. $ Name $ B $ the projection of $ A $ upon $ d. $
[b]a)[/b] Are $ A,B,P,Q, $ coplanar?
[b]b)[/b] Knowing that a perpendicular to $ \beta $ make with $ AB $ an angle of $ 60^{\circ} , $ and $ AB=4, $ find the area of $ BPQ. $
2025 NCJMO, 5
Each element of set $\mathcal{S}$ is colored with multiple colors. A $\textit{rainbow}$ is a subset of $\mathcal{S}$ which has amongst its elements at least $1$ color from each element of $\mathcal{S}$. A $\textit{minimal rainbow}$ is a rainbow where removing any single element gives a non-rainbow.
Prove that the union of all minimal rainbows is $\mathcal{S}$.
[i]Grisham Paimagam[/i]
2011 VTRMC, Problem 4
Let $m,n$ be positive integers and let $[a]$ denote the residue class$\pmod{mn}$ of the integer $a$ (thus $\{[r]|r\text{ is an integer}\}$ has exactly $mn$ elements). Suppose the set $\{[ar]|r\text{ is an integer}\}$ has exactly $m$ elements. Prove that there is a positive integer $q$ such that $q$ is coprime to $mn$ and $[nq]=[a]$.
2022 IMO, 4
Let $ABCDE$ be a convex pentagon such that $BC=DE$. Assume that there is a point $T$ inside $ABCDE$ with $TB=TD,TC=TE$ and $\angle ABT = \angle TEA$. Let line $AB$ intersect lines $CD$ and $CT$ at points $P$ and $Q$, respectively. Assume that the points $P,B,A,Q$ occur on their line in that order. Let line $AE$ intersect $CD$ and $DT$ at points $R$ and $S$, respectively. Assume that the points $R,E,A,S$ occur on their line in that order. Prove that the points $P,S,Q,R$ lie on a circle.
2023 Bulgaria JBMO TST, 3
Find all natural numbers $a$, $b$, $c$ and prime numbers $p$ and $q$, such that:
$\blacksquare$ $4\nmid c$
$\blacksquare$ $p\not\equiv 11\pmod{16}$
$\blacksquare$ $p^aq^b-1=(p+4)^c$
2024 Czech and Slovak Olympiad III A, 4
There were $10$ boys and $10$ girls at the party. Every boy likes a different 'positive' number of girls. Every girl likes a different positive number of boys. Define the largest non-negative integer $n$ such that it is always possible to form at least $n$ disjoint pairs in which both like the other.
2015 India PRMO, 10
$10.$ A $2\times 3$ rectangle and a $3 \times 4$ rectangle are contained within a square without overlapping at any interior point, and the sides of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square $?$
1976 Dutch Mathematical Olympiad, 4
For $a,b, x \in R$ holds: $x^2 - (2a^2 + 4)x + a^2 + 2a + b = 0$. For which $b$ does this equation have at least one root between $0$ and $1$ for all $a$?
1936 Eotvos Mathematical Competition, 2
$S$ is a point inside triangle $ABC$ such that the areas of the triangles $ABS$, $BCS$ and $CAS$ are all equal. Prove that $S$ is the centroid of $ABC$.
2010 USA Team Selection Test, 4
Let $ABC$ be a triangle. Point $M$ and $N$ lie on sides $AC$ and $BC$ respectively such that $MN || AB$. Points $P$ and $Q$ lie on sides $AB$ and $CB$ respectively such that $PQ || AC$. The incircle of triangle $CMN$ touches segment $AC$ at $E$. The incircle of triangle $BPQ$ touches segment $AB$ at $F$. Line $EN$ and $AB$ meet at $R$, and lines $FQ$ and $AC$ meet at $S$. Given that $AE = AF$, prove that the incenter of triangle $AEF$ lies on the incircle of triangle $ARS$.
2016 Germany Team Selection Test, 2
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient \[ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} \] is an integer. Prove \[ 2n \leq q_1+q_2+\dots+q_n < 3n. \]
2021 Latvia Baltic Way TST, P4
Determine the smallest positive constant $k$ such that no matter what $3$ lattice points we choose the following inequality holds:
$$ L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}} $$
where $L_{\max}$, $L_{\min}$ is the maximal and minimal distance between chosen points.
2012 EGMO, 7
Let $ABC$ be an acute-angled triangle with circumcircle $\Gamma$ and orthocentre $H$. Let $K$ be a point of $\Gamma$ on the other side of $BC$ from $A$. Let $L$ be the reflection of $K$ in the line $AB$, and let $M$ be the reflection of $K$ in the line $BC$. Let $E$ be the second point of intersection of $\Gamma $ with the circumcircle of triangle $BLM$.
Show that the lines $KH$, $EM$ and $BC$ are concurrent. (The orthocentre of a triangle is the point on all three of its altitudes.)
[i]Luxembourg (Pierre Haas)[/i]
2012 AMC 10, 13
An [i]iterative average[/i] of the numbers $1$, $2$, $3$, $4$, and $5$ is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
$ \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} $
2021 Sharygin Geometry Olympiad, 8.3
Three cockroaches run along a circle in the same direction. They start simultaneously from a point $S$. Cockroach $A$ runs twice as slow than $B$, and thee times as slow than $C$. Points $X, Y$ on segment $SC$ are such that $SX = XY =YC$. The lines $AX$ and $BY$ meet at point $Z$. Find the locus of centroids of triangles $ZAB$.
2022 Durer Math Competition Finals, 4
$ABCD$ is a cyclic quadrilateral whose diagonals are perpendicular to each other. Let $O$ denote the centre of its circumcircle and $E$ the intersection of the diagonals. $J$ and $K$ denote the perpendicular projections of $E$ on the sides $AB$ and $BC$ . Let $F , G$ and $H$ be the midpoint line segments. Show that lines $GJ$ , $FB$ and $HK$ either pass through the same point or are parallel to each other.
2024 AMC 12/AHSME, 10
A list of 9 real numbers consists of $1$, $2.2 $, $3.2 $, $5.2 $, $6.2 $, $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) infinitely many}\qquad
$
2021 China Team Selection Test, 3
Find all positive integer $n(\ge 2)$ and rational $\beta \in (0,1)$ satisfying the following:
There exist positive integers $a_1,a_2,...,a_n$, such that for any set $I \subseteq \{1,2,...,n\}$ which contains at least two elements,
$$ S(\sum_{i\in I}a_i)=\beta \sum_{i\in I}S(a_i). $$
where $S(n)$ denotes sum of digits of decimal representation of $n$.
1997 Rioplatense Mathematical Olympiad, Level 3, 3
Prove that there are infinitely many positive integers $n$ such that the number of positive divisors in $2^n-1$ is greater than $n$.
2002 Indonesia MO, 5
Nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ are going to be inputted to the empty cells in the following table:
$\begin{array} {|c|c|c|} \cline{1-3}
10 & & \\ \cline{1-3}
& & 9 \\ \cline{1-3}
& 3 & \\ \cline{1-3}
11 & & 17 \\ \cline{1-3}
& 20 & \\ \cline{1-3}
\end{array}$
such that each row sums to the same number, and each column sums to the same number. Determine all possible arrangements.