Found problems: 85335
2010 IFYM, Sozopol, 3
Through vertex $C$ of $\Delta ABC$ are constructed lines $l_1$ and $l_2$ which are symmetrical about the angle bisector $CL_c$. Prove that the projections of $A$ and $B$ on lines $l_1$ and $l_2$ lie on one circle.
KoMaL A Problems 2022/2023, A. 856
In a rock-paper-scissors round robin tournament any two contestants play against each other ten times in a row. Each contestant has a favourite strategy, which is a fixed sequence of ten hands (for example, RRSPPRSPPS), which they play against all other contestants. At the end of the tournament it turned out that every player won at least one hand (out of the ten) against any other player.
Prove that at most $1024$ contestants participated in the tournament.
[i]Submitted by Dávid Matolcsi, Budapest[/i]
2018 Stanford Mathematics Tournament, 1
Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.
1965 IMO Shortlist, 4
Find all sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.
May Olympiad L1 - geometry, 2012.3
From a paper quadrilateral like the one in the figure, you have to cut out a new quadrilateral whose area is equal to half the area of the original quadrilateral.You can only bend one or more times and cut by some of the lines of the folds. Describe the folds and cuts and justify that the area is half.
[img]https://2.bp.blogspot.com/-btvafZuTvlk/XNY8nba0BmI/AAAAAAAAKLo/nm4c21A1hAIK3PKleEwt6F9cd6zv4XffwCK4BGAYYCw/s400/may%2B2012%2Bl1.png[/img]
1949-56 Chisinau City MO, 30
Through the point of intersection of the diagonals of the trapezoid, a straight line is drawn parallel to its bases. Determine the length of the segment of this straight line, enclosed between the lateral sides of the trapezoid, if the lengths of the bases of the trapezoid are equal to $a$ and $b$.
2015 Purple Comet Problems, 4
Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.
2022 Sharygin Geometry Olympiad, 9
The sides $AB, BC, CD$ and $DA$ of quadrilateral $ABCD$ touch a circle with center $I$ at points $K, L, M$ and $N$ respectively. Let $P$ be an arbitrary point of line $AI$. Let $PK$ meet $BI$ at point $Q, QL$ meet $CI$ at point $R$, and $RM$ meet $DI$ at point $S$.
Prove that $P,N$ and $S$ are collinear.
2015 Thailand TSTST, 2
Let $\mathbb{N} = \{1, 2, 3, \dots\}$ and let $f : \mathbb{N}\to\mathbb{R}$. Prove that there is an infinite subset $A$ of $\mathbb{N}$ such that $f$ is increasing on $A$ or $f$ is decreasing on $A$.
2023 Ecuador NMO (OMEC), 1
Find all reals $(a, b, c)$ such that
$$\begin{cases}a^2+b^2+c^2=1\\ |a+b|=\sqrt{2}\end{cases}$$
PEN A Problems, 18
Let $m$ and $n$ be natural numbers and let $mn+1$ be divisible by $24$. Show that $m+n$ is divisible by $24$.
1998 China Team Selection Test, 3
For a fixed $\theta \in \lbrack 0, \frac{\pi}{2} \rbrack$, find the smallest $a \in \mathbb{R}^{+}$ which satisfies the following conditions:
[b]I. [/b] $\frac{\sqrt a}{\cos \theta} + \frac{\sqrt a}{\sin \theta} >
1$.
[b]II.[/b] There exists $x \in \lbrack 1 - \frac{\sqrt a}{\sin \theta},
\frac{\sqrt a}{\cos \theta} \rbrack$ such that $\lbrack (1 -
x)\sin \theta - \sqrt{a - x^2 \cos^{2} \theta} \rbrack^{2} +
\lbrack x\cos \theta - \sqrt{a - (1 - x)^2 \sin^{2} \theta}
\rbrack^{2} \leq a$.
2003 Croatia Team Selection Test, 2
Let $B$ be a point on a circle $k_1, A \ne B$ be a point on the tangent to the circle at $B$, and $C$ a point not lying on $k_1$ for which the segment $AC$ meets $k_1$ at two distinct points. Circle $k_2$ is tangent to line $AC$ at $C$ and to $k_1$ at point $D$, and does not lie in the same half-plane as $B$. Prove that the circumcenter of triangle $BCD$ lies on the circumcircle of $\vartriangle ABC$
2008 Canada National Olympiad, 3
Let $ a$, $ b$, $ c$ be positive real numbers for which $ a \plus{} b \plus{} c \equal{} 1$. Prove that
\[ {{a\minus{}bc}\over{a\plus{}bc}} \plus{} {{b\minus{}ca}\over{b\plus{}ca}} \plus{} {{c\minus{}ab}\over{c\plus{}ab}}
\leq {3 \over 2}.\]
2015 Caucasus Mathematical Olympiad, 3
Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her?
2011 Albania Team Selection Test, 1
The given parabola $y=ax^2+bx+c$ doesn't intersect the $X$-axis and passes from the points $A(-2,1)$ and $B(2,9)$. Find all the possible values of the $x$ coordinates of the vertex of this parabola.
1981 AMC 12/AHSME, 13
Suppose that at the end of any year, a unit of money has lost $10\%$ of the value it had at the beginning of that year. Find the smallest integer $n$ such that after $n$ years, the money will have lost at least $90\%$ of its value. (To the nearest thousandth $\log_{10}3=.477$.)
$\text{(A)}\ 14 \qquad \text{(B)}\ 16 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 20 \qquad \text{(E)}\ 22$
2022 HMIC, 4
Call a simple graph $G$ [i]quasicolorable[/i] if we can color each edge blue, red, green, or white such that
[list]
[*] for each vertex v of degree 3 in G, the three edges incident to v are either (1) red,
green, and blue, or (2) all white,
[*] not all edges are white.
[/list]
A simple connected graph $G$ has $a$ vertices of degree $4$, $b$ vertices of degree $3$, and no other vertices, where $a$ and $b$ are positive integers. Find the smallest real number $c$ so that the following statement is true: “If $a/b > c$, then $G$ must be quasicolorable.”
2021 JHMT HS, 2
David has some pennies. One apple costs $3$ pennies, one banana costs $5$ pennies, and one cranberry costs $7$ pennies. If David spends all his money on apples, he will have $2$ pennies left; if David spends all his money on bananas, he will have $3$ pennies left; is David spends all his money on cranberries, he will have $2$ pennies left. What is the least possible amount of pennies that David can originally have?
MIPT Undergraduate Contest 2019, 1.2
Does there exist a strictly increasing function $f: \mathbb{R} \rightarrow \mathbb{R}$ that takes on only irrational values?
2009 China Girls Math Olympiad, 7
On a $ 10 \times 10$ chessboard, some $ 4n$ unit squares are chosen to form a region $ \mathcal{R}.$ This region $ \mathcal{R}$ can be tiled by $ n$ $ 2 \times 2$ squares. This region $ \mathcal{R}$ can also be tiled by a combination of $ n$ pieces of the following types of shapes ([i]see below[/i], with rotations allowed).
Determine the value of $ n.$
2014 NIMO Problems, 2
In the Generic Math Tournament, $99$ people participate. One of the participants, Alfred, scores 16th in Algebra, 30th in Combinatorics, and 23rd in Geometry (and does not tie with anyone). The overall ranking is computed by adding the scores from all three tests. Given this information, let $B$ be the best ranking that Alfred could have achieved, and let $W$ be the worst ranking that he could have achieved. Compute $100B+W$.
[i]Proposed by Lewis Chen[/i]
2021 CCA Math Bonanza, T1
How many sequences of words (not necessarily grammatically correct) have the property that the first word has one letter, each word can be obtained by inserting a letter somewhere in the previous word, and the final word is CCAMT? Here are examples of possible sequences:
[center]
C,CA,CAM,CCAM,CCAMT.
[/center]
[center]
A,AT,CAT,CAMT,CCAMT.
[/center]
[i]2021 CCA Math Bonanza Team Round #1[/i]
2019 Romanian Masters In Mathematics, 1
Amy and Bob play the game. At the beginning, Amy writes down a positive integer on the board. Then the players take moves in turn, Bob moves first. On any move of his, Bob replaces the number $n$ on the blackboard with a number of the form $n-a^2$, where $a$ is a positive integer. On any move of hers, Amy replaces the number $n$ on the blackboard with a number of the form $n^k$, where $k$ is a positive integer. Bob wins if the number on the board becomes zero.
Can Amy prevent Bob’s win?
[i]Maxim Didin, Russia[/i]
2008 Danube Mathematical Competition, 3
On a semicircle centred at $O$ and with radius $1$ choose the respective points $A_1,A_2,...,A_{2n}$ , for $n \in N^*$. The lenght of the projection of the vector $\overrightarrow {u}=\overrightarrow{OA_1} +\overrightarrow{OA_2}+...+\overrightarrow{OA_{2n}}$ on the diameter is an odd integer. Show that the projection of that vector on the diameter is at least $1$.